Superstrings ’89: proceedings of the Trieste Spring School, 3-14 April 1989 9789810201388, 9810201389

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SUPERSTRINGS g

rnrn

SUPERSTRINGS ~

rnrn

Proceedings of the Trieste Spring School 3-14 April 1989

Edited by M. Green R. lengo S. Randjbar -Daemi E. Sezgin A. Strominger

b World Scientific

UII

'

Singapore· New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. POBox 128, Farrer Road, Singapore 9128 USA office: 687 Hartwell Street, Teaneck, NJ 07666 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

SUPERSTRINGS '89 Copyright © 1990 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photo· copying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN 981-02-0138-9

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v

PREFACE

The 1989 Spring School followed the tradition of recent years of concentrating on areas of theoretical physics closely associated with String and Superstring theory. As in previous years the lecturers were among the leading researchers in this area. The main emphasis Was on developments in two-dimensional conformal field theory (with "introductory' lectures by Moore) and its relationship to statistical mechanics (lectures by Ginsparg) and to 2 + 1- dimensional Chern -Simons theory (lectures by both the Verlindes). The particular area of N - 2 superconformal field theory and its relationship to the construction of superstring theories was covered in lectures by Warner, Gepner, Dine and Cande1as. Issues relating to the integration over world-sheet geometries that is needed in string theory were discussed by Fujikawa (in the context of the BRST method) and Tseytlin (who described string loop corrections to sigma models) while Schwartz covered topics in the geometric description of superstring theory in terms of universal moduli space. The lectures on baby universes by Strominger surveyed a rapidly evolving subject that addresses issues in quantum gravity that superstring theory is not yet able to address.

vii

CONTENTS

Preface

v

Lectures on RCFT G. Moore & N. Seiberg Some Statistical Mechanical Models and Conformal Field Theories

130

P. Ginsparg Lectures on N = 2 Superconformal Theories and Singularity Theory N. P. Wamer

197

Lectures on N = 2 String Theory

238

D. Gepner Compactification of Superstrings

303

MDine Rolling among Calabi -Yau Vacua P. Candelas, P. S. Green & T. Hubsch

366

Conformal Field Theory and Geometric Quantization H. Verlinde & E. Verlinde

422

Universal Moduli Space and String Theory A. S. Schwarz

450

Reparametrization BRS Cohomology in Two-Dimensional Gravity and Non-Critical String Theories

457

K. Fujikawa Sigma Models and Renormalization of String Loops

487

A. A. Tsey tlin Baby Universes

A. Strominger

550

Lectures on RCFT

*

Gregory Moore

Institute for Advanced Study Princeton, NJ 08540, USA and

Department of Physics Yale University, New Haven, CT 06511-8167,USA and Nathan Seiberg **

Institute for Advanced Study Princeton, NJ 08540, USA and

Department of Physics and Astronomy Rutgers University, Piscataway, NJ 08855-0849, USA

* Given by G. Moore in the Trieste spring school 1989 and by N. Seiberg in the Banff summer school 1989. ** On leave of absence from the Department of Physics, Weizmann Institute of Science, Rehovot 76100, ISRAEL.

2 We review some recent results in two dimensional Rational Conformal Field Theory. We discuss these theories as a generalization of group theory. The relation to a three dimensional topological theory is explained and the particular example of the Chern-SimonsWitten theory is analyzed in detail. This study leads to a natural conjecture regarding the classification of all RCFT's.

3 1. Introduction - a trip to the Zoo

The fundamental principles of string theory are not yet known. Since conformal field theory [1] plays a crucial role in string theory, many researches believe that a detailed study of conformal field theory will bring us closer to the concepts underlying string theory. It is hoped that a better understanding of the mathematical foundations of conformal field theory will lead to interesting and relevant generalizations of CFT, which might in turn lead to progress in string theory. There are other good reasons to study CFT, on the one hand, the study of CFT might eventually be useful in identifying 2D critical phenomena in nature and on the other it has lead to beautiful results and applications in pure mathematics, and promises to lead to more. Motivated by the desire to understand better the mathematical structure of conformal field theories one turns to the problem of classifying theories. We are not so much interested in the final list of theories as we are in the techniques used to obtain such a list, and the mathematical structures characteristic of members on that list. General conformal field theories have not yet been attacked in any meaningful way, but the study of an interesting subclass of theories has been very successful in the past two years. In order to motivate and define these theories let us recall that some theories have the beautiful properties that their correlation functions, partition functions etc. have very simple analyticity properties in the moduli. The prototype of such behavior is the holomorphic factorization of determinants on Riemann surfaces:

detaa ~ IF(r)1 2

which plays a key role in the Belavin-Knizhnik theorem of string theory. Should we focus on this criterion? No: the theories which have this property are too simple - they are basically free theories (on the world sheet!). Holomorphic factorization admits a generalization which leads to a very rich class of conformal field theories, namely, the rational conformal field theories (RCFT). These may be characterized by saying that all correlation functions, partition functions, etc. can be expressed in terms of finite sums of analytic times anti-

4 analytic functions: N.

For the primaries in 'H, and

1i k we have

where

~

is the conformal dimension of field. We can compute matrix elements between

descendants using the Virasoro algebra and the rule

This only defines

tf1

on Verma modules. Demanding that

quotients forces some of the constants

II tf1{k II

tf1

is defined on the irreducible

to vanish .

• Exercise 2.1 Null vector, at work. a.) Suppose

O.

For Im(zl - Z2)

< 0 we

have, in

general a different matrix B( -). If the sign is omitted, we refer to B( +) . • Exercise 2.3 Relation to BPZ. Compare the above discussion with section four of [lJ and show that the definition of conformal blocks as matrix elements of ~ corresponds with that of BPZ.

All of this has been derived in the simplified notation appropriate for the minimal models, but these considerations apply to arbitrary RCFT's. In the general case, when the space of three-point couplings

V/

k

is a vector space of dimension larger than one we

have linear transformations

The other part of the algebra of the

~

operators follows from the operator product

expansion. We have

L

~!lQ(Z2)(QI~~k(ZI2)lk)

QE'H. v Summari.ell the theoretic

reprf!l~n'Ationconh~n'

of the operatur

sum over descenda.nts

(2.4)

11

• Exercise 2.4 Defining the F Matriz. Prove that the operator product expansion of two

~

operators has the form

{Hint: Write out the operator product expansion with arbitrary coefficients. Use translation and scaling invariance to determine some of the structure of the coefficients. Now take the operator product expansion with a third operator ~ and demand consistency with braiding.}

Now going back to our blocks :F;;kl we see that we can insert the operator product expansion and define a new basis of conformal blocks, which we may denote pictorially:

J

Ie

i---L_--1.I_~

~]

p

In the general case we have a linear transformation Fpq [ii k] l .. Vi;p

,co. '¢I

VklP

---t

q Viql "" V;k '¢I

The F, B transformations are the basic duality transformations. The reader may well ask why these objects are of interest. We may answer with two immediate consequences of these considerations. Fir~t

point: Already the

ezi~tence

of B, F have interesting consequences. Since they are

defined by a change of basis, the transformation

12 is an isomorphism. Therefore, matching dimensions, we have

p

p

This defines Verlinde's fusion rule algebra: [21] • Exercise 2.5

Fu~ion

Rule Algebra. Using the fact that Band F define isomorphisms

show that the matrices

form a commutative associative algebra. This algebra is known as Verlinde's fusion rule algebra .

• Exercise 2.6. Ezample, of Fu,ion Rule Algebra,. a.) Show that the fusion rule algebra for the rational torus (see section 10) is 7l./N71.. b. )Write out the algebra for the Ising model. Try to determine all physically acceptable fusion rule algebras with three self-conjugate primaries. c.) Show that the FRA for the WZW model SU(2)k (the subscript denotes the level) is generated by elements

tPl, l

= 0,1/2, ... , k/2 with min(l. +l"k-(l. +l,»

tPl.tPl, =

L

Il.-l,1

tPl

by considering the null vector J~~2l+1ll; l). (See, e.g. [22].) d.) Consider the WZW model SU(3k Show that the fusion rule algebra for the six integrable representations 1,3,3*,6,6*,8 can be determined purely from the known group theoretic decompositions and consistency conditions on the FRA. Note in particular that Nsss = 1 whereas in group theory it is equal to two.

Second point: Next, the matrix B2 is not an identity matrix, precisely because of the cuts.

In fact, B2 is exactly the monodromy matrix for the analytic continuation of Zl around for the vector of blocks :F~jkl(Zl' Z2)' That is, if 1'(8) is the following curve:

Z2

13

Y(OI

A curve in

Zl

plane surrounding

Z2

Then one can compute the monodromy as in • Exercise 2.7 Monodromy of the

block~.

Show carefully that upon analytic continua-

tion we have:

Now, the monodromies of conformal field theory are related to the mutual locality factors and therefore to the conformal weights. Thus, the primitive hope is that nontrivial identities on B, F matrices are so restrictive that one can solve them and thus classify RCFT's. This is too naive, but it is on the right track. At any rate, with this hope in mind it is clearly wise to get better acquainted with B as in the following exercise: • Exercise 2.8 Band F with the unit operator. By setting various external representations in the four-point function to be the identity we obtain a computable three-point function. Use this observation to evaluate the F and B matrices in the special cases that one of the fields is the identity. Notice that

defines a linear map

14 which may be interpreted as the square-root of a mutual locality factor (compare the previous exercise). Show that

Therefore

e= ±1. In simple RCFT's like the discrete series eis always +1. In other theories e can be -1. For example, in S'U(2) KM, the sign e corresponds to the symmetry or where

antisymmetry of the tensor coupling the representations. Show that in this example

where the representations are labeled by their spin (which is integer or half integer). For simplicity, we will limit ourselves in some of the formulae below to the case

e= 1.

From this discussion it is clear that we need to understand the identities on B, F. A number of questions arise: How can we obtain nontrivial identities? What is the full set? What is the minimal set of independent relations? To understand these identities we should understand better what a

evo

is. Therefore, let us broaden our viewpoint

on chiral vertex operators so that we see more clearly the S3 symmetry of three-point couplings which is fundamental to duality. Instead of choosing the state should consider a

~ingle

f3 to define q; we

linear operator

that commutes with contour deformation of the chiral algebra. We would like to give this operator a geometrical interpretation. Namely, suppose we have representation spaces on three circles as in the following picture:

15

three-holed sphere with rep spaces on the three holes Placing one of the holes about the point at acting on the Hilbert space 'Hi at

00

00

we can define the Virasoro generators

by:

But, since T is analytic, these can be deformed to generators around zero and z

(2.5)

The chiral vertex operators commute with contour deformation, so we look for operators that satisfy:

L,,(oo)

jk. [(" L... (n+k 1) (jki) .(,8®,,),)= (i)

) ( ) ] z n+l-k L (Z) k - 1 ,8 ®"),+,8®L,,O,,),

(2.6) for any states ,8,,,),. This equation can be interpreted as follows. Think of L,,(z) as a set of Virasoro operators acting on a Hilbert space at z, 'H •. Then L,,(z) ® Lm(O) acts on the Hilbert space 'H. ® 'Ho. The operators L,,(oo) act on the tensor product 'H. ® 'Ho. They satisfy the Virasoro algebra with the same value of the central charge as L,,(z). Therefore, equation (2.5) defines a map fl. from the Virasoro algebra, A to A ® A

fl.(L,,) = 1 ® L"

+L 00

k=O

(n+1) k

Z

n+l-k

Lk -

1

®1.

16 This "comultiplication" allows us to take tensor products of Virasoro modules with given central charge. Then

evo's are "intertwining operators" for this notion of tensor product.

(More on this below.) The above considerations generalize to arbitrary chiral algebras. We must specify the z-dependence of these operators completely and this leads to the condition that (2.7) In RCFT's there is a finite-dimensional space,

VA of operators satisfying (2.6) and (2.7)

and we take these equations as our final definition of the

evo's.

The connection to our

previous description is that

• Exercise 2.9 Prove the equivalence of these two definitions. The superiority of our final definition is evident since we can now understand more clearly in terms of the formula (2.6) the statement that the

evo is an operator associated

to a 3-holed sphere. Furthermore, it suggests a natural generalization, since we can consider more complicated situations - say, a 4-holed sphere. There will be a finite dimensional vector space

V/kl of operators

which commute with contour deformation on the surface:

00:.

i

4-holed sphere with representations at each hole

17 The space of these operators is the same as the space of conformal blocks. This must be true since they are determined by the same equations (which follow from contour deformation arguments) used in more standard descriptions of conformal field theory [1][23]. The new spaces

VAl can be understood in terms of the simpler spaces of 3-point couplings.

Geometrically, we can represent the 4-holed sphere as sewn 3-holed spheres. Analytically, we can use completeness of states to write operators in

V/kl as compositions of evo's.

Each sewing has a corresponding composition of evo's and a corresponding decomposition of

V/kl into simpler spaces: K

J..

Jo

~i

Ie

J

i

L

r

V;~I ~ ffiV;~ ® V{;

)-

k

!2.

~i

k

t

f

V;ikl

~

ffiVip ® V;~

~

1r ~

"l

18

k

~

~

11.

• 1

Note that each of the sewings corresponds to a different asymptotic region of Teichmiiller space. The general construction is the following - any

,p3

diagram can be thick-

ened to give a surface - we can put FN length/twist coordinates on that surface and the region with small lengths corresponds to a region in Teichmuller space. In the asymptotic regions of Teichmuller space where the length coordinate goes to infinity the Riemann surface looks like a ,p3-diagram. In this limit the amplitudes of the conformal field theory and the conformal blocks have poles. The leading singularity corresponds to keeping only one'intermediate state in the corresponding channel.

Thus different sewings simply correspond to different bases for V/~l'

The braid-

ing/fusing isomorphisms express the relationships between these sewings. They are computed from the projectively fiat connection on moduli space - according to the picture of the Friedan-Shenker modular geometry [241.

Finally we need the following remark - The compositions described so far only give us 9 = 0 surfaces. For evo's of type

~i we can sew to get:

19

one holed torus obtained by sewing The space of such conformal blocks with channel i will be the space Vi~' and the space of all one-point blocks is EBi

Vk

In formulas, if we put a state

f3 at a puncture on the torus

we may form

(2.8) Here z is a point on the complex plane, but the trace essentially identifies z we actually compute a torus amplitude. If

~

qz so that

f3 is a Virasoro primary these blocks form a

representation of the modular group with the matrix:

(2.9) In terms of sewings we are relating the following two diagrams

I:v~ ® Va

--->

dim R, .

We have a map of a one-dimensional vector space to itself, which is, canonically, a complex number. One can compute the value of this number by decomposing in terms of intertwiners, and one finds the answer

fri .

• Exercise 5.1 Deligne'l condition in terml of F,. By considering the sequence

and decomposing the tensor products into irreducible representations, show that

~ F,

= dim R,

• Exercise 5.2 A nother proof of Deligne', Condition in ternu of Fi . Consider the "group theoretic one-loop two-point function":

53

Group theoretic two point function Consider the "monodromy" under i32

-+ pj,(g)i32,

Where (;~) denote intertwiners.

Using the basis of tensors:'

_ - L -_ _

k"

show that the monodromy is just:

--,"--ltV

r----Ie" --~

J

54 Take the limit 9

-+

1. Show that the other terms vanish and deduce that

and hence

The nontrivial result is that these axioms now characterize group theory. This is due to Deligne [30] and, in a slightly different form to Doplicher and Roberts [31] [32]. More precisely, suppose we are given the following:

Axioms for a Tannakian Category Data:

1. An index set I with a distinguished element 0 and a bijection of I to itself written

2. Vector spaces: VA i,j,k E I, with dimV/h = Njh

< 00

3. Isomorphisms:

n~k : VA ~ V~i

F[1: ~:] :9r V/,'r ® VJ,h, ~ 9.V.i~, ® v,:" ConditionA: 1. (i)"= i and 0"= O.

2. V;i ~ Sii C

V;~ ~ SijC

V/h ~ v,~

r

(V/h ~ VIii

3. n}hn~i = 1 . 4. The identities:

F(n ® l)F = (1 ® n)F(l ® n) F23F12F23

5. The normalization condition:

=

P23 F 13 F 12

(5.6)

55 From such a set of axioms we can reconstruct a group for which

V/" are the intertwin-

ers, F, the Racah coefficients, etc. The proof of this result is rather involved, but it would probably be worthwhile to sketch the main ideas of reconstruction which proceeds as follows: a) Define vector spaces Ri = CO";, obviously. (In category theory these correspond to simple objects which we must realize with honest vector spaces.) b) Define the space of intertwiners (morphisms) to be:

H om(R; ---+ Ri)

= CO

Hom(Ri ---+ Rj )

= 0 i

of j

and extend by linearity to H om( EIlR ---+ EIlR). c) Define tensor products: Ri ® Rj ~ EIlV;' ® R". That is,

Vi' is a set of intertwiners.

Now we define the set Rep = {all sums, products, quotients, duals of the

R.}.

d) Finally define the set of families of linear transformations:

9

= {(>'z)zERepIV:z:,>'z ::z: ---+:z: i3 an invertible linear trans/ormation. >'z®y = >'z ® >.y

T : :z:

-+ y

an intertwiner ~ T>.z = >.yT}

9 is a group: This is the group we want! One might naturally wonder whether, had one started with a group G, produced the objects F, n etc. and formed the group Aut, one would have recovered the same group G. This is settled in the following exercise:

• Exercise 5.3 On Recon3truction [30][33J. Suppose one begins with a compact group

Gand constructs the spaces V/" as above.

We will indicate why the reconstructed group

9 defined by the abstract procedure given here is exactly the original group G. a.) Note first that G

c g.

Note that every 9 E G defines a family {.\X}XERep via

>'x(g) = px(g), where Px is the representation defined by X.

b.) Show that if v E X is fixed by all of G, i.e., if Vg E G: px(g)V=

v

56 then it is fixed by all of

g, i.e.,

for any family satisfying the defining axioms of g. (Hint: Show that

z

~

zv is an intertwiner C

~

are no "broken" generators in

ARo

= 1, and that

X.) If G is a continuous Lie group we conclude that there

9 / G and hence that 9

= G.

c.) More generally suppose that G egis a proper subgroup. Then there is some Ax E End(X) which is not in the set {px(g)lg E G}

c

End(X). Use the fact that G

is compact to show that there must exist a polynomial P on End(X) which vanishes on {px(g)lg E G}, but not at Ax. Show that the space S of polynomials of degree ~ deg(P)

on End(X) is a representation of G. Note that PES violates '(b), to conclude that 9 = G. In the above characterization of a Tannakian category we have worked directly with the data

Vjk

etc. Alternatively we could have defined the category more directly in terms

of objects, with a tensor product of objects satisfying pentagon and hexagon conditions identical to (5.2) and (5.3) , and with some axioms relating to the unit object and dual objects. This is the definition one finds in the literature. The situation arising in RCFT is more complicated than the one we have described for the Tannakian categories. In RCFT the index set I is finite. Moreover n 2

=1=

1. This is

crucial: it is the characteristic that leads to interesting monodromy and hence interesting braid representations. The pentagon relation remains but there are two hexagon relations involving nand n- 1 . The category so defined (equivalently, the category defined by axioms on objects and morphisms of objects) is closely related to what is known as a "compact braided monoidal category" which was studied in [34J. Different definitions differ slightly on such details as whether

is involutive, or whether the set I should be finite or not.

Thus, roughly speaking, the duality properties of RCFT's on the plane are characterized by "compact braided monoidal categories." Well defined RCFT's have more structure and must be defined on all Riemann surfaces. By the completeness theorem it suffices to define S(p) : $ V;i ~ $ V;i according to (4.3) and impose the relations of the modular group. We

will call the category defined by these axioms a modular tenaor category. More precisely we have

57

Axioms for a Modular Tensor Category Data:

1. A finite index set I with a distinguished element 0 and a bijection of I to itself

written i

>-->

i-.

2. Vector spaces: V/k

i,i,k E I, with dimV/k = Njk


0- 00

.

ED;,j R; ® R; ® R; ® R; ®

1

---> --->

(5.8)

.

ED;,; Rj ® R; ® R; ® R; ®

ED;R; ® R;

Similarly one may use 0 to define the data ±e;,...o.j as a morphism R;

-+

R; and from

this define T on EDR; ® R,· and impose a relation on S2 (relating it to 0) and the relation

(ST)3 = S2. The name modular ten,or category was suggested by Igor Frenkel and we will adopt it. We thank him for discussions on this subject and for urging us to express the definition of S, (4.3) , in terms of simple objects, along the lines of (5.8). As we have mentioned, the above axioms are sufficient for establishing the relation

Sa = bS. Thus we may summarize the main result of [13][15] in the statement that a modular tensor category (henceforth MTC) is equivalent to a modular functor. As in section four we may ask whether all MTC's are associated to some RCFT, and to what extent an MTC characterizes the original RCFT. From the analogy of Tannakian categories and MTC's one naturally wonders whether there is a reconstruction theorem for MTC's analogous to Deligne's theorem. This is not known at present, but there is some good evidence that such a statement exists. First, there is an analog of the integrality condition in RCFT. From the proof of Verlinde's formula one finds

We have already noted that classically the quantity on the LHS is related to the dimension. The quantity on the RHS has been interpreted as the "relative dimension" of the

59 representation spaces. Note that [35J

"dimH;" trHo qL o-c/24 So; -..:::.!''-:'':-----,.,-,"dimHo" q-l trHo qLo-c/24 Soo All this strongly suggests that some axioms additional to the above polynomial equations .".------'.". = lim

in fact characterize RCFT's - and that classifying solutions to these equations is the same as classifying RCFT's. The relation between the axioms of RCFT as discussed above and the Tannaka-Krein approach to group theory becomes more complete in a certain limit of RCFT. Some RCFT's are labeled by a parameter k such that they simplify considerably in the k

--+ 00

limit. In

this limit the conformal dimensions of all the primary fields approach zero. More generally, there is a subset of the primary fields with a closed fusion rule algebra (namely, if i and

i

are in the subset then NJj

f= 0 only for

an integer in the k

limit. We define this limit as the clauical limit of the RCFT.

--+ 00

I in the set) whose conformal dimensions approach

Examining our axioms at genus zero in this limit we see that they simplify. In particular, since the relevant fl.'s are integers,

(5.9) Therefore, there are no monodromies in the classical theory and the two hexagons are the same equation. In this limit the axioms of a RCFT are identical to those of group theory in the Tannaka-Krein approach. Since classical RCFT is the same as group theory, it is natural to conjecture that quantum RCFT i~ a generalization of group theory. We'll return to this conjecture below. For the moment we note the following correspondences between group theory and conformal field theory: Group

Chiral algebra

Representations

Representations

Clebsch-Gordan coefficients/Intertwiners

Chiral vertex operators

Invariant tensors

Conformal blocks

Symmetry of couplings

n

Racah coefficients (6j symbols)

Fusion matrix

It is also interesting to examine a larger class of CFT's. We refer to them as "quasir-

ational eFT's." In these theories the chiral algebra has an infinite number of irreducible

60 representations. However, the fusion rules are finite; i.e. forgiven i,j, Ni~ is non zero only for a finite number of representationsk. Because of this condition, the formalism of the CVO and the duality matrices on the plane is still applicable. Consequently, the polynomial equations on the plane (the pentagon and the two hexagons) are satisfied. One can still define S(p) by (4.3) but since the number of irreducible representations is infinite, the torus polynomial equations are not obviously present. The category of representation spaces of the chiral algebra of a quasirational conformal field theory is also a generalization of a tensor category. A well known example of such a theory is the Gaussian model at an irrational value of the square of the radius. Finally, we must not lose sight of the fact that many interesting irrational (nonquasirational) CFT's exist and that the challenge to understand their structure remains unanswered.

61 6. Combining left movers with rightmovers. OFT is not just the study of chiral algebras and their representations. In order to have a consistent conformal field theory, we need to put together left and right-movers to obtain correlation functions with no monodromy. The left and right chiral algebras A, and A are the algebras of purely holomorphic and anti-holomorphic fields. We can decompose the total Hilbert space of the theory into irreducible representations: Hr ® Hr , so t he partition function is:

TrH qL o-c/24 qLo-C/24 =

L hrrXr(q)Xr(q). r,r

The nonnegative integers hrr characterize the field content of the theory. We can write the physical conformal fields in terms of the chiral vertex operators as

';'im,jm(z z) = 'I' ,

"Jii] _ ~,m(z)¥i.!.m(z) L..., (ji}(iI,iI) ,iI ,,. .

(6.1)

i,k

We assume for simplicity that there is only one field with representation (i,"i) in the theory. Below we'll show that this assumption is always satisfied. Now the physical correlation function must be independent of the choice of blocks, so there are certain conditions on the d-coefficients. For example, from invariance of the partition functions under T : 7"

-+ 7"

+ 1, we see that

hoi = 0 unless

~i -~.

E Z Proceeding

more systematically we could have deduced this from an analysis of 2 and 3 point functions. Moving on to the four-point function, we must have the same correlator from either basis of blocks:

k

i

I

J.

or"l

~

sand t channel blocks relevant for the four point function

R..

62 this implies " Jil) Jp,) ~ W)(p,) (~~)(/I)

F.

pq

[ii Ie] I

F.

,q

[Jt k] I

=

Jil) Jqq) (qq)(/I) (i3)(H)

(6.2)

p,p

• Exercise 6.1 Monodromy invariance. a.) Write out the conditions on d following from locality of the three-point function. b.) Show that the invariance of the physical correlator under B is guaranteed by the condition of part (a) together with the equation for F (6.2). By using the operator product expansion for chiral vertex operators together with (6.2) we may deduce that ",j,m;1,"'(z z)",/o,n;li,ft(z z) 'I'

,'I'

,

-

"d(p'fJ') ~ (i3)(U) p' ,IJ'

"

~

q,P',P';P',p,(w , ill)

PE"'" ,P' E?ip'

(6.3)

(P'Ic)~:';(z - w)ln)(P'Ic)~:';(z - w)ln), Again there is a nice analog of this equation in group theory. Recall that for a compact group the Hilbert space of L2 functions on the group has an orthonormal basis given by the matrix elements D:v in the irreducible representations R. The operator U(D:v) on L2(G) given by multiplication of functions may be represented in terms of intertwiners as (6.4)

Where we sum over a basis of intertwiners and ci is a basis dual to a. The algebra of functions on the group manifold is given by

Thus we see that in the the fields with A

-->

Ie

--> 00

limit of WZW models the operator product expansion of

0 becomes the algebra of functions on the group G, thus providing an

explicit example of an old idea of Dan Friedan's for the reconstruction of manifolds from

63 the operator product expansion of CFT. In fact, as described later, in the specific example of

c~rrent

algebra the above ope for finite Ie is closely related to the algebra of functions

on a quantum group. For further discussion of these and related ideas see [36]. We now show how the above equations can be used to deduce some general theorems about the operator content of rational conformal field theories .

• Exercise 6.2 No repre4entation appear4 more than once. Consider a RCFT where some representations occur more than once (either hrr zero for l'

> 1 or

both hrr and h rr , are non

i= 1").

a. Add indices in equation (6.1) to describe this situation. b. Rewrite equation (6.2) for this case. c. Study the four point function of

(rprpr/lrp') where rp and rp' transform the same under

A (the representation r) but they are different conformal fields (they might or might not transform the same under A) and assume for simplicity that all the representations are self conjugate. Use (3.5) to bring

F to the other side of the equation and study it for the

case where the intermediate representation is 0 on both sides. Simplify the equation by using the fact that the A (A) includes all the holomorphic (antiholomorphic) fields i.e. the identity operator is the only primary field under A ®

A which is holomorphic.

The ope of

rprp contains the identity operator and rprp' does not contain the identity operator. Use this fact to show that one side of the equation vanishes. The other side is proportional to Fr and does not vanish. Therefore, we are led to a contradiction and no representation can appear more than once. Notice that in proving this result one uses only the equations on the plane and not the equations on the torus.

Hence, this result applies not only in RCFT but also in

quasirational theories. On the other hand, this result is not true in theories which are not quasirational [11]. A Z2 orbifold of the Gaussian model at an irrational value of the square of the radius is not quasirational - the ope of two twist fields includes all the untwisted representations. Since the previous proof does not apply, we are not surprised to see the same representation appearing more than once in the spectrum.

64 Similarly there is an equation for the d's following from the modular invariance of the 9 = 1, one-point functions .

• Exercise 6.3 Equation for d from

genu~

one. Write the equation for invariance under

S(p) for every p. Remember that the characters of the one point function on the torus are

defined as differential forms i.e. they have a z-dependence ~ (dz / z )i!.(p) (otherwise they are not invariant). Therefore, there is a phase relating S of the left-movers to S of the right-movers. At this point one may wonder whether there will be further constraints on the d coefficients from duality invariance of correlation functions on other Riemann surfaces. The answer is no. Since duality matrices defining an MTC allow us to define duality matrices on all surfaces we know that the conformal blocks are duality covariant. To check invariance of left-right combinations of blocks we merely have to check invariance under the generators of duality transformations. Since an MTC defines a modular functor, the generators can be taken to be those duality transformations represented by F, B, S. Thus the above duality invariance conditions suffice to guarantee invariance on all surfaces. A similar conclusion was reached independently in [37] . • Exercise 6.4 Every repre~entation of A occur~ in the ~pectrum. Show that S(O) is unitary. Use this to show that one of the equations of the previous exercise can be written as

(6.5) Use

hOi

= hiO = .5;0, i.e. A (A) includes all the holomorphic (antiholomorphic) fields, to

show that there is no r such that h ri = 0 for every j. Hence, no representation can· be omitted. From the last exercises we conclude: If the chiral algebras, A and

.A are

extended, h r ,¥ must be a permutation matrix. We are now ready to tackle

maximally

65

• Exercise 6.5 The left mover~ are paired with the right mover~ by an automorphi~m

0/ the

fu~ion

rule algebra. Use Verlinde's formula relating the fusion rules to Sand (6.5)

to prove this.

We conclude that FRA(A) = FRA(A) and the pairing of the left movers and the right movers is an automorphism of the fusion rule algebra:

where

The main point here is that the classification of RCFT's is a two-step process. First we classify all chiral algebras and their representation theory, then we look for all automorphisms of the fusion rule algebras.

• Exercise 6.6 No New Condition~ on F. For a unitary diagonal (i.e. hi, = 6i,) theory, assuming F is real and the fusion rules are zero and one, show that the operator product coefficients may be written d2 _ F.Ok [ii ii) iik - F [i i) leO i i

a.) Use the polynomial equations to show that d is totally symmetric. b.) Substitute the above equation back into the full set of equations for diik on the plane. Show that the resulting identities are guaranteed by the polynomial equations.

• Exercise 6.7 Open Problem. How general is the result of the previous exercise? Do the equations for the torus one-point function follow from the other identities? (Felder and Silvotti [38] have shown that for the discrete series the answer is yes, by direct calculation.) What about non-unitary theories? What about arbitrary fusion rules? Is this true for the

66 non-diagonal theories - when a non-trivial a.utomorphism is used to pair left and right movers?

• Exercise 6.8 Modular Invariance of A~l) Character~. a.) Find the automorphisms of the fusion rule algebra for the level k SU(2) WZW model. b.) Impose other necessary conditions, e.g. the monodromy invariance of the twopoint function. c.) Using the above point of view interpret the other modular invariants of A~l) characters.

• Exercise 6.9 for

Nij1t

Automorphi~m~

and Kac's formula for

of Kac-Moody Fu,ion Rule,. Using Verlinde's formula

Si;,

show how automorphisms of the extended Dynkin

diagrams can define automorphisms of the fusion rule algebra. An application of this fact can be found in [39J.

• Exercise 6.10 the d-coefficient, and gauge invariance. How does d transform under the gauge transformations of rescaling the chiral vertex operators? Show that the equations for d are gauge invariant.

• Exercise 6.11 Modular

invariant~

for the rational

below, the Gaussian model at radius squared R2 =

fq

toru~.

As we will see in section 10

has a chiral algebra which depends

only on the quantity pq. Compute the automorphisms of the fusion rule algebra of the rational torus and show that they define the different models for which pq = p' q', but p/q

of. p' /q'.

67 The analogy between conformal field theory and group theory continues to hold for the combination of left movers with right movers. We can add to the table at the end of section 5 a few more rows: Functions on the group

Physical fields

Product of functions on the group

Operator product expansion

Average over the group

Physical correlation function

of a product of functions

• Exercise 6.12 Analogy with group theory. Explain the table. Show that it corresponds to the diagonal theory. The equations for the ope coefficients d can be interpreted as defining a metric [24] on the vector space of the conformal blocks. Therefore, if all the d's are real and positive (and therefore we can pick the gauge d = 1), the vector space of the conformal blocks is a Hilbert space. This interpretation will play an important role in the following sections where this Hilbert space will appear in the quantization of a quantum mechanical system.

68 1. 2D Duality

VS.

3D General Coordinate Invariance

Many people have noticed that RCFT's lead to knot invariants [20][40J [41 J [27][42J [43J. One way of producing knot invariants is to view the B matrices as "transition amplitudes" of conformal blocks, then defining an appropriate trace (Markov trace) on these amplitudes the resulting polynomials are, in fact, knot invariants. There is an alternative formalism, used in [40J and elaborated upon in [42][43J which dispenses with the need for a trace at the cost of introducing some new moves. With these new moves the knot invariant becomes the transition amplitude for proceeding from the "null block" to itself with an intervening knot projection. We will present these results from our point of view using the formalism of the previous sections. Consider the planar projection of a knot from S3, e.g.

A projection of a knot on a plane We assign a number to this figure by using the graphical formalism described above. For this, we label every line by a representation of a chiral algebra and also label the areas bounded by the lines by such representation. We assign factors of B to

69

\ 1

~ J

Graphical rules for computing a knot invariant The knot that we consider is a "framed knot." It looks like a ribbon and hence

A non-trivial operation on a framed knot The operation in the figure corresponds to a factor of e 27fi Jl., in the knot invariant. We also need to introduce two new operations on lines for pair creation/annihilation:

.0 Pair creation and annihilation moves The factors for these operations are determined by requiring that:

70

/

/

(2. )

(3.) Consistency conditions on pair creation and annihilation We make the anlatz

and deduce from the first consistency condition that

Q;{3;

1 =-

F;

Since for a closed graph there is always an equal number of Qi and {3;, we can set, without loss of generality,

Q;

= {3i =

)Pi'

This result leads to a new interpretation of Deligne's condition discussed earlier. It is simply the requirement that the value of a circle is a trace. Hence it should be an integer in group theory.

71

o

cl;..., R· J = J

fl·

Deligne's condition In RCFT it is the relative dimension, as explained in the above. We will see below how this follows from the three-dimensional viewpoint .

• Exercise 7.1 No more

con~idency condition~.

Show that consistency conditions (2)

and (3) are automatically satisfied by using the polynomial equations discussed above and this value of

Ah and elk'

The non-trivial problem in knot theory is to prove that this procedure leads to a knot invariant. In other words, different projections of the same knot to two dimensions lead to the same result for the knot invariant. From the discussion in the previous sections and these exercises, it is clear that the polynomial equations guarantee this fact and we indeed find a knot invariant from every RCFT .

• Exercise 7.2

Reidemei~ter Move~.

must check the Reidemeister moves

In the combinatorial approach· to knot theory one

72

\

I.)

F

I

2.) 2>. ) The three Reidemeister moves. Check these using the above formalism. Note that the first move is only satisfied up to phase. This may be fixed by discussing framed links or by introducing the writhe, following Kauffmann [44].

. *

The analysis can easily be generalized to graphs with vertices, which are the analogs of the fusing move of conformal field theory. Define fusing and defusing moves

~

1

11'\

1\

k

-/

=

F.

1 II

Fusing and defusing

k] =f)1c [r>t le1 . F rI1

1

1n

t1. J(

-1.,0

-

j ~ ',0(.

[

-1... (!

3j symbol for coupling three spin 1 representations. and we will denote the Racah coefficient by Aj. Clearly the above formula may be regarded as defining an algebra for the To operators, the structure constants being defined by the 3j symbols for three spin 1 representations and the Racah coefficient Aj. That is, we may write:

L [~~1-'11] T{3T-y = AjT., {3,-y

For example, for Uq (sl(2» one may easily compute:

q-l/2T+To - ql/2TOT+ = AjT+ T+L - LT+ = (ql/2 - q-l/2)T:

+ AjTo

q-l/2ToT_ - ql/2T_To = AjT_ for any value of q. This is precisely the algebra derived in [62J. The reason for this is that graphs are computed with quantum Racah or 6j symbols. But, upon analytic continuation

93 away from

iqi

= 1 the 6j symbols have large spin limits which are precisely 3j symbols.

More precisely we have [40]

. = I t.ma.--+oo

= [2j

{a+(1 j

1 [1

+ Ijl/2

-(1

Thus Witten's lassoing and limiting procedure produces the algebra of 3j symbols.

94 9. Chern-Simons-Witten gauge theory - Quantization The discussion in section 7 was quite general. It can be made much more explicit in a particular field theory - the CSW theory[27]. This is a particular example (we will later mention a conjecture that this is essentially the only example) of a topological field theory. The theory is a gauge theory based on the gauge field A = A~ TO. dz" in some Lie algebra 9 with action

s= ~ r Tr(AdA + ~A3) 411" }y 3

for a three manifold Y. For simplicity we limit ourselves here to SU(N) gauge theory with a trace in the fundamental representation (TrTo.T b = _5 Gb ). Clearly, the action is independent of the metric on Y. To prove that the theory is indeed topological, one needs to show that the measure of the functional integral is also independent of the metric. In what follows, we will assume that this is the case 3 • Perhaps the easiest way to understand the theory is by canonical quantization. Suppose we have a Riemann surface

~

and consider the theory on the 3-dimensional manifold

Y=~xlR..

If we canonically quantize the theory we obtain a space of physical states ciated to the surface

~.

H(~)

asso-

Witten showed that these states have a natural interpretation in

terms of the WZW model for g-current algebra at level k. Specifically: vector space of conformal block for partition function on ~ surface pierced by Wilson line in Representations iI, ... jn

=>

conformal blocks for n-point function on HE for n fields in the representations: iI ... jn

Moreover for 3-manifolds interpolating between two surfaces gives a transformation H(~d

---> H(~2)'

~l

and

~2

the path integral

Witten shows that these transformations are

3 In [63] Witten showed that the existence of the central charge in two dimensions is related to some dependence on the metric on Y - the theory depends on the "framing on

Y".

95 just the duality transformations on the space> of blocks. Why is it true? We will explain these matters in a simple physical way. Choose Ao = 0 gauge: If I: has no boundary then

S = -k 471'

J"f·1TrA·-A· d • dt

1

We then have a first order Lagrangian and therefore, the phase space is the space of gauge fields on I:. The symplectic structure on this space leads to the commutation relations

where

J O(2)(Z -

w)~z = 1. It is convenient to pick a complex structure

T

on I: and to

write

The wave functions in holomorphic quantization are holomorphic functions of A., .,p =

.,p(A.). The Hilbert space is the space of all these functions. The physical space is the subspace of the Hilbert space which is invariant under the Gauss law . • Exercise 9.1 Gau" , law. Show that 'k U(f) = ~

471'

J

Tr(eF)

generates an infinitesimal gauge transformation by

f:

[u(f), A] = -Df [U(fl),U(f2)] = u([el' f2l)

By integrating U(f) the operator generating a finite transformation 9 = e< is

U(g) = e"«) 50

96 Now how does it act on physical states? We certainly must have:

to find

I,

we impose the group law:

U(h)U(g) = U(gh) and find:

I(Ajgh) = I(Aj h)

+ I(A\g)

mod 27rik .

The solution is:

So

This is the key equation. From it we may get the independent physical states as follows. Physical states are invariant under the Gauss law - so we are looking for linearly independent solutions to the equation

Now, given any functionaltPo we can generate such a solution by

..pl'hy. = i.e. we can write:

tPl'hy.(A.)

=

J

DgU(g)..po

J

Dg

eikS(g;A ..

O)..po(A~)

.

We will now carry this out for three examples: E = T2, the torus; E = 52 pierced by Wilson lines and E = Disk.

97

From general principles we expect that Hr. will be the space of characters of the affine Lie algebra. The easiest thing to do is choose a complex structure z = .,.1

+ T.,.2

so we

represent the torus by a parallelogram as usual. Define A. = T'!.'~:2. So

(In the equations above the factor ImT was in the definition of the delta function.) Now we use a basic fact: we can always gauge A. to the constant Cartan:

with h in the complexification of the gauge group where a is constant in the Cartan sub algebra. So - by the Gauss law it suffices to know the values ,p[a.] because ,p[A.] = e-ikS(h, ..

,O),p[a]. Now if we take the family of testfunctions for

J~l where

J

in the Cart an subalgebra,

,p!(A) = e* !TrA.J then the corresponding physical states are

where 5(g, A., A!) is the gauged WZW action:

J -~~ J

5(g, A., AI) = ik 411"

Trg- 1 8gg- 1 ag Tr[Ag- 1 ag

+ ikr

+ A8gg- 1 + gAg- 1 A -

The value of this path integral is well-known, it is just

where

,p).(a) = e-

l i = .. ' h

-ilmT

X>.(r, -h-a)

AA]

is a constant

98 where x~ are the Weyl-Kac characters. Thus - as we vary

J

we sweep out a space of states

spanned by the characters . • Exercise 9.2 The Weyl Alcove. Consider quantization of the Chern-Simons-Witten gauge theory on the torus with a real polarization, that is, t/J = t/J[AI(:z:)]. Take the gauge group to be connected, simply connected and simply laced. a.) Derive the Gauss law and show that t/J has support on those Al which are components of a fiat connection. Thus the wavefunction is determined by its value for Al constant and in the Cartan sub algebra. b.) Show that the Gauss law for the gauge transformations preserving the constant Cartan force t/J to be a periodic delta function whose support is at Aweight /W ~ where

Aweight (Aroot)

kAroot

is the weight (root) lattice and W is the Weyl group. The elements

of this coset are in a natural one-to-one correspondence with the integrable highest weight representations of level k of the associated Kac-Moody algebra .

• Exercise 9.3 Moduli Space of Flat Connection3. a.) In his original paper Witten first imposed the constraints and then quantized the resulting phase space. Show that this phase space is just the moduli space of fiat connections on

~.

b.) A fiat connection is characterized by its holonomies, up to conjugation. Show that the real dimension of the resulting phase space is (2g - 2)dimG for the gauge group G on a surface of genus g

> 1.

c.) Use the WKB approximation to show that the number of physical states grows as k(g-l)dimG

and compare with exercise 3.8.

99 I::= S2 punctured by Wilson lines The Wilson lines for finite transition amplitudes are

where the Wilson line carries some representation j and m /' mk are states in the representation j as in the following figure

two sphere's with Wilson lines Since the Hamiltonian of the theory is zero, finite time amplitudes are the same as overlaps of wavefunctions. So we see that the wavefunctions in the case with punctures are simply wavefunctionals valued in the tensor products of representations:

m, We know how Wilson lines transform under gauge transformation, so it is clear that the action of the Gauss law is just:

As before, we may use the basic fact that we can gauge away A. Le. A. = -8.hh- 1 Thus physical wavefunctions are completely determined by their value at A. == 0:

fphY'[A. Now

fo

= 0] =

J

Dg

eikS(g)

®i p(g-I(Pi))fo( -8gg- I ).

is an arbitrary functional of the holomorphic current, so, by the holomorphic KM

Ward identities we obtain a basis of physical states:

100

From this example we see that the transition function given by the path integral for braided Wilson lines is indeed the appropriate duality matrix . • Exercise 9.4 Knizhnik-Zamolodchikov equation8. a.) From the discussion of wavefunctions above write the Gauss law for the case of the sphere with sources as:

u(~) =

!!.-JTrfF+ ""Ttfa.(P;} 4rr ~ .

We would like to see how the wavefunctions change as the positions Pi of the sources change. b.) Show that

for 0 = pi(T")A;(P;}. c.) Writing physical states as path integrals show

8ii[Aj Pi]lA=O = Pi(T")A;(Pi)

For simplicity (and WLOG) take

J

ik(S-t- jTrA9-189) ®i Pi ( g-1(Zi,Zi) ) ",-oIA=O

Dge'

io to be a constant tensor.

d.) We must define the singular product of operators at Pi. We do this by point splitting, then making an appropriate subtraction, which will be uniquely determined from self-consistency. Use the conformal field theory operator product relation (for a proof see

[23J.): ja.(()Pi(T")g-1(Zi,Zi)

=

(~iZi + (k + h)8i g- 1(Zi'Z;) + 0(( -

z;}

where h is the dual Coxeter number and C i = C 2 (Vi;) is the Casimir of the representation

Vi;, to deduce that we must define the singular product of operators by

: Pi(T")j"(Zi)Pi(g-1(Zi,Z;)): ==

J~[Pi(Ta.)ja.(()Pi(9-1(Zi,Zi)) - ;; Ci _ _ h8i9 - 1(Zi,Z;)] ~

- z"

101 e.) Plugging in this definition and using the Kac-Moody Ward identities for that physical states satisfy the Knizhnik-Zamalodchikov equations [231

J show

102

I;=Disk=D. Finally, we consider the case of I; with a boundary. In the case where I; is a disk, HE is the chiral algebra of the theory[27] . We consider the path integral on D x

m..

Let us try to "evaluate" the path integral

J

DA e''s vol G .

In order to do that we must decide on the appropriate boundary conditions. These are determined by demanding no boundary corrections to the equations of motion:

55 =

~

{ Tr(MA) 411" J8DXR

+ 2k 11"

{

Tr(MF)

JDxR

.

So we choose Ao = 0 on the boundary. The gauge group appropriate for these boundary conditions is

G = {g

:D x

m. . . . GIgl8DxR = I}

Now let's decompose A into time and space components:

so

Next, do the integral over Ao giving

We can solve this to get

A = Ju

U- 1

for U : D ...... G, since D is simply connected. Moreover, one can argue that there is no Jacobian

DA5(F) = DU

103

• Exercise 9.5 No Jacobian. Show that in the change of variables

J

DA.5(F)O(A) =

J

DUO(-U-1dU)

for gauge invariant functionals O. Finally, we plug

A=

-dUU- 1 back into the Lagrangian to get:

where'{) is the angular coordinate on the rim of the disk, and r stands for the Wess-Zumino functional. As is well known, this does not depend on the values of U on the interior - so we can divide out the volume of the gauge group to get the path integral

where

U : aD x ffi.

-+

G.

Quantization of this system is well-known to give the chiral algebra of the WZW model [2] . • Exercise 9.6 A Di,le with a 'ouree. Work out the analogous change of variables for the case of a disk with a source in a representation A. Represent the source by a quantum mechanics problem with the action [64]

J

dtTrAW-1(ao + Ao)w(t).

Integrate over Ao to find a constraint on

A.

Show that the holonomy of the flat connection

around the source is determined by the representation of the source. Find the effective action on the boundary of D x ffi.. Its quantization leads to the representation A of KacMoody [65]. Use this Lagrangian to find the set of A'S which lead to inequivalent effective field theories and hence to the set of integrable representations of Kac-Moody.

104

• Exercise 9.7 Two ~ource~ on 52. Repeat the analysis of the previous exercise for this case and prove that the Hilbert space is one dimensional if one source is in the conjugate representation to the other source and it is empty otherwise.

From these remarks we see. that we can also learn about descendents from the 2 + 1 dimensional viewpoint. Moreover, note that the quantization on the disk allows us to define a 2 + 1 dimensional analog of a chiral vertex operator. Consider the following solid pants diagram threaded by three Wilson lines joined together with an invariant tensor a:

Solid pants diagram The different boundaries are meant to reflect corresponding boundary conditions on the gauge field. From the above exercises we see that the path integral defines an operator from 'H. j ® 'H.k to 'H.;. Moreover, from general principles of CSW theory this operator has the braiding and fusing properties of a chiral vertex operator. Thus it is natural to suppose that it

i~

a chiral vertex operator at some canonical value of z, but this has not yet been

demonstrated. Not all aspects of RCFT have been understood from the 2 + 1 dimensional viewpoint. We end with the following exercise, part (c) of which is an open problem:

• Exercise 9.8 Nontrivial Modular Invariants. a.) Show that the natural inner product on quantum wavefunctions for CSGT with connected and simply connected gauge group defines a pairing of representations corresponding to the diagonal modular invariant. b.) Give the 2 + 1 dimensional interpretation of the unitarity of the matrix 5.

105 c.) Find a natural interpretation of the nentrivial modular invariants especially exercise 6.5 from the 2 + 1 dimensional viewpoint.

106 10. Chern-Simons-Witten gauge theory - Other RCFT's In the previous section we saw how KM theories can be reconstructed from connected and simply connected gauge groups in three dimensions. It is therefore natural to ask if other RCFT's can be similarly related to CSW theory for different gauge groups. Here we will show that all known examples of RCFT arise from CSW theory for some gauge group. Among the other known RCFT's there are three kinds: 1. Extended algebras. Examples include the rational torus, chiral algebras of D"

modular invariants(W-aigebras), and other modular invariants obtained by orbifolds of WZW theories. 2. Coset models. Examples include various discrete series 3. Orbifolds of the above. The holomorphic part of each of these theories can be given a CSGT interpretation:

1. Extended KM algebras

Most chiral algebras include high spin fields. Some of them can be obtained by adding extra holomorphic operators to a KM algebra. Theories not finitely decomposable in terms of KM or Virasoro representations might be finitely decomposable with respect to this larger algebra. For example, to form extended' algebras one usually uses the "spectral flow" transformation associated to automorphisms of extended Dynkin diagrams. Thus, if we wish to extend level kg-current algebra we begin with 8 E Center(G) and write 8

= e2 ..,.

for some weight vector p,. (For simplicity we take G

= SU(n), the discussion can

be generalized.) The integrable level k representations are given by the points in the Weyl alcove

Aweight/W ~ kA root . The transformations A --+ A+kp, is equivalent, via the affine Weyl group to a transformation

A

--+

p,(A) of highest weight representations. For example for SU(2) level k the spin j

representation transforms by j

--+

k/2 - j.

Equivalently, we may consider the change in the currents obtained when the boundary

107 conditions are twisted by the multiple-valued "gauge transformation"

(10.1) which acts by

(10.2) In modes we have:

(10.3) L .. ---+L .. + 0; H~

+ ~k028n,0

(E,H correspond to simple roots and Cartan elements, respectively) and in the special case of SU(2) this becomes: 3

3

J .. ---+ J ..

J: L ..

---+ ---+

k

+ 28n o

J:±l L ..

(10.4) 13

k

+ 2J.. + 28..0

In general, for any subgroup Z C Center(G) we can "mod out" by this action thus obtaining the extended chiral algebra

A well known example is the rational torus. The toroidal c = 1 model with a boson 4J ~ 4J + 211"R has a U(l) KM symmetry generated by J

= 84J

when R2

= lq

is rational

there are extra holomorphic fields generated by

which generate a large algebra. It can be shown that this process of extension of the algebra:

corresponds in CSW gauge theory to a change in the gauge group. Namely we can have an Abelian gauge field wi th action

S = ik JAdA 811"

108 but it makes a big difference if the gauge group is ill. or ill./Z = U(l).

If the gauge group is ill., the allowed gauge transformations are A e:Y

--->

--->

A - dC(:l:) where

ill. is a well-defined function. In that case:

1.) We can scale k out of the action 2.) The observables in the theory are the Wilson lines

Recall that the value of a defines a representation - this corresponds to a continuously infinite set of representations in CFT. 3.) No two Wilson lines are equivalent. On the other hand, if the gauge group is U(l) then around non contractible cycles e is only well-defined modulo 21!', and this leads to some consequences:

1.) The theory only makes sense for k = 0 mod 4 2.) The observables are

W,,(C) = ei " fc

A

neZ

3.) Two Wilson lines can be equivalent

• Exercise 10.1 Level k U(l) Current Algebra. a.) Compute explicitly the expectation values of Wilson lines in S3 for the abelian case:

J

DAe 14 h

J AdA IIi"; J: A = e J'c; .

e:l:p [21!'i k "~ ninji)ij ]

1

'&,1

where i)ij is the linking number. i)ii is ambiguous-but may be regularized and defined up to an integer.

b.) Show that the cross terms are invariant under the change n

--->

n

+ ~.

Show that

the invariance of the self-linking number requires k = Omod 4. c.) Perform a (singular) gauge transformation A

--->

A

+ d¢>

variable around some Wilson line. Show that this changes W"

--->

where t/J is an angular

W

1 L' "+2~

This illustrates

109 how changing the gauge group from lR to U(l) = lR/Z brings about an identification of Wilson lines. d.) If k = 4N we refer to the corresponding OFT as U(l)N, "level N U(l) current algebra." Show that the conformal field theory is just the holomorphic part of the rational torus R2

= p/2q where pq = N.

e,) The Wilson line W f which is a non· trivial operator if the gauge group is lR behaves like the identity operator when the gauge group is U(l). The reason for this is the following. In the U(l) theory one needs to sum over U(l) bundles. The non-trivial bundles can be characterized by an insertion of an 'tHooft operator [66] in the functional integral of the lR theory. Using part c of this exercise, show that the 'tHooft operator is equivalent to W t. Since we have to sum over the insertions of such operators, the value of the functional integral is not modified if we add another one. Hence, this operator behaves like the identity operator. The two dimensional analog of this is the fact that the representation ~ eztendJ the lR KM chiral algebra. This field becomes a descendent of the identity operator (under the larger chiral algebra) and its conformal blocks are the same as those of the identity. f.) Show that the above considerations extend to any even integral lattice. g.) Quantize the theory by canonical quantization on T2 as in the previous section. Find the different states as the different representations of U(l)N and write their wave functions in terms of theta functions of higher level [67]. h.) Quantize the theory on a manifold with boundary. Find the extended chiral algebra by quantization on the disk (hint: because of the boundary conditions, there are non-trivial bundles corresponding to the insertion of ~ in the lR theory) and the different representations by quantization on a disk with a source. i.) Show that the center of A(U(l)N) is simply 71../2N71... (Hint: We normally think of the gauge group of a U(l) gauge theory, which is generated by

for smooth functions

E

as an abelian group. However we now allow functions like

Ep

~

tP

110 for t/J an angular coordinate centered at any point P. Show that

so that the group becomes nonabelian. Note that the elements of the center are in one-one correspondence with the representations of the rational torus chiral algebra.) Interpret the existence of this center from the two dimensional point of view (hint: the chiral algebra contains charged fields) .

• Exercise 10.2 G = SO(3) = SU(2)j71.2 a.

Show that the only representations which survive have odd dimension. Show

moreover that to avoid global anomalies, or to have the extending Wilson line be invisible we must have k = Omod4. b. Show that by the singular gauge transformation we can prove equivalence of the Wilson lines

c. Show that the Wilson line W k / 4 is in fact not the simplest operator in the theory, rather we have W k / 4 = 0+

+ 0-

where the operators

o±

cannot be simply expressed in

terms of Wilson lines. d. Find an expression for

o± in terms of SU(2)

theory. (Hint: consider a three point

vertex of Wilson lines with one in the representation kj2.) Quite generally one can show that all known extended algebras are obtained from 3 dimensional CSW gauge theories by changing the gauge group by G~GjZ

where Z is a subgroup of the center of G. In going from G to

G=

G j Z, three changes in the possible representations take place:

a. Selection rule: of the representations of G current algebra only those which are invariant under Z should be kept.

111 b. Identification: different irreducible representations of G related by the spectral fl ow operation are combined into one

G irreducible representation.

c. Fixed point: if the spectral flow has a fixed point, there are different

G representa-

tions which are the same as G representations. These three rules generalize the three parts of the previous exercise . • Exercise 10.3 The three

rule~

from canonical quantization. Derive these three rules

from canonical quantization on the torus. Hints: 1. Rule a follows from gauge transformations which wind around one cycle 2. Rule b from gauge transformations which wind around the other cycle. 3. Rule c is the most subtle. Twisted bundles on the torus are labeled by the subgroup Z used to divide the universal cover to obtain G. These bundles may be defined by cutting

out a disc and using the transition function g( .. is A). = ).2\~~~)' If the spectral How is generated by the representation 1', the extending representation is kl' and its dimension is

AklJ.

= k~\kk'++"~P). The condition on k is that this number should be

an integer. The same result has been obtained by other considerations in [69J.

2. Coset models G / H They may be obtained as follows: we take gauge fields

A B , Aii'eLie(G)

ii denote directions in

Lie(G)/Lie(H)

BGELie(H)

and action

We must be careful to take the gauge group (G x H)/Z where Z is the common center of H embedded in G. To see that this prescription is correct consider the quantization on the disk D x JR, and let us reconsider the boundary conditions. Variation gives

8S =

~:

J

8DxR

Tr(5AA) -

~:

J

8DxR

Tr8BB

+

bulk terms

113 One possibility is to choose Ao

= Bo = 0 which leads to a G

x H theory. However, when

He G and k2 = lkl (l is the index of the embedding) we may choose instead the boundary conditions: A~ = BQ

{ Au = 0 . Performing the change of variables we had before we write (we have chosen l

1 for

simplicity)

A = -dUU 1

B = -dVV- 1 and get, as before

J

DJ..DU DV exp [ikS..,z..,(U) - ikS..,z..,(V) +ik

J

Tr>.(8",UU- 1 -8",VV- 1 )]

where>. is a Lagrange multiplier enforcing the boundary condition AG = BG. Making the change of variables U path integral

->

gV, -8",VV-l

->

a"" and>'

->

at we get the

J

dadgeikS(g,G.p,a,)

which is the gauged WZW model, which is well-known [70] to be the path integral representation of the coset models. Actually, it is quite easy to see why this must be so. The phase spaces are the coadjoint orbits of the pair of (; and

II

representations (A, >.):

(LG/T) x (LH/T)* which, upon quantization give the space of states: 'H.A ® 'H.~. Now we may impose the fir3t cla33

constraints: 1f'H(8",UU- 1 )

-

8", VV- 1 (1f'H is a projection from G to H) which is an

H-current algebra with k = 0 to obtain the physical states:

where the final symbol is the space of states in the coset model, defined by the decomposition 'H.A = Ef7>. 'H.A,>. ® 'H.>..

114

• Exercise 10.4 Ezample

0/ a

Co~et.

a.) Show, using CFT, that the coset model U(l)N x U(l)MIU(l)N+M for the case that N,M have no common factors is equivalent to the rational torus U(l)L for L =

NM(N+M). b.) Consider the expectation values of Wilson lines in S3 for the action:

N JAdA+ M JBdB _ N +M JCdC

2~

2~

2~

where A, B, C are three abelian gauge fields. Show that the expectation value is consistent with the result of part (a). c.) Show that the quantization of this theory on T2 leads to the correct answer only . 1·f the gauge group IS

U(l)XU(lL> 4 [1].

Another class of 2d lattice models is provided by the solid-on-solid (SOS) models. These involve an integer "height" variable h j E 71. at each site of a square lattice, and a local interaction among heights. Such models describe the position of the interface

134 between two different media, with the interaction energy taking into account the surface tension. At high temperature, it costs little for the surface to fluctuate wildly, whereas at low temperature the dominant contributions to (1.1) would come from flat surfaces, i.e. with hi independent of the site i. A subclass of these models, known as restricted-solid-on-solid (RSOS) models, involve a height variable that takes only a restricted set of values, e.g. fj = 1, ... , m. Exactly solvable versions of these models were constructed in [12]. There the heights at nearest neighbor lattice sites (ij) are constrained to satisfy f j =

fj

± 1, and the

interactions are taken among the heights at the four sites around each plaquette of the lattice. Since the partition function can be written as a product of the Boltzmann weights associated to each face, these models are frequently termed "face" or IRF (for "interactions-round-a-face") models. The weights are defined so that each model has a second order phase transition at It self-dual point. At their critical points, these models were noted [13] to provide particular realizations of the c

< 1 discrete series [14,15] of

unitary representations of the Virasoro algebra, with central charge c

=1-

6

m=3,4, ....

--;---::7

m(m+ 1)

(1.3)

The central charge c is the coefficient of the leading singular term in the operator product expansion of the energy-momentum tensor with itself,

c/2

T(z)T(w) = (Z-W)4

2

1

+ (z_w)2T(w) + z_w 8T(w),

(1.4)

as will be discussed further in section 2. (In general the continuum limit of a lattice model, describing the physics at distance scales very large compared to the lattice spacing, will be a massive field theory. At a critical point (2nd order phase transition) the long distance physics is described by an effective massless field theory, the lack of mass gap following from the presence of fluctuations on all length scales in the lattice theory. In 2 dimensions, as already remarked, the effective field theory is moreover a conformal field theory.) The above constraint, fj

= fj ± 1, on nearest neighbor heights can be diagrammat-

ically represented as

1

2

m-l

m

0-0- - - - --0--0 ,

(1.5)

where the nodes are labelled by the values of the height variables, and a link between two nodes indicates that those values are allowed at nearest neighbor sites. We recognize (1.5) as the Am Dynkin diagram. Furthermore the critical partition functions

135 of these models defined on a torus are naturally labelled by the A series in the ADE classification of modular invariant combinations of Virasoro characters given in [16]. For these reasons, the models of [12] are frequently termed the Am models. (More specifically, minimal series modular invariants are labelled by a pair of affine su(2) invariants, which in turn are associated to simply-laced lie algebras. The ABF models are labelled by the pair of diagonal invariants (Am, Am-I)') In [17], generalizations of these models based on the Dm and E 6 ,7,8 Dynkin diagrams were constructed, and indeed their critical toroidal partition functions realize the remaining D and E members (i.e. the (A, D) and (A,E) invariants) in the classification scheme of [16]. After some further preliminaries we shall recapitulate the construction of [17] in subsection 1.4. The notion of duality also generalizes [18] to these models, where it plays a role similar to that of the orbifold in conformal field theory. A final class of models we mention here are the "ice"-type models. Ice, we recall, may be viewed as a lattice of H 2 0 molecules. A simplified two dimensional version consists of Oxygen atoms at the sites ofa square lattice and Hydrogen atoms on the links. If we allow the H atoms to reside at just two positions, closer to one or the other end of the link, their locations can be equivalently specified by placing an arrow on each link. Due to charge conservation it is natural to allow exactly two H's on the four links incident at each site to be close to the 0 atom there. This allows the six following arrow configurations at each site,

xx xx

X.X ..

...

1,2 = a

3,4= b

(1.6)

...

5,6 = c

where for later convenience we have rotated the vertices by 45 0 • The partition function is defined as the product, over the sites of the lattice, of the Boltzmann weights associated to each vertex. Such models are generically known as "vertex" models, and the model with vertices in (1.6) is known as the six-vertex (6V) model. Ordinarily one imposes a symmetry under reflection of all the arrows so that there are only three distinct weights a, b, c, each for a pair of vertices, as depicted in (1.6). When the weights a and b are

set equal, it is also called the F-model (and serves as a model ferro-electric). The 6V model is critical in the region

I( a

2

+ b2 -

c2 )/2abl ~ 1, in which its correlation functions

decay algebraically.

If we add non-vanishing Boltzmann weights for the source and sink vertices,

xx v

7,8 = d

(1.7)

136 we arrive at the eight-vertex (SV) model which provides an off-critical extension of the 6V model. Baxter has found an exact expression of the bulk free energy of this model in the infinite volume limit (see e.g. [1]), and for this reason the model is referred to as "exactly solvable". On its critical line, it possesses continuously varying critical exponents.

1.2. Algebraic reformulation To treat some of the models mentioned above in somewhat more detail we begin by reformulating the Potts model in a more algebraic form (following [19,1]). For definiteness, we take a square lattice with M rows and N columns,

i time

M

(loS)

.

N

and impose cylindrical boundary conditions: last row identified with the first in the 'time' direction (i.e. periodic), and no identification along the 'space' direction (i.e. free). As described earlier, we take site variables Ui = 1, ... ,Q, but now generalize to an anisotropic model with partition function

' " -E/T

Z=~e

",'

=~e

{,,}

E(i,j).

Kv C,,' "i

+ E(i,j). Kh C,,' "i

,

(1.9)

{,,}

where ( )h,v denotes nearest neighbors in the horizontal and vertical directions. \Ve wish to write this partition function as a trace of a product of transfer matrices. The transfer matrices are defined to act on the QN dimensional vector space 1i (CQ)@N of vectors

Iuo ... UN-i)'

=

These represent the Hilbert space of constant 'time'

sections, which we choose as the horizontal rows of the lattice,

w ~------~v--------~

N

v

(LlO)

137 The QN

QN transfer matrices V and W, acting on this space with indices u {uo, ... , uN-d and u' = {u~, ... , uN_l}, are defined to have matrix elements X

v:(f,U' = e

K

"N-1 h LJi=O

8'" ,"'+1

n

N-l

c Ou;,crj

(1.11)

j=O

V and W implement respectively the horizontal and vertical couplings, as indicated in (1.10), so that

(1.12) The Hilbert space is analogous to a time slice in a path integral, and the transfer matrices propagate the system through time. To abstract the algebraic content of these matrices, acting on each tC Q we choose a convenient basis of matrices, 1 ~ i,j,a,/3 ~ Q ,

that satisfy E;j Ekl

Acting on 'H.

1~i

~

= (tCQ)®N

(N

= number

= 8jk

Ei/ .

of horizontal links), we define the matrices e;,

2N -1,

(1.13)

Q

e2j-l

=

Q-l/2

1

.

= 0,

where we have written the components va of the eigenvectors above the corresponding node of the diagram. (These are easily checked by noting from the analog of (1.28) that the eigenvalue multiplied by any given component should equal the sum of the components on neighboring nodes.) The largest eigenvalue is (3 = .j2, in accord with the identification of the Ising model as the (Q

= (32 = 2)-state Potts model.

(Note also

from (1.24) and (1.3) that m = 3 and c = 1/2.) In (1.30b), we have used the components of the eigenvector with largest eigenvalue to define the statistical weights of the model. What role do the other eigenvectors play? We shall see later that they can be used to define scaling operators in the theory whose Boltzmann weights are specified by the sa,s. Alternatively, the components va of some other eigenvector can be used to define weights as in (1.30b), but they are no longer positive definite and the resulting lattice model is not unitary.

These can be used

to define critical models that realize[17,28,29) examples of the non-unitary minimal conformal field theories of [30). The choice of eigenvector in defining the weights is thus closely related to the choice of asymptotic ground state on the lattice, and to the background charge at infinity in the Feigin-Fuchs prescription for the minimal conformal field theories. In closing this subsection, we remark briefly on the formulation of these models In

"Lagrangian" form. From this point of view, the partition function Z is defined

148 equivalently by assigning a "weight" c

(1.35)

W(db)=bOd ac , a

to each face of the lattice, where a, b, c and d are the heights going cloekwise around the face starting from the lowermost comer. It takes the form

L II w~~b)

Z =

(1.36)

,

{heights} fa.ces

where the sum is over all allowed height configurations and the product is over all faces. (The sum, of course, depends on the boundary conditions.) From (1.30b), we find that the Boltzmann face weights take the explicit form (db)

Wac

";sasc = vac + \,e (b) ac = vac + \,vbd - - bs C

I:

C

I:C

The earlier cited Perron-F'robenius theorem, insuring that the

.

sa

are all positive, assures

positive Boltzmann weights, and a physical model. 1.6. Origin of integrability

As we have discussed, the TL algebra (1.14a-c) is a slightly more restrictive version of the Heeke algebra (1.14a, b},(1.23). Face transfer matrices that obey the Heeke algebra define weights that obey the star-triangle, or Yang-Baxter, relations. We show directly here how the Hecke structure leads to the attendant commuting transfer matrices. Let

IL

= I1 j W2j(€)

and T(€') = W2j+1(€') be the products of face transfer matrices along successive rows. Now if there exists an object R such that R 2j+lW2j(€)W2j+l(€') =

T(€)

W2j(e)W2j+l(€)R 2j, then with periodic spatial boundary conditions we would have T(€)T(O = T(€')T(€) .

(1.37)

This is because the periodic boundary conditions allow us to insert an RR- 1, and then pulling one or the other through gives pictorially

R

149 To find an R with the desired properties, recall that wee) = l+e e, w'(e') = 1+e' e. If we take as ansatz R(x)

= 1 + xe, the Yang-Baxter relation Rww' = w'wR requires

By the Heeke relations (1.14a) and (1.23), this reduces to two occurrences of the identity

x

+ e + xee + (:Jxe = e ,

which can always be solved for x in terms of

e, e'. Thus objects ei that satisfy a Heeke

algebra can be used to generate solutions Wi(e)

= 1 + eei to the Yang-Baxter equation.

When the resulting weights obey the star-triangle relations, then, transfer matrices at different values of

e commute as in (1.37), giving an infinite number of conserved

quantities. Models with this property are frequently called integrable, which after some technical assumptions usually means they are 'exactly solvable' in the sense that the free energy per site ('" In Z/ N M) can be calculated in the infinite volume limit.

2. Orbifold conformal field theories To make contact between the statistical mechanicals discussed thus far and the conformal field theory description of their critical points, we give a lightning review here of some necessary aspects of conformal field theory. For a more extensive introductory treatment, see for example [7] (from which much of this section is shamelessly extracted). The reader familiar or impatient with this standard material should feel encouraged to skip directly to section 3.

2.1. Some free bosonic field theory Our model system is one of the simplest conformal field theories, that of single massless free boson, also known as the gaussian model. The euclidean space action takes the form

J -

1 s=ax ax , 271"

where we shall use complex coordinates, z = x

(2.1)

+ iy, on our two dimensional base space,

and the integral is over the complex z plane. The normalization is chosen so that X(z, z) has propagator (X(z, z)X( w, iii))

= -t log Iz - wi·

For solutions of the equations of motion, we find that X(z, z) splits into two pieces, 1 X(z,z) = 2(x(z)+x(z)) ,

(2.2)

150 with only holomorphic and anti-holomorphic dependence respectively. (These are the massless left- and right-moving modes.) On the plane, these pieces have propagators (x(z)x(w»)

= -log(z -

(x(z) x(iii») = -log(z - iii) .

w),

(2.3)

Note that the field x(z) is not itself a conformal field, but its derivative, ax(z), has leading short distance expansion ax( z) ax( w)

=

1

(Z-W)2

(2.4)

+ ... ,

inferred by taking two derivatives of (2.3). The stress-energy tensor T( w) for the holomorphic part of the theory is defined via the normal-ordering prescription T(w)

= -'21 :ax(z)ax(w): ==

1 --2 lim [ax(z)ax(w) z-w

(2.5)

+ (Z-w 1 )2]

Its operator product with itself is easily verified to satisfy (1.4) with c Virasoro generators Ln are the modes in the Laurent expansion T(z)

=

1. The

= EnE71 z-n-2 Ln'

Analogous properties are satisfied by the anti-holomorphic part of the theory. We see from the scaling properties of the right hand side of (2.4) that ax(z) is a (1,0) conformal field. In general an operator (z,z) of conformal weight (h,ii) by definition has a leading operator product expansion

(z,z)¢>(w,w)~(

1)2h 1 z- w (z- iii)2h

+ ...

(2.6)

h and ii are respectively the eigenvalues of the Virasoro generators Lo and Lo acting on an asymptotic state created by the operator . The conventional scaling weight and spin of the state, .6.

= h + ii and s = h - ii,

are determined from the identifications of

Lo ± Lo as the dilatation and rotation generators. The operator: exp iax( w): , for example, has conformal weight h

= a 2 /2, ii = 0, as

can be inferred from the 2-point function (

ei"'X(Z)e-i",X(W»)

= e",2(x(z)x(w»

=

1

(z - w)'"

2

(2.7)

(The first equality is a general property of free field theory, and the second equality follows from the specifically two dimensional logarithmic behavior (2.3).) We see that

151 the logarithmic divergence of the scalar propagator leads to operators with continuously variable anomalous dimensions in two dimensions, even in free field theory. We shall now compactify both the base and target spaces of (2.1). First we take z to live on a torus, 1m z T

T+1

(2.8) 1 Re z o thus defining a conformal field theory on a genus 9 = 1 Riemann surface. The periods

in z are conventionally taken to be z == z

+ 1 and z == z + T,

where

T

is a complex

parameter known as the modular parameter. The torus (2.8) is invariant under different choices of

T

that give the same lattice

of periods. These different choices are related by the group of modular transformations, equivalently the group of disconnected diffeomorphisms of the torus (generated by cutting along either of the two non-trivial cycles, then regluing after a twist by 211"). These transformations are of the form

+b cT+d

(~ ~)

aT

T-+ - - -

E SL(2,71)

(2.9)

(i.e. a, b, c, dEll, ad-be = 1), known as the modular group. (Since reversing the sign of all of a, b, c, d in (2.9) leaves the action on

T

unchanged, the action of the modular group

is actually PSL(2,71) = SL(2,71)171 2 .) A convenient set of generators for this group is given by T : T

-+

T + 1 and S: T

-+

-liT, which satisfy the relations S2 = (ST)3 = 1.

A conformal field theory should not depend on the particular parametrization of the torus, i.e. it should be invariant under these modular transformations. (In general a conformal theory should be sensible on an arbitrary Riemann surface, and thus be modular invariant to all orders. This turns out to be guaranteed by duality of the 4-point functions on the sphere together with modular invariance of all I-point functions on the torus[31]. Intuitively this results from the possibility of constructing all correlation functions on arbitrary genus surfaces by "sewing" together objects of the above form. Thus all the useful information about conformal field theories can be obtained by studying them on the plane and on the torus, one of our motivations for considering the torus here.)

152 We shall also take our theory with action (2.1) to have the bosonic coordinate X compactified on a circle of radius r, X == X the functional integral

+ 27rr.

That means when we calculate

Jexp -S, we need to consider all "instanton" sectors n' 0

boundary conditions X(z

with

n

+ T, Z + T) =

X(z, z)

+ 27rrn'

X(z

+ 1, z + 1) =

X(z, z)

+ 27rrn

.

Since we are dealing with a free field theory, the functional integral is easily calculated by summing over these different sectors and including the detenninant of the quadratic fluctuation operator for each. The result is

Z eire () r

where q

=

J

-S

e

1

1/1/

l(.!!l q 2 2r

00

= =-

'""' L...

+ nr)2 - l(.!!l 2 2r

nr)2

q

(2.10)

,

m,n=-oo

= (exp27rir), and 1/(T) = q14 II::"=l(l -

qn) is the Dedekind eta function.

The result (2.10) is easily verified to be modular invariant. The sums over m and

n in (2.10) correspond respectively to the sums over momenta and winding sectors, and the inverse 1/'s give the contribution of the bosonic oscillators in each such sector. Recognizing (2.10) as tr q L O-l/24 qL o-l/24, we read off the confonnal dimensions (h, h) of the operators associated with winding and momentum states in (2.10) as

_ (12(m +

(h, h) =

2r

nr

)2, 21(m2r - nr)2)

Invariance of (2.10) under the simultaneous interchanges (r

(2.11)

. H

i;:), (m

H

n) results

in the duality (2.12) At the self-dual point r = 1/v'2, we find from (2.11) states with Lo and Lo eigenvalues

t(m ± n)2. For m

= n = ±1 there are two (1,0) states, and for m = -n = ±1 two (0,1)

states. In operator language these states are created by the operators and

(2.13a)

with conformal weights (1,0) and (0,1). (At radius r, the operator under x

-+

x

+ 27rr, so these become suitably single-valued at

eix(z)/r

is well-defined

the radius r = 1/v'2.) ·At

arbitrary radius, on the other hand, we always have the (1,0) and (0,1) operators and

(2.13b)

153 The operators J±, J3 in (2.13a, b) are easily verified to satisfy the operator product algebra J+(z) r(w) ~ J3(z) J±(w)

and similarly for

J±,J3.

e h12(x(z)-x(w»

(

z-w

~ V2

z-w

)2

~ (

iV2

1 z-w

+- 8x(w) z-w

)2

,

J±(w) ,

If we define J± = ~(Jl ± iJ 2 ), then this algebra can be

written equivalently as Ji(Z) Jj(w) =

8

ij

(z-w)2

+ iV2e

ijk

z-w

Jk(w) .

(2.14)

(2.14) defines what is known as the algebra of affine Kac-Moody SU(2) at level k = 1 (level k would be given by substituting 8ij -+ k8 ij in the first term on the right hand side of (2.14)). We see that the circle theory Zcirc(r) at radius r = SU(2) x SU(2) symmetry. The modes

Jj in the expansion

usual SU(2) commutation relations, [Jj, JJ]

=

Ji(z) =

1/V2 has an affine L:n J~ z-n-l satisfy

iV2e ijk Jt, and thereby generate an

ordinary ("horizontal") SU(2) symmetry. 2.2. Orbifolds in general

The action (2.1) possesses the symmetry x

-+

-x. Symmetries of conformal field

theories can frequeritly be used to construct new conformal field theories, called "orbifold" conformal field theories. Orbifolds arise in a purely geometric context as generalizations of the notion of manifolds that allow a discrete set of singular points. Consider a manifold M with a discrete group action G : M iffor some 9 E G (g

-+

M. This action is said to possess a fixed point x E M

# identity), we have gx =

x. The quotient space M/G constructed

by identifying points under the equivalence relation x

~ gx

for all 9 E G defines in

general an orbifold. If the group G acts freely (no fixed points) then we have the special case of orbifold which is an ordinary manifold. Otherwise the points of the orbifold corresponding to the fixed point set have discrete identifications of their tangent spaces, and are not manifold points. A simple example is provided by the circle, M = SI, coordinatized by x with group action G = 71..2 : SI

-+

SI defined by the generator 9 : x

action has fixed points at x = 0 and x = topologically a line segment,

7rT,

-+

== x + 27rT,

-x. This group

and we see that the SI/71..2 orbifold is

o~I

(2.15)

154 If we mod out instead by the action generated by the rotation 9 : x

o

find

--+

0

-+

x

+ 27rr/n,

we

(2.16)

In this case, there are no fixed points and the result is a manifold, the circle at reduced radius r/n (pictured for n

= 6).

In conformal field theory, the notion of orbifold acquires a more generalized meaning. It becomes a heuristic for taking a given modular invariant theory 'T, whose Hilbert space admits a discrete symmetry G consistent with the interactions or operator algebra of the theory, and constructing a "modded-out" theory 'T /G that is also modular invariant [32). Orbifold conformal field theories occasionally have a geometric interpretation as u-models whose target space is the geometrical orbifold discussed in the previous paragraph. This we shall confirm momentarily in the case of the SI/71.2 example. There are also examples however where the geometrical interpretation is either ambiguous or nonexistent. Consequently it is frequently preferable to regard orbifold conformal field theories from the more abstract standpoint of starting from a given modular invariant theory and modding out by a Hilbert space symmetry. (Historically, orbifolds were introduced into conformal field theory [32) (see also [33]) via string theory as a way to approximate conformal field theory on "Calabi-Yau" manifolds. Even before the "phenomenological" interest in the matter had subsided, orbifold conformal field theories were noted to possess many interesting features in their own right, and in particular enlarged the playground of tractable conformal field theories.) Currently, the notion of orbifold plays a role in conjectured[34) classification schemes for so-called rational conformal field theories, which suggest that they are saturated by coset conformal field theories[15) and orbifolds thereof. As will be discussed in subsection 3.8, the orbifold idea also generalizes away from the critical point of many of the lattice face models introduced earlier[18). The construction of an orbifold conformal field theory 'T /G begins with a Hilbert space projection onto G invariant states. It is convenient to represent this projection in Lagrangian form as

1~ILgD, gEG 1 where 9

D

(2.17)

represents boundary conditions on any generic fields x in the theory twisted

1

by 9 in the "time" direction of the torus, i.e. x( Z+ T) = gx( z). In Hamiltonian language such twisted boundary conditions correspond to insertion of the operator realizations

155 of group elements 9 in the trace over states, and hence (2.17) represents the insertion of the projection operator P =

for E9E~g.

But (2.17) is evidently not modular invariant as it stands since under S : T for example we have gn

0,

gO -+ 10.

Under 1 9 so we easily infer the general result

T

-+

T

+n

we have moreover that

-+

-liT

10 -+ 9

9

gO-+ gBhbn

g~

h

under

+b cT+d' aT

(2.18)

T-+ - - -

for g, hE G such that gh = hg. To have a chance of recovering a modular invariant partition function, we thus need to consider as well twists by h in the "space" direction of the torus, x(z

+ 1) =

hx(z),

and define

ZT/G=

L I~I LgO= I~I L h

hEG

gEG

(2.19)

gO·

g,hEG

h

The boundary conditions in individual terms of (2.19) are ambiguous for x(z 1) unless gh = hg.

+T +

Thus in the case of non-abelian groups G, the summation in

(2.19) should be restricted only to mutually commuting boundary conditions gh

hg.

=

From (2.18) we see that modular transformations of such boundary conditions

automatically preserve this property. Moreover we see that (2.19) contains closed sums over modular orbits so it is formally invariant under modular transformations. (In the following we shall consider for simplicity only symmetry actions that act symmetrically on holomorphic and anti-holomorphic fields, so modular invariance of (2.19) is more or less immediate.) We note that the orbifold prescription, changing only boundary conditions of fields via a symmetry of the stress-energy tensor, always gives a theory with the same value of the central charge c. For G abelian, the operator interpretation of (2.19) is immediate. The Hilbert space decomposes into a set of twisted sectors labeled by h, and in each twisted sector there is a projection onto G invariant states. A similar interpretation exists as well for the non-abelian case, although then it is necessary to recognize that twisted sectors should instead be labeled by conjugacy classes Cj of G. This is because if we consider fields hx(z) translated by some h, then the 9 twisted sector, hx(z manifestly equivalently to the h-1gh twisted sector, x(z

+ 1)

+ 1)

= ghx(z), is

= h-1ghx(z). Now the

number of elements 9 E Ni C G that commute with a given element .h E Cj C G depends only on the conjugacy class Cj of h (the group Nj is known as the stabilizer group, or little group, of Cj and is defined only up to conjugation). This number is given

156 by

INil = IGI/ICil,

where

ICil

is the order of Ci. In the non-abelian case, we may thus

rewrite the summation in (2.19) as (2.20)

manifesting the interpretation of the summation over 9 as a properly normalized projection onto states invariant under the stabilizer group

Ni in each twisted sector labeled

by Ci . While we have discussed here only the construction of the orbifold partition function (2.19), we point out that the orbifold prescription (at least in the abelian case) also

allows one to construct all correlation functions in principle (see e.g. [35]). 2.3. 5 1 /7L2 orbifold We illustrate the general orbifold formalism introduced above with the example of a G

=

7L2 orbifold conformal theory based on the 9 : X

--+

-X symmetry of the free

bosonic field theory (2.1). The general prescription (2.19) for the T IG orbifold partition function reduces for G = 7L2 to

Zorb(r) =

~ (+9 + -9 + +Q + -Q)

= (qq)-1/24tr(+)

1

-

(2.21)

2(1 + g)qLoqLo

1 + (qq)-1/24 tr 2.) Deformations

of a conformal field theory, preserving the infinite conformal symmetry and central charge c, are generated by fields V; of conformal dimension (1,1) [37]. To first order, the perturbations they generate can be represented in the path integral as an addition to the action, flS

= flg i J dzdZV;(z, z),

or equivalently in the correlation function of

products of operators 0 as fl( O} = flg i J dzdZ( Vi(Z, z)O). It is clear that deformation by a conformal weight (1,1) operator is required to preserve conformal invariance of the action at least at the classical level. In the case of the circle theory (2.1), we have the obvious (1,1) operator V = aX8X. We see that perturbing by this operator, since it is proportional to the Lagrangian, just changes the overall normalization of the action, which by a rescaling of X can be absorbed into a change in the radius r. The operator V, invariant under X

--+

-X, evidently survives the 112 orbifold projection in the untwisted sector, and

remains to generate changes in the radius of the orbifold theory (2.22). (See [38] for further details concerning the marginal operators in c = 1 theories.) The mere existence of (1,1) operators is not sufficient, however, to result in families of conformal theories. An additional "integrability condition" must be satisfied [37] to guarantee that the perturbation generated by the marginal operator does not act to change its own conformal weight from (1,1). In the case of a single marginal operator V as above, this reduces in leading order to the requirement that there be no term of the form Cvvv(z -

wtl(z - wtl V in the operator product of V with itself. An operator

V that is exactly marginal to all orders generates a one-parameter family of conformal theories. The affine SU(2) symmetry

po~sessed by the circle theory at r

= 1/>12 can be used

to demonstrate that the circle theory at radius r = >12 and the orbifold theory at radius r

= 1/>12 are identical, and consequently that (2.23)

Equivalences such as (2.23) show that geometrical interpretations of the target spaces of these models, as alluded to earlier, can be ambiguous at times. The geometrical data

158 of a target space probed by a conformal field theory (or a string theory) can be very different from the more familiar point geometry probed by maps of a point (as opposed to loops) into the space. To establish the equivalence (2.23), we consider two possible ways of constructing a 712 orbifold of the theory Zcirc{I/v'2). Under the symmetry X _ -X (so that

-x, X _ -x) discussed in detail earlier, we see that the affine SU(2) generators

x -

(2.13) transform as J± x and

J~, J3 _

x by the same amount)

_ J3. The shift X _ X

+ 27r/{2v'2)

(shifting

is also a symmetry of the action (2.1), and instead has

the effect J± _ -J±, J3 _ J3. But by affine SU(2) symmetry, these two symmetry actions are equivalent, one corresponding to rotation by 7r about the I-axis, the other to rotation by 7r about the 3-axis. Modding out the circle theory at radius r by a tl n shift X _ X

+ 27rr/n in general

reproduces the circle theory, but at a radius decreased to r/n (as pictured in (2.16)). From the Hilbert space point of view, the projection in the untwisted sector removes the momentum states allowed at the larger radius, and the twisted sectors provide the windings appropriate to the smaller radius. Modding out Zcirc{I/v'2) by the 712 shift X

-+

X

+ 27r/{2v'2)

thus decreases the radius by a factor of 2, giving Zcirc{I/2v'2),

which by r .... 1/2r symmetry is equivalent to Zcirc (v'2). Modding out Zcirc (1/ v'2) by the reflection X -

-X, on the other hand, by definition gives Zorb{1/v'2). Affine

SU(2) x SU(2) symmetry thus establishes the equivalence (2.23) as a full equivalence between the two theories at the level of their operator algebras. The picture of the moduli space of c = 1 conformal theories that emerges from the arguments reviewed thus far may be found in [38,39,40], with various specific points of interest highlighted.

Part of the motivation for studying c = 1 systems is that

they represent the first case beyond the classification of [14]. For systems with N = 1 superconformal symmetry, the corresponding boundary case between a (classified) discrete series and an (unclassified) continuum lies at c

= 3/2.

A treatment of the

moduli space for this case may be found in [411.

2.5.

rc

SU(2) model8

We now finish providing the background for our treatment of the relation between orbifold models and some of the statistical mechanical models introduced earlier. The remaining conformal field theory aspects of these models have already been treated in relatively pedagogic detail in [39], so we will be mercifully brief in that respect here and turn quickly to aspects of their lattice realizations in the next section.

159 The r

Z2

orbifolding that took us from the affine SU(2)

= .../2 on the

subgroups

r

X

SU(2) point to the point

circle line generalizes. Indeed we can mod out by any of the discrete

of the diagonal SU(2). It is easiest think of this in terms of subgroups

of SO(3) acting simultaneously on the vectors Ji(z),

J\z). Then the generator of the

symmetry group C n , the cyclic group of rotations of order n about the 3-axis, is specified by the action X -> X + 27r/(n../2) (i.e. J± -> e±21 J3, and similarly for the J's). The additional generator adjoined to give the dihedral group Dn corresponds to X

->

-X (J3

->

-J3, J± -> JT). Modding out by the Cn's thus gives points on the = l/(n../2), which by the duality (2.12) is equivalent to radius

circle line at radius r

r = n/Vi. Similarly, modding out by the Dn's gives the points at the radius r on the orbifold line.

= n/Vi

Something special happens, however, for the tetrahedral, octahedral, and icosahedral groups, T, 0, and I. For these it is easy to see that the only (1,1) operator that is invariant under the full discrete group is V = Z::~=1 JiJi, which is hence the only marginal operator that survives the projection. But recalling that our affine SU(2) currents satisfy (2.14), we easily verify that C yyy = -2 for V = Z::~=1 JiJi. This means[39] that the SU(2)/r orbifold models for r/z 2 = T,O,I are isolated points in the moduli space for c = 1 conformally invariant theories. This absence of truly marginal operators is intuitively satisfactory for these cases since we are modding out by symmetries that exist only at a given fixed radius, the SU(2) x SU(2) radius r =

l/Vi,

and hence

modding out by the symmetries effectively freezes the radius. (Further properties of the SU(2) orbifold models are discussed in [39,42].)

3. The stat mech

H

cft connection

In the remaining section here, we discuss some relations between the statistical mechanical models and conformal field theories of the previous two sections. 9.1.

A2n - 1

H

gau88ian connection

Locally, the A2n - 1 models can be written as 6V models. This is because there are only six different height configurations for each face, modulo a fixed height k,

k-l

k+l

k 0 k- 2

kOk+2 k+l

£:-1

X

. a=l

X

k-l

k+l

k-l

k+l

kOk

kOk

kOk

kOk

k+l

k-l

k-l

k+l

X ... X b=e

X. X c=l+e

(3.1)

160 (the heights are labeled modulo2n). Associated with each face is one of the 6V model vertices (1.6) on the dual lattice, determined as shown in (3.1) by rotating the arrows counterclockwise by 7r/2 about their midpoints. (And conversely, any 6V configuration uniquely determines a height configuration on the dual lattice, up to a global constant height shift, given by reversing the procedure in (3.1).) The face weights in (3.1) are easy to write down, because the ..42n -

1

diagram of (1.26a) has a Perron-Frobenius vector

with all components s6 = 1. From (1.30b), the weight in (1.35) takes the value Cac+eC6d, from which we determine the 6V weights indicated below the vertices in (3.1). The above map, while manifestly defined locally, may involve additional subtleties in finite geometries. Cylindrical boundary conditions, for example, require identifying the heights in the top and bottom rows, and this will not be satisfied by the height transcription of arbitrary 6V configurations. Allowed

..42n - 1

configurations comprise

only a subset of all 6V configurations on the cylinder or torus. We also recall that on the plane or with appropriate boundary conditions on the cylinder, the

..42n - 1 models

all have the same partition function (it depends only on (3 = 2 in (1.14». As usual, however, their different operator content is signalled by differing partition functions on the torus. The vertex/face model connection described above has various group theoretic generalizations[21,43-45]. The two states on the links in the vertex model can be identified with the spin-1/2 representation of SU(2). The heights in the face model are identified with representations of SU(2). The constraint on allowed nearest neighbor heights can then be interpreted to be dictated by the branching rules for the representation at a site crossed with the (2) representation on the intervening dual link. We discuss this point of view further in subsection 3.6. One evident generalization is to take some other representation of SU(2) on the links in the vertex model. Again there is a "zero-flux" condition on the states of the four links incident on any vertex, requiring that their J 3 eigenvalues sum to zero. Choosing the spin-1 representation of SU(2), for example, gives the 19 vertex model. A further generalization is to take a representation of SU(n) on the links of a vertex model, with a constraint that their weight vectors sum to zero at each vertex. The heights in the dual face model remain identified with the representations of the group, now SU(N), and the constraint on nearest neighbor heights is dictated by the branching rules with respect to the representation chosen on the dual links. (Strictly speaking, in the above height models one chooses not representations of SU(n) but rather of its affine extension SU(n) at some level k, which has only a finite

set of highest weight representations. For example, the branching rules for SU(2) level k

161 representations tensored with the spin-1/2 representation are given by the AHI Dynkin diagram of (1.25a). The face model based on this incidence matrix has a critical continuum limit described by the coset conformal field theory SU(2h x SU(2)k_l/ SU(2)k. A face model described by the coset theory SU(2)l x SU(2)k/ SU(2h+l is defined via the incidence matrix that codes the branching rules for SU(2) level k + 1 representations tensored with the spin-l/2 representation. For WeN) at level k, the allowed representations have highest weights>..

= L,:~/ >"aAa

(Aa are the N - 1 fundamental

weights of SU(N)) with components restricted to satisfy L,:~I >"a :5 N - 1 + k. The generalization[46] of (1.3Gb) is

>-

Ui

>-+ ••

0

>-+.c

).+eb+ed

where Sbd(>") = sin( k';N (eb - ed)' >..) arid e a are the weights of the vector representation

= AI, eb = Ab - A b- b eN = -AN_I)' ~hese Ui are straightforwardly verified to satisfy the Hecke relations (1.14a,b),(1.23) of the e;'s with (3 = 2 cos k';N'

of SU(N) (el

and can thus be used (see subsection 1.6) to define an integrable model. We note that the representation theory of affine W(Nh is closely related to "quantum" SU(N) with· deformation parameter q = e 21fi /(HN). (See for example [21,47]. Many special properties of quantum groups for q a. root of unity are related to the "rationality" of the corresponding conformal field theories. FUrther connections to the 2+1 dimensional Chern-Simons approach may be found in [48].) The RSOS models with incidence matrices derived from the fusion rules of affine algebras typically have integrable extensions away from criticality. The local height probabilities for these models take the form of affine characters which depend on the parameters of the Boltzmann weights of the model rather than on the modular parameter of a torus (see [4]). In these models, we thus have a deep group theoretic reason underlying their integrability. Later on here I will employ the simpler case of models based on branching rules of discrete groups to illustrate some of the affine and quantum group technology in a simpler context, namely in the q = 1 "classical" limit.) In the region in which the 6V models are critical, i.e. for which the correlation functions decay algebraically, it is natural to assume that the model renormalizes onto a gaussian model in the continuum limit. (Models similar to the 6V model can be mapped onto a lattice electric Coulomb gas, which block-spin calculations show renormalizes onto a gaussian model (for a review, see [3])). The continuum gaussian model is a free field theory of a single boson c.p which takes values on a circle of some radius R, c.p == c.p + 271" R.

162 We have just seen that the

A2n - 1 models are locally 6V models, and renormalization is

a local operation, so they too should be described by free field theory in the continuum limit, where the height variable renormalizes onto the field '(k)v(k) = V'(i!il + v,(k-t) The 2n eigenvalues and eigenvectors of the incidence matrix AA,n_t are thus given by (3.20) with w(k) =

(k

(k = 0, ... , 2n -1) and (

has eigenvector For an

==

e27ri/2n.

The largest eigenvalue f3 =

>'(0)

= 2

s= V(O) = (1,1, ... ,1), as shown in (1.26a).

A2n - l

model, the operator (3.18) thus becomes (3.21a)

1 We regret using latin indices i,(k) in (3.18) both to label the sites i in the lattice model and to label the different operators ~(k) - the parenthesized subscript for the

latter should avoid any ambiguity in what follows.

169

Recalling from (3.4) that X

= 72£ (£ = 0, ... , 2n -

1), together with (2.2) we see that

this operator renormalizes in the continuum limit onto

~ (x(z)

ik::liX 'P(k)(i) -+ e

n

+ x(z))

(3.21b)

=e

The scaling dimension of this vertex operator in free conformal field theory is easily determined. We recall from the discussion before (2.7) that the operators respectively have conformal weights

h=

a 2 /2 and

Ii =

eicu:(z), eiaz(i)

a 2 /2. The conformal weight of

(3.21b) is thus

(3.22) Operators with these conformal weights indeed appear in the SU(2)dC n

= Zcirc(r =

n/V2) model, as can be read off from the partition function (2.10). Now we consider the operator algebra in this (the abelian) case. Using the property

Pa Pb =

Dab

Pa of projection operators, we find _ " akp bi n 'P(k)'P(I) - L.J W a W .q

= "a(k+i)p L.J W a = 'P(k+l)

(3.23)

•

a

a,b

This is indeed the selection rule for abelian orbifolds, representing the conservation of the C n charge of the different twist operators. For the remaining c = 1

ADE

models, the general result for the scaling weight of

operators of the form (3.18) is (3.24) with the values of mk and h given in (3.8). (3.24) can be derived most easily from (3.22) by means of a certain universality result[20,50j (see also [52]). In the calculation of the lattice partition function sketched after (3.7), the effect of insertions of operators of the form (3.18) is easily shown to modify the weight associated to the cluster containing the operator by a factor depending only on the eigenvalue

\k)'

Since the dependence

is only on the eigenvalue, the calculation of the 2-point function, for example, is effectively mapped onto the case considered above, and the result (3.24) follows immediately from (3.22). (This reduction of the calculation to the abelian case for any given operator sector mirrors the method used to calculate the orbifold partition functions in the argument leading to (3.7a-e).) Before turning to consider the operator algebras of the remaining

ADE

models

and the corresponding non-abelian c = 1 orbifolds, we mention some related results

170

< 1 models, with incidence diagrams given by the. unextended Dynkin diagrams

for c

(1.25a-e). For these, as mentioned in subsection 1.5, the largest eigenvalue is given by

= 2cos7r/h, where h = C A/1jJ2 is the dual Coxeter number of the ADE Lie algebra (h = n + 1, 2n - 2, 12, 18, 30 respectively for An, Dn, E 6,7,S). The remaining eigenvalues

f3

are given by A(k) = 2 cos 7rmk/h, where the integers mk are the exponents of the ADE Lie algebra (and can be found tabulated, together with the eigenvectors V(k) for example in [17,20]). The scaling weights of the operators defined according to (3.18) for these models are given by

hk (except for mk =

m~-l = -hk = ...,-~-:,---

(3.25)

4h(h - 1)

h -1, in which case the operator renormalizes onto the null operator).

These coincide with the values hpq, with p = q, of allowed conformal weights of the

h-

(m =

l)'st member of the c < 1 discrete series (1.3) (see for example [7]). (3.25)

follows[20,52] from the same sort of universality result sketched following (3.24). (The appearance of

mi -

1 in (3.25) can be traced to the occurrence of

sa '"

1 in (3.18),

related in turn to the presence of background charge in these models. The operator in (3.21) can also be regarded to create an electric charge e

= 2mk/h in the conventional

normalization of a Coulomb gas. There the dimension of the operator. would be given by h

= h = e 2 /4g for 9 = 4, again in agreement with (3.22),(3.24).)

For the AD models,

(3.25) can also be verified from the exact solutions[12,13,25]. (The above results for the eigenvalues and eigenvectors of the An case are as well contained in recent work on the ''fusion algebra" of conformal field theories[53]. The incidence matrix AAn for the diagram in (1.25a) describes the fusion of representations of

= n-l. If we write the fusion algebra as (Aa)*(Ab) = ~c Nab c (Ac), then N(2)a b, i.e. (2) * (Aa) = ~b Aab (Ab), where (2) is the basic spinor representation.

W(2) at level k Aab =

In [53] it was shown that v(k) = eigenvalues of (Na)b c

8 ck

and Ack) =

8(2)k/80k

give the eigenvectors and

= Nab c, where 8 is the unitary matrix of modular transformations

of the affine characters (0 above specifies the identity representation). This form for the (unfused) incidence matrix generalizes to WeN) type models as Aab = N(N)a b, where

(N) is the fundamental vector representation.) 9.5. More group theory for the non-abelian case Earlier we exhibited the equality between the toroidal partition functions (3.7a-e) of the

ADE lattice models[50] and the c =

1 orbifold conformal field theories[39]. In this

subsection we illustrate further some of the relations between their operat or algebras.

171 In an orbifold conformal field theory constructed from modding out by a discrete group

r, the twist operators 'Pi are associated to the conjugacy classes Ci of r

(as in the

decomposition (2.20) of the partition function). The selection rule for the composition of twist operators is given in general by the multiplication rule for classes of the group I' [35], i.e. whether the product of representative elements in Ci and Cj is some element

in the class Ck. This is easily visualized by considering the effect of twists by group elements gI' g2 E 1', created by two operator insertions 'PI' 'P2 in the conformal plane. With respect to a third point, the net twist is determined by taking a path that leads close to 'PI (picking up some factor hEr), around it (picking up a factor of gI), then back (picking up h-I)j and then similarly for 'P2 (picking up h'g2h'-I for some h' E r). We see that the result hgIh-Ih'g2h,-I can depend in general on the details of how the two twist operators are brought together, but must be contained in the product of the two classes C1 and C2 containing

gl

and

g2.

(In the abelian case, each group element

forms a class by itself, and the class multiplication reduces to group multiplication). This picture is closely related to the analogous picture for the combination of point defects in two dimensions, or line defects in three dimensions (see for example [54]). To analyze the properties of the lattice operator algebras, we now recall some standard properties of discrete groups. We assume the group I' to have order 11'1 and n classes denoted Ci , i = 1, ... , n. We let M( .. )(g) be a matrix representation of dimension d(a),

and

X(a)

be the character in representation (a), i.e.

X( .. )(g)

=

trM( .. )(g).

It will be convenient to work with the group algebra Ar, for our purposes a

Irl

dimensional vector space over the reals with basis vectors labelled by group elements. Each vector takes the form L:~~I algt. Because the class containing a given group element 9 contains as well all elements of the form hgh-I, the quantity Gi

= Lg gEe,

satisfies hGih-I = Gi. The Gi'S consequently commute with all other elements of the group algebra. (In fact they provide a basis for the center of the group algebra.) In a matrix representation, M(a)(Gi) commutes with all representation matrices M(a)(g)j by Schur's lemma it is thus proportional to the identity, (3.26) Taking the trace of both sides gives ICil elements in the class Ci and

xi . ) =

d( ..

)'xi. ), where

ICil is the number of

xi .. ) is the character X( ..)(g) evaluated for any 9

E Ci.

Solving for ,x, we find (3.27)

172 (recall that (a) specifies an irreducible representation of r, and i a conjugacy class). In the multiplication of any two classes Ci and Cj it is easy to see that the set CiCj consists of complete classes, each some integer number of times. The product GiGj therefore satisfies (3.28)

GiGj = LnijkGk, k

where the nijk, symmetric in ij, are non-negative integers giving the number of times the class Ck is contained in the product CiCj; and the sum is over all the classes of the group. It follows from (3.28) that a matrix representation M(a) satisfies

M(a)(Gi)M(a)(Gj) = LnijkM(a)(Gk) , k

and (3.26) then implies

Aia) Ata) = L

(3.29)

nijk Ata) .

k

We see that the Ata) 's can be considered as eigenvectors of the matrix (ni)jk = nijk with eigenvalue Aia)' (Knowledge of the nijk'S and ni's could be used to find the Aia) 's, and then the characters X(a) by going backwards through (3.27).) Now recall that the incidence matrix A associated with

an extended Dynkin di-

agram gives the branching rules for any representation of the associated subgroup

r

C SU(2) times the (2) representation: (2) x r a = Aab rb. It follows immediately

that

xb) Xia)

= Aab Xib) for any class i. We see that the eigenvectors V(i) of the inci-

dence matrix A are just the characters Xia) of the SU(2) subgroup, with eigenvalues

A(;) given by the character Xi2) of the fundamental two-dimensional representation. (In terms of the Cartan matrix J{ab = 28 ab - Aab of the extended Dynkin diagram, we have

J{ab Xib) =

(2 -

Xb»)Xia)' The zero eigenvector of

equal to the dimensions

d(a)

J{

has components X~a) = X(a)(e),

of the irreducible representations.) So if we associate to

each conjugacy class the operator (3.30)

we find from (3.27),(3.29), and the relation PaPb = 8ab P a, that 'P(i)'P(j)

=L

Aia)Aib)PaPb

a,b

=

L k

=L a

nijk 'P(k) •

Aia)Ata)Pa = L k,a

nijk Ata)Pa (3.31 )

173 We see that the algebra (3.31) of operators (3.30) agrees identically with the class algebra (3.28), encoding the selection rules of the orbifold conformal field theory. Up to the (irrelevant) normalization factor

lCd,

we recognize the operators (3.30) identically

as the scaling operators introduced in (3.18). We note finally that the group class algebra is not identically the orbifold fusion algebra. While each twisted sector is labelled by a conjugacy class of the group, different primary fields

4>10)

of the orbifold fusion algebra associated to a given twisted sector

i are labelled additionally[55] by representations (a) of the stabilizer group Ni of the

class Cj (as defined before (2.20». The linear combinations of primary fields 2 (3.32)

satisfy identically the class algebra (3.28) and their diagonal holornorphic/anti-holomorphic combination can thus be identified with the operators (3.30). In (3.32) d10l is the dimension of the representation rio) of the stabilizer group Ni, whose order is

INil = 2:(o)(d1o»2. (For more on fusion algebras from an abstract group theoretical point of view, see

[56].) 9.6. Vertex / IRF correspondence In the preceding subsections, we have observed the equivalence between the

SU(2)/r orbifold models and the critical face models based on the branching rules of

r.

Here we wish to present some formalism in which results such as these are more

manifest, and generalize to other models. We begin by describing the correspondence between the vertex and IRF presentations of models. Consider first the Hilbert space for a vertex model, whose degrees of freedom live on the links,

(3.33)

We denote the space of states for an individual link by Hr, where r specifies the representation of some group G. In the case of the six-vertex model (1.6), for example, where the links can take either of two values denoted by an arrow, we have Hr where 2

t is the (two dimensional) spin-1/2 representation of G = SU(2).

I thank R. Dijkgraaf for this comment.

= H1 = CC 2 ,

The full Hilbert

174 space for the model with N links is 1-(. =

1-(.1f!N

(we restrict to the homogeneous case, in

which all links possess the same degrees of freedom), and the global group G : 1-(.

-->

1-(.

is realized diagonally as 9 ® ... ® 9 for 9 E G. The linear operators on this Hilbert space consist of all linear maps. (endomorphisms) M : 1-(. Now let us specify a discrete subgroup

-->

1-(..

reG, and consider a restricted model

whose operator algebra includes only those operators M E End(1-(.1f!N) that commute with the diagonal action, i.e. for which (3.34) (This defines the "fixed-point" algebra, or "commutant", which has many interesting properties from the Von Neumann algebra point of view.) We now change to a Hilbert space basis in which properties of the restricted operator algebra are more evident. The direct product of N copies of r can always be decomposed into a sum of irreducible

r

representations, r ® ... ® r

= E!)a

nar a ,

(where na is the multiplicity of the representation ra). This means that the Hilbert space decomposes as (3.35) where Va is a vector space of dimension na. In this basis, the linear operators that commute with the action of all 9 E

r

are easily characterized. By group invariance,

they block diagonalize into a direct sum M = E!)aMII where each Mil:

VII --> VII

is realized

as an na x na matrix. We see that a state in the height description of the theory should be identified with

a "fusion path" from the identity to a given representation r ll ,

(3.36)

The different ra define different Hilbert space sectors, of dimension nil, for the model with N spatial links, where na is the number of independent ways that the representation Ta

appears in the tensor product of N copies of the

T

representation.

A vertex model whose link states (3.33) are in some representation

T

of a group

G "contains" in a sense a large number of different face models. These are defined by restricting the operator algebra to the commutant with the diagonal action of a given subgroup

reG. The change of basis to an IRF height model associates irreducible

175 representations of r to heights, and an adjacency restriction on nearest neighbor heights given by the "incidence" properties of r in the representation ring. We shall see some examples of "incidence diagrams" based on discrete groups in the next subsection. One of the virtues of the IRF version of the model is that it naturally encodes the desired restrictions on the operator algebra (imposed by hand in the vertex language). Each basis vector of the space

Va

in (3.35) can be identified with a path of repre-

sentations of length N from the identity to r a , where the adjacency rules for the path are summarized by the incidence diagram. The number of paths of length N

+ 1, i.e. the

number of states associated to an (N + I)-link Hilbert space, is determined by tensoring on an additional copy of 'Hr to (3.35). In terms of a generalized "incidence matrix" A (r), whose ab th entry specifies the number of times r a occurs in r

states n~ for the (N

X rb,

the number of

+ IHink space is given by n~ = ~b A~~ nb.

The states in the Hilbert space can thus be associated to paths in a certain diagram known as the Bratteli diagram. This is a diagram which has at level N a single node for each possible representation

ra

in the N-link Hilbert space (i.e. not duplicated to

take into account the multiplicity na). There is a link connecting each node N to a node rb at level N

level

+ 1 if rb

ra

at level

appears in the decomposition of r ® ra. If we take

°to have a single node associated to the identity representation, then the number

of paths to node

ra

at level N is evidently equal to

n a,

and each such path on the

Bratteli diagram specifies a state in the N-link Hilbert space. (We also see that the Bratteli diagram encodes the embedding of the Hilbert space of the N -link model into that of the (N

+ I)-link model.)

Each path can further be identified with a projector

operator of the form (3.9), ultimately allowing one to reconstruct the full commutant (fixed-point algebra (3.34)) in path form, so we see how this structure descends from the information contained in the original incidence/Dynkin diagram. Further algebraic aspects of these constructions may be found in [57], and their application to lattice statistical mechanical models is described in [20,21]. (A given sector of the Hilbert space of states of an N-link model can also be identified with a specific space of conformal blocks in a rational conformal field theory, r

r

r

r

r

r

r

(3.37)

1 where we fuse N times with the r representation of the chiral algebra.)

176

9.7. FU8ed model8 Consider first a vertex model whose links carry the spin-I, rather than the spin1/2, representation of SU(2). For this three dimensional representation, we have three possible links, (3.38) and there are a total of 19 vertices with zero net flux (the middle link above, with 13 = 0, carries no flux) generalizing the six vertices (1.6). A set of Boltzmann weights for the 6V model can be used to induce weights for the 19V model by a procedure known as fusion[58,4]. 19V weights derived from 6V weights that satisfy the star-triangle relations as well satisfy these relations and thereby define an integrable model. The continuum limit of the unadorned 19V model (i.e. with no "sewing" term/ modified trace/background charge) at criticality is a c = 3/2 conformal field theory (see [28] for details). Depending on the boundary conditions, it maps somewhere onto the c = 3/2 moduli space of N

= 1 superconformal theories constructed in [41].

The fused

version of the modified 6V construction that gives the c < 1 discrete series leads to a set of 19V models whose continuum limit is described by the N = 1 superconformal discrete series with central charges c =

!(l- 8/m(m + 2»). In general, the "lth-fused" models,

with the SU(2) spin-l/2 representation on the links in the vertex description, provide generalized vertex models whose continuum limit is described by the coset theories SU(2)l x SU(2)k/ SU(2)k+l' In what follows, we shall construct fused models from a

slightly different point of view. To describe the analog of the construction of subsection 1.5, in which incidence diagrams were used to define the models, we continue to use as illustration the simplest SU(2)-based models. Consider then a vertex model whose links carry the spin-l rep-

resentation of SU(2), and restrict the operator algebra to the commutant with respect to some discrete

r

C SU(2). The argument of the previous subsection again provides

us with an IRF model whose allowed heights are associated with representations of

r,

but since we now fuse with the spin-l representation as we pass through dual links, the "incidence diagram" is determined by the branching rules associated with the spin-l representation. More specifically the diagram that specifies allowed nearest neighbor adjacencies is constructed from the branching rules that follow from tensoring with the (in general reducible)

r

content of the spin-l representation of SU(2).

Some relevant examples of "incidence diagrams" for representations of the subgroups

r

C SU(2) tensored with a distinguished representation r are easily constructed.

These are (not necessarily connected) diagrams whose nodes are labelled by the irreducible representations of the group, with a link connecting node r a to node

rb

for each

177 time the representation rb occurs in the'decomposition of ra tensored with r (if r is complex, then the link is directed with an arrow). These diagrams for r taken to be the 2 dimensional representation of the non-abelian subgroups

rc

SU(2) are identically

the Dynkin diagrams (1.26b-e) in the correspondence (3.5). A typical non-abelian example is given by T

c

SU(2), embedded so that the spin-

1/2 representation of SU(2) decomposes to the 2 of the binary tetrahedral subgroup T. The branching rules for 3

X

ra follow from the relation 2 x (2

X

ra) = (3 + 1) x r a , and

the incidence diagram for the 3 representation of T is given by

SU(2h/T

1

I'

(3.39)

I"

The links indicate the representations that occur in the decompositions of the 3 crossed the representation labelling the node, e.g. 3 x 2 = 2 + 2' + 2", 3 x 3 = 3 + 3 + 1 + I' + I" , etc. There are two disconnected components to (3.39) because the spin-1 representation of SU(2) is vectorial in character, and tensoring by it preserves the Z2 vector/spinor grading of the representations. From a practical standpoint this means that (3.39) defines two distinct models, one comprising only the heights 1,1'1",3, the other with only 2,2',2". We shall see in a moment that these two models share some properties. Denoting henceforth the SU(2) representations by their dimensionality (spin-£/2 (£+1»), the decomposition of arbitrary representations with respect to their

rc

-+

SU(2)

content is easily determined by repeated use of the SU(2) relation (2) x (k) = (k-1)

+

(k+1). The decomposition of the first few SU(2) representations with respect to the

irreducible representations (1, 1', I", 2, 2', 2",3) of T are

SU(2):::> T:

+ 2"

(2) = 2

(6) = 2 + 2'

(3) = 3

(7) = 1 + 3 + 3

(4)=2'+2"

(8) = 2 + 2 + 2' + 2"

(5) = 3 + I' + I"

(9) = 1 + I'

+ I" + 3 + 3 .

178 The incidence diagram for the case of (4)

= 2' + 2"

(spin-3/2) is a single connected

diagram 1

2'

2"

I"

I'

SU(2h/T

(3.40)

2

since we are again tensoring by a spinorial type representation. The diagram is bicolorable as in (1.26c), and possesses the same 7l a symmetry. The incidence matrices for the representation content associated with the decomposition of higher spin SU(2) representations are similarly determined. For example, we define the matrix B

= A(a)

by

(3.41a) Then from the definition 2 x ra

= Aa6 r6, we see that A!6 r6 = 2 x 2 x ra = ((3)+ 1) x r a,

from which it follows that B

By iteration, the incidence matrices

= A2 -1.

A(n)

(3.41b)

for the branching rules of the higher represen-

tations (n) are all easily represented in terms of the basic A

= A(2)'

To exhibit the incidence matrices for the diagrams of (3.39),(3.40), we use the even/odd grading of the heights of (1.26c) to write A = 3

:' (~ 2"

1

I'

C" aT), where

I"

1 1

(3.42)

1

(3.43)

179 are the incidence matrices for the odd and even height diagrams in (3.39). For the (4) representation, we write C

== A(4)

= (1' 1'T) = A 3 -

2A, so that

1 'Y=OtOt T Ot-20t= ( ;

1

2 1 1

~)

(3.44)

determines the incidence matrix for the diagram in (3.40). A case with more structure is given by decomposing the first few SU(2) representations with respect to the irreducible representations (1,2,3,4,5,6,4,2',3') of the binary icosalledral subgroup I,

=2 (3) = 3 (4) = 4 (5) = 5 (6) = 6

(2)

SU(2):::) I:

= 3' + 4 (8) = 2' + 6 (9) = 4+ 5 (10) = 4 + 6 (11) = 3+3' +5.

(7)

For fun in fig. 1 we depict the incidence diagrams for the first few of these representations. (We notice that these incidence diagrams have a greater symmetry for higher level. This is related to the existence of a 71.2 outer automorphism of I that is not evident from (1.26e). It is most easily exhibited by recalling that I

Rj

SL(2, 71.5). In this

language, the 71.2 automorphism is represented as conjugation by the matrix (~ -;). It has the effect of interchanging the representations 2 +-+ 2' of I. In passing we recall here also the isomorphisms, T

Rj

SL(2,71.3) and 168

Rj

PSL(2,71. 7 ), useful on occasion.)

In subsections 3.1-4, we noted that the IRF models based on the incidence diagrams (1.26a-e), related to vertex models carrying the spin-1/2 representation of SU(2) on links, had critical partition functions that coincide with the SU(2h/r orbifold models. To what conformal field theories do the critical IRF models based on higher incidence diagrams correspond? As we have seen, these are the models that naturally encode the operator algebra of a vertex model carrying the spin-R representation of SU(2) on its links, restricted to the commutant with the diagonal action of r

c SU(2).

This

restriction leads to a height model whose nearest neighbor adjacencies are determined by the branching rules of the direct sum of r representations contained in the SU(2) spin-R representation. Since the unadorned vertex model with spin-R on its links has a continuum limit described by the SU(2) level k

= 2R conformal field

theory (see e.g.

[28)), with c = 3k/( k + 2), it follows that the higher incidence diagrams again lead to orbifold theories thereof. Specifically, a model based on the incidence diagram of

180

3'

SU(2)dI 2

SU(2h/I

3

2'

4

~4 3 Va,

2

1

4

6

SU(2h/I

2'

SU(2)s/I

2'

Fig. 1.

Icosahedral fusions

2'

181 branching rules for the decomposition of spin-l with respect to some

r

C SU(2) has a

continuum limit at criticality described by the orbifold conformal field theory

SU(2)u/ r

.

(3.45)

This explains the labelling at the right of the incidence diagrams in (3.39),(3.40), and fig. 1. We see that the height transcription of the model, as described in the previous subsection, automatically encodes what in orbifold language would be the effect of twisting by possibly non-linearly realized operators along the two cycles of a torus. By methods similar to those employed earlier here, we can construct scaling operators in these fused models and identify their operator algebras. This allows their placement in the relevant moduli space of conformal field theories. (The generalization of the studies of the c

= 1 and c = 3/2 moduli spaces undertaken in [39,41]

to general

SU(2h has not been carried out, although for some partial results with an emphasis on N = 2 conformal models, see [59].) Further suggestions for generalizing these results to

groups G other than SU(2) may be found in [39,45,60].

It is evident that the incidence diagrams for the decomposition of even dimensional (spinorial) representations of SU(2) are connected whereas those for the odd dimensional (vectorial) representations have two disconnected pieces. Let us consider some of the properties of the two models that appear for a given odd representation (3), (5), ... ), by studying the incidence matrices

A(2k+1)

for the (2k

+ 1) representation.

If we start

from an unfused SU(2)-based model with no odd heights and ne even heights (e.g. with incidence diagram any of (1.26a-e)), we have previously noted that the incidence matrix takes the form

where a is an ne x no matrix and aT is no x ne. Writing the eigenvectors in this basis as v = (v o , v e ), the eigenvector equation for A is avo = >.ve , aT Ve = >'Vo . It is evident that non-zero eigenvalues come in pairs

±>., associated to eigenvectors (v o , ±ve ). = C-I) that satisfies {( _l)F, A} = 0,

(Equivalently we could say there exists a (_l)F so that v

= (v o, v e ) and (_l)F v = (v o, -ve ) have opposite eigenvalues under A.)

Since

aTa Vo = >.2vo and aa T Ve = >.2ve , the non-zero eigenvalues of the no x no matrix aTa

and of the ne x ne matrix aa T coincide identically, and the larger of the two matrices has an excess of

Ino -

nel zero eigenvalues. It is natural to define an index of A by

indA

= dimkera -

dimkera T

= no -

ne .

(3.46)

182 The "fused" incidence matrix

Bk

= A(2k+l) for the representation content of the (2k+1)

representation is expressible entirely in terms of the even powers and the identity 1, i.e.

Bk =

(,80,8J

=

E!=l c n A2n + col.

("T")~""T)n)

A2n =

It follows from the above

discussion that the eigenvalues of f30 and f3e coincide except for lind AI = Ino - nel additional ). = Co eigenvalues possessed by the larger of the two matrices. The argument sketched at the end of subsection 3.2 can be generalized to calculate the partition functions of the fused models. Again they depend only on the eigenvalues of the defining incidence matrix of the model. Thus we should expect a relation between the partition functions of models (those defined by the incidence diagrams in (3.39), for example) whose incidence matrices have coincident eigenvalues except for the lind AI extra>.

= Co eigenvalues.

Let us focus on the 19-vertex models, whose links carry the

spin-1 representation (the first-fused case) and whose continuum limits are described by the SU(2) level 2 orbifolds with c = 3/2. The incidence matrices are given by

(3.41b), so that Co = -1. For this case it can be argued as follows that the difference between the continuum limit toroidal partition functions of the two disconnected models is simply a constant, equal to no - ne. To match eigenvalues according to the earlier argument there now enters an additional (trivial) model, with two nodes and a vanishing incidence diagram. The two disconnected components of the first-fused version of this model have constant partition functions, normalized to unity, and incidence matrices with a single {-I} eigenvalue. According to (3.41b), the eigenvalues of the incidence matrices (3.43) for the two diagrams in (3.39), for example, are {3, 0, 0, -I} and {3, 0, O} respectively. Either connected component of the first-fused incidence matrices

(A3h, and (Alh respectively has eigenvalues {3,0,O}, {3, -I}, and {3}. the partition functions of the

(E6h 1

models of (3.39) is thus -

Z = 2(2Z3 where

(Ash,

The result for

+ Z2 -

-"

Zl ± 1) ,

Zn is the critical partition function of either of the two disconnected models that (A2n - 1 h diagrams.

occur in the first-fused

The continuum liInits of these models all have N = 1 superconformal symmetry, and the relevant superconformal field theories, the SU(2h/r orbifold models, were studied in [41]. There it was seen that two possibilities indeed exist for many of these orbifold models, each consistent with modular invru:iance, and differing by the choice of projection in the Ramond sector. The toroidal partition functions for the two choices consequently differ by tr( _l)F in this sector, where (_l)F is an operator defined to anticommute with the supercharge. By the standard superconformal argument, this trace receives non-cancelling contributions only from the Ramond ground states, and

183 is thus a constant. It turns out that tr(-l)F in the superconformal continuum limit of these lattice models is determined by the index (3.46), i.e. the difference between the number of odd and even heights in the basic (unfused) models, (3.47) From the incidence diagrams (1.26b--e), we see this latter number takes the values no - ne

= 3,1,2,1

respectively for

r = 'D2n ,T,O,I,

in agreement with the results

for trR(-I)F calculated in [41] directly for the SU(2h/r orbifold models. The partition functions, coincident with the continuum limits of the critical lattice models based on first-fused

ADE

Dynkin diagrams (depicted in figs. 1-4), are

Z [SU(2h/C 2n ] == Zn ± 0 Z[SU(2h/'Dn]

= { t(~n + 2~2 - ~1 ± 0) 2"(Zn + 2Z2 - ZI ± 3)

n odd n even

-

(A2n- 1 h

(3.48a)

(Dn+2h

(3.48b)

(E6h

(3.48c)

Z[SU(2h/T]

1 = "2(2Z 3 + Z2 -

Z [SU(2h/O]

= ~(Z4 + Z3 + Z2 -

ZI ± 2)

(E7h

(3.48d)

Z[SU(2h/I]

= ~(Z5 + Z3 + Z2 -

ZI ± 1) .

(Esh

(3.48e)

ZI ± 1)

We see that a remnant of the superconformal symmetry persists in indA even away from the critical point. 9.8. Some stat meek orbifolds In section 2, we reviewed the conventional orbifold construction of conformal field theory. In [18], it was shown how this notion of orbifold can be generalized to statistical mechanical models whose Boltzmann weights possess some discrete symmetry at or away from their critical points. In simple models such as the Ising model, with a 712 symmetry, this non-critical orbifold construction reduced to the Kramers-Wannier duality alluded to in section 1. More generally it gives a new model whose toroidal partition function can be expressed in terms of that of the original model by an equation analogous to

(2.19). In the case of an abelian 7l n symmetry generated by some 9 with gn

= 1, for

example, the resulting partition function is n-l

Z-

=;;1

'"' L..J Z(r,s) , r,,=O

(3.49)

184 where Z(r, s) represents the toroidal partition function for the original model with boundary conditions twisted by gr and g' along the two cycles of the torus.

The

construction of [18] preserves the star-triangle relations, so that the "orbifold" of an exactly solvable model remains exactly solvable. It was thereby used to relate certain exactly solvable non-critical models, such as the A 2n the

A2n - 1

1

and D n +l models of [12,25] and

and DnH models of [26,50] via their Z2 symmetries. Finally, at a critical

point the construction reduces to the conventional orbifold notion of conformal field theory. The incidence diagram for the "orbifold" IRF model is determined directly from that of the original model by the action of the symmetry. First we identify all heights in a given orbit of the symmetry group to a single height, and then we "replicate" any height that is invariant under the symmetry so that it becomes an orbit of the symmetry group in the new model. If any two heights are linked in the original model, then their images in the new model are also linked. Various examples in addition to the ones mentioned above may be found in [18,61]. Here we give further examples using incidence diagrams of some of the fused models to illustrate additional subtleties that may arise, specifically in the case of incidence diagrams with multiple links connecting nodes (which were largely avoided in [18]). The analysis of [18] was performed using a "Lagrangian" formulation of the models. Using the ideas described in subsection 3.6, it is straightforward to construct a "Hamiltonian" description of the non-critical orbifold procedure, identify the Hilbert space sectors in terms of paths from 1 to the appropriate r a , and construct the interesting operators. (This point of view is more or less implicit in [20,43], and further related ideas may be found in [62].) The basic idea is to employ an "intertwiner" V between two models whose incidence diagrams (and Boltzmann weights) are related by a symmetry. Take two models based on incidence matrices Al and A 2 , respectively nl

x

nl

and

n2

x

n2

matrices, and assign each diagram a distinguished node denoted 0

(related under the symmetry). Then V is an

nl

x

n2

matrix of non-negative integers,

with no vanishing rows, that satisfies (3.50) and has VOj = 8jo. Note that AI> A2 share the same largest eigenvalue since if 8 is the Perron-Frobenius vector for A2 (A 2 8= (38), then by (3.50) V8 is the Perron-Frobenius vector with same eigenvalue (3 of AI. (The related critical conformal field theories thus share the same central charge.) The intertwiner V can be used to induce an action on the path representations of the Hilbert spaces of the two models, as described in the

185 previous subsection, thereby relating the operators that act on them. Due to space limitations, we shall not pursue this point of view further here. (We point out, however, that the behavior of the operators (3.18) under the symmetry of a diagram can be read off directly from the eigenvectors symmetry of the A3 diagram in (1.34), for example, the respectively even and odd. From (3.25) (with

mk

V(k)'

Under the

712

v'2,0 eigenvectors are

oX =

= 1,2; h = 4), these are the (0,0) and

(-h, -h) operators of the Ising model and this 712 is a symmetry of its fusion algebra. A similar

712

symmetry generalizes to all of the A 2n -

thereby to the fusion rules of the c

1

and

A2n - 1 lattice models, and

< 1 discrete series and c = 1 models; modding out

by it leads respectively to the Dn+1 and

Dn+2 models (see e.g.

[7,18]).)

The simplest examples involve the C2n orbifolds. (Recall for the SU(2h theory,

Zcirc(r = 1/vI2), modding out by C2n gives the circle theory at radius r = n/v'2.) We embed the abelian subgroups C2n C SU(2) so that the spin-1/2 representation decomposes to the (reducible) (+ (-1 representation of C2n • (We label C2n representations by their characters, and (

== exp(27ri/2n).)

The branching rules for this representation are

represented in (1.26a). The first few representations of SU(2) decomposed with respect to the ilTeducible representations (1, (, (2, ... , (2n-2, (2n-1 = (-1) of its C2n subgroup are

(2) = (+ C 1 SU(2) :) C2n

:

(3) = 1 + (2 (4)

= (+ C 1 + (3 + C 3

(5) = 1 + In general, we have (2k) =

+C 2

(2

+ C

2

+

(4

+ C4

2::7=1 ((2 i -1 +(-(2i -1»), and(2k+1) =

•

1+ 2::7=1 ((2i +(-2i ).

In fig. 2, we depict the incidence diagrams corresponding to the decompositions of the spin-.e SU(2) representations, .e = 1/2, ... ,5/2, for the groups C2 , C4 , C6 , and Ca. The incidence diagrams for the group C2 , contained in the center of SU(2), define models whose critical continuum limit is described by the diagonal (unorbifolded) modular invariant combination of SU(2) level k = 2C characters. We recognize the double link in the SU(2h model, for example, to represent the two possible links in a 6V or 8V model. We only show one of the two sets of incidence diagrams that appear at each even level -

the other has the same form and is given by shifting all representations (m

-+

(m+1.

The C2n models can all be regarded as tl n orbifolds of the C2 model (either at or away from criticality). A node fixed under the tl n symmetry is "replicated" to an orbit of n nodes, each identified with a tl n representation w m (w = e 21ri / n ; m = 0, ... ,n - 1) which we term "charge m". A multiple link fixed under the symmetry, however, may

186

Cs

~ ~

SU(2h

SU(2)a

D

SU(2)s

Fig. 2.

C2

+-+

C4 ,6,8 (Z2,3,4 orbifolds)

187 have its different components transform non-trivially under the

~n

symmetry. A link

that transforms as the trivial Zn representation, 1, has its n images continue to link a replicated node to its equal charge partner (or to itself in the case of a tadpole link), whereas a link that transforms in a non-trivial representation, wP , is replicated to links between charge m nodes and their charge m + p partners. In the SU(2h case for example, the incidence diagram at left has a node connected to itself by a triple link (representing the three possibilities (3.38». Under

~n,

one of the links transforms

trivially, and the other two transform as w and w- 1 , leading to the orbifolded diagrams to the right. In conformal field theory, the construction of an orbifold by a group G with a normal subgroup H can proceed in two steps -

first we mod out by H and then by

the quotient group G / H. This possibility as well extends away from the critical point. The the

~4 ~2

orbifolds of fig. 2, for example, can be considered

Z4/~2

=

~2

orbifolds of

orbifolds. We further illustrate this by considering the incidence diagrams for

the binary tetrahedral group T and its normal subgroup, the binary dihedral group

'D 2. Since T /'D 2 = ~3 the SU(2)k/T orbifolds may be constructed by modding out the SU(2h/'D2 orbifolds by a ~3 (and vice-versa). To construct the corresponding lattice IRF models, we need to decompose SU(2) representations with respect to T (as given before (3.40» and as well with respect to 'D 2. The first few representations of SU(2) decomposed with respect to the irreducible representations (1,1',1",1"',2) of its 'D 2 subgroup are

SU(2) :::> 'D2 :

(2) = 2

(5) = 1 + 1 + I' + I" + I'"

(3) = I' + I" + I'"

(6) = 2 + 2 + 2

(4)=2+2

(7) = I' + I" + I'" + I' + I" + I'" + 1 .

In general for even dimensional representations, we have (2k) = k . 2, and for odd dimensional representations (4k

± 1) = k· (I' + I" + 1111) + (k ± 1)· l.

In fig. 3, we depict the incidence diagrams corresponding to the spin-l/2 through spin-5/2 SU(2) representations. These define face models whose critical limits are SU(2) levels 1 through 5 modded out by'D2 and T (with two models at each even level). The involutive

~3

that acts to relate the incidence diagrams is evident, and we observe as

well the various possibilities for multiple links. In the level 3 case, for example, going from 'D 2 to T the two links connecting the nodes labelled by 1 and 2 transform as the two non-trivial representations,

w and w-I,

of Z3. In the level 5 case, there is as well

a third link that transforms as the trivial ~3 representation. Again, any Boltzmann weights that respect the Z3 symmetry of either set of incidence diagrams can be used to

188

level 1"

l'

~ l'

l'

\V

1'"

1"

W! 1'" +--+

1

@!

l'

1"

~rA· l'

~ l'

1"

+----+

2

1'"

1

2

1"

~ 2'

211

3

3

.,."

.~rA,,,

4

l'

l'

~ l'

1"

'~'"

1"1

Fig. 3.

1"

1)2 .... T (Z3 orbifolds)

5

189 level I"

I'

A

4

2

,

1

3

2'

I'

I'

3~3' ~2'

I"

3

I'

,¥," ,·As. ' 1

2

I'

1"

4

I'

5

I"

I'

Fig. 4.

T

+-+

0 (Z2 orbifolds)

190 induce Boltzmann weights for the other set, defining non-critical lattice models whose toroidal partition functions are related by (3.49). Another such example is given by the quotient group 1)2/C4 = ]l2. The C4 diagrams in fig. 2 and the

1)2

diagrams on the left

hand side of fig. 3 are easily deformed to manifest the ]l2 symmetry that relates them. Our last example is given by the normal embedding of the tetrahedral in the octahedral group, with quotient group 0/7 = ]l2. With respect to the irreducible representations (1,1',2,2',3,3',4,2) of 0

c

SU(2), the first few SU(2) representations

decompose as

(2) = 2 (3)

SU(2):::> 0:

=3

(6) = 2' + 4

(7) = I' + 3 + 3'

= 2 + 2' +4

(4) = 4

(8)

(5) = 3' + 2

(9) = 1 + 2 + 3 + 3' .

We use these decompositions to construct in fig. 4 the incidence diagrams again corresponding to the spin-1/2 through spin-5/2 SU(2) representations. Also depicted are the associated T-based diagrams now deformed to reflect the relevant ]l2 symmetry by which we orbifold. The examples here have been given to illustrate some of the basic principles of IRF models and their applications. Generalizations to groups G larger than SU(2) can be considered, and the resulting models enter in the classification program of modular invariant combinations of characters of the affine

G (e.g.

[60]). Models may be con-

structed from incidence diagrams based on the fusion algebra of

SU(2»

G (as

in (1.25a) for

or on the subgroups of G (as in (1.26».

Appendix A. Perron-Frobenius theorem In this appendix, we establish some properties satisfied by the eigenvectors of the incidence matrices A. It is easiest to consider first real matrices Kab, all of whose off-diagonal elements are non-positive. By subtracting the identity times the smallest eigenvalue, we can equivalently consider matrices whose smallest eigenvalue is identically zero. It is easy to see (consider a diagonal basis) that the real quadratic form Kab xax b is then positive semidefinite, and vanishes only at x a proportional to a zero eigenvector

s

of K, i.e. for which Kab sb = O. Note that if sa are the components of a zero eigenvector, then so are

Isal.

which sasb

< O. But since

This is because Kab Kab

Isallsbl and KabSasb differ only in those terms for i= b, we have 0 S Kab Isallsbl S KabSasb = 0,

S 0 for a

and the S is replaced by an equality.

191 Next we show that if the quadratic form Kab xax b is connected, i.e. doesn't separate into separate sums of different variables (or equivalently the matrix K is not block diagonal), then the zero eigenvalue

s

is non-degenerate and its components are

either all positive or all negative. To show this, we first note that if there existed more than one zero eigenvector, we could always take differences to find a zero eigenvector with components sa = 0 for a greater than some m. But then we would have :Ea>m,b;S;m Kab

Isbl

= O. In other words Kab = 0 for a > m, b ~ m, and the quadratic

form would be disconnected. Now since the ness of

s

Isal comprise a zero eigenvector, the unique-

up to overall normalization implies that its components must all be of the

same sign. For an incidence matrix A ("-' c1-K), all of whose off-diagonal elements are nonnegative, the above results imply that the large8t eigenvalue is non-degenerate and its

associated eigenvector has non-vanishing components all of the same sign. To construct models defined by partition functions (1.32), we find the PerronFrobenius vector for a given incidence matrix and define a Temperley-Lieb representation according to (1.30). The planar partition function depends only on the eigenvalue

f3 and, as mentioned in subsection 1.1, such models will have second-order phase transitions (and thus conformal continuum limits) only if f3 ~ 2. To classify incidence matrices whose largest eigenvalue is less than or equal to 2, we can equivalently consider the Cart an-like matrices K = 21 - A, whose smallest eigenvalue should be greater than or equal to zero. The problem then emerges as very similar to the Cartan classification of the simply-laced Lie algebras, in which the K positive-definite case is exhausted by the ordinary ADE Dynkin diagrams, and the K positive semi-definite case by the extended

ADE

diagrams. It is first of all easy to

see that adding nodes or links to a given incidence diagram will always decrease the smallest eigenvalue of the associated matrix K. For a Carlan matrix associated to a given extended Dynkin diagram, decreasing any entry of Kab by adding a link, for example, causes the quadratic form K xx to become indefinite, signaling that the matrix has picked up a negative eigenvalue. Thus any diagram that contains an

ADE

diagram

will have an incidence matrix A whose largest eigenvalue f3 is greater than 2. In this way it is straightforward to argue (see e.g. [63]) that the ADE and all possibilities for incidence diagrams that lead to distinct

f3

~

ADEdiagrams exhaust 2 representations of the

Temperley-Lieb algebra (1.14). (There are as well diagrams with tadpoles satisfying the above criteria, e.g.

1

2

...

0---0- - - -

n-1

n

--o--cO

192 but these do not lead to new statistical mechanical models. The above diagram, for example, is easily seen to define the same model as the A 2n diagram. (It can as well be considered a 712 orbifold of the latter, the orbifold procedure leading to an equivalent model in the absence of a fixed-point in this case.)) In subsections 3.7 and 3.8 and figs. 1-4, we have of course seen a variety of additional incidence diagrams that are used to define models with 2nd order phase transitions. In these cases, however, the partition function (1.32) is not based on ei's that satisfy a Temperley-Lieb algebra. In the first-fused case of subsection 3.7, there is instead a Hecke algebra structure, and more generally for higher fusions there would be a BirmanWenzl-Murakami algebra (see e.g. [64] for more on these algebras).

193 References [1] R. J. Baxter, Exactly 80lved model8 in 8tati8tical mechanic8, Academic Press, N.Y. (1982). [2] J. Cardy, "Conformal Invariance," in Domb and Lebowitz, Pha8e Tran8ition8 and Critical Phenomena, Vol. 11, Academic Press (1987); "Conformal invariance and statistical mechanics", Les Houches lectures (summer, 1988), to appear in Les Houches, Session XLIV, 1988, Field8, String8, and Critical Phenomena, ed. by E. Brezin and J. Zinn-Justin (1989). [3] B. Nienhuis, "Coulomb gas formulation of two-dimensional phase transitions," in Pha8e Tran8ition8 and Critical Phenomena, Vol. 11 op. cit. [4] E. Date, M. Jimbo, T. Miwa, and M. Okado, "Solvable Lattice Models," lectures at AMS summer institute Theta Function8, July, 1987, RlMS preprint. [5] H. De Vega, "Yang-Baxter algebras, integrable theories and quantum groups," LPTHE 88-26 (8/88). [6] J.-B. Zuber, "Conformal field theories, coulomb gas picture and integrable models", SPhT/88-189, Les Houches lectures (summer, 1988), op. cit. [7] P. Ginsparg, "Applied conformal field theory", HUTP-88/ A054, Les Houches lectures (summer, 1988), op. cit. [8] P. Ginsparg, "Conformal invariance and statistical mechanical systems," lectures given at Methode8 en Theories des Champ8 et de8 Corde8, Seminaire de Mathematiques Superieures, Seminaire Scientifique OTAN, 27th session, Montreal (June, 1988), unwritten. [9] "My dinner with M. Green," unpublished. [10] E. Lieb, T. Schultz, and D. Mattis, Rev. Moc;l. Phys. 36 (1964) 856. [11] J. Polchinski, Nucl. Phys. B303 (1988) 226. [12] G. E. Andrews, R. J. Baxter, and J. P. Forrester, J. Stat. Phys. 35 (1984) 193. [13] D. A. Huse, Phys. Rev. B30 (1984) 3908. [14] D. Friedan, Z. Qiu, and S. Shenker, Phys. Rev. Lett. 52 (1984) 1575. [15] P. Goddard, A. Kent, and D. Olive, Comm. Math. Phys. 103 (1986) 105. [16] A. Cappelli, C. Itzykson and J.-B. Zuber, Comm. Math. Phys. 113 (1987) 1, and references therein. [17] V. Pasquier, Nucl. Phys. B285[FSI9] (1987) 162. [18] P. Fendley and P. Ginsparg, "Non-critical orbifolds," HUTP-89/ A004, to appear in Nucl. Phys. B324 (1989). [19] H. N. V. Temperley and E. H. Lieb., Proc. Roy. Soc. (London) A322 (1971) 251. [20] V. Pasquier, J. Phys. A20 (1987) 5707. [21] V. Pasquier, Nucl. Phys. B295[FS21] (1988) 491. [22] V. F. R. Jones, Inv. Math. 72 (1983) 1. [23] R. J. Baxter, S. B. Kelland and F. Y. Wu, J. Phys. A9 (1976) 397.

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195 [46] H. Wenzl, "Representations of the Heeke algebra and subfactors," PhD thesis, Univ. of Pennsylvania (1985). [47] G. Moore and N. Reshetikhin, "A comment on quantum group symmetry in conformal field theory," IASSNS-HEP-89/18 (1989); L. Alvarez-Gaume, C. Gomez, and G. Sierra, "Duality and quantum groups," CERN-TH.5369/89; Phys. Lett. 220B (1989) 142; Nucl. Phys. B319 (1989) 155. [48] E. Witten, "Gauge theories and integrable lattice models," IASSNS-HEP-89/ll, to appear in Nucl. Phys. B; "Gauge theories, vertex models, and quantum groups," IASSNS-HEP-89/32. [49] P. Di Francesco, H. Saleur, and J.-B. Zuber, J. Stat. Phys. 49 (1987) 57; Nucl. Phys. B285[FS19] (1987) 454. [50] V. Pasquier, J. Phys. A20 (1987) L1229. [51] J. McKay, Proc. Symp. Pure Math. 37 (1980) 183; B. Kostant, Proc. Natl. Acad. Sci. USA 81 (1984) 5275. [52] O. Foda and B. Nienhuis, "The coulomb gas representation of critical RSOS models on the sphere and the torus," THU-88-34 (8/88). [53] E. Verlinde, Nucl. Phys. B300[FS22] (1988) 360. [54] N. D. Mermin, "The topological theory of defects in ordered media," Rev. Mod. Phys. 51 (1979) 591; S. Coleman, "Classical lumps and their quantum descendants," (Erice Lectures, 1975) reprinted in S. Coleman, Aspects of Symmetry, Cambridge Univ. Press (1985). [55] R. Dijkgraaf, C. Vafa, E. Verlinde and H. Verlinde, Comm. Math. Phys. 123 (1989) 485. [56] T. Kawai, Phys. Lett. B217 (1989) 247; M. Caselle and G. Ponzano, Phys. Lett. B224 (1989) 303. [57] A. Ocneanu, "Quantized groups, string algebras, and Galois theory for algebras," Penn. State Dept. Math. Research report 88007, in Operator Algebra and Applications, London Math. Soc. Lecture Note Series 136. [58] P. P. Kullish, N. Yu. Reshetikhin and E. K. Sklyanin, Lett. Math. Phys. 5 (1981) 393; E. Date, M. Jimbo, T. Miwa, and M. Okado, Lett. Math. Phys. 12 (1986) 209; E. Date, M. Jimbo, A Kuniba, T. Miwa and M. Okado, Nucl. Phys. B290[FS20] (1987) 231. [59] S.-K. Yang, Phys. Lett. 209B (1988) 242. [60] P. Di Francesco and J.-B. Zuber, "SU(N) lattice integrable models associated with graphs," Saclay preprint SPhT /89-92 (6/89). [61] P. Fendley, "New exactly solvable orbifold models," HUTP-89/A014, submitted to J. Phys. A. [62] Ph. Roche, "Ocneanu cell calculus and integrable lattice models," Ecole Poly technique preprint A900.0689 (6/89).

196

[63] F. M. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter-Dynkin diagram3 and towers of algebras, Chpts. 1,2, IHES/M/87/6. [64] V. F. R. Jones, Ann. Math. 126 (1987) 335; J. Birman and H. Wenzl, "Braids, link polynomials, and a new algebra," Columbia Univ. preprint (1987); Y. Akutsu and M. Wadati, Comm. Math. Phys. 117 (1988) 243.

197

Lectures on N = 2 Superconformal Theories and Singularity Theory

N.P. Warner

*

Mathematics Department Massachussetts Institute of Technology Cambridge, MA 02139

It is shown how singularity theory can be used to classify the N = 2 superconformal

field theories that appear at the fixed points of the renormalization group flow of LandauGinzburg models. The properties the chiral rings of general N = 2 superconformal theories are discussed, and it is shown that if these theories can be considered as Landau-Ginzburg fixed points then the chiral rings can be identified with the local ring of the superpotentials. This identification enables one to relate the central charge of the conformal theory to the singularity index of the superpotential. These ideas are then employed to derive the relationship between Calabi-Yau compactifications of strings and models constructed from tensor products of discrete series representations of N = 2 superconformal algebras.

* Work supported in part by the NSF under grant No. 87-08447, and also by a fellowship from the Alfred P. Sloan foundation.

198 1. Introduction

My intention is to show how the idea.s of singularity theory (and its somewhat disreputable cousin, cata.strophe theory) can be used to understand a great deal about N = 2 superconformal field theories (S.C.F.T.'s). I shall approach this in two different ways, first by gleaning a.s much a.s I can from the mean-field theory of N = 2 statistical mechanical systems, that is, by working with N = 2 Landau-Ginzburg models. The second approach will be to start with the N = 2 superconformal algebra and work back to the mean-field theory. I will then go on to discuss how these techniques can be applied to Calabi-Yau compactifications, and, in particular, derive the relationship between such compactifications and tensor products of N = 2 minimal models. The references upon which these lectures are ba.sed are [1], [2] and [3]. Closely related work has appeared in [4] and [5], while more recent developments are discussed in [6], [7], [8] and [9]. For the singularity theory, the best references are the books by Arnold et al. [10]. The original papers by Arnold [11] are more condensed but are still easily readable [12]. Starting with the (two dimensional) statistical mechanics, the simplest such model to consider is the Ising model. It describes the behavior of a two state spin field. The a.ssociated Landau-Ginzburg theory (mean-field approximation) is given by one scalar field

¢>( x) with action (1.1) where

(1.2) For a < 0 there are two distinct minima, and hence two possible vacuum states (all spins up, or all spins down). For a :?: 0 there is one vacuum state. The critical point of the system occurs when the two minima come together, or when the single minimum splits or bifurcates (i.e. when a

-->

0). At the critical point one has a = 0, and the fields have long

range fluctuations whose correlation functions correspond to some conformal field theory (C.F.T.). There are two important points to note: 1. The action, S, is obviously not conformally invariant, and so we will have to clarify

in what sense it defines a conformally invariant model. 2. I may have given the impression that a Landau-Ginzburg theory completely specifies the conformal theory at the critical point. One should realize that it is the original statistical model, underlying the approximating Landau-Ginzburg theory, that specifies the conformal theory at a critical point. One ha.s no a priori reason to believe

199 that the Landau-Ginzburg form will precisely determine the fixed point. However, one can take the attitude that the Landau-Ginzburg action defines, in its own right, a two-dimensional field theory, and one can therefore seek infra-red fixed points of this theory. I will take this viewpoint here, but one should bear in mind that since one has lost some information about the original statistical mechanical system, this loss may well manifest itself later as some ambiguity in determining the fixed points of the Landau-Ginzburg model. I will discuss these points further after I have introduced the N = 2 Landau-Ginzburg theories. There are, of course, obvious generalizations of the Landau-Ginzburg action. One could take n

V( behaves asymptotically as e- m1zl for large Itl. Thus, in the infra-red limit the correlation functions are simply delta functions -

the theory is trivial. In other words, the particles freeze out

when the mass scale of the theory becomes much less than the particle masses. This means that given (m + n) fields cJ>A, A = 1, ... , m

+n

and a superpotential W( cJ>A) of the form m+n

W(cJ>A) = W(cJ>i)

+

L

(cJ>A)2 ,

A=n+l

where i = 1, ... , n, then the conformal field theories corresponding to W and W are identical. Thus we can not only change variables, but we can change dimensions by adding quadratic pieces to W.

205 Therefore, we want to classify all potentials, W, with the multi-critical structure described earlier, up to the addition of extra "quadratic dimensions" and changes of variables. This classification is the text-book definition of stable singularity theory [10]. (The word stable refers to the possibility of changing the dimension by addition of quadratic terms.) One can now go to a book on the subject and learn a great deal about N = 2 superconformal field theory. Before doing this we note there is an additional physical condition that we need to impose. The non-renormalization theorems tell us that W is invariant up to wave function renormalization, whereas, the F-term is unrenormalized. Under scaling z have 8

-+ A- 1 / 2 8

and thus

J Iflzlfl8 -+ A-I J Iflz1fl8.

Hence

W(4)i)

-+ A-I z,

we

must scale according

to: (2.12) at the fixed point. However, wave function renormalization means that (2.13) for some numbers Since the

4)i

Wi.

The numbers

Wi

are called the anomalous dimensions of the

are scalars, the conformal weights of 4)i are given by

For a generic potential,

W(4)i),

hi = hi =

with some power expansion about

4)i

4)i.

!!ff. = 0, the only

way to reconcile (2.12) and (2.13) is if the coefficients of the power series are viewed as coupling constants that also scale so that when this scaling is combined with (2.13) one obtains (2.12). For example, given gl -+ AI - 3w gl

and

g2 -+ AI - 5w g2.

W(4»

= gl4)3

+ g24)5

with

4) -+

AW 4), we must have

If all the conformal dimensions are strictly positive

it follows that the lowest order terms in the power series of

W(4)i)

will dominate as we

approach the critical point. At the critical point the F-term must be scale invariant, and in the infra-red limit this means that all the higher order terms must vanish, and the scaling dimensions of the fields must be such that the lowest order part, W o, of the superpotential, W, satisfies

(2.14) with no scaling of the remaining coupling constants. In the example given above, as A -> g2 =

00,

and in order to get a scale invariant superpotential we must take

0 and then

gl

g2 / gl -> W

0

= 1/3,

does not scale. In general, at the conformal point, the superpotential of

the theory must satisfy (2.14) (with no scaling of the coupling constants) for some choice of the

Wi.

Such a potential is called quasihomogeneous with weights

Wi.

Note that these

206 weights are precisely the anomalous dimensions of the ~i at the conformal point. The

Wi

are rational numbers, and are obviously completely determined by the form of Wo(~i), provided that Wo has an isolated critical point at ~i = O. The notion of the "lowest order part", Wo, of W, is, in general, ambiguous. Moreover, even if the critical points of W are isolated, those of Wo may not be. These difficulties may give rise to indeterminacies in the renormalization group flow -

there could be several

fixed points, or surfaces, to which the theory might flow. It is conceivable that there might even be strange attractors. On a more down-to-earth level, the presence of flat directions in Wo could reflect some kind of spontaneous 'decompactification' of the theory. While the consideration of these possibilities might contain some extremely interesting physics, we shall take the simpler course and remove the problem by requiring that the starting superpotential, W(~i), be semi-quasihomogeneous. In other words, we require that W has the form W

= Wo + W l

where Wo is quasihomogeneous, and has a single, isolated

multi-critical point at ~i = 0, and Wl only contains terms of scaling dimension strictly greater than one.

4

Finally, one should observe that at the conformal point, W l scales

away leaving us with Woo Therefore we would have obtained precisely the same result if we had started with Wo in the first place. As a consequence of the foregoing considerations, we will, from now on, only consider a quasihomogeneous superpotential W, satisfying (2.7)-(2.10) and

(2.15) Observe that this fixes the conformal weights of the fields ~i to be: -

hi = hi =

Wi

2'

(2.16)

and these are all strictly positive. This means that the D-terms of (2.5) can only contain terms of conformal dimension strictly greater than 2, which are irrelevant. There are thus no marginal operators in the D-terms of the Landau-Ginzburg theory. This restriction to quasihomogeneous superpotentials simplifies things considerably, but it turns out that it is not as severe a restriction as one might expect. Many of the stable singularity types have quasihomogeneous (or semi-quasihomogeneous) representatives. The whole point of the discussion above is to point out that we are explicitly excluding a 4 The scaling dimensions of the fields ~i, and hence of the terms in W l are determined by Woo

207

number of potentially interesting Landau-Ginzburg theories that do not necessarily have well-defined or unique fixed points (or surfaces). Some of the excluded class are very interesting from the point of view of singularity theory. Returning to the main subject, given a superpotential W satisfying (2.7)-(2.10) and

(2.15), I will prove later on that the central charge of the conformal theory at the fixed point of the flow is given by c=

The quantity

613 ;

13 =

t (~ -Wi) .

(2.17)

.=1

13 is called the 8ingularity index of W. For a general superpotential, W, the

singularity index is defined by the asymptotic expansion of the integral (

JRn as h

-+

e~W("')dnx ~ h(~-~)

O. By using this very suggestive definition one can make a path integral argument

to establish that c =

613 [1]. I will not do this here as there is a much more precise, and

extremely simple argument based on the superconformal algebra. A8ide. -

There was a conjecture in singularity theory that the singularity index is upper semicontinuous [11] in the sense that if one is given a function

f

whose singularity index is

f3t,

then there is a small neighbourhood of the function f (in the space of all analytic functions)

f3t is the maximum value of the singularity index on that neighbourhood of f. 13 is a locally non-increasing function on the space of analytic functions. Given the identification (2.17) between 13 and c, we see that the foregoing conjecture is merely a such that

That is,

statement ofthe Zamolochikov c-theorem [14]. The surprising thing is that while the upper semi-continuity of 13 is valid for a broad class of singularities, there are counterexamples to the conjecture [17] [10]. I do not believe that this means that there are counterexamples to the Zamolodchikov c-theorem. The counterexamples to the upper semi-continuity of

13 are not semi-quasihomogeneous, and so the lack of upper semicontinuity presumably reflects an ambiguity in the conformal fixed point. It would be interesting to investigate this further. There is another extremely useful and natural structure that one introduces in singularity theory. It is the notion of the local ring of W. Consider the ring of power series of functions expanded about ~i = O. Call this ring 'P. Let J denote the ideal of this ring

208 generated by all the first partial derivatives, ~::, of W. The ideal

.J consists of all linear

combinations of power series that have one (or more) factors of one of the ~::. The local ring

n of W

is defined by

n

=

P/.J

(2.18)

and it consists of all power series taken modulo the partial derivatives (~::). The dimension of

n is finite and is, in fact, equal to the multiplicity, p"

of the singularity.

As an example, consider

This is quasihomogeneous with weights

Wx

= 1/3,

Wy

= 1/5. The partial derivatives are

2

3x and 5y4. Hence n consists of all polynomials taken modulo x 2 and y\ and is thus

equal to all all linear combinations of

which is an eight dimensional ring. Physically the local ring,

n, of a superpotential, W, corresponds to what is called the

chiral ring of the superconformal theory. (N.B. In these lectures I shall never use the term "chiral" in the sense of being either holomorphic or anti-holomorphic, instead I will always use the term chiral in the sense of N = 2 supersymmetry, that is, it will mean that the constraints (2.2) have been applied to all the superfields.) In the conformal field theory all the chiral fields of the theory must be obtainable from polynomials in the ;pi (i. e. elements of P). The partial derivatives, ~:: of Ware, by the equations of motion, proportional to some superderivatives acting on some combination of ~\ and ;pi's. Thus ~~ are polynomials in ;pi that represent descendant fields. 5 Moreover the set of (~~) and their products with all polynomials in ;pi generate all polynomials of the ;pi that are descendant fields. Hence n is precisely the ring of chiral, primary fields in the theory. We will see later (and I believe that it should also follow from the non-renormalization theorems), that the operator product algebra of the chiral, primary fields is in fact the polynomial algebra of n. That is, there are no powers of z by which one needs to rescale, and the operator product vanishes if and only if the leading term is a descendant field. In other words, the local ring 5 For present purposes a combination of fields will be thought of as a descendant if it can be rewritten as some other combination of fields that involves derivatives, or superderivatives. This definition will coincide with the usual one at the conformal point.

209 of Wand the ring of chiral, primary fields (with multiplication being the operator product) are isomorphic as rings. As a result of this, one can conclude that the conformal dimension of any monomial in

n is simply the sum of the conformal dimensions of the fields in the fa. The

monomial. Thus in our example above we know that hx = twx = ~, hy = tWy = conformal dimensions of the monomials of n are thus field

field

dimension 0

y3

10

xy xy2 xy3

30

x y y2

6

30 6

30 12

30

dimension 18

30 16

30 22

30 28

30

Note that the denominator, 30, is the Coxeter number of E 8 , while the numerators are the degrees of the Casimirs of Es minus two, and that the dimension of the ring is 8, which is the rank of E 8 • This is not a coincidence. To apply the results of singularity theory we need to introduce three other concepts: versal deformations, modality and the Poincare polynomial. Given an analytic function in

en,

f :U

~

e where U is some neighbourhood of the origin

then an analytic function F : U x V .,...

(i) V is some neighbourhood of the origin in

e is said to be a versal deformation of f

if

em, with m < 00.

(ii) F(x,O) = f(x), for x E U. (iii) There is a small neighbourhood, F, of bourhoods of the origin in of variable

y( x)

en

f in the space of analytic functions on neigh-

such that for every 9 E F there is an analytic change

such that for x in some neighbourhood of the origin in

en

we have

g(y(x» = F(x, A) for some value of A near the origin in

em.

In other words, in suitably small neighbourhoods of the origin, every analytic function near

f can be obtained by analytic changes of variable of F(x, A) for some value of A. If f has a singularity of multiplicity J.I at the origin then it can be shown (the versality

theorem [10]) that the following is a versal deformation: IL

F: U x

elL

~e;

F(x,A) = f(x)

+ LA"r,,(x) v=l

6

Remember that since the superfields are scalars, we have h =

h.

,

(2.19)

210 where r,,(x) are a basis for the local ring of

f.

In other words, the local ring provides a

universal class of deformations of f(x). It is also useful to note that for general values of the parameters A", the function

F(x, A) morsifies f(x). That is, F completely resolves the singularity of f into /I distinct critical points at which det (a:~IxJ) =f. o. 7 It turns out that the foregoing deformation of miniversal. That is, for a function of multiplicity /I, a versal deformation must always have at least /I parameters, and thus the deformation (2.19) has the minimum number of possible parameters. The question naturally arises as to whether one requires a continuum or discrete set of values of the parameters A so as to get to everything in the neighbourhood of f via a change of variable. A function,

f, is said to have a singularity of modality zero if every f is coordinate transformation equivalent to a versal

function in the neighbourhood of

deformation F(x, A) of f(x) for a finite set of values of A. This means that only a finite set of choices of A are needed to cover a small neighbourhood, F, of

f

via the action of

changes of variables. Note that no matter how small we make F, a function of modality zero always has a finite set of choices for A that enable one to cover F by the orbits of the corresponding finite set of functions F( x, >.) under the diffeomorphism group.

A function f has a critical point of modality m if F can be covered by the action of the diffeomorphism group on F(x, >.) for finitely many m parameter families of deformation parameters A, and m is the least such number of parameters. These parameters are called modal parameters, or moduli, of the singularity. Physically there is a very simple picture of a modal parameter. In general a deformation will break up a singularity, resolving it into two or more singularities of decreased multiplicity. A modal deformation is one that does not do this. Thus a modal deformation is one that preserves the multiplicity or multi-critical structure, but cannot be thought of as a change of variable. Thus it represents the fact that the statistical system has some kind of physical modulus. One might be tempted to conclude that such modal deformations might correspond to moduli of the conformal theory. This is not so: some of the statistical mechanical moduli can correspond to irrelevant operators at the conformal point. For a quasihomogeneous function (with an isolated critical point) the characterization of modal deformations is extremely simple. We have

7 Technically, a morsification of f( x) must also take different values at each of its critical points.

211 and we can also choose a basis of the local ring 'R to be a set of monomials in

c)i.

Each

such monomial may thus be assigned a scaling dimension. Those of dimension greater than or equal to one (i.e. h = exactly one (i. e. h =

Ii:::: 1/2) represent modal deformations [10]. Those of dimension

Ii = 1/2) represent moduli of the conformal field theory since adding Ii = 1) to

them to the superpotential would introduce a dimension two operator (h = the component action after performing the

integrations. Thus modal deformations are

(J

precisely the marginal and irrelevant operators in 'R. The modality of the singularity is therefore easily counted by computing 'R. As one would expect, a relevant deformation breaks up the singularity, decomposing it into singularities of lower multiplicity. Suppose that we have a quasihomogeneous function, W, of n variables,

c)j,

with

weights Wj. Let Pj and N be (non-negative) integers such that Wj = pj/N. For definiteness, let N and Pj be reduced so that they have no overall common factor.

The Poincare

polynomial of W is defined to be:

pet) where

nk

(2.20)

is the number of monomials of scaling dimension kiN in a basis for the local

ring, 'R, of W. It is relatively easy to see that [10]:

n( n

pet)

=

1- tN-Pi

1_

tPi

(2.21 )

)

j=l

The proof is simply to observe that the terms in the denominator generate the partition function of the free polynomial algebra in the

c)j,

while the numerator subtracts out the

contribution of everything generated by ~:; (whose scaling dimension is (N - pj)/N). There are two immediate consequences of the foregoing result. First, from the definition of pet) we see that p = P(l), but, applying l'Hopital's rule in (2.21), we get (2.22) where Wj = pj / N are the weights. Now consider the limit as t -+ that the highest power of tin pet) must be n

M

= 2)N-2pj) j=l

tM

00.

From (2.21) it follows

where n

NL(1- 2wj) j=l

2N;3,

212

and moreover the coefficient of this term in pet) is one. Therefore there is an unique element, p, of maximal scaling dimension in the local ring,

n of W,

and the maximal value

of the scaling dimension is 2{3. Thus p must have conformal weights: (2.23) One can also see that p must be the polynomial: (2.24) Observe that pet) has a (Poincare) duality:

pet) = t 2Nfi P(l/t) . (This follows trivially from (2.21).) Under this duality the state p interchanges with the unique state with h =

Ii = 0, that is, with the vacuum.

Finally, it is interesting to note that given an arbitrary set of integers Pj, and an integer N, then a quasihomogeneous function with weights Wj = pj/N can only exist if the function pet) defined by (2.21) is a (finite) polynomial. If (2.21) is a finite polynomial then such a quasihomogeneous function still might not exist, but the interesting observation here is that the set of weights of a quasihomogeneous function are, in general, a highly restricted set of rational numbers. With this basic introduction to singularity theory, we are in a position to quote some of the theorems. All singularities with modality S 2 have been classified up to stable equivalence [10J. All modality zero singularities are stably equivalent to one ofthe following [10]: Ak

x k+1

k:?: 1

Dk

x k- 1 + xy2

k:?:2

E7

x3 +y4 x 3 + xy3

Es

x3 +y5

E6

c= 3-

k!l

c = 3 - 2(k~1)

c=3-fi c=3-fs c=3-fo

where c is the central charge of the corresponding conformal theory, and has been computed using (2.17). Note that the labelling of the singularities with the labels of the simplylaced, finite Lie algebras is no accident. First observe that the central charge is given by

213

c = 3 - ~ where N is the Coxeter number of the algebra. One can check that the local ring has dimension I, where 1 is the rank of the algebra, and there is a basis for the local ring in which the dimensions of the fields are

¥

where d; are the degrees of the Casimirs

of the Lie algebra. These coincidences are not mysterious numerology. There is a close connection between the topology of the bifurcation varieties of these functions and the braid group of the Weyl group of the corresonding Lie group. On a more down to earth level, suppose W is some generic versal deformation of W, and one tries to find integral curves of V'W, i.e., solutions of (2.25) and suppose one seeks solutions running between the critical points of W. Represent the existence of such curves by drawing a dot for each critical point and a bond for each integral curve. Then up to some mathematical niceties, the generic result is the Dynkin diagram for each Lie algebra [18]. These integral curves have a beautiful physical interpretation. Observe that if (2.25) is satisfied then

Now recall that

V = IV'WI2 is the bosonic potential of the theory and that

these integral

curves represent rolling motions on the inverted bosonic potential well. Thus they represent semi-classical tunelling solitons of the theory. The Dynkin diagram thus tells you about the vacuum states and the tunelling solutions. The list of modality zero singularities singularities corresponds precisely with the list of all the modular invariant N = 2 superconformal discrete series models [19] [20]. Indeed, the discrete series models are known to have c = 3 - ~ where N is the Coxeter number of a simply-laced Lie algebra. In general the chiral fields in the theory have J h = h = 2N;

j =0,1, ... ,N-2.

Such theories correspond to the AN-l series discussed above. For N even, there is a Z2 orbifold of these AN -1 theories, which gives rise to the D Jf+! series. There are also three exceptional models in which many more fields have been projected out, and which give rise to the E 6 , E7 and Es theories. There are several important things to note:

214

1. Modular invariance of the Landau-Ginzburg models is guaranteed; there is no chiral splitting and there is always left-right symmetry and thus no global world sheet anomalies can appear. 2. Given a quasihomogeneous superpotential W(~j) with weights Wj, write the weights in the form Wi = Pi/N where Pi and N are integers as above. It is clear that the Landau-Ginzburg theory has a ZN symmetry

S

~j -+ e 21fip;/N ~i

(2.26)

~i -+ e-21fip;jN ~j.

For the discrete series models this symmetry has been discussed in [19). 3. The table above strongly suggests isomorphisms

E6 - A2 ®A3

(2.27)

Es - A2 ®A4 and indeed this is true up to ZN twists. 4. There are many, many more singularities beyond the modality zero ones described above. There are vast tables up to modality 2, and of course it is elementary to write down higher modality potentials. For example, the singularity of the function

has multiplicity J.i = 6k, and modality k - 1. (The

ai

are the modal parameters.)

Some singularites give rise to conformal theories that are obviously tensor products of simpler ones such as the discrete series, while other singularities correspond to completely new conformal theories. For more recent results on this see [3)[5). There are several major virtues of the perspective that the application of singularity theory affords one. First it yields new models. Secondly, the ring of chiral, primary fields captures some simple intrinsic structure of the theory that tells one when two theories can or cannot be isomorphic. The example of Es

~

A2 ® A4 is typical. Thus it seems hopeful

that one might use the structure of the chiral ring to classify the N = 2 superconformal s If one considers the action of this Z N on the Ramond sector, then this symmetry may well become a Z2N symmetry.

215

theories. The problem with such a programme is that not all N = 2 superconformal models correspond to Landau-Ginzburg theories. The simplest counterexample is a complex boson and a complex fermion compactified on a 2-torus. This is an N = 2 theory, but has no Landau-Ginzburg form since the chiral fields are not left-right symmetric. There are many other counterexamples, some of which will be discussed later. However, it turns out that in many of these counterexamples one can turn the theory into a Landau-Ginzburg theory by choosing the moduli appropriately and dividing by a discrete symmetry. A generic property of many of the known counterexamples is that the chiral ring is too small and the number of moduli is too large. Division by discrete symmetries usually freezes out some of the moduli and introduces twist fields that fill out the chiral ring. So one might hope that if one twists a theory sufficiently then it will have a Landau-Ginzburg form. It is not known whether this is true in general. We now start from the other end of the renormalization group flow and see what can be deduced about chiral rings and the Landau-Ginzburg formalism from the properties of the superconformallimit.

3. N = 2 Superconformal Algebras

At a fixed point of the renormalization group flow of an N = 2 Landau-Ginzburg theory we obtain an N = 2 superconformal model. This means that the states of the theory form a (modular invariant) unitary (and, in general, reducible) representation of the N = 2 superconformal algebra (S.C.A.).

9

In this algebra, the Virasoro algebra is

extended by two supercurrents and a U(l) current. The commutation relations of the generators of the N = 2 S.C.A. are:

(m - n)Lm+n + 1c2m(m2 - l)om+n,O c

[Jm,Jn]

3" mOm+n,o

[Ln,Jm ]

-mJm+ n

[Ln,G~±a]

G-

[In,G~±a]

± G~+n±a

{G~+a, G;'_a}

(m ± a))

(3.1)

G~+n±a

2Lm+n + (n - m

+ 2a)Jn+m + ~

[(n

+ a?

-~] om+n,O ,

9 For a discussion of the properties of the representations of N = 2 S.C.A.'s see, for example, [21] [22].

216 where m and n are integers and a is a real parameter. Corresponding to these generators are the conformal fields: 00

T(z)

L n Z -n-2

L

n=-oo 00

G±(z)

L

G~±a

z-(n±a)-3/2

(3.2)

n=-oo 00

J(z)

L

Jn

Z

-n-l

.

n=-oo

The foregoing commutators maybe expressed in terms of operator product expansions:

~e 2J(w) + 2T(w) +8w J(w) (z-w) (z - w)3 + (z - w)2 G±(w) ±-( - ) + ... z-w

J(z)J(w) T(z)J(w)

!e

3 )2+ ... z-w J(w) + 8w J(w) (z - w) (z - w)2

(

!G±(w) (z-w)2

!e

T(z)T(w) = (z-w)4 where, as usual,

+ ...

+ ...

(3.3)

+ ...

+

8w G±(w) (z-w)

+ ...

+

2T( w) (z-w)2

+ (z-w) + ... ,

8w T( w)

means plus terms that are finite in the limit as z

~

w.

In this algebra, a is a free real parameter that dictates the branch cuts in G(z). In the Ramond sector a E Z and in the Neveu-Schwarz (N.S.) sector a E Z

+!.

Shifting a

by an integers yields isomorphic algebras. The S.C.F.T.'s arising from Landau-Ginzburg theories have an N = 2 superconformal algebra for the left-moving and right-moving sectors. The generators of the left-moving algebra will be denoted by G, L and J, and will correspond to holomorphic world-sheet coordinates, while the right-moving algebra will be denoted by G,

L, and J

and will corre-

spond to anti-holomorphic coordinates. In the sequel, I will largely suppress all discussion of the right-movers as their treatment exactly parallels the discussion of the left movers. I shaH, for the present, work only in the Neveu-Schwarz sector, but one should bear in mind that there is an exactly parallel discussion for the Ramond sector. I shall. also no

217 longer restrict myself to the consideration of Landau-Ginzburg theories, but will consider an arbitrary (2,2) theory. Let the indices r, S,

. •.

take values in Z

G;:

+ !.

A primary state

1111 > is defined by

1

1111>

0,

r> -2-

Ln 1111> = I n 1111>

0,

n

~

(3.4)

1

(with similar conditions for the right-movers). The conditions Lnlll1 >= J n lll1 >= 0, n follow from the condition G r ±11I1

>

°

> = 0, r > 0, by unitarity and (3.1). Thus the second

two conditions in (3.4) are redundant. The states of the theory can be diagonalized into eigenvectors of Lo and J o

hi 111 >

(3.5)

qlll1> , where h is the conformal weight of

1111 > and q is its U(l) charge. 1111 >, there is a primary conformal field lI1(z) such

Corresponding to a primary state that:

T(z) lI1(w) J(z) lI1(w) G±(z) lI1(w)

h 111 (z - w)2 (w) q

-( - ) lI1(w) z-w 1 ± -(- - ) A (w) z-w

ow ll1 (w) + ... w)

+ (z + ...

(3.6)

+ ...

for some field A±(w), The fields A±(w) are the superpartners of lI1(w). A primary, chiral state,

111 >, is defined to satisfy (3.4) and

G~1/2 111 > =

c:r:. 1/ 2 111 > =

0.

(3.7)

111 >. There is, of course, the 0::111 > = 0, r ~ -1/2, for the right-moving algebra. These constraints are basically the superconformal analogues of (2.2). More precisely, 111 > is the lowest This constraint removes half the superpartners of the state

parallel constraint

component of a chiral superfield, the other components will be obtained by acting with G=1/2 and a=1/2 on

or

111 >. Equation (3.7) means that 111 > is chiral because it has no 8+

e+ superpartners. The corresponding primary, chiral field 11(z) therefore satisfies (3.6)

with A+(w) == 0. That is,

G+(z) 11(w) ~ finite.

(3.8)

218 Now observe that for any state lilt> we have, by unitarity,

o ~

=

>, and observe that

2C] < ~I [2Lo - 3Jo +"3

which implies that (2h - 3q + ¥) ~ O. Using q = 2h, one obtains h

< ~ -

(3.11)

6

and h = c/6 if and only if

G~S/2 I~ > =

o.

9.1. Spectral Flow.

We now observe that given a representation, 'Ha of the N = 2 S.C.A. for some value of

a, we can slowly change the value of a, and continuously change the representation 'Ha as a function of a. If we change a by an integer, then the algebra flows back to an isomorphic copy of itself. However, some of the representations 'H a, 'H a±l, 'H a±2 etc. may well be

219 distinct. Let Us denote the flow operator taking ?-la to ?-la+8' Given an operator Da on ?-la, there is an operator Da+s acting on ?-la+S:

In particular,

Us G;±a Ui

1

Us I n Ui

1

In

Us Ln Ui 1

Ln

G;±(a+8)

+ ~3 elin, 0 + eJn + 6'C e2 lin,o.

(3.12)

One can explicitly construct the operator Us by splitting ?-la into two pieces: a pure U(l) Hilbert space, and the U(l) neutral space that is the orthogonal complement of the pure U(l) Hilbert space. That is, introduce bosons .w,~i. Let (1 = X and

-~6/~1 (~1 )-P1/P2

o

o The determinant of this is

This is independent of ~1 and hence of X if and only if

1)

L-:-PJ n

(

= 1,

j=1

or equivalently

3(n-2).

(4.3)

229 The left-hand-side of this identity is the central charge of the Landau-Ginzburg theory at the conformal point. Now ignore the kinetic term and consider the path integral in the new variables:

! dX ax ( gdeidei)

IJ((i)Iexp

2

(! ~()+

W((41 1)Pl/P; ei)

+

! ~()-

W((¥I)pl/P; ei) ). (4.4)

Using the quasihomogeneity of W we see that: (4.5) Substituting (4.5) into (4.4), and providing that J is independent on X and

X, one can

easily integrate out the variables X and X to obtain:

! IT dej dej

IJ(01-2 8(W) .

(4.6)

j=2

In other words we get a path integral over the vanishing set of W evaluated in the projective coordinates 1,6,··.,

en.

The coordinates 6, ...

,en can be regarded as a patch on (n -1 )-dimensional weighted

projective space, WCPn-t, in which

(4.7) The patch, of course, excludes the point 41 1 == O. Other patches can be obtained by making different changes of variable. The delta function imposes the constraint

(4.8) in this projective space and thus restricts the path integral onto a complex (n - 2)dimensional hypersurface. We have seen that the foregoing is only true if the Jacobian, J, is independent of X or, equivalently: c = 3(n-2).

(4.9)

230 However, this is precisely the central charge of a conformally invariant N = 2 supersymmetric string moving on a complex (n - 2)-dimensional manifold (each complex boson contributes 2 to c and each complex fermion contributes 1 to c). Thus, in order to perform the path integral over X and X we had to arrange that (4.9) be satisfied, but this is precisely the requirement that we get the correct central charge for a string moving on the algebraic hypersurface defined by (4.8). In addition to this, it is a straightforward exercise in algebraic topology to check that the hypersurface (4.8) in the weighted projective space (4.7) has vanishing first Chern class if and only if J((;) is independent of

(I

=

x.

In other words our Jacobian condition not

only is equivalent to getting the correct central marge, it is also equivalent to arranging that the algebraic hypersurface is an (n - 2)-dimensional Calabi-Yau manifold. The results that I have just obtained can obviously be extended to any Calabi-Yau space defined by a single polynomial in a single weighted projective space. Moreover, the foregoing argument can be further generalized: given an arbitrary complete intersection Calabi-Yau manifold that is expressed as the vanishing locus of a set of polynomials in products of weighted projective spaces one can often reverse the procedure above and obtain a corresponding Landau-Ginzburg theory. To accomplish this one may have to integrate out several variables to return to the Calabi-Yau space (one integration per delta function) and one may have to introduce some redundancy into the description of the Calabi-Yau manifold (such as tensoring in CPo's, i. e. points) so as to pass to the LandauGinzburg form. There are, however, Calabi-Yau manifolds for which it is not known how to construct equivalent Landau-Ginzburg theories. An example [24] is the surface defined by: X2Y2

o

+ X~ + xt)

0,

XIYI

YI(X~

+ x! -

X~)

+

Y2(Xt

+

(4.10)

where Xl, ... ,Xs and YI, Y2 are homogeneous coordinates in C P4 and C PI respectively. The difficulty lies in finding a Landau-Ginzburg superpotential with an isolated singularity (i. e. one for whim the multiplicity, /1-, is finite). To find a Landau-Ginzburg form corresponding to the Calabi-Yau manifold (4.10) would prove extremely interesting as this manifold has Euler maracteristic -168 and its space of moduli has a common boundary with the space of moduli of the quintic in CPs [24], whose Euler marcteristic is -200, and for which the corresponding Landau-Ginzburg theory is known [15][4][2]. Because of the difference in the Euler characteristics, something interesting should happen in the transition between the conformal field theories corresponding to these Calabi-Yau manifolds.

231 Putting aside the question of generalizations, there are still some important technical points to be resolved in the original argument. First, the delta function, b(W), in (4.6) is a super-delta function. Its lowest, B independent, component forces the bosons onto the algebraic surface, while the higher components in B force the fermionic parts of the superfield to be tangent to the algebraic surface. Secondly, we must return and consider the role of the kinetic term in what, so far is a highly suggestive, but formal calculation. In the foregoing we ignored the kinetic term completely and one cannot really do this in practice. However, if one were to include it, one could still perform the the X and X integrations by including all the appropriate propagators. The result would no longer be a delta function, but an extremely complicated Gaussian concentrated on the surface W == 0 and whose characteristic width is

t, where A is the length scale of interest. In the infra-red

limit this Gaussian would become a delta function, while the kinetic parts that lie tangent to the algebraic surface would survive. Thus the correct way of interpreting the foregoing argument is not that the Calabi-Yau manifold and the Landau-Ginzburg theory are the same, but that they lie in the same universality class. That is, their infra-red fixed points are the same and they yield the same N.= 2 S.C.F.T .. The final technical point is to note that the change of variables (4.2) is only single valued provided that we identify (4.11) where Wi = l/pj is the scaling dimension of qii. Equivalently, in order to interpret the variables

ei as defining a a patch in WCP

n- b

we have to make the identifications (4.11).

This means that we have to twist the original Landau-Ginzburg model by the symmetry (2.26) in order to get the Calabi-Yau manifold. This is precisely Gepner's U(l) projection [20]. Note also that because the resulting orbifold has a quantum symmetry of at least ZM, where M is the least common mulptiple of the Pi, we obtain the Calabi-Yau manifold

at a rather symmetric point in its space of moduli [7]. Therefore, given a Calabi-Yau manifold to which we can apply the path integration argument and thereby obtain an equivalent Landau-Ginzburg theory, it is not strictly true that the Landau-Ginzburg theory gives the entire Calabi-Yau manifold. It appears that we only get the Calabi-Yau manifold at some point in its space of moduli. However, we are now erring too far on the side of modesty, and we can usually do considerably better than this. One should remember that the superpotential of the Landau-Ginzburg theory can

232 have many independent, dimension one, modal parameters. These correspond to moduli of the S.C.F.T., and will not be projected out by the U(l) projection (4.11) precisely because the corresponding operators have dimension one. Thus they will be moduli of the theory on the Calabi-Yau manifold.

In many instances the number of moduli of

the Landau-Ginzburg potential and the Calabi-Yau manifold are the same, and thus the Landau-Ginzburg model may well cover the entire Calabi-Yau moduli space. When the number of modal parameters in the potential is less than the number of Calabi-Yau moduli, then the deficit of the Landau-Ginzburg model will be made up from the twisted sectors under the U(l)-projection (4.11). It is perhaps most instructive to consider a simple example of the foregoing results.

One of the modality one singularities is

W(x,y,z) = x 3

+

y3

+

z3

+

axyz

(4.12)

where the parameter a is a complex modulus of the singularity. By the integration arguments above we see that the Landau-Ginzburg theory with this superpotential corresponds to the algebraic surface

in C P2 (where the

ei are projective coordinates).

It is straightforward to see that this

surface is simply a complex torus, and the parameter a determines its complex structure. The corresponding N = 2 S.C.F.T. must therefore be a c = 3 superconformal theory that is equivalent to a torus compactification of a single N = 2 superfield. This result can easily be demonstrated directly. As we have already seen, a Landau-Ginzburg theory with an x 3 superpotential corresponds to a c = 1 S.C.F.T that can be described by a single boson compactifled on a lattice.

Therefore the Landau-Ginzburg theory with superpotential

(4.12), with a = 0, can be described by three compactified bosons. Indeed, we can take

G±(z) =

~ V3"

t

exp(ivj . ¢(z» ,

j=1

where

Make an orthogonal transformation of the bosons ¢(z), so that the Vj have the form:

VI

= (l,a),

V2

= (1,iJ),

V3

= (1, -(a + $»,

233

where a and

i3 are the simple roots of SU(3).

still orthogonal and have length

v'3.)

(One can easily check that ii}, V2 and Va are

Observe that in this new basis,

where

and

Note that

J+(z) r(w) = ( where + ... denotes terms that are finite as z

1

z-w

-->

)2

+ ... ,

w. Therefore we can introduce new bosons

Xl and X 2 such that, ( 4.14) and hence

which is the standard N = 2 supercurrent on the 2-torus. The states in the N.S. sector of this N = 2 S.C.F.T. have i-momenta on the lattice

H(nl VI + n2v2 + n3v3) : ni E Z}. While these states are local with respect to G±(z), they, in general, have

Zl/3

branch cuts with respect to ifJ±(z) and J±(z) separately. Thus to

obtain all the states of this S.C.F.T. one must apply a Z3 twist to the free bosons Xi and to the free fermions ifJ i . Indeed the twist fields are precisely the Landau-Ginzburg fields themselves:

Because re-bosonization (4.14) ofthe currents (4.13) in terms of the

ax i merely represents

a change of basis in the SU(3) Kac-Moody algebra it follows that the entire Hilbert space of the Landau-Ginzburg S.C.F.T. is a Za orbifold of a superstring compactified on the maximal torus of SU(3). Thus, up to this Za twist, we get a the expected torus compactification, but note that the torus has a very specific size and shape (i.e. that of SU(3)).

234 One should also note that because of the identifications (4.11) implicit in the change of variable argument, we should also have expected this Za twist. Finally, a string compactification on a 2-torus has two complex moduli, one of which corresponds essentially to a combination of the scale of the torus and the expectation value of the B-field, while the other modulus corresponds to the complex structure of the torus. Upon dividing by Za, the modulus corresponding to the complex structure survives, while the other modulus is frozen out. In the Landau-Ginzburg theory, the remaining modulus is represented by the modal defomation of the singularity, and corresponds to the conformal field:

x(z) y(z) z(z) = expGCvl

+ V2 + va)· ~(z)) =

¢+(z),

which has conformal weight one half. The super-partner of this field is J-(z), which has conformal weight one. This, together with its right-moving counterpart, can be used to make the (super-)marginal deformation of the S.C.F.T. that corresponds to changing the complex structure of the 2-torus.

5. Conclusions. The question naturally arises as to whether every N = 2 S.C.F.T. has a LandauGinzburg description. The simple answer is no. The 2-torus and other general Calabi-Yau manifolds provide counterexamples. However, we have seen that at special points in the moduli space of some of these theories one can twist them and the twisted forms of the theory are Landau-Ginzburg. It is thus tempting to suggest that all N = 2 superconformal theories might be Landau-Ginzburg theories up to orbifoldizations and marginal deformations, this is true of a large class of Calabi-Yau compactifications, but it is not clear whether it is true for all Calabi-Yau compactifications, let alone general N = 2 superconformal theories. Even if one does not have a Landau-Ginzburg description of an N = 2 superconformal field theory, the chiral ring, R, of that theory is still an extremely useful object in classifying the theories. If the chiral rings of two theories are isomorphic, it probably does not guarantee isomorphism of the two theories (unless there is some Landau-Ginzburg potential). If, however, the two rings are different the two theories cannot be isomorphic. Thus, the chiral rings of N = 2 S.C.F.T.'s are simple, and from experience, easily computable sub-structures of the theory that at least partially characterize it.

235 Chiral rings appear to be natural "topological" objects. There is an obvious formal similarity between chiral rings and the Dolbeault complex, with G~1/2 and G+ 1/ 2 playing =+ -the role of 8 and {) (while 8 and {) correspond to the right moving G -1/2' G+ 1 / 2 ). In section 3 we saw that under spectral flow to the Ramond sector, the chiral, primary fields flow to zero-modes the Dirac-Ramond operators. One also usually finds that the dimension, ft, of the chiral ring is equal to the index of the Dirac-Ramond operator of the theory. For level one, N = 2, coset models on hermitian symmetric spaces [25] one can show [3] that the correspondence with the Dolbeault complex is exact. If one grades the ring, R, of chiral, primary fields according to their charge (or conformal weight) then there is a one to one correspondence between the chiral, primary fields charge q and the elements of Hq,q(G/H) (with q suitably normalized)P Putting this another way, the Poincare polynomial of the chiral ring R is identical to the Poincare polynomial of the cohomology ring H*( G/ H). In addition, the ring structure of H*(G/ H) appears to coincide with that of R, though this has not been checked in general. In [3] it was also shown that the level one, N = 2 coset models on hermitian symmetric spaces are, in fact, Landau-Ginzburg models. On the other hand, general N = 2 coset models are not Landau-Ginzburg models. 12 However, one can show that elements of the chiral rings of these models correspond to non-trivial elements of the Lie algebra cohomology of the loop manifold of the coset. One can also show that the dimension of the chiral ring is equal to the index of the DiracRamond operator. On a more mundane level, these models have large chiral, primary rings that can be completely characterized in terms of the representation theory of the relevant Kac-Moody algebras. The details may be found in [3]. Thus I hope that I have presented some compelling reasons as to why the chiral ring should be regarded as an intrinsically topological object that can carry a great deal of information about an N = 2, conformal field theory. If the N = 2 theory arises as the fixed point of some Landau-Ginzburg model, then this the ring, or the corresponding superpotential, completely characterizes the theory.

Acknowledgements: I am extremely grateful to my collaborators: B. Greene, W. Lerche and C. Vafa, without whom this work would not have been possible. I would also like to thank the ICTP in Trieste for its hospitality and the opportunity to organize the material presented here into a, hopefully, coherent set of lecture notes. 11

12

For Hermitian symmetric spaces HP,q(M) = 0 when p =f q. It is possible that some twisted form of these theories might be Landau-Ginzburg.

236 References [1) C. Vaia and N.P. Warner, Phys. Lett. B 218 (1989) 51. [2) B. Greene, C. Vaia and N.P. Warner, Calabi- Yau Manifolds and Renormalization Group Flows, Harvard preprint HUTP-88/ A047, to appear in Nucl. Phys. B. [3) W. Lerche, C.Vaia and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Har~d preprint HUTP-88/ A065, Caltech preprint CALT-68-1540, to appear in Nucl. Phys. B. [4) E.J. Martinec, Algebraic Geometry and Effective Lagrangians, Chicago preprint EFI 88-76. [5) D. Gepner, Scalar Field Theory and Superstring Compactijication, Princeton preprint PUPT-1115. [6) C. Vaia, String Vacua and Orbifoldized LG Models, Harvard preprint HUTP-89/ A018. [7) C. Vaia, Quantum Symmetries of String Vacua, Harvard preprint HUTP-89/A021. [8) E.J. Martinec, Criticality, Catastrophes and Compactijications, preprint (1989), to appear in V.G. Knizhnik memmorial volume, edited by L. Brink et al.. [9) S. Cecotti, L. Girardello and A. Pasquinucci, Non-Perturbative Aspects and Exact Results for the N = 2 Landau-Ginzburg Models, ICTP Trieste preprint SISSA 5189/EP, University of Milan preprint IFUM 364/FT. [10) V.1. Arnold, Singularity Theory, London Mathematical Lecture Notes Series: 53, Cambridge University Press (1981); V.1. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of Differentiable Maps, volumes 1 and 2, Birkhiiuser (1985). [11) V.1. Arnold, Russian Mathematical Surveys, 28 (1973) 19; 29 (1974) 11; 30 (1975) 1. [12) M.B. Green, gratuitous reference. [13) A.B. Zamolodchikov, Sov. J. Nucl. Phys. 44 (1987) 529. [14) A.B. Zamolodchikov, JETP Lett. 43 (1986) 731; Sov. J. Nucl. Phys. 46 (1987) 1090. [15) Kastor, Martinec and Shenker, EFI preprint 88-31. [16) M. Grisaru, private communication. [17) A.N. Varchenko, Functional Anal. Appl. 10 (1970) 175. [18) S.M. Gusein-Zade, Functional Anal. Appl. 8 (1974) 10. [19) Z. Qiu, Phys. Lett. B 188 (1987) 207; Phys. Lett. B 198 (1987) 497; D. Gepner and Z. Qiu, Nucl. Phys. B 296 (1988) 757. [20) D. Gepner, Phys. Lett. B 199 (1987) 380; Nucl. Phys. B 296 (1987) 380. [21) W. Boucher, D. Friedan and A. Kent, Phys. Lett. 172B (1986) 316. [22) P. Di Vecchia, J.L. Petersen and M. Yu, Phys. Lett. 172B (1986) 323; P. Di Vecchia, J.L. Petersen, M. Yu and H.B. Zheng, Phys. Lett. 174B (1986) 280; A. Schwimmer and N. Seiberg, Phys. Lett. 184B (1987) 191. [23) G. Waterson, Phys. Lett. 171B (1986) 77.

237 [24] P. Candelas, P.S. Green and T. Hiibsch, Rolling Among Calabi- Yau Vacua, University of Texas preprint UTTG-1O-89; P. Candelas and E. Martinec, private communications. [25] Y. Kazama and H. Suzuki, New N=2 Superconformal Field Theorie8 and Super8tring Compactijication, Tokyo preprint UT-Komaba 88-8,88-12.

238

Lectures on N = 2 String Theory

DORON GEPNER

Joseph Henry Laboratories Princeton University Princeton, New Jersey 085,44

ABSTRACT Starting from an arbitrary N

= 2 superconformal field

theory it is described

how a fully consistent, space-time supersymmetric heterotic-like string theory in an even number of dimensions is constructed. Four dimensional theories which arise in this construction have a gauge group which contains Es x E6 with chiral fermions in the 27 and f7 representations of E 6 , and thus are phenomenologically viable. The explicit massless spectrum is studied for particular solvable examples. It is shown that such spectra are the typical ones expected from the field theory

compactification on manifolds of vanishing first Chern class. It is concluded that all 'N =2 string theories' describe string propagation on such manifolds. An explicit

calculation of the Yukawa couplings 27 3 is described for all the string theories in which the N

= 2 superconformal theory affords a scalar description.

this calculation is shown to be geometric.

The result of

239 1

INTRODUCTION

The study of four dimensional string theory is central to the idea that strings might provide a framework for unification. Initially, it was hoped that the internal consistency of first quantized string theory would be sufficiently restrictive to pinpoint the correct theory, much as it is for ten dimensional string theory. However, it was soon realized that this is not the case, and at least, for closed bosonic strings, any internal conformal field theory can be used in compactification. A number of compactification schemes based on free field theory were proposed. The simplest one is the toroidal compactification [1]. More realistic theories are obtained by projections of toroidal theories (orbifolds) [2]. Models based on free fermions were studied in ref. [3]. There are a number of reasons for studying string theories constructed from non-free conformal field theories. First, the space of conformal field theories in two dimensions is huge, and the free field theories are only a small subset of it. Thus string theories based on interacting conformal field theory offer a much richer set of possibilities to explore for realistic model building. Second, the general compactification on such conformal field theories can offer a unified understanding that would otherwise be quite obscure in disconnected examples. Last, the study of non-trivial string theory may enhance the understanding of conformal field theory itself, an important goal by its own right. As was demonstrated, for example, by the study of closed bosonic strings propagating on group manifolds [4], a consistent closed bosonic string theory is more or less guaranteed to arise by the structure of conformal field theory. The most important restriction on the possibilities comes from the modular invariance of the one-loop amplitude. Modular invariance is obeyed for bosonic compactifications if one takes the full spectrum of the conformal field theory for the construction of the string. For phenomenological reasons one would actually like to study heterotic-like string theories in four dimensions, in which the left moving Minkowski degrees

240

of freedom are fermionic strings (with total central charge c cone gauge) and the right moving are bosonic (c

=

=

12 in the light-

24). Another requirement

one might wish to impose on the theory is that of space-time supersymmetry in four dimensions. Space-time supersymmetry can solve various phenomenological questions (such as the hierarchy problem) in addition to its theoretical advantages like the absence of tachyons. Both the heterosis and the space-time supersymmetry seem to be in apparent contradiction with the requirement of modular invariance in a general conformal field theory. The only generic way to achieve modular invariance in a nonfree conformal field theory is to have a left-right symmetric spectrum. However, the heterotic-like string is asymmetric from its very nature, where even the trace anomalies of the left movers and the right movers are different. Also, space-time supersymmetry requires the appearance of superpartners in the spectrum, spoiling the left-right symmetry. There is a more or less unique solution to both issues. For any N perconformal field theory with c

= 9 one can construct

=

2 su-

a fully consistent closed

superstring theory with space-time supersymmetry [5]. The space-time supersymmetry generator is given by Q = exp(i4» where 4> is the U(l) boson of the N

=2

algebra. Such a generator was first seen in the leading 1/ R term (where R is the radius) of Calabi-Yau manifolds [6]. In ref. [7] it was noted that .the internal part of this field may be represented as a twist of the N

=

2 algebra ('spectral

flow'). The implementation of supersymmetry follows from a new supersymmetry projection [5] in which one discards all states for which the total U(l) charge is not an odd integer. If one imposes at the same time supersymmetry in the spectrum, modular invariance is restored. We review the construction of this projection in section (2). The solution to the problem of heterosis is provided by a map that takes any consistent superstring-like theory in any even dimension to a consistent heteroticlike string theory with the gauge groups Es x SO(8+d) or SO(24+d) where d is the

241

number of transverse dimensions [5J. In the supersymmetric case and d

= 2 (four

dimensions) the gauge group becomes Es x E6 or 80(26). This map is discussed in section (3). The resulting supersymmetric heterotic-like string theories, which will be referred to as N

=

2 string theories, have chiral fermions in the 27 and 27 of E6

and are thus phenomenologically viable. To construct explicit examples of such theories we turn to solvable conformal field theory in two dimensions. A relatively simple family of such theories is provided by the minimal realizations of the N

=2

superconformal algebra [8,7, 9J, for which the trace anomaly is 3k

c=-k+2'

for k

= 1,2,3 ....

(1)

The minimal theories are discussed in sections (4) and (5). Using the minimal theories as building blocks, the supersymmetry projection and the map to heterosis, gives a consistent solvable string theory for any combination of the minimal models with the correct central charge. The partition function of these minimal N

= 2 string theories are considered in section (6).

The massless spectra of such theories are studied in section (7). Using discrete symmetries as a tool in classifying states, it is shown that the resulting spectra agree exactly with manifolds of vanishing first Chern class. Compactifications of the field theory limit of the heterotic string on such manifolds were initiated in a beautiful work by Candelas et al. [1 OJ and were subsequently studied extensively [l1J. By making a comparison with the results of Candelas et a1. it is established that all N

= 2 string theories describe string propagation on manifolds of vanishing

first Chern class. This section is based on ref. [12J. In section (8) we discuss scalar field theory formulations of N

= 2 supercon-

formal field theories. It is shown that the structure constants for the chiral fields in the theory can be computed exactly for any such theory. Using this result we compute the Yukawa couplings of the type 27 3 • It is shown that these couplings agree exactly with a field theory geometrical formula [13J.

242 2

SPACE-TIME SUPERSYMME'l'RY AND SUPERCONFORMAL INVARIANCE

Consider the most general superstring compactification to D

= d + 2 dimen-

sions. We assume that D is even, d is the number of transverse dimensions. The total central charge of the theory in the light cone gauge is c

= 12. The space time

theory is composed of d free bosons and d free fermions on the world sheet. Since each free boson contributes c

= 1 to the trace anomaly and each fermion contributes

1/2, the trace anomaly of the space-time degrees of freedom is c

= d +d/2 = 3d/2.

The trace anomaly of the internal theory is thus 12 - 3d/2. In particular, in order to compactify to four dimensions we need an internal theory with the trace anomaly c

= 9.

The consistency of the superstring theory requires that the internal theory is not only conformally invariant, but also has an N

= 1 superconformal invariance in

two dimensions. In what follows, we shall usually assume that the internal theory actually has an N

=

2 superconformal symmetry. The reason is that this will

enable us to achieve space-time supersymmetry.

= 2 superconformal algebra contains in addition to the usual moments of the stress tensor T( z) = L: L n z- n - 2 , two fermionic superpartners, G; and a U(1) current whose moments we denote by I n . G±(z) = G;z-n-3/2 and J(z) = The N

Jnz- n- 1 • The values of the indices n depend on the boundary condition assumed for the superstress tensors,

(2)

= 0 corresponds to the Neveu-Schwarz sector and "1 = ! to the Ramond Accordingly, the indices n in G; take values in Z + ! ±.", where Z denotes

where "1 sector.

the integers.

243 The commutation relations of the N

= 2 algebra are given by,

= (m - n)Lm+n + 1~(m2 - m)c5n +m ,o, [Lm, JnJ = -nJm+n, [Lm, G;=J = (; - r )G;'+r, [Jm, JnJ = -nJn+m, [Jm, G;=J = ±G;'+r' {Gt, G;-} = 2Lr+. + (r - s )Jr+. + ~(r2 - ~ )c5r +.,o. [Lm,LnJ

(3)

The first equation is the usual Virasoro algebra for the moments of the stresstensor. The second and third commutation relations simply imply that G%(z) and J(z) are primary fields of the Virasoro algebra with the dimensions! and 1, respectively. The fourth relation implies that J( z) is a free boson U(1) current, J

= iA8:o4>

where 4> is a canonical free boson; the In's are the usual bosonic

creation and annihilation operators. The fifth equation implies that G%(z) have the U(1) charges

±1.

The only new commutation relation is the last one involving

the superstress tensors. All the indices take value in the integers, except rand s which take their values in Z

+ ! ± TJ, as

explained above.

For every value of TJ we get a different algebra, 0

~ TJ

< 1. Algebraically, al-

though not physically, the algebras obtained for different values of TJ are isomorphic to each other {14J. To see this one makes the change of variables,

(4)

It is easy to check that the primed variables obey the same commutation relations

as the unprimed ones, with a shift of the moding by TJ.

244

The isomorphism of the algebras for different boundary conditions has a natural interpretation. Consider the U(l) current algebra generated by J(z). As noted earlier J may be bosonized,

(5) where

~

is a canonical free boson. Now, an arbitrary field in the theory,

f, with

the U(l) charge q may be written as

(6)

j

where the field

is neutral. It follows from eq. (6) that all the fields in a unitary

superconformal field theory obey

t::.. > 3q2

(7)

- 2c' where the equality holds for the fields for which

j = 1.

If f is a primary field of

the U(l) current algebra,

J(z)f(w)

then the field N

=

j

qf(w) =- + regular terms, z-w

(8)

commutes with the U(l) current algebra. In other words, every

2 superconformal field theory, with the central charge c, is a product of a

U(l) current algebra with c = 1 and some quotient theory whose central charge is c - 1. In particular, the fields G±(z) may be written as,

(9) where {;± is a field in the c - 1 conformal field theory.

245 Now, for every field,

I,

in the Neveu-Schwarz sector ('1

= 0)

we can write a

field in the sector twisted by '1,

(10) It is straightforward to check that the O.P.E. ofthe field I,.,(z) with G±(w) contains

the terms (z - w)n±,." where the n are integers, implying that the field f,., is in the sector twisted by '1. The dimension and U(l) charge of the field f,., can be read from those of the field

I.

We find,

q'

= q + '1c/3,

(11)

Comparing eqs. (11-12) with eq. (4) we see that these two expressions are identical. We conclude that the isomorphism eq. (4) corresponds to nothing but a product with a free boson exponential. Consider now some N

= 2 superconformal field theory with the central charge

c. The Hilbert space of the theory, 1£, decomposes into a set of representations of the left and right N

= 2 algebras.

The number of representations to which 1£

decomposes may be finite or infinite,

(13) where p and q label the left and right representations. It is actually convenient to split every representation of the N

= 2 algebra into two representations of the

subalgebra generated by an even number of G±. The reason for this will become clear in the sequel. Since acting with an even number of G changes the dimension of a field by an integer and its U(l) charge by an odd integer, we can assume without loss of generality that all the dimensions of the fields in the representation

246

1ip differ from one another by an integer, and, similarly, the charges differ by an even integer. We shall denote by

~pmod

1 and Q p mod2 the common dimension

and charge of the fields in the representation 1tp • For each representation, 1tp , we may define the character as the function

(14)

which is a generating function for the number of states in the representation with a given dimension and U(I) charge,

Xp( r, z, '1£)

= e- 2lriu 2: mu1t( q, ~)e211'izg+211'iT(.:'\.-C/24),

(15)

g,.:'\.

where mult( q,~) is the number of states in the representation with the charge q and dimension

~.

The I-loop partition function of the theory is

Z

= 2: Np,gXp( r, 0, O)Xg( r, 0, 0)"

(16)

p,g

where Np,g denotes the number of times the representation 1tp,g appears in the spectrum. Under modular transformations, r -+ .:~t~, the partition function

Z( r) must stay invariant. In particular, the typical situation is that the characters Xp( r, 0, 0) form a unitary representation of the modular group,

XI'

ar + b ) = M (a nb) Xg(r, 0,0), (---,0,0 mr+n p,g

where the matrix M is unitary.

m

(17)

247 Define an action of the modular group on the variables (T, Z, u) by, aT + b Z Cz 2 ) T,z,uIM= ( ---,---,u+ ( ) mT + n mT + n 6( mT + n)

,

where M = ( : : ) , (18)

and c is the central charge of the theory. An important point is that the full character Xp( T, Z, u) transforms under the action of the modular group defined in eq. (18) in precisely the same way as the specialized characters, eq. (17), 2 aT + b z cz Xp(---,---,u+ 6( » mT+n mT+n mT+n

=S

In the sequel, when discussing examples of N

(a m

b) Xq(T,Z,U). n p,q

(19)

= 2 superconformal field theories

we shall see that indeed eq. (19) is satisfied. We will give below a justification for the transformation law eq. (19) in the general N Consider a general N c in the Neveu-Schwarz

= 2 superconformal field

theory.

= 2 superconformal field theory with some central charge (17 = 0) sector. Let 1-£p denote some representation. As

discussed earlier, from each field in this representation,

I,

we can get a new one

by multiplying it with an exponential of the U(l) free boson, eq. (10). This new state is in the representation of the 17-twisted N

= 2 superconformal algebra.

full character of the resulting 17 twisted representation,

The

1-£;, is,

where we used eqs. (11-12). Consider now the behavior under the modular transformation T -+ -liT of

X;( T, 0, 0). From the transformation law, eq. (19), we find,

(21)

248

which can be written as

(22)

The partition function, eq. (22), has a very natural physical interpretation. It is the torus partition function in the representation 1lp with a charge operator inserted in the time direction. The charge is defined as ." times the U(1) charge. Thus, this is the partition function for the sector twisted by ." in the time direction. On the other hand,

xj( T, 0, 0) is the partition function for the theory twisted by ." in the

space direction. That the two partition functions get exchanged under the modular transformation

T -+

-t is indeed precisely the correct behavior. Thus we see that

the transformation law, eq. (19), is compatible with the modular transformations of the torus partition functions in the twisted sectors of the theory. Assume now that the conformal field theory under discussion has the central charge c

=

12, and that it isa tensor product of a d dimensional superstring

with a 12 - 3d/2 internal N

=

2 superconformal field theory. The superstring

degrees of freedom describe a conformal field theory with c of d transverse bosons, XI', and d transverse fermions,

= 3d/2

composed out

,pl'. When d is even, this

= 2 superconformal invariance. Grouping XI' and ,pI' into d/2 complex fermions and bosons, the N = 2 currents can be written as,

conformal field theory has an N J =

,pi,pi, G+ = -v'2,pio"Xi and G- = -v'2,p; X;, where i = 1,2, ... ,~. Thus

the total c = 12 theory also has an N

= 2 superconformal invariance.

The field

Q=

.£

~

(23)

et1v i,

where 4> is the bosonized total U(1) current of the total (c

=

12) theory, is a

space-time fermion, as we shall see in the sequel. Consider the action of the field n

Q

- - n'" = e'.'1v'f 3

(24)

249

on the character Xp(T, z, u). From eq. (20) we find,

(25) From the partition function XP we can form a 'supersymmetrized' partition function by summing over all states related by the action of the supersymmetry charge, Q. Explicitly, the supersymmetrized partition function is,

Now, consider the action of the modular transformation persymmetrized partition function

X~um.

T --+

-~ on the su-

From the transformation law, eq. (19),

we find,

X~um( -~,O,O) = Spq L( -ltXq(T, ~,O).

(27)

nEZ

The sum on the r.h.s of eq. (27) has a very simple interpretation. Using the fact that all the states in the representation 1iq have the same U(l) charge up to an even integer, Qq mod 2, we find,

L( -l)qXq(T'~' 0) = ~re2ri(T-C/24) L( _lte"'inJo = Spqc5(Qq)Xq(T, 0, 0). nEZ

P

(28)

nEZ

where we denote by 6(Qq) a delta function which vanishes unless the U(l) charge,

Qq, is an odd integer, in which case it is equal to one. In other words, the supersymmetrized partition function, X~um transforms under

T

--+ - ~

into exactly the

same sum of partition functions as the original partition function, missing only all the representations with a total U(l) charge which is not an odd integer. Eq. (28) shows also that the supersymmetrized partition function

X~um

is

actually a certain sum of characters of the untwisted theory, since it is a modular transform of such a sum. Thus it is a bona fide partition function of a conformal

250 field theory. In addition, the minus signs in the definition eq. (26) are precisely the correct ones to guarantee spin-statistics. The fidd Q takes a space-time boson to a space-time fermion and vice versa. The space time fermions must have a negative sign in the partition function, and indeed from eq. (26) we see that multiplying by

Q precisely flips the sign in the partition function. As we saw above, the supersymmetrization and the condition of having only odd integral U(l) charges are dual under the modular transformation T --+ -liT. Thus, we can get a partition function which is invariant under T --+ -~ by imposing both conditions at the same time. Precisely for c

= 12 the field Q changes the total

U(l) charge by an even integer and thus it is consistent to do so. Consider then the partition functions, 6(Qp)X;um. Under the S modular transformations these partition functions transform with the same unitary matrix as before, Spq. Thus we may form a fully modular invariant partition function by imitating the original partition function, eq. (16), Z susy

~N. sum sum· = L..J pqXp X" ,

(29)

p,,,

where the sum extends over all the left and right representations with the U(l) charge which is an odd integer. Finally, to insure that the full partition function

ZIUSY

need to check only the invanance under the generator we find that x.;um transforms under

T

--+ T

T

is modular invariant we

--+

T

+ 1.

From, eq. (14),

+ 1 as, (30)

~(_lte211'i(~..-r.)e1l'inQ.. e-211'i*x~um(T'0, 0)

= e211'i(~.. -r.)X~um(T, 0, 0),

nEZ

where we have used, crucially, the fact that c c

= 12

(actually, c

T

--+ T

+ 1 in

= 12 mod 24)

= 12.

So we see that precisdy for

the partition functions

X~um

transform under

precisely the same way as the original partition functions. We

251

thus conclude that the supersymmetrized partition function

ZSlUY

is fully modular

invariant and describes a physical, d dimensional superstring theory. We will now prove that the string theory described by the partition function, eq. (29), is indeed space-time supersymmetric. In particular, at each mass level of the string there should be an equal number of bosonic and fermionic excitations. Consider the field Q, eq. (23). The total N internal N

= 2 algebra,

U{l) current, J •. t . and

Ji

C•• t.

= 3.

= 2 algebra is

composed of the

whose U{l) current we denote by Ji and the space-time

= 1/J;1/J,..

For a compactification to four dimension,

Ci

=9

We can also bosonize the space-time and internal U{l) currents,

= iv'3oz4>i and J•. t. = ioz¢>•. t .•

From J

= Ji + J•. t. it follows

that,

4> = v'34>i + ¢>•• t .• 2

(31)

Substituting into eq. (23) we realize that Q may be written as, !!a.1.. !.i&.

Q=e • e •

= 8e !.i&. • ,

(32)

where 8 is the spin field of the 80(2) current algebra. Similarly, the complex conjugate field can be written as,

Qt

- •• vs

= 8te'.

(33)

Similar formulas hold in dimensions greater than four, where 8 stands for the highest weight component of the spin field. Now, the field Q = eiI/J may be fermionized. Up to irrelevant co cycle factors Q is a free Dirac fermion on the world sheet. The fermion number is equal to half the

U(l) charge. Since all the fields in the theory have odd integral U(l) charges, it follows that all the fields are in the Ramond sector of this free fermion theory. (This R sector should not be confused with the Rand NS sectors of the N

= 2 algebra.)

The spectrum of the theory is invariant under the action of Q and Qt {defined in

252 the operator product sense; equivalently we may define this action on the states via the moments of Q and Qt which form the usual Ramond algebra). This follows from the expression of the partition function, eq. (26), and the invariance under the

U(1) current J

= 2i8z

and Se-it/>. When multiplying these left movers with the field 81:Xv from the right moving sector we get the usual N

=

1 supergravity multiplet in four

dimensions. From the product 'ljJ1"8I XV, we get the vertex operators of the graviton (the symmetric part) anti-symmetric tensor field (the anti-symmetric part) and the dilaton (the trace). Multiplying by the SO(2) spinors gives their fermionic superpartners. Similarly, multiplying these fields with the Es the vertex operators of the Es

X

X

SO(10) currents, Ja(z), gives

SO(10) gauge group. The SO(2) spinors give their

fermionic superpartners. The invariance of the right moving sector with respect to the right moving U(1) current algebra, 8·d> and the right moving supersymmetry charges,

(50) where Sa and Sa are the spinor and anti-spinor representations of the S0(10) current algebra, implies that the identity field appears in the same conformal block as these fields. In other words, the left moving v + s

+ s representations of SO(2)

multiply the additional right moving fields 8 1 ?>i, Q and Qt. It is easy to see that indeed these fields have the dimension one and charge zero, and thus appear in the massless spectrum. Since these fields depend only on Z, they are anti-holomorphic fields of dimension one. From the associativity of the operator product algebra such fields always correspond to a current algebra for some semi-simple Lie algebra. The Lie algebra generated by the Es X SO(10) currents, Ja, together with the fields 8 i ?>, Qa and

Q1 is, in fact, the Es

X

E6 current algebra at level one. The additional fields

complete SO(10) to E6. The number of fields is 45 (from the adjoint of SO(10)) 16 (spinor of S0(10)) another 16 (anti-spinor) and 1 singlet, giving a total of 78 currents which is indeed the dimension of E 6 • These fields are precisely the ones needed for the vertex operator construction of the Es one.

X

Es current algebra at level

262 To see this consider the root space decomposition of SO(10). The roots of

S0(10) are given by

±(E, ± Ej)

where

E"

= 1,2, ... ,5 are orthogonal unit vectors.

i

To the Carta.n subalgebra of SO(10) we add the generator

014>, the right moving

U(I) current, which commutes with the S0(10) currents. Denote the root space vector associated with the U(I) by the unit vector dimensional Cartan-subalgebra are,

±E, ± Ej

K.

The eigenvalues of this six

(from the adjoint of SO(10)), and,

form Q and Qt, where 5,

= ±1

and 5152 ... 56

These are precisely the roots of E6. The simple roots are given by E3 -

E4, E4 -

E5, E4

+ E5 and -HE1 + E2 + ... + E5) + ¥K.

= 1.

(51)

E1 - E2, E2 - E3,

It is easy to check that

all the positive roots are sums of simple roots. Also, the length of each vector is 2 and the scalar products are either 0 or -1. The extra simple root has a scalar product -1 only with the simple root

E4

+ ES.

Thus the Dynkin diagram of this

algebra is that of SO(10) with an extra node attached at one end. This is precisely the Dynkin diagram of E 6 • Thus we proved that the actual gauge symmetry of the theory is Es x E6, and that the spectrum contains the gauge multiplet of this gauge group. The invariance of the entire theory under this gauge group follows from the invariance of the theory under the U(I), as well as the invariance under the supersymmetry charges Q and Qt. We conclude that from the unit field we get the N as well as the gauge multiplet of the group Es

X

= 1 supergravity multiplet

E 6 • The same argument applies,

mutatis mutandis, in other dimensions. In 6 dimensions the resulting gauge group is Es X E7 and in 8 dimensions it is Es X Es. We leave the verification of this as an exercise for the reader. For N

=2

theories where the algebra is further extended, more fields might

appear in the identity conformal block. To appear in the massless sector, such fields must have dimension one and thus would correspond to some additional current

263 algebra symmetry, leading to more gauge bosons. In this case the gauge symmetry would be Es

X

E6

X

G where G is the extra 'enhanced' gauge symmetry. We would

see examples of such enhanced gauge symmetries in the sequel. Similarly, more spinor fields might appear in the left moving sector. Such fields would correspond to additional supersymmetries. However, since in four dimensions the existence of more supersymmetries rules out chiral fermions, such theories are not interesting from the phenomenological viewpoint. Let us consider now the right moving vector representation. From table 2 the internal dimension for the vector fields is

.6.i =

!

and the charge is

Qi = ±1.

.6. = IQI/2, and are thus chiral or anti-chiral fields of the = 2 algebra with charges ±1. Denote such a field with Q. = 1

Again, these fields obey right moving N by

G = Cexp( i4>/ va),

where

C is

a neutral field. The vertex operator for the

massless fields in the vector representation of 80(10) is VpC exp(i4>/va), where VI" p,

= 1,2, ... 10 represents the vector of 80(10) at level one.

to be 10 free Majorana fermions.)

(VI' can be taken

Acting on this field with the right moving

supersymmetry generator Qt we obtain a massless spinor field, 801.0 exp( -i4>/2.../3), where 801. is the spin field of 80(10). Acting once more with Qt gives the massless singlet field

C exp( -2i4>/ va).

Counting states we find 10

+ 16 + 1 = 27.

Indeed

these fields together give the 27 representation of E6. It can be easily checked that the weights of these fields are the correct ones for the 27 of E6 (as we did for the adjoint), and that this is precisely the vertex operator representation for the 27 of

E6 at level one. Similarly, the right moving vector fields with tion of E6 when acting with Q twice, 2-7

Qi = -1 give the 27 representa-

= 10 + f6 + 1. It can be further seen that

these are all the possible fields in the right moving sector of the theory. How are the right movers in the 27 and 27 representations of E6 connected together with the right movers? The only possible fields in the right moving sector that these fields can multiply are the spinor and anti-spinor multiplets of 80(2). We thus have four possibilities: space-time left fermions which are 27 (Qi

= Qi =

264

1), right fermions which are 27 (-Qi

(Qi

= -Qi = 1)

= (Ji =

1), left fermions which are 2-7

and right fermions which are f7 (Qi

two are CPT conjugates of the first two (Qi

-+

= Qi =

-1). The last

-Qi along with Qi

-+

-Qi). We

conclude that the matter content of the theory consists of a number of left handed fermions in the 27 of E6 and a number of left-handed fermions in the 2-7 of E6 • The

27 fields correspond to left and right chiral fields, (c, c), whereas the 2-7 correspond to the fields which are left chiral and right anti-chiral, (c, a). In general the number of 27 fields, N27 would be different from the number of f7, NtT, giving rise to a net number of chiral generations in the theory, N

4

= N27 -

NiT.

MINIMAL SUPERCONFORMAL FIELD THEORIES

So far we have been discussing the structure of a four dimensional heterotic string theory based on any N

= 2 superconformal field theory.

actual examples of such theories we need to find N

=

In order to construct

2 superconformal field

theories. The simplest non-trivial such theories are the so called minimal N theories ref. [8,7,9]. These are the only unitary N

=2

=2

theories with central

charge c < 3. The central charge of the k'th minimal model (where k is any positive integer) is 3k

c=--. k+2 There is a simple construction of the N

(52)

= 2 minimal theories [8,18] by adding

one free boson to the Z", parafermionic field theories [20,19]. The Z", parafermionic field theories contain the parafermion tPI and its hermitian conjugate tP!. The fields obey the OPE,

where

hI = (k - l)jk

is the dimension of the field

tPI

and c

=

2(k - l)j(k

+

2) is the central charge of the theory. For every positive integer k there is one such theory. The Z", parafermionic theories are intimately related with the SU(2)

265 current algebra [20,19]. The fields of the theory, 4>~, are labeled by two integers I and m, 4>~, where I is an isospin index, 0 ~ I ~ k and m relates to the Zk charge. Similar indices label the right movers, so we will denote the full parafermionic field by 4>~.m. The Zk charge is given by (m + m)/2modk. The left dimension of the 'primary' field 4>~ is given by

II

when

Iml < I.

=

2

1(1 + 2) m 4(k + 2) - 4k'

(54)

The dimensions of the fields 4>~m and 4>~±!" are the same. Similar

formula holds for the right movers. The field

1/JI

may be identified with 4>~:~.

The Zk charge conservation is consistent with the following operator product,

00

1/Jl(Z)4>!..(w)

=

L

(z - w)-i-+i- 1 A(m+l)/k-i4>!..(w),

(55)

(z - w)m/k+i-l A1I _ m)/k_;4>!..(w),

(56)

i=-oo and 00

1/Jt(z)4>!..(w)

=

L

;=-00

where the A and At are operators acting on the field

4>.

Consider now a conformal field theory composed out of the Zk parafermionic system and one free boson, 4>(z, z)

= 4>(z) + 4)(z).

(4)(z, z)4>( w, tV))

The energy momentum tensor is Ttl> for such a system is T = Ttl>

+ Tk,

The field 4>(z, z) obeys

= -21n Iz - wi.

= - !oz4>oz4>.

The total energy momentum

and the central charge is c = 3k/(k

+ 2).

266 In the combined free boson-parafermion system we may construct the fields,

G+(z) G-(z) J

J~ = J ~ tPl:e=

=

k

2 tPl: eiq,y'!¥:,

k

2

iJ

k

i

(57)

q,y'!¥:,

~ 28.4>.

From eq. (53) it is easy to check the operator product,

G+( z )G-( w ) -_ ( 2c/3 )3 z-w which implies the N

=

2J(w)

+ (z-w )2 +

2T(w) + J'(w) z-w

+ ...

(58)

2 algebra anticommutation relations, eq. (3). Similarly,

the other relations in eq. (3) can be verified. It follows that the fields J, G± and

T generate an N

= 2 superconformal field

theory with the central charge given in

eq. (52). The primary fields of the N

= 2 superconformal algebra are all of the form (59)

The fields in the Neveu-Schwarz sector are the ones which are local with respect to G±. The primary fields obey ±

G (z)V(w)

1 = 0(--) + .... z-w

(60)

Similarly, the Ramond sector fields are semi-local with respect to G± (pick up a minus sign when circled around G±) and the primary fields are at most singular as

OCIL.).

Using eqs. (55-56) it is easy to see which fields correspond to the

267 primary fields of the Neveu-Schwarz and Ramond sectors. We find, m-

am

=

am

=

(! sign(O) + a)k Jk(k

= -1.

The U(1) charge is

,

m - (!sign(O) + 1 + a)k Jk(k + 2) ,

where in the Ramond sector a and sign(O)

+ 2)

m

= ... ,1- 2,1.

m

= 0 and sign(O) = ±1

= 1,1 + 2, ... ,

(62)

and in the NS sector a

=!

The conformal dimension of the field V is,

J~am.

(61)

t:.. = t:..(4)!..) +a~/2.

It follows that the dimensions and U(1) charges in

the NS sector are,

1(1 + 2)

m2

m

t:.. = 4(k + 2) - 4(k + 2)'

Q = k + 2'

(63)

for 1 = 0,1, ... , k and m = -1, -1 + 2, ... ,1. In the Ramond sector the dimensions and charges are

~ 4(k+2)+S'

t:.. _ 1(1 + 2) _ (m ± 1? -4(k+2)

5

(64)

PARTITION FUNCTIONS FOR THE MINIMAL THEORIES

Using the connection of the minimal N

= 2 superconformal field

theories with

the Z,. parafermions allows us to compute the one loop partition functions of such theories. As in section (2) denote by X!..(T,Z,U.) the full character of the

= 2 algebra over the representation 1t!.. which is defined to be the representation of the N = 2 algebra containing the field 4>~ exp(iam 4». For convenience, we shall split each N = 2 representation in two, grouping together states related by

N

the action of an even number of G%. The field G+, when it acts on the field

= 4>!.. exp(iam 4», produces in the leading term of the operator product the field 4>!..+2 exp(i(aq + J(k + 2)/k)4». In addition, we can act arbitrarily with the U(1)

V

268

current J, implying the U(1) invariance of the spectrum. Also, we can act on the representation with any combination of the parafermions which has a net Z1c charge zero, since all such combinations appear in the operator products of even numbers of G±. Thus the representation of the N

= 2 algebra includes

all the

fields obtained from V by shifting the Z1c charge and simultaneously shifting the

U(1) charge. Applying 2k times the field G+, the parafermionic piece cancels, and we are left with a net shift in the U(1) charge, which shows that the bosonic piece is given by a theta function at levd k(k

+ 2).

The splitting of the representations

into two is most conveniently described by introducing an index s which is defined modulo 4 and is even in the NS sector and odd in the R sector. Denote the partition function ofthe s sector by X~)(T, z,u) defined as in equation (14). Then from the arguments above, 1(.)( ) Xm T,Z,U

=

L

(65)

c!..+4j-,( T)0 2m+(4j-.)(1c+2),21c(1c+2)(T, 2kz, u),

jmod1c where in the R sector we have replaced m

-+

m

+ s.

c~ are the characters of the

parafermionic field theory which are given by, sign( z )e 2,..i1'[(1c+2) .. ·-1c,,.1,

(66)

and 17(T) is the Dedekind's function. The classical theta functions of SU(2) at level m are defined by

on,m (T,Z ,u)-- e- 2,..iu

""' L-,

e2,..i1'mj·+2,..ijz: ,

(67)

jEZ+n/2m

where n is defined modulo 2m. The character X~) is invariant under m and m

-+

m

+ 2(k + 2)

defined modulo 2(k

-+

m

+4

which shows that indeed s is defined modulo 4 and m is

+ 2).

Also X~)

= 0 if 1+ m + sf:. Omod2.

The character is

269 also invariant under the simultaneous interchange, 1 - t k - 1 and q

-t

q

+ k + 2.

The action of the modular group on the classical theta functions is given by 0 n,m( r

+ 1, z, u) = e,..in'/(2m)0 n,m(r, z, u),

(68)

and

o

(-!.r'r'.:. u+~) = _1_(_ir)1/2 2mr.J2m

'"" e-,..i'n/m0, (r z u). L.J

,m

lmod2m

,

,

(69)

From the modular properties of the characters one can prove the identity,

X~)( r, z, 0)0 m ,A:+2(r, -2z, 0) = A'(r, 0, 0)0.,2(r, 2z,0),

(70)

mmod2(k+2)

where A' are the characters of the SU(2) current algebra at level k,

A1=

0,+1,1:+2 - 0-1-1,1:+2

(71)

01,2 - 0-1,2

The identity eq. (71) has a very important consequence. It implies that under modular transformations, the 1, m and s indices transform independently, where the 1 index transforms as a level k SU(2) partitions function, the m index transforms like a level k

+2

theta function and the s index transforms as a level 2 theta

function. Explicitly,

. (1+1)(1'+1))

sm

1r

k+2

"imm' _.nu'

1'(,')(

e"TiT"e' X m,

)

r,O,O,

",m',.' I'+m'+.'=O mod2

(72) and l(.}

Xm (r+l,O,O)=e

2 . /(.) 1(.) ""7... Xm (r,O,O),

(73)

where 2 1(1 + 2) m2 8 1m - 4(k+2) - 4(k+2) +"8'

1(.) _

(74)

which follows from the modular transformations of the theta functions, eqs. (68-

69). C is some constant determined by unitarity. From the modular properties

270 of the theta function it can be seen that, indeed, the full character transforms according to eq. (19). We leave this as an exercise for the reader. The fact that the modular transformations of the N

= 2 characters factorize

in the way described above implies that we can write a modular invariant partition function for the minimal theories starting from any modular invariant of the 8U(2) current algebra, and the two theta function systems. Take Z A = :E"r N"lA' AI*, 2 ZA:+2 = 1111- :Em,,,. L m ,,,.0 ,A:+2 0 ;",k+2 and Z2 = :E.,,8.,,0.,20;,2 to be any m

modular invariant partition functions for the affine 8U(2) and theta function systems. Then, a modular invariant partition function for the N

= 2 minimal theories

is given by,

W

" 1(.) 1(,)* = 2"1 'LJ N rLm ,,,.8.,JXm X". . I,m,. 1,_,3

(75)

"

The problem of modular invariance for the 8U(2) current algebra was introduced in ref. [4]. A level by level classification of the partition functions, via a direct decomposition of the relevant representations of the modular group, was described in [21]. These partition functions are listed in table 3. They include two infinite sequences corresponding to the 80(3) and 8U(2) WZW field theories, along with three sporadic solutions at levels k

= 10,16,28 t.

The complete list of modular invariant partition functions for the theta function

system at level m is given by a theorem proved in ref. [19]. These include the leftright symmetric partition function Z

= :En mod 2m 0n,m0~,m

and its projections

by an arbitrary subgroup of the Zm symmetry group. Thus the complete list of partition functions for the N

= 2 minimal theories is obtained by choosing arbitrary

modular invariants for the affine and theta systems. In particular, this analysis shows that the k'th minimal theory has a Zk+2 discrete symmetry.

t The Ii = 28 solution was later described in ref: [22]. A theorem giving the complete affine SU(2) modular invariants was proved in [19].

271 Table 3. The SU(2) current algebra partition functions[ 21 1.

SU(2)

~k+l 1=1 A1A*1

k~l

SO(3)

~~;od~=IIAI + A4;+2_11 2 + 2I A2;+11 2

k = 4j

SO(3) S10 S16

~4;-1

1odd=1 IA112 + ~4;-2 1even=2 A1A*4;-1

IAI IAI

k = 4j - 2, j

+ A712 + IA4 + A812 + lAs + All 12

k = 10

+ A1712 + lAs + A1312 + IA7 + All 12

k = 16

~

2

+IA912 + (A3 + AlS)A; + c.c IAI

S28

+ All 12 + IA19 + A2912

+IA7

k = 28

+ A13 + A17 + A2312

Our next question is to identify the chiral fields for a given N

=2

minimal

~

and the

theory. These are the fields in the NS sector for which the dimension

U(l) charge Q obey,

~

= Q/2.

From eq. (64) it is easy to check that the only such

possible fields are 4{~ ei(l#l,fi)j Jk(k+2) for some 1 and 1. Similarly, the anti-chiral fields, which obey ~

=

-Q/2, are given by i):~,_le-i(I#1,fi)/Jk(k+2). Whether

any of these fields appears in the spectrum depends on the particular modular invariant chosen. Choosing the theta function invariants to be the diagonal ones, and the SU(2) affine invariant, implies that there is precisely one such field for each value of 1 = 1, 0 ~ 1 ~ k, a total of k + 1 fields. Similarly, chosing any of the other invariant gives a chiral field for each left-right symmetric term in the affine partition function. Thus for example, for the SlO solution the allowed values of 1 = 1 are 1,4,5,7,8,11.

272

6

MINIMAL STRING THEORIES

We can get aD

= d + 2 (D = even) dimensional superstring-like string theory

using the minimal models as building blocks. We take a collection of minimal models, kI, k 2 , •• • , kr with the total central charge,

(76)

and joining to it d free bosons, XI', and d free fermions, .,pI', which represent the Minkowski space degrees of freedom (including both the left and right movers). The partition function for a given minimal model is given by

ZI

.- 1 ~ N -

2"

~

1(.) 1(.)·

l,rx.q x.q

,

(77)

l,l,q,.

where N"l is one of the SU(2) affine modular invariants (table 3). We assume here the left-right symmetric modular invariants for the q and s indices. The fermions with the space-time indices form a SO( d) current algebra. Denoting by .x the level one theta functions of the algebra (>. ranges over the singlet, the vector or the two spinor representations) the partition function of the current algebra part is,

(78)

To form a consistent string theory in this way we cannot simply multiply the partition functions of each sub-theory. The reason is that we must preserve the N

= 1 supersymmetry on the world sheet, since it is local in the string.

To do so

we must make sure that the Neveu-Schwarz states in each of the sub-theories will be coupled only to one another and would not mix with Ramond states. Similarly, the Ramond states should couple only to each other.

273 We may rewrite the partition function Zi, eq. (77), as follows. Define

NS± - 1 ""' N -' 1(0) ± 1(2»( 1(0) ± 1(2». - 2" LJ 1,l\Xq Xq Xq Xq ,

(79)

I,l,q

1 ""'( 1(1) ± 1(3»( 1(1) ± 1(3». R ± -_ 2" LJ Xq Xq Xq Xq .

(80)

1,1,q

The partition functions N S± and R± may be interpreted as the partition functions on the torus with boundary conditions in the time direction which are the modular transformation

R-

-+

R+. Under S:

and R-

-+

T

-+

T

-+ -~

T

+ 1 we have,

we have, NS+

-+

NS+

-+

NS-

NS+, NS-

-+

-+

±1.

Under

NS+, R+

R+, R+

-+

-+

NS-

R-. This implies that the partition function, (81)

is modular invariant. Indeed this partition is identical to the one given in eq. (77). We can form similar combinations for any N

=

2 theory (actually, N

=

1

supersymmetry is sufficient), which would then transform in the same way. Note that the second terms in eqs. (79-80) correspond to the action of G on the first terms. Thus the generalization of these equations is,

N S±

= L(Xp ± G(Xp»(Xq ± G(Xq»*,

(82)

pq

where p and q range over the representations in the Neveu-Schwarz sector and each representation is obtained by the action of an even number of G±. Similarly we can define R± as the same sum over the R sector representations. In particular, for the SO (d) current algebra, we can form NS± =

IBo ± Bvl 2 and

R± =

lB. ± B,12.

It is now evident how to form a partition function for a product of n theories in which the NS states are coupled only to each other and similarly the R states.

274 Define

(83) where NSf and

Rr

are the partition functions for the i'th sub-theory. Then the

full partition function, (84) is clearly modular invariant. In addition, the condition of coupling NS only to NS and R only to R is obeyed. It would be useful to introduce some notation for products of characters of the

minimal models. Let us group the l, q and and

v=

8

indices into vectors

(80,81,82, .•. ,8r,q1!~, ... ,qr)' where

level one and qi and 8i, i

~

80

f = (h, l2,.'"

lr)

= Vo is a weight of SO(d) at

1 are q the 8 indices of the various minimal models.

Define the character, '.(v.) = Bv. II Xv.+> , r

Z;;i

(85)

i=l

which represents a product character for the various theories. Define also r

N,~l = ,

l. , II N,.(Al.) "'

(86)

i=l

the product of multiplicities for each of the SU(2) invariants. We can rewrite the partition function Z, eq. (84), as

(87) where the sum over the vector p. ranges over all the elements in the lattice spanned by the vectors

iii = (von 0, and the vector

2 on i, 0 elsewhere),

for i= 1,2, ...

v obeys vo = Vi mod 2 for i = 1,2, ... , r.

,r,

(88)

275 This partition function corresponds to a consistent four dimensional string theory. However it is not supersymmetric. To get supersymmetry we need to use the general supersymmetry projection described in section (2). Define the vector

ilo = (s,l,l,l, ... ,l). It can be seen that the operation of acting with Q

characters is equivalent to shifting the vector

(89)

=

v by ilo.

exp(icP) on the minimal Thus, as described in

section (2), we get a supersymmetric partition function by summing over states related by the action of

Po and in addition eliminating any states in the spectrum

for which the left and right total U(l) charges are not odd integers. To get a heterotic-like string theory we use the map from superstring-like theories into heterotic-like strings described in section (3). This is implemented by replacing the characters of SO(d) in the right sector by the characters of Es x

SO(8

+ d)

or SO(24

+ d).

The full partition function for the supersymmetric

heterotic-like string theory (without the contribution of the transverse bosons) is then,

(90)

where II- is the vector, II-

= (von 0,

0 elsewhere) (implementing the change of

SO(d) representations to SO(8+d) x Es ones; for the right movers, B stands for the characters of this group), the sign is determined by spin-statistics, Q is the lattice spanned by the vectors

Pi and Po and the sum is limited to representations for which

the left and right U(l) charges are odd integers, and which are all either in the R sector or all in the NS sector. This partition function represents a fully consistent, modular invariant space-time supersymmetric heterotic-like string theory.

276 7

MASSLESS FIELDS IN THE MINIMAL STRING THEORIES AND MANIFOLDS

Let us tum now to the discussion of the explicit spectrum of the minimal string theories described in the previous section. As discussed in sections (2-3) the massless spectrum of any N contains the usual N

= 2 string theory (without enhanced supersymmetry)

= 1 supergravity multiplet.

In four dimensions, in addition,

there are the gauge bosons for the gauge group Es x E6

X

G where G is a possible

enhanced symmetry group. The rest of the spectrum consists of a number of chiral fermions in the 27 representation of Es ('generations'), a number of chiral fermions in the 2-7 representation of E6 ('anti-generations') and a number of Es singlets, along with their superpartners. The matter fields are all singlets of Es but can transform non-trivially under the enhanced gauge group. Particular string theories might also have discrete symmetries. For example, the k'th minimal model has a discrete symmetry group which is Zl:+2. The charge of the field 4>~,'q (in the NS sector) is given by (q

+ q)/2 mod k + 2.

Denote by

k 1 k2 ... kr the string theory made from the kl, k2, to the kr minimal theories. The discrete symmetry group of such a product is Zk, +2 x Zk.+2 x ... Zk.+2. Actually, since the discrete symmetry is embedded in the U(I) charge, and the total U(I) charge is an odd integer, the element 9

= {I, 1, ... , 1} E G acts trivially on all the

fields in the theory. Thus the actual symmetry group of the product of minimal theories is G/(g), the quotient group of G by the subgroup generated by g. Denote by (P!'P2, ... ,Pr) the charge ofa field transforming as V

-+

exp(21ripn8n/(kn+2))V

under the element {8 n } E G. In addition, if a given minimal model appears more than once in the product, we have the freedom to permute the various copies, giving rise to permutation symmetries in the spectrum. Not all the discrete symmetries commute with the supersymmetry generator. The condition for the element

{8!, 82, ... ,

8r} to commute with supersymmetry is

r

'LJ "'

;=1

8i . k-"""2 = mteger.

•

(91)

+

Other generators of G are R symmetries (i.e., different fields in a supersymmetry

277

multiplet transform differently under these). The odd permutations are also R symmetries. The generations in a given N

= 2 string theory correspond to the chiral fields

of the internal conformal field theory (section 3). Similarly, the anti-generations come from the fields of the type (c, a) (left chiral and right anti-chiral). The chiral fields in the supersymmetric string theory are of two types: 1) Fields which are chiral fields in the original N

= 2 theory before

the projection . Any chiral field

in the original theory with left and right charges equal to one gives rise to one such field in the string theory. 2) Fields which arise from the supersymmetry projection. The chiral fields in the minimal model are given by fields of the type Xl

= 4>:::eil (t/>+4i)/y'A:(k+ 2)

where 0 ~ I ~ k and I appears in the SU(2) modular

invariant chosen. There are no fields of the type (c, a) or (a, c) in the minimal models. The chiral fields of charge one in the product ofthe k 1 , k 2 , • •• , kr theories are thus given by x~,l)x~:) ...

xt) where the

Ii obey 0

Ii

r

2::r.-2 = 1. i=l

The multiplicity of this field is

•

~ Ii ~ ki and (92)

+

N~rwhere

N is the affine modular invariant chosen

(in the notation of section (6)). The discrete symmetry charge of these generations is (l},1 2 , ... ,lr ). The full massless spectrum of a given minimal conformal field theory may be computed by expanding the partition function eq. (90). Such an enumeration of states can be quite tedious to perform by hand, but can easily be computerized. Let us first consider in detail one example. Take the theory obtained from k + 2 copies of the k'th minimal theory with the SU(2) modular invariant. The central charge ofthis theory is (k + 2)&

= 3k.

Thus this theory can be used to compactify the

string to 10 - 2k dimensions. The case of k

= 3 is

particularly interesting since it

gives a compactification to 4 dimensions. Enumerating states for the theory 35 we find that the generations are all given by fields of the type

Xl, Xl • ••• Xl.

which obey eq. (92). (This is actually true for

278 any k). There is one anti-generation and 330 E6 singlets. The gauge group of this theory is Es x E6

X

U(1)4.

Consider the manifold MIc (Fermat surface),

V(Z i )

°

= Z IIc+2 + ZIc+2 2 + . .. ZH2 1c+2 = ,

(93)

where the Zi are complex variables, modulo the identification of fields {Zd

==

{WZi} where w is any complex number. This is a hyper-surface in C pHI. The complex dimension of this manifold is k. The first Chern class of this manifold vanishes as can be seen by writing the holomorphic (k,O) form on this surface

(94) Note that this form is well defined in CPIc. The case of k

= 1 corresponds

to a

torus (the only complex curve with vanishing first Chern class) in the shape ofthe

SU(3) maximal torus. For k

= 2 this is the maximally symmetric shape of the K3

manifold. M3 is the quintic hypersurface in CP4. The discrete symmetry group of the manifold MIc is as follows. We may first multiply any of the Zi by a complex phase which is a k +2 root of unity, Zi where w:+2

= 1.

Thus there is a

--+

WiZi

z:ti symmetry on the surface. Due to the Cplc

identification, however, the overall phase is irrelevant. Second, we may permute any of the variables Zi into one another, Zi

--+

ZP(i) where p E SIc+2 is any permutation.

We conclude that the symmetry group of the manifold MIc is

(95) The Euler number, XIc, of the manifold MIc can be easily computed by the method (for example) of multiple covers of CP". We find Xl etc.

= 0,

X2

= 24,

X3

=

-200,

279

Note that the discrete symmetry group of the string theory kH2 is precisely the same as that of the manifold M,.. Thus, it is not unlikely that the two are related. The compactification of the field theory limit of the heterotic string on manifolds of the type M x K, where M is a four dimensional manifold and K is a six dimensional manifold, was considered by Candelas et al. [10]. By requiring unbroken N

= 1 supersymmetry in four dimension, these authors showed that the field

equations lead to a three dimensional complex manifold with a Ricci flat metric. Such manifolds are called Calabi-Yau manifolds. The only topological obstruction to finding such a metric is the vanishing of the first Chern class [23]. Thus, for example, the manifold M3 admits such a metric. As discussed by Candelas et al. the gauge group in the field theory is Es x E6 with a number of generations and anti-generations in the 27 and f7 of E 6 • The number of generations is equal to the number of harmonic (2,1) forms, h 2,1, and the number of anti-generations is equal to the number of harmonic (1,1) forms, hI,I. For the manifold M3 these numbers are h 2 ,1

= 101 and hI,I = 1.

These are

precisely the number of generations and anti-generations in the string theory 35 ! There are 102 E6 singlets in the field theory limit which implement the change of radii and complex structures. In addition, in the field theory there are 224 singlets associated with HI(End T), which come from an index theorem for the octet ofthe SU(3) holonomy [24]. Thus, for a field theory compactification on the manifold M3

we would expect a total of 326 singlets, which is 4 less of what we find for the string theory 35 • Recall, however, that the theory 35 has the enhanced gauge symmetry

U(1)4. Thus there must be additional 4 'Higgs bosons' E6 singlets in the spectrum needed to give the U(l) gauge fields a mass when deforming this theory. Thus the total number of singlets expected from field theory considerations is indeed 330, precisely the number found in the theory 35 • The 101 harmonic (2,1) forms (which give the generations in the field theory) correspond to polynomials which can be added to the defining equation (93). These

280 are polynomials of the type

Z;' Z;· ••• Z~·

where

° r. ~

~ 3 and ~.

r. =

5. The

discrete symmetry charge of this polynomial is (rl, r2, ... , r5)' These polynomials match precisely the generations in the string theory 35 • From equation (92) these are given by the fields

Zr, Zr•••• Zr.

where

° r. ~

~ 3 and ~

r. =

5. The discrete

symmetry charge of this field is (rI, r2,"" rs). Thus, not only the number of generations match precisely, but their charges under all the discrete symmetries are the same in the string theory and on the manifold. Similarly, it can be seen that the charges for the anti-generations and the E6 singlets are precisely the correct ones. There is an isomorphism ofthe entire massless spectrum in the theory 35 into their corresponding polynomials and elements of Hl(End T) (which can be represented by some tensors) which preserves the

75,000 element discrete symmetry group. The singlets along with their isomorphic polynomials and tensors are shown below

(20)

(0,8,0) (0,0,0)3 (2,2,0)

zlz;

(30)

(0,8,0)(0,0,0? (1,1,0)2

ZlZ4 Z 5

(20)

(0,0,0)3 (1,7,0) (3,3,0)

P44555

(30 x 2)

(0,0,0)2 (1,1,0) (1,7,0) (2,2,0)

P43455 P53445

(30 x 2)

(0,0,0)2 (1,1,0) (2,6,0) (3,3,0)

P34555 P43555

(30)

(0,0,0)2 (2,6,0) (2,2,0)2

(20)

(0,0,0) (1,1,0)3 (1,7,0)

(20 x 3) (20)

Z3z1z~ Z2 Z 3 Z 4 Z ;

(0,0,0) (1,1,0)2 (2,6,0) (2,2,0)

P23455 P32455 P42355

(1,9,0)3 (2,0,0)(3,3,0)

(5)

(1,1,0)4(2,6,0)

(5)

(1,9,0)4 (2,4,0)

P45555

4

ZlZ2 Z 3 Z Z 5 P12345 P21345 P31245 P 41235

Kahler + 4 extra

(These are the (1, q, 8) numbers for the right movers in each of the sub-theories. The numbers above correspond to spinors which are singlets ofSO(lO).)

281

We are thus bound to conclude that the string theory 35 describes string propagation on the manifold M 3 • Similarly, it can be seen that the discrete symmetries and spectra of the kk+2 theory are the same as the ones expected for the manifold

Mk. Many other examples of such identifications can be made. One example of physical interest is the string theory 11 16 3 with the for all the k

=

S16

modular invariant used

16 theories. This string theory has 35 generations and 9 anti-

generations. The manifold which fits the massless spectra of this theory is the hypersurface

Z~ + Z: + Z~ + Z!

= 0, (96)

in C p4 x C p3, as can be seen by an analysis analogous to the one described above [25]. This theory leads to a three generations model via a quotient with a Z3

X

Z3

symmetry. Recent study [26] shows that this theory has excellent phenomenological prospects. The reader might wonder, are all the N

= 2 string theories in correspondence

with manifolds of vanishing first Chern class? We shall argue that the answer to this question is yes [12]. To see this it is useful to first consider compactifications to 8 and 6 dimensions. To compactify down to 8 dimensions we use an N theory with the central charge c

= 3.

field theories with central charge c

=2

We will prove that all N

=3

superconformal field

= 2 superconformal

and integral U(l) charges are supercon-

formal field theories on a torus. Namely, they can be realized as a theory of one complex boson and one complex fermion on some even self-duallattice. Indeed the torus is the only manifold of vanishing first Chern class in one complex dimension. Denote by if> the U(l) free boson, J

= iOzif>(z).

Similarly the right current is

282 j

= i8J tP.

Now, since G± has charges plus and minus one they can be written as

(97) where G± are some fields which commute with H. In the case of c

= 3 the fields

G± can be represented as one complex boson and its complex conjugate. This follows from the OPE of G+ with G_, eq. (58). For G± this OPE translates into,

(98) and similarly G±(z)G±(w) = regular. These are precisely the OPE relations of a complex free boson H. Using also the holomorphicity of these fields, we can identify

(99) Now, the free boson

tP

can be fermionized into one complex fermion,

and its complex conjugate (neglecting co cycle factors). Thus, G+

t/J = :exp itP:

= V2t/J8z H

and

G- = V2t/J*8z H*. The U(l) current is then J = i8z tP = t/Jtt/J. Finally, the stress energy tensor can be found from the OPE of G+ and G-, eq. (58), (100)

= 2 algebra of a theory of one complex fermion other words, the N = 2 superconformal algebra at

This is the canonical form of the N and one complex boson. In c

= 3 is

identical to the algebra of one complex boson and one complex fermion

after a simple change of variables. Thus any such conformal field theory must contain these fields. What about the rest of the fields in the theory? Assuming that the U(l) charges are all integral means that each field in the theory has an integral fermion number and thus can be expressed in terms of these free fermions and free bosons. (In other words, the

283 free boson fjJ lives precisely at the radius one which can be fermionized.) All that remains is to specify the values of the zero modes (momenta) for the complex free boson. Due to the closure of the operator product algebra these momenta form an additive group - a lattice. Due to modular invariance this lattice must be an even self dual Lorenzian lattice with signature (2,2) [1]. To summarize, we proved that any c

= 3 N = 2 superconformal field

theory with integral U(I) charges is

equivalent to some Narain type toroidal theory on a (2,2) lattice. From the general theorem described above, it follows that, in particular, when we tensor N

=2

minimal models to get a c

=3

theory and impose the general

supersymmetry projection, the resulting theory describes string propagation on a torus. There are three such possibilities, 13 , 22 and 1141. The first and the last correspond to the SU(3) torus and the 22 theory corresponds to the SU(2)2 torus. Consider now the case of compactification down to 6 dimensions. In this case we need a conformal field theory with c

= 6.

There are 17 combinations of cen-

tral charges from the discrete series that give this value (using only the left-right symmetric SU(2) and affine modular invariants). One theory, 13 22 can be seen to correspond to string propagation on the SU(3) x SU(2)2 torus. The spectra of the other 16 theories are listed in table 4. From table (4) we see that all these theories have 20 spinors in the 56 of E7, no anti-spinors and a number of E7 singlets equal to 130 plus twice the dimension of the enhanced gauge symmetry group. What is the explanation for this? The only two dimensional complex manifolds of vanishing first Chern class are 4-tori, T 4 , and the K3 manifold. The number of generation expected for a field theory

compactification on K3 is equal to the number of harmonic (1,1) forms, hI,I

= 20,

giving indeed 20 generations. The number of singlets expected is 130 where 40 come from deformations of the complex structure and radii and 90 come from Hl(EndT). The extra gauge bosons are Higgs singlets. Thus the numbers in this

table match precisely the spectrum expected for the K3 manifold. We conclude that all these theories correspond to string propagation on the K 3 surface. Since,

284

Table 4. Theory

8

16 1441 122110 1 1242 112 241 115140 1 1161221 117116 1 118 113 1 1110 2 24 213118 1 214110 1 2162 3281 43

in 56 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

8

in 56

E7 singlets

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

140 140 136 136 146 134 134 134 134 134 136 134 134 134 134 134

Gauge U(1)5 U(1)5 U(1)3 U(1)3 SU(2?xU(1? U(1)2 U(1)2 U(1)2 U(1? U(1)Z U(1)3 U(1)2 U(1)Z U(1)Z U(1)2 U(1)2

the theories made from various minimal models along with their very large number of possible projections are not in the least special in the space of N field theories, we must conclude that any N

=2

= 2 conformal

string theory in 6 dimensions

describes string propagation on a manifold of vanishing first Chern class. Finally, it can be seen by various indirect methods that the four dimensional N

= 2 string theories

also correspond to string propagation on manifolds of van-

ishing first Chern class. For example, there is always a 1-1 map from 27's and f7 in the spectrum to the singlets. This map preserves all the discrete symmetries [27] for any N

= 2 string theory (not necessarily a minimal one). The map gives

the singlets associated with the change of radii and complex structures. Since the harmonic (2,1) and (1,1) forms give both the generations and anti-generations and the singlets that deform the complex structure and radii, such a map is indeed expected for a string theory on a Calabi-Yau manifold. It can be seen that the potential for such singlets is perturbatively fiat, as it should be [28].

285 Another piece of evidence comes from the study of projections of the minimal string theories. If the theory corresponds to string propagation on the manifold M, the theory projected by the discrete symmetry group Z would describe string propagation on the quotient manifold M/Z, where we identify points related by the action of Z on the manifold. Let us digress on this. In the first step we would like to suggest a formula for the Euler number of a Calabi-Yau manifold, H, obtained through a quotient of some other manifold, G, by some discrete automorphism group, Z, acting on G. In other words, H

= G/Z.

The interesting case is when the group Z does not act freely on the manifold. In this case H is not a smooth manifold, but rather has some singularities. In many cases of interest, these singularities can be resolved by cutting some small disks around them, and then gluing some smooth non-compact manifolds. First, we suggest a criteria when this can be done. The singularities can be resolved, and the resulting manifold,

iI is a

smooth manifold of SU(n) holonomy,

if and only if the group Z acts trivially on the holomorphic (n, 0) form. Next we give a formula for the Euler number of !t,

X(H)

1 = iZT

'"' L.J x(g, h),

(101)

g,hEZ

where

IZI

is the order of Z, and X(g,h) is the Euler number of the points of M

left fixed by both elements 9 and h. Eq. (101) was described in ref. [2], for Calabi-Yau manifolds obtained by blowing up orbifolds. We suggest, that it is valid for all manifolds of vanishing first Chern class and in particular all CalabiYau manifolds. A variant of eq. (101) gives the Euler number of the manifold H before the resolution of singularities. In the case of Calabi-Yau manifolds, this number is defined to be, X(H)

= 2(hl,1 -

h2,1), where hl,l (h2,1) is the number of

(1,1) «2,1)) forms invariant under all elements of Z. We then have,

X(H)

= I~I

L gEG

x(g, 1).

(102)

286 In all cases where a procedure for resolving the singularities is known, eq. (101-

102) can be seen to give the correct answers. Later we will see such examples. If a string theory for a manifold is known, these equations can be compared to the number of generations in this string theory, before and after a projection by some symmetry group. Consider, for example, the quintic hypersurface M 3 • String propagation on this manifold is described by the theory 35 • Concentrate on automorphism groups which are cyclic, and involve only the phases. There are three inequivalent such automorphism groups which act trivially on the (3,0) form. These are generated by the elements

{0,1,2,3,4}

{O,O,O, 1, -I},

{O, 0, 2, -1, -I}

where we denoted by {rn} the transformation Zn

-+ e21rir.. /5 Zn.

(103)

The first element

9

= 1 and h = 1.

Eq. (101-102) then gives the correct answer,

°

= unless X(H) = x(il) =

generates a freely acting Z5 group. For freely acting groups, X(g, h)

x(G)/IZI· The third group is generated by the element {O,O, 0, 1, -I}. The submanifold left fixed by any nontrivial group element is

zf + Z~ + zg = 0. This is a Riemman surface with genus 6 and Euler number X

(104)

= -10.

Substituting

this into eq. (101-102) we find that the Euler number of the twisted manifold is

x(il)

= -88 and for

the singular manifold X(H)

= -48.

As explained earlier, the automorphism group of the manifold manifests itself as discrete symmetries in the 35 string theory. Now, the process oftaking a quotient manifold has an equivalent in the string theory. Propagating a closed string on

Q/Z is equivalent to projecting out all states which are not invariant under Zj this is the closed string sector. In addition, there are winding sectors which correspond to a closed string in Q/Z, which are open strings when lifted to Q, where the end

287 points differ by a non-trivial element of Z. By implementing the formulas of [12], the exact spectrum in these sectors can be computed. The massless spectrum of a heterotic string theory on Q/ Z is then found to consist of 49 generations (27 of

E6),5 anti-generations (f7 of E6), and 220 singlets of E 6. Of these, 25 generations, and one anti-generation come from the closed string sector, and each of the four winding sectors gives 6 generations and 1 anti-generation. We see that indeed eqs. (101-102) give the correct results. The net number of generations in the exact spectrum is -44 corresponding to Euler number -88. In the closed string sector, we find X = -48. Again, in agreement with the topological calculation. The real importance of eqs. (101-102) is in a different situation. Namely, when we do not have a candidate manifold for some abstract string theory. The three Euler numbers appearing in eqs. (101-102) correspond, as explained above, to the net number of generations in the original theory, the closed string sector and the quotient theory. Assume that a particular theory describes string propagation on an unknown manifold. From this assumption alone, even without having an explicit manifold candidate, we can derive powerful predictions. For simplicity, consider the .case of cyclic twists by some Zp group, where p is a prime number. The Euler number of the fixed point set a

= X(g, h)

is some

fixed integer for all g, h E Zp, except for X = X(O,O) which is different. Now, in the quotient theory the net number of generations is

!Xt, of which !Xo come from

the closed string sector. Using eqs. (101-102) these can be expressed as (105)

XO

=

X

+ (p -l)a P

.

(106)

From eqs. (105-106) we can compute a,

a

- X - X • = PXt 2 = PXo = Integer. P -1 p-1

(107)

288

Thus starting from an arbitrary N

= 2 string

theory, we can compute X as the

number of generations appearing in the original string theory, Xt as the number of generations of the theory twisted by Z and XO as the number of generations in the closed string sector. Assuming a geometrical interpretation for the theory implies that eq. (107) holds for these three numbers. Also, each of the winding sectors must contribute exactly the same number of generations. These are all highly nontrivial relations, which can easily be checked. From the viewpoint of conformal field theory, having such relations appears utterly mysterious. The only conceivable explanation for such relations is that indeed the theory is geometrical. We have checked these relations in many examples. They always work. We conclude that there is conclusive evidence for the following [12]: All N

=2

string theories describe string propagation on manifolds of vanishing first Chern class. Perhaps most importantly, the results described above show irrefutably that Calabi-Yau compactifications exist as conformal field theories and are exact solutions of the string equations of motion, despite the breakdown of conformal invariance at the four loop level of the sigma model [29], and the possibility that non-perturbative effects on the world sheet would destabilize the vacuum. The fact that we find in the spectrum all the modes corresponding to the deformations of the complex structure and the radii, shows that for any complex structure, and radii the string theories are conformally invariant, space-time supersymmetric and fully retain the topological and geometrical properties of the field theory formulation.

8

YUKAWA COUPLINGS FOR ANY SCALAR FIELD THEORY

Consider a two dimensional N is chiral, n+cp iO~o%,

= 2 scalar superfield cp( z, 0+,0- ,0+,0-) which

= D+cp where the covariant derivatives are defined by n± = OIJ± ±

D± = o{j± ± iO~OI.

A typical supersymmetric lagrangian involving such a

field is given by (108)

289 The conservation of the U(l) charge implies that the potential V is of the form V = eli k+ 2 where k is some integer. More generally, we could have a number of scalar fields of various U (1) charges, such that the potential V has a U (1) charge 1, and is thus a quasi-homogeneous function. As a generalization of scalar field theory for N = 0 and N = 1 minimal theories [30J, Kastor et al. have noticed [31J that the equation of motion of the theory eq. (108) with the superpotential V(eli) has the same form of an operator product of the k'th N

= elik+2

= 2 minimal theory.

Thus,

it is likely that the scalar field theory has a fixed point which is described by the minimal model. The correspondence of chiral fields in the two theories is eli l = ZI, where

Zl

is the l'th chiral field in the minimal theory using the notation of section

(7). If we group together k

+ 2 copies

of the k'th minimal theories, the resulting

potential has the form, 10+2

V(eli.) =

:E eli~+2 .

(109)

.=1

Intriguingly, the potential, eq. (109), has precisely the same form as the manifold which corresponds to this string theory, as we saw in section (7). More generally [32J if V is the potential of a given collection of minimal theories, k1 x k2

L. eli~·+2

V =

X •••

x kr'

can describe the Calabi-Yau manifold,

(110) where the eli. are now regarded as complex variables modulo the identification 1

= 5 this is a Calabi2 Yau manifold. For r < 5 one may add trivial eli theories. (The case of r > 5 seem

eli.

==

wiii+""> eli. and w is an arbitrary complex constant. For r

to correspond to a number of embedding spaces.) By a case by case comparison [32J it can be seen that the resulting CY manifold has the correct Euler number to give the number of generations in the string theory. Such scalar field theory realizations are not limited to the minimal series [33J, but can describe also theories of the Kazama-Suzuki type [34J. It is, in fact, not

290

unlikely that all N = 2 superconformal field theories can be realized as projections of scalar field theories. Our aim in this section is to describe how the structure constants among the chiral fields may be computed for an arbitrary scalar field theory [33J. We then use the result to compute the Yukawa couplings of the type 27 3 in the string theory compactified on any scalar field theory. The result of the calculation is then seen to agree precisely with the field theory formula for the 27 3 Yukawa couplings for the manifold V

=

0, where V is the superpotential. This result establishes, in

particular, that indeed an N

= 2 string

theory based on any scalar field theory

describes string propagation on a manifold of vanishing first Chern class, as well as a quantitative identification of the superpotential and the manifold for all the complex structures.

= 2 superconformal field theory. The chiral fields in the theory obey the equation t:J. = Q/2 where t:J. is the dimension and Q is the U(1) charge. Consider an N

All other fields in the theory obey t:J. >Q/2. Let Ci and Cj be two chiral fields in the theory with charges qi and qj. The charge of fields appearing in the operator product of the two fields Ci and Cj is qi + qj. Consequently, the dimension of such a field, t:J. is greater or equal the sum of the dimensions of Ci and Cj, t:J.

~

t:J.l +t:J.2,

with equality holding only for a chiral field. Thus we have the operator product,

Ci(Z, z)Cj(w, w)

= f~CIe(w, w) + regular terms,

(111)

where the Cle are chiral fields of charge qi + qj and the f's are some constants. We can thus define a product structure on the set of chiral fields as the (non-singular) limit of the operator product [33J,

(112) Since this operator product is non-singular, it follows that the product so defined

291 is associative,

~ 1i~/r., = ~ f[lel'J,. Ie

(113)

Ie

In addition, we can add two chiral fields (as usual) and multiply by a complex number. It follows, that the set of chiral fields forms an associative commutative algebra over the complex numbers. (An algebra is a vector space over a field which forms a ring. The ring and vector space structures are connected together by some relations. For more detail see, for example, [35].) This algebra is graded by the

U(l) charge; the U(l) charges add when multiplying. Since the U(l) charge is at * most c/3, the algebra is finite dimensional . The structure constants of the algebra are the operator product coefficients of the chiral fields. The anti--chiral fields are the complex conjugate of the chiral ones, and thus form the complex conjugate algebra with the structure constants

If; *.

Similar

algebra can be obviously defined for the fields which are left-chiral and right antichiral, (c,a) or their complex conjugates (a,c). Suppose that the maximal chiral field (114) is in the spectrum of the theory. Here t/> and 4> stand for the left and right U(l) bosons. (Equivalently, from locality, all the fields of the theory obey qz - qr

=

integer where q, and qr are the left and right charges.) We can then define a transposition operation on the chiral fields. For a given chiral field C, define the field (115) It can easily be checked that the field C t is chiral. Thus the transposition operation is a one-to-one map of order two, namely, (Ct)t

* It is

= C,

on the set of chiral fields,

interesting to note that in all the known N = 2 conformal field theories the U(l) charges are rational. This is certainly true for scalar field theories and their projections. We conjecture that it is true in general.

292 which takes an element with the charge q to an element with the charge c/3 - q. (This map is not an algebra automorphism though, since it does not preserve the product structure.) This implies, in particular, that the dimension of the vector subspace of chiral fields with dimension q and e/3 - q is the same. Consider now a scalar field theory with the superpotential V( ~1' ~2' .•. , ~n)' The fields

~i

are some chiral fields in the theory. We can form the product (in

the above sense) of, say, the fields denote by :~~' ~;'"

:~1 ~2:'

~1

and

~2'

This is a chiral field which we

Continuing, in this fashion, we may form, by induction, the field

... ~:''': defined as the chiral field obtained by repeatedly multiplying

fields. It is absolutely crucial that the algebra of chiral fields is associative and commutative. Otherwise, we would get different answers by changing the order of the multiplications. Thus we have an algebra homomorphism from the algebra of polynomials in n variables P[:l:b :1:2,""

:l: n]

into the algebra of the chiral fields.

The map, r, simply takes the polynomial :I:~' :c;'" field

:~~' ~;'"

... :c:'''

to the normal ordered

... ~:''':. This map is an algebra homomorphism (i.e., preserves the

addition, product and multiplication by a complex number) as is clear from the associativity and commutativity. So far the discussion was general to any N

=2

theory, without assuming that it is a scalar field theory. In a scalar field theory, we expect all the chiral fields to be given in this fashion, since the fields in the theory are composite operators of the

~'s

and its derivatives, but chiral fields cannot

contain derivatives. Thus, in a scalar theory the map

r

is onto. The kernel of map

r, Ker(r), consists of all the polynomials p for which r(p)

= O.

What is Ker(r)?

From the equations of motion,

(116) it is clear that the normal ordered fields given by the derivatives of the potential vanish. Similarly, any polynomial containing such derivatives vanishes. Thus, Ker( r) is the ideal generated by the derivatives of the potential, Ker(r)

av av av = (a-'-a ''''-a ). :1:1:1:2 :l: n

(117)

293 It follows (from the first isomorphism theorem) that the algebra of chiral field is

isomorphic to the algebra P[z}, Z2, ••• , zn]/Ker( r), the algebra of polynomials in n variables modulo the ideal generated by the derivatives of the potential. Most

crucially, the structure constants among the chiral fields are given by the product of polynomials. For example, consider the simplest potential V( 4»)

= 4)k+2.

The chiral fields

are given by 4)1 where 0 ~ 1 ~ k. The operator product of two such fields is

:4)I:(Z):4)m(w):

= :4)I+m(w): + h.o.t

(118)

where the coefficient in front of 4)1+m is one. Using the relation to the k'th N

=2

minimal model, 4)1 = k,Z , where the k,'s are some normalizations, implies

(119) where the structure constants G',m obey, (120) Eq. (120) is a highly non-trivial relation for the structure constants of the N

=2

minimal theory. The structure constants G"m may be computed directly in the minimal model (e.g., ref. [36].) Indeed, they turn out to be of the form eq. (120) with the normalizations,

(121)

There are in general many different ways to write the maximal chiral field Gmax • Since it is the unique chiral field in the theory with charge

c/3,

any polynomial

with this charge has to be proportional to it. The constant of proportionality for each polynomial may be computed, up to one overall constant, using the equations 8V. ..piifi"i.

= 0.

294 From this description of the operator algebra we can compute also the transpose element,

ct, for a given chiral field C.

This field may be defined as the unique

chiral field which obeys the operator product

ctc = CIflAX . Thus to compute this

field with the correct normalization we simply take any maximal polynomial which contains C and divide it by the polynomial which represents C, C t

= CIflAXIC.

For any three chiral fields Ci, Cj and Ck, consider the following structure constant,

(122) From the description of the operator algebra given above in terms of polynomials we may compute the structure constant !ijk. The result is as follows. Form the product polynomial CiCjCk. IT the total U(I) charge of the product is not equal to c/3, the structure constant !ijk vanishes. IT the U(I) charge is equal to c/3 then the product polynomial is proportional to the maximal polynomial. The proportionality constant is the structure constant !ijk, CiCjCk

= !ijkCIfIAX'

This

proportionality constant may be computed as explained above. Suppose that the scalar field theory under discussion has a central charge c

=

9. We may then form a supersymmetric heterotic string theory by following the procedure described in sections (2-3). At the massless state sector of the theory we then encounter a number of space time spinor multiplets in the 27 representation of

Ee (generations), spinor multiplets in the 2-7 of Ee (anti-generations) and spinors which are singlets of Ee. As discussed in sections (2-3), the generations correspond to fields in the internal theory with the dimension 1/2 and left and right U(I) charges equal to 1. In particular, all chiral fields in the original scalar theory, with U(1) charge equal to 1 would give rise to some generations. In general there might be more generations coming from other fields in the theory, namely, fields appearing as a result of the general supersymmetry projection. Denote by

tP and 4> the left and

right moving U(1) free bosons. Let Ci be any

chiral field in the theory with charge one. The vertex operator for a generation

295

which is a space-time scalar and a singlet of 80(10) is (123) The vertex operator for a space-time fermion in the vector representation of 80(10) can be written as, (124) where 8a. is a space-time spinor and Va represents the vector of 80(10). In addition, in the covariant gauge, there are ghost factors. Consider now the Yukawa couplings of three generations to one another, 27 3 • For various examples these couplings were calculated [36,37,25]. The 27 3 couplings may be extracted from the structure constants (125) which can be written (up to an overall constant) as (126) The exponential in this formula is the field O!n.x. Thus lijl is identical to the structure constant appearing in the operator product, .

.

0'0'

I t

= !;jl( G ) .

(127)

This is precisely the structure constant we computed earlier, eq. (122) for any scalar field theory. We conclude that the Yukawa couplings for three generations which come from the chiral fields of the scalar field theory are given as follows: simply multiply the corresponding polynomials. This is then proportional to the maximal chiral field 01D1J.X. The proportionality constant is the Yukawa coupling, which can be computed using the equations :Pi~:

= o.

296 Let us consider now the field theory limit of a heterotic string propagating on the manifold M, V(~i)

where

~i

(128)

are now considered to be complex variables living in a weighted projec-

tive space defined by .the quotient of {aqi~;},

= 0,

where

qi

en with the identification of points {~;} =

is the U(l) charge of the field

~i

and a is an arbitrary complex

number. Assume also that the theory has the trace anomaly c

= 9 and that there

are five generating chiral fields in the potential V. If the number of generators is less then five than we add to the theory trivial ~2 theories, which do not change the conformal field theory, so as to make the number of generators equal to five. The manifold M is then a Calabi-Yau manifold. The generations on the manifold correspond to the cohomology group of antiholomorphic one forms with values in the tangent bl1ndle, Hl(T), whose elements we denote aP., where JL is a tangent bundle index. These one forms correspond also to the polynomial perturbations of the defining equation for the surface, eq. (128). For each polynOlnial that may be added to eq. (128), p, there is a corresponding one form (129) where X~ is the extrinsic curvature of the manifold M. The forms so defined are closed, daP.

= o.

Some of the polynomials which may be added to eq. (128) do not

give a different surface, but rather can be absorbed in a redefinition of the variables

~, XL --> -XL,

is a gauge symmetry. Again the corresponding vacua are to be identi-

fied, there are no domain walls, and so on. The spectrum, just as in the case of the bosonic string, also exhibits a symmetry under which the roles of left and right are reversed. However, the operation

xio--> xio, xi? --> -xi? is not a symmetry of the

theory. To see this, note first that due to the local right moving supersymmetry, we must also transform ",,10

-->

Ramond sector, since it takes

_",,10. But this changes the GSO projection in the

r ll --> _rll.

Thus the duality transformation maps

states allowed by GSO to states forbidden by GSO. This may be seen another way. If one bosonizes the space-time fermions,

(2.12)

then the fermion vertex operator (in the (-1) picture) is proportional to (2.13) with, say, an even number of plus signs (GSO). The duality transformation, however, flips the sign of ",,9 and hence of .

2

A~!A~! tP~!IO >. The U(I) here is the 2

2

symmetry which completes 0(10) to Es.

2

The corresponding Kac-Moody

current is

3. Gauge bosons in the "other" E8: A~! A~! tP~1 10 > 2

2

2

4. Graviton, axion, dilaton: €P.IJJt~ltP~! 10 > 2

5. Moduli associated with deformations of the original torus. Note that while the original torus was simply a product of three two dimensional tori, there are many tori (continuously connected to the original one) which permit a Z3 action. These include dilations of the separate tori, as well as deformations which mix the three directions. Corresponding to each of these deformations is a massless particle. There are nine such states, as well as goldstone-boson like partners: a~l tP~! 10 > and i ~

J.

(These can be thought of as zero

2

modes of the internal graviton and antisymmetric tensor fields.) 6. Matter fields in the 10 of S0(10). There are nine such fields (and their antiparticles): A~.1. A~.1. tP~! 10 >. Note that under 2

2

iL (eqn. (3.3)) these have

2

charge +1. 7. Nine matter fields which are singlets of 0(10): A~!A~!tP~110 >. Note that under

iL (eqn. (3.3)) these have charge

2

2

2

-2.

Recall that the 27 of E S decomposes under 0(10) x U(I) as 101 + 1-2 + 16_1. 2 We have found the first two sets of states in this decomposition. The others lie in the NS sector for the first sixteen fermions and the Ramond (R) sector for the second. Similarly, we have found 46 of the 78 generators of the adjoint of E6. The other 32 generators transform as a 16

+ 16

of 0(10). These are also found

in the NS-R sector of left movers. The construction of vertex operators for the various states we have listed here is completely straightforward. For example, for

317

the matter fields in the 10 one has, in the -1 ghost number picture:

(3.4) where ¢J is the bosonized superconformal ghost. To construct the vertex operator for the matter field in the 16 of 0(10), one needs to replace the left moving part of VlO by a (1,0) operator in the spinor representation. This is most easily constructed by bosonisation. Group the 10 fermions, Aa , into five complex fermions, and bosonize them. This gives five left moving bosons, ~i. Bosonize as well the internal fermions,

(3.5) Then the necessary (1,0) operator is E- a --

e~(±~1 .. ·±~·±x'±x2±x3)

.

(3.6)

The signs here must be chosen so that the full vertex operator is invariant under (up to an 0(10) transformation); thus, for example, one can take a plus sign in front of Xl and minus signs in front of X2 and X3. The net number of plus and Z3

minus signs is fixed by the GSO projection. The construction of the gauge boson vertex operators in the 16 and 16 is

k

x

x

ak.

equally simple. Call = Xl + 2 + 3 ; then iL = Note that under Z3, k -+ k + 211'. Then the currents associated with the spinor representations are given by

j

= eH±~1 ... ±~2±ii)

.

(3.7)

The integrals of these currents are the corresponding gauge charges. The required vertex operators then have the form

(3.8) In a similar way, one constructs the vertex operators for states which are space-time spinors. For example, to construct the gaugino vertex operators, one must replace

318 the operator

1/11' in eqn.

(3.8) by a (0,1) operator which is invariant under Z3. To

do this, bosonize both the internal right moving fermions, 1/1 i , and the remaining free fermions,

1/1° ... 1/1 3 :

(3.9) Then the required operators are

(3.10)

where H = Xl

+ X2 + X3,

JR = 8H, and again, GSO requires that the net number

of plus signs (say) be even. The integrals of these currents are the four space-time supersymmetry generators. It is an instructive exercise to construct the vertex operators for the remaining states. This theory has two left moving and two right moving superconformal symmetries. On the right, the generators of the super conformal algebra are two fermionic currents of dimension

i: (3.11)

a U(I) current, (3.12) and the stress tensor

(3.13)

Note that all of these operators are invariant under the Z3 symmetry, and thus survive the process of modding out.

319 To compute the operator product algebra, recall

< x(z)x(w) >= -In(z - w)

1

< t/;(z)t/;(w) >= - - z-w

(3.14)

It is then straightforward to show that

a(+)(z)GH(w)

=

c + 1 iR(W) 4(z-w)32(z-w)2

iR(Z)G(±) =

+ !T(w) + 18w iR(w) + .... (z-w)

±G± (z - w)

--

1G± 28G± T(z)G± = (2 )2 + -(z-w --) z-w Here

c=

(3.15)

(3.17)

(3.18)

6.

Because of the left-right symmetry, there is an identical algebra on the left, with t/;i replaced by Ai, and 8 replaced by

8.

On the right, one important conse-

quence is space-time supersymmetry. Indeed, we have seen that the supersymmetry generator is constructed from the current iR and the free fermions associated with the remaining flat directions. Similarly on the left, we have seen that the current

iL allows construction of the full set of E6 generators. Without this current, the gauge symmetry would be only 0(10). This argument can also be run in reverse; it is possible to show that space-time supersymmetry requires the existance of this larger algebra. l l

If we stopped at the projection on invariant states in the untwisted sector, the resulting theory would not be modular invariant. In flat ten dimensions,recall that modular invariance is achieved, at one loop, by the particular sum over spin structures indicated in Fig. 3. The plus and minus signs indicate periodic and antiperiodic boundary conditions on the appropriate fermions in the functional

320

integral.

In a Hamiltonian approach, this particular sum over spm structures

corresponds to

(3.19) Here the first two factors refer to the two groups of left moving fermions, AA and

AA, while the third factor refers to the right moving fermions,

,pl'.

Note that these

terms give precisely the usual GSO projections: on the left one keeps states of even world-sheet fermion number, on the right states of odd world-sheet fermion number. The projection onto Z3 invariant states corresponds to adding to this expression terms where the boundary conditions on all of the fields are twisted in the "time" direction by hand h2, where h denotes the Z3 transformation. This is indicated in fig. 4. But this combination of terms is clearly not modular invariant by itself; in particular, it is not invariant under T ~ -~. To obtain a modular invariant result, it is necessary to add terms twisted in the "space" directions, the "twist sectors." In these sectors, the boundary conditions on a general field, c/J( u, T) are twisted, ¢>(u+7r, T) = h4¢>(U), where in this case a = 1,2. More generally, when one mods out a conformal field theory by the action of a discrete symmetry, it is necessary to include sectors with the u boundary conditions twisted by a discrete group element. The precise coefficients of these terms are completely determined by modular invariance. 12 These in turn determine the required projections in the twisted sectors. To see how this works, first note that the normal ordering constant in Lo (to) for a boson (fermion) with moding shifted by 1/, 0 < 1/ < 1 is given by 7

(3.20)

This can be derived in various ways. For example, one may calculate the sum over modes using (-function regularization (which has the feature that it preserves conformal invariance for this problem). An alternative derivation will be given

321

below. With this result, it is a simple matter to calculate the various terms in the partition function. For example, the partition function for right movers of fig. 6 is given, with q

= e21rir ,

by (including the contributions from

'l/Ji

only, since only

they are twisted) (3.21) This can be written in terms of standard

e functions.

In particular, using

and the definition of the Dedekind "I function,

(3.23) the expression of eqn. (3.21) becomes (3.24) Now we can use the modular transformation properties of the 8- functions (these can be derived using the Poisson resummation formula): (3.25)

(3.26) Unravelling all of this, the first term in fig. 6 is mapped into the second, with a net factor of -1. Working out the corresponding transformation for the term with antiperiodic boundary conditions in the

0"1

direction gives, on the right (3.27)

Carrying through the analysis for the rest of the terms, one finds that, in the singly twisted sector one must project onto invariant states which carry even fermion number on the right and odd fermion number on the left.

322

Using the formula, eqn. (3.20), the net shift in Lo for the right movers, in the singly twisted sector, with NS boundary conditions is OJ for the left movers, it is

-4- Thus we can easily write down the massless states in this sector: (3.28) (3.29)

+ 1 in

These fields look like they make up the 10

the decomposition of the 27.

But what about the U(I) charges? For these to work out, it is necessary that the twisted ground state have a non-zero charge. We can, indeed, show that the twisted ground state is charged, in various ways. We can simply compute

< Q >,

using a suitable regulator both in the twisted and untwisted sectors, and take the difference. Alternatively we can consider the operator which creates the ground states in the twisted sector of the conformal field theory. This operator is called a "twist field." Its construction is discussed in some detail in Dixon's lecture notes. 9 In general, the twist fields change the boundary conditions on the two dimensional

fields, and as a result introduce cuts in Green's functions. They are of the form (3.30) where

(T

twists the bosonic fields, while TL and TR twist the fermionic fields. The

construction of the bosonic twist fields is in general somewhat complicated, but it is a simple matter to write down the fermionic twist fields. Bosonize the left moving fields according to (3.31) Then

.i1!>.

TL(Z) = lIe'

(3.32)

3

This introduces cuts, as it should, in fermion correlation functions. For example, (3.33) TL

has dimension

is, which agrees with the shift in Lo obtained from eqn.

Note in using this formula that the fields are in the NS sector, i.e. Tf

= :.

(3.20).

323 But now it is easy to see from eqns. (3.31) and (3.32) that the U(l) charge of TL is +1, so the charge of the ground state in the twisted sector is also +1. One can, in a similar way, check that the 16 states in the Ramond sector have charge -~. Thus we obtain a full 27 in each of the twisted sectors; this gives 27 additional

generations. There are several other massless states in the twisted sectors of this model which should be mentioned. We have already discussed the nine moduli associated with the deformations of the original torus. There is an additional modulus in each of the twisted sectors. These are associated with "blowing up" the orbifold singularity to obtain a smooth manifold. Their vertex operators are easily constructed. To do this, it is useful to return to the untwisted sectors and note that there is a pairing between moduli and matter fields. Recall that the matter field vertex operators were (in the -1 ghost number picture) (3.34) The vertex operators for the moduli can be obtained by removing the field ,\4, and acting on the remaining operator with the left moving supersymmetry generator, (3.35) to give the operator (corresponding to the state in item (5) above)

This pairing, as we will show later, is quite general. In the twisted sectors, we can perform a similar construction. From the 10's of 80(10) we thus obtain the states (3.36) These fields are moduli in the sense that they have no potential in space-time (equivalently, they are exactly marginal operators of the conformal field theory). In addition, in each of the twisted sectors, there are eight other massless gauge singlet states, (3.37)

324

4. General (2,2) Compactifications Much of the structure of the orbifold example generalizes to other models of compactification. In general, one has a product of a free two dimensional field theory, involving :r;1' and 1/;1', f.L

= o... 3, some free fields on the left, and an internal

conformal field theory with c = 9 for the right moving Virsasoro algebra.' Vertex operators (states) are constructed from products of operators from the internal field theory with the various free fields. Particularly interesting are theories with at least two right moving supersymmetries. The existance of two such symmetries is a necessary and sufficient condition for the theory to exhibit N = 1 supersymmetry in spacetime.ll In such cases, we will be able to derive a number of striking results. We will show that there are no tadpoles to any order of string perturbation theory (with one set of exceptions), and that massless fields, even those not protected by any symmetry in space-time, remain massless. We will also be able to obtain some non-perturbative information about these theories. The key to obtaining these results will be to examine the low energy effective lagrangian,

Leff'

for the massless modes in any

particular compactification. The power of this approach lies - as always, in the fact that

Leff

is highly restricted by the symmetries of the system. In particular,

the lagrangian must respect all unbroken gauge symmetries; any breaking of these symmetries must show up as spontaneous breaking in the effective lagrangian. This is familiar from theories like the standard model. In compactifications which possess space-time supersymmetry in some leading approximation, there is a light spin-i particle. Such a particle can only be described consistently in a theory with a gauged supersymmetry, i.e. a supergravity theory. Thus for

Leff

to be consistent

and make sense, it must be supersymmetric. So, for example, if supersymmetry breaking is to occur at some order in perturbation theory due to quantum effects, this must show up as spontaneous breaking in L eff . If tadpoles, mass shifts and the like are to occur, they must be permitted by the various symmetries of the problem. That said, let us very briefly review the content of four dimensional supersymmetric field theories. There are, in fact, three types of supermultiplets that will be , Of course, in the orbifold example all fields are free, but it is only the internal fields which are subject to the projections and twists.

325 of interest to us. The first are called chiral fields, and consist of a complex scalar,

1>, a Weyl fermion 'I/J,

and an auxiliary field, F (a field which does not appear with

derivatives in the lagrangian). There are vector or gauge supermultiplets, which consist of gauge fields,

A~,

chiral fields Aa , and auxiliary fields, D a . Finally, there

is the supergravity multiplet, consisting ofthe graviton and gravitino, and a variety of auxiliary fields. The interactions in such a theory consist, first, of the usual gauge invariant terms for all of the fields. In addition, there are Yukawa couplings of the gauginos, matter fields, and matter scalars, given by

( 4.1) Here T a denotes the gauge generator in the representation of 1>i. There are terms involving the auxiliary D fields,

( 4.2)

Here the term in brackets is called a Fayet-Iliopoulos D term. By gauge invariance, such a term can only appear for a U(l) generator. The auxiliary D fields can be eliminated by their equations of motion, giving rise to a potential for the scalars. The Fayet-Iliopoulos term, when present, can lead to a breakdown of gauge symmetries, supersymmetry, or both. Apart from these terms, there is another source of terms in the scalar potential (as well as masses and Yukawa couplings for fermions). This is the "superpotential," W( 1». W is a holomorphic function of the fields; it gives rise to the additional terms in the lagrangian:

(4.3)

Again, the auxiliary fields Fi may be eliminated by their equations of motion to yield a potential for the scalars. In global supersymmetry, unbroken supersym~ry

326 requires that the vacuum energy vanish. At the classical level, this means

It is usually not difficult to satisfy the D4

= 0 equations;

the ~:; equations are

typically harder. If supersymmetry is unbroken at the tree level, the effective lagrangian must be sllpersymmetric. In field theory, one can prove certain nonrenormalization theorems for quantum corrections to the action. In particular, one can show that the F terms are not renormalized. 13 We will see that the same is true in string perturbation theory.14 As a consequence, supersymmetric vacua will remain good vacua to all orders of perturbation theory, and massless states will remain massless. In particular, this will guarantee the "finiteness" of the string perturbation expansion about these vacua, i.e. there will be no tadpoles for massless particles. The low energy effective lagrangian which describes the orbifold model, for example, must be superymmetric, and this tightly constrains the possible phenomenon which can occur in this compactification. We would like, however, to consider more general compactifications. We will focus here on solutions which preserve at least N

=

1 supersymmetry in space-time. This is

not only for "phenomenological" reasons, but also because, as mentioned above, these solutions are quantum mechanically stable, at least in perturbation theory. The first models of this type to be discovered were the so-called "Calabi-Yau" spaces. 15 They can be constructed in either of two equivalent ways. First, if one assumes the compactification radius is large compared to the characteristic string scale, it is reasonable to study an effective lagrangian for the light fields in ten dimensions: the graviton, antisymmetric tensor, gauge fields, etc. In principle, this lagrangian contains terms of arbitrarily high dimension, arising from iutegrating out the massive string modes. In practice, to obtain these terms one computes scattering amplitudes in string theory and determines an effective action which reproduces these amplitudes. One can then study solutions of the equations of motion obtained from this lagrangian. This approach to obtaining solutions of the string equations is discussed extensively in ref. 16. There is an alternative approach to constructing these solutions which is more two dimensional in character. Consider the action for a bosonic string propagating

327 in a gravitational background, ( 4.4) From a two dimensional point of view, this action describes a non-trivial, interacting field theory, a non-linear sigma model. Consistent string propagation requires the existance of a Virasoro algebra, i.e. that the two dimensional field theory be a conformal field theory with the correct value of the central charge. Alternatively, at the classical level, there is a simple argument that a conform ally invariant two dimensional field theory corresponds to a solution of the classical string equations of motion. This is simply the fact that on the plane S L(2, C) invariance implies that vanishing of the zero, one and two point functions.1 7 ,15,18 In particular, the vanishing of the one point function implies that tadpoles for massless fields vanish, which in turn implies that the equations of motion are satisfied. In the case of the sigma model at large radius, R, the conditions for conformal invariance can

b.

be studied perturbatively in To see this, write gij = R2g?j, where g?j is some standard metric. The R2 sits out front of the action of eqn. (4.4), i.e. is the

b

loop expansion parameter of the model. Similar sigma models describe string propagation in the other string theories. In the heterotic string, for example, including only background metric and gauge fields, for simplicity, the sigma model has the form

Here w(:c) is the background spin connection, A(:c) is the background gauge field, and F is the associated field strength. These background fields can be though of as coupling constants of the two dimensional theory. The various terms in the action above then all have naive dimension two. (The theory as we have written it above is not quite renormalizablej it is necessary to include as well the couplings to the anti symmetric tensor field and of the dilaton to the two dimensional curvature.) To obtain a conformally invariant theory, we require vanishing of the beta functions

328 for the various couplings, including that for the dilaton (which is closely related to the problem of obtaining the correct value of the central charge).18 At large radius (weak sigma model coupling) we want to find a g, A and B which yield a conformally invariant theory. In this language, the construction of Calabi-Yau spaces 15 amounts to the following ansatz. As in the case of the orbifold, we take the two dimensional theory to be a product of a free theory associated with the ordinary

M4 degrees of freedom and an internal conformal field theory. We divide the bosonic fields X I into a set of complex, interacting fields, zi and z', corresponding to string propagation on some internal manifold, K, and a set of free fields, zl'. Similarly the right moving fields are decomposed as vi,

.,p' and .,pl'.'

From the 32

left moving fields, AA and AA, we take six of the AA's to be complex, interacting fields, Ai and Ai, and the rest to be free, Aa , a In the lowest approximation in

= 1, ... 10.

b, we treat the left and right moving internal

fields symmetrically, i.e. we require that the internal conformal field theory have a left-right symmetry (parity). To obtain N = 1 supersymmetry in space-time, the world sheet theory must have two right moving supersymmetries. l1 This requires that the metric, gii be Kahler, i.e. that locally one can write the metric as

gil

= 8i8;K .

(4.6)

The spin connection for such a metric is necessarily a U(3) matrix. To lowest order, the calculation of the ,a-function for g is the same as in the purely bosonic theory. This gives the condition that the Ricci tensor, Rii, vanish. The spin connection for a Kahler manifold in three complex dimensions is necessarily a U(3) matrix. It is a simple matter to show that if the Ricci tensor vanishes, the U( 1) piece of

the spin connection must vanish. Vanishing ,a-function for the gauge fields can be achieved in the following way. 0(16) has an obvious SU(3)xE6 subgroup. Take the background gauge field to lie in this SU(3), and to be equal to the spin connection. The internal sigma model is then completely symmetric between left and right; all of the ,a-functions vanish to lowest order. Because of the left-right symmetry, the

*

Here we are being rather cavalier in our treatment of complex manifolds. A more careful discussion, especially of the important global issues, can be formulated following the treatment of refs. 15 and 16.

329 model possesses two left moving and two right moving supersymmetries, or (2,2) world sheet supersymmetry. All of this represents a treatment only to lowest non-vanishing order in the sigma-model coupling constant. From a two dimensional point of view, it is by no means clear that we can continue to make the ,a-functions vanish while preserving the full (2,2) supersymmetry to higher orders. For example, renormalizability requires that one include antisymmetric tensor field backgrounds. Also, at high enough order, the ,a-function receives corrections which are not proportional to the Ricci tensor. However, as we will see shortly, it is possible to make quite general statements about these compactifications, not only to all orders in perturbation theory, but exactly, by exploiting their features in space-time, and in particular by using the fact that the low energy effective lagrangians which describe these solutions are supersymmetric. Any violation of supersymmetry must appear as a spontaneous effect in this lagrangian. The question, for example, of whether there is any obstacle to constructing an exact classical solution of the theory, with the Calabi-Yau solution as our starting point, can be thought of as the question: do we, in some order of approximation, obtain a tadpole for some massless field. We will answer this question by studying the effective action. To understand space-time supersymmetry in this model, we should first consider the superconformal algebra. To lowest order, the construction of the superconformal generators is almost identical to the construction in the orbifold case. The calculations are essentially identical, since to lowest order the propagators of the various fields can be taken as free propagators. The right-moving superconformal generators are given by

(4.7)

C-(z)

1

.•

1

.

= -g··ax''Ij;J = -ax''Ij;'

J2

'J

J2'

(4.8)

and the right moving U (1) current is given by

(4.9)

330 Similarly, the left-moving generators are given by

-+

G

1 -

= -8:r;·)'"

.

(4.10)

y'2'

The construction of the supersymmetry generators proceeds as in the orbifold case. Since the right moving current is conserved, it can be bosonized,

iL

=

8H .

( 4.11)

The space-time supersymmetry generators are then constructed precisely as in eqn.

(3.10). To analyze

Leff'

we must first determine what fields it contains, i.e. what is

the massless spectrum. Despite the fact that we do not know the explicit metric which enters in eqn. (4.5), many features of this spectrum follow from quite general considerations. Since our theory, in this approximation, is conformally invariant, we can ask this question by constructing vertex operators. There are, first of all, a number of vertex operators which can be constructed out of the free fields,

:r;I', )..a,

etc. These include the vertex operators for the four dimensional graviton, dilaton and antisymmetric tensor field (axion) (in the -1 superconformal ghost number picture),

( 4.12) where the €'s are appropriate tensors! The vertex operators for the gauge bosons of 0(16) x 0(10) are equally easy to write down. For example, for 0(10):

(4.13) The other E6 bosons are also constructed exactly as in the orbifold theory. The

U(l) boson, for example, is created by ( 4.14) where

iL

is the U(l) current in the left moving superconformal algebra.

The

spinorial generators are constructed by bosonizing this current, as well as the free

t

The dilaton vertex operator also contains a term involv:ing the world-sheet curvature

331

fields Aa , and then writing the generators in terms of exponentials of these free fields. There are other massless fields, whose vertex operators involve interacting two dimensional fields. These are of three types. The first are "moduli" of the compactification. The equation R i , = 0 does not fully determine the metric; there are typically a multiparameter set of solutions. For example, if we have a Ricci flat

..

Kahler metric, g~~), we can always obtain a new one simply by multiplying by a constant. This means that we can add to the conformal field theory a term oR

v,.,

where

(4.15) obtaining a new conformally invariant theory. v;. is thus a marginal (in fact an "exactly marginal" or "truly marginal" operator); it is a good vertex operator ofthe theory, and corresponds to a massless particle, the "radial dilaton." This result can be described another way.

v,.

is an operator of naive dimension 2. Its anomalous

dimension is calculated, as usual in field theory, by adding this operator to the lagrangian with a small coupling, and calculating the ,a-function of the theory to first order in the coupling. Requiring that the anomalous dimension vanish means that

lin = O. However, since the vanishing of the ,a-function is equivalent, in string

theory, to the equations of motion, the vanishing of this anomalous dimension is

..

equivalent to the statement that g~) itself solves the linearized equations of motion . An N = 1 supersymmetry multiplet contains two real scalar fields·. Thus supersymmetry requires that there should be another scalar field in the supermultiplet with r. The corresponding particle is easily found; it can be thought of as arising from the anti symmetric tensor field; we will refer to it loosely as the "world-sheet axion," b. b has vertex operator (4.16)

At this point we can pause and prove a striking result in sigma model perturbation theory.2o Since gil

= Bi8,K, at zero momentum v" is a total derivative, (4.17)

In perturbation theory, one includes only topologically trivial field configurations,

332

so

f d2zv;, vanishes,

i.e. the b field decouples at zero momentum. But we have

already said that b lies in a supermultiplet with the field r. The scalar component of the corresponding chiral superfield is R

= r + ib.

(analytic) function of R, it is independent of R.

Since W is a holomorphic

But changing r changes the

overall radius of the complex space, i.e. it changes the sigma-model coupling. So the superpotential is independent of the coupling and is not modified in sigma model perturbation theory! Because the superpotential is not modified, one cannot obtain tadpoles for massless fields. This means that, starting from the zeroth order solution, one can construct a solution of the classical string equations to all orders in perturbation theory. By more closely examining the structure of string vertex operators, one can in fact show that the superpotential for the matter fields (to be discussed below) is purely cubic. 21 This result does not mean that the original Calabi-Yau solution itself is an exact solution to the string equations. Indeed, the ten dimensional graviton and gauge bosons couple to massive string modes, so their expectation values should give rise to vev's for massive states. These, in turn, should lead to further shifts in the background metric, etc. The only important question, however, is whether there are tadpoles for the massless fields in four dimensions. This can be seen either from a space-time or a two dimensional viewpoint. 22 Consider, first, space-time. Suppose one has some massive field, H, with lagrangian (at zero momentum)

( 4.18) Here

f

represents a small tadpole for the heavy field. The equations for H then

give

6H

= __f_ M2

(4.19)

Thus tadpoles for massive fields are innocuous; the fields simply undergo small shifts. Tadpoles for massless fields are much more .serious. There is no guarantee that there will be, in such cases, a nearby vacuum. This argument makes clear that we should focus, for such questions, on the effective lagrangian for massless fields. This does not mean that we ignore the massive fields; only that we carefully integrate them out. This same point can be seen from a world sheet viewpoint.

333

View the internal field theory as a regulated theory with a cutoff.

There are

three types of operators, in general, which can appear in such a theory. First there are operators, Tn, of dimension less than two. Such operators are called "relevant." From a space-time point of view they correspond to tachyons, and are absent in supersymmetric compactifications. The second type are operators, On, of dimension two. These operators are called "marginal" (in more old-fashioned terminology they are called "renormalizable"). From a space-time point of view, they correspond to massless fields. Finally, there are operators, Hn, of dimension greater than two. Such operators are called "irrelevant," and describe massive fields in space-time. In string theory, classical solutions are equivalent to fixed points of the renormalization group. However it is not necessary to work at the fixed point; it is enough to work at a critical point. A critical point differs from a fixed point by the presence of irrelevant operators. Under the renormalization group, the theory at such a critical point will flow into the fixed point in the infrared; this is the origin of the term "irrelevant." From a space-time point of view, the presence of irrelevant operators corresponds to expectation values for heavy fields. We see, in this way, that including (small) background values for heavy fields has no effect on physics. These ideas are nicely illustrated by a problem in Calabi-Yau compactification. In lowest order, we have seen that there are some massless fields. There are also massive fields with masses of order ~ = 9 2 , where 9 2 is the sigma model loop expansion parameter. At four loop order, there is a non-vanishing ,a-function if

RIJ = 0. 19 However, one can show that the counterterm only involves massive states. 19 ,23 These results initially caused some confusion, because in perturbation theory the divergences are logarithmic, whereas we expect effects associated with non-renormalizable (irrelevant) operators to vanish as powers of the cutoff. The point here is simply that the lowest order masses (anomalous dimensions) are proportional to 9 2 , so the effects associated with these operators go as _g' _

2

A - I - 9 lnA

+ ....

( 4.20)

It is necessary to slim these logarithms; this is done using the renormalization

group. A counterterm for a massive field, Hn, corresponds to a renormalization

334

group equation of the form ( 4.21) where t = lnA. The solution of this equation is (4.22) Here '"tn, the anomalous dimension, is just the mass of the field Hn, from a spacetime viewpoint, while

rn

corresponds to a tadpole. Thus as t

---> 00,

Hn tends to

the fixed point value suggested by our field theory discussion above. The lesson, again, is that massive states take care of themselves; to determine if one has a solution of the equations of motion, it is enough to study the massless states. Our argument above shows that, in the effective lagrangian from which heavy fields have been properly integrated out, there are indeed no tadpoles for the field R, and the Calabi-Yau configuration can be extended to a solution to all orders in perturbation theory. Beyond perturbation theory, the argument we have given is not valid, since the symmetry under shifts in b breaks down. We will demonstrate, however, that Calabi-Yau spaces do correspond to exact solutions of the classical string equations. Before doing so, let us first obtain the other massless states. Consider, first, the Ramond sectors for the right movers and the left movers in the first Es, and the NS sector for the left movers in the "other" Es. On both the left and right, the contribution of the various free fields to the normal ordering constant is zero. The internal field theory has both left and right moving superymmetries. For very large radius, we expect that the spectrum contains a set of states with masses of order

-il,

and other states with energies of order one. Thus, to obtain the light

states, it should be enough to work in a sort of Born-Oppenheimer approximation, dropping all of the u dependence of the fields. 24 This leaves a (supersymmetric) quantum mechanics problem, involving variables qi(T), Ai(T), and '¢i(T). There are four super symmetry generators, given by (4.23)

335

where IT..

= -if;q

is the conjugate momentum to qi, etc. The fermions obey the

anticommutation relations

(4.24) These are just creation and annihilation operator commutation relations. By convention, we can take ~i and .,pi to be creation operators. As in any supersymmetric quantum mechanics, the anticommutators of the supersymmetry generators give the Hamiltonian. Zero energy states are states annihilated by the Q's and Q's. A general state can be constructed by building a "Fock space" using the ~i's and

.,p"s

acting on a "vacuum" state

10 >

which is

annihilated by the Ai's and .,pi's. We can construct states of definite four dimensional helicity by noting that the four and six dimensional helicities are correlated by the GSO projection. The internal helicity operator is just the internal fermion number. Thus the general state of helicity (-) is

(4.25) The coefficient functions A, B, etc. can be thought of as differential forms. To determine the condition that the states be annihilated by the various Q's, note that QR adds a barred index to a state and takes the exterior derivative,

80.

(JR, on the other hand, removes an index by taking the divergence. The corre8, *8, and are called the

sponding operations on differential forms are denoted by

Dolbeaut operators. For the left moving supersymmetries we have corresponding operators

a and *0.

Thus zero energy states - which from a space-time point of

view correspond to massless states, are in one to one correspondence with globally defined forms annihilated by these operators. The problem of obtaining these differential forms is the problem of Dolbeaut cohomology; a good introduction is

336 provided by the text of Green, Schwarz and Witten. 16 The number of such forms with p indices of type i and q of type t is denoted by hpq , and called the Hodge number. These numbers satisfy ( 4.26) where N is the dimension of the space, and ( 4.27) By convention, we can take the B" terms in eqn. (4.25) to correspond to 16's of

SO(10). The Cii,fc term then corresponds to 16's. Thus the number of generations of E6, which is the difference of the number of 27's and 27's, or the difference of the number of 16's and 16's is

N gen -- h2 ,1

_

h1,1

( 4.28)

where X is the Euler number of the manifold (this relation is derived, for example, in ref. 16).The (0,3) forms give rise to gravitinos and gauginos. By similar reasoning we can find more massless states; these are the moduli. The states we have just constructed are in the NS-R sector for the left movers. Their vertex operators for take the form (4.29) Here ~i' i

= 1, ... 5 are the ten

bosonized left moving free fermions (Am); 4>1 and

4>2 are the bosonized free right moving fermions of eqn. (3.9).

~

is an operator in

the internal conformal field theory, ofleft and right moving dimension

i, and with

left and right moving U(l) charges indicated by the superscripts. This structure is easily checked for the orbifold case. From

VIS

we can construct an NS operator on the left, either by acting with

the generators in the spinor representation of 0(10), or by spectral flow. One can

337 do the same thing on the right, if one wishes, using the space-time supersymmetry generators. This gives operators of the form (again check this for the orbifold)

(4.30) Now from this we can construct vertex operators for massless fields which are gauge singlets. These will turn out to be the moduli of the compactification. First remove the free field Aa from the vertex operator. Then act with c~-2 (C(-) = 8xiAi) to 2

give a new operator; in the sigma model this has the

(4.31) Indeed, this prescription applies to any (2,2) conformal field theory; the moduli and the 27'5 of E6 are paired. This pairing was noted in the original work of ref. 15, and also in ref. 25. Note that we can think of this pairing another way.22 A (2,2) conformal field theory is a good background for the type II theories (in these theories the original left and right moving local supersymmetries require that the conformal field theory have at least one left moving and one right moving supersymmetry). The analysis we have just described of the supersymmetric quantum mechanics model now gives fields in the R-R sector of dimension

(i, i).

Acting with the

space-time supersymmetry generators associated with the left and right movers gives NS-NS fields in some supermultiplet. Written in the (-1, -1) superconformal ghost number picture, these are good heterotic string vertex operators, provided that on the left one drops the superconformal ghost and either replaces it with a free field, Aa, to give the vertex operator in the 10, or acts with the left moving supersymmetry generator, to give the modulus. Apart from the moduli and the matter fields in the 27 and 27 which we have described up to now, there are sometimes additional massless fields. 20 ,16 From the u-model point of view, these states are associated with deformations ofthe tangent bundle. These fields are not generic, and are not readily treated ill the framework above. As we will see shortly, these fields, if present to lowest order at large radius, generically gain mass once non-perturbative effects in the u-model are included,26 unless they are protected by symmetries. Such states, protected by symmetries, indeed appear in the models of Gepner. 25

338 So far we have been describing the spectrum to lowest order in sigma model perturbation theory. The non-renormalization theorem we proved earlier guarantees that these states will remain massless to all orders in perturbation theory. However, as we remarked before, this non-renormalization theorem is not valid once one considers nonperturbative effects in the sigma model, and in particular world sheet instantons. In general the superpotential is renormalized. 26 One might worry that this would lead to tadpoles for massless fields, since they are not protected by any symmetry. This would indicate that the Calabi-Yau spaces are not solutions. Even if this is not the case, one might expect massless field to gaill mass, and other features of the lowest order analysis to break down. In fact, there is a simple argument that the lowest order Calabi-Yau solutions generalize to ezact solutions of the string equations. One needs to show that the fields we have called moduli are indeed moduli, i.e. we must show that they have no potential. In the language of conformal field theory, we must show that they are exactly marginal operators. To demonstrate this, we can use either space-time 22 or world sheet 9 reasoning. The space-time argument goes as follows. Consider these theories as backgrounds for the Type II theories. In spacetime, the low energy

N = 2 supersymmetry is extremely restrictive; it is essentially impossible to write any potential. Thus one can give the various moduli expectation values at no cost in energy, obtaining new vacua. Correspondingly, one obtains new conformal field theories by perturbing the original conformal theory with the vertex operators for the moduli. These new conformal field theories again have (2,2) superconformal invariance. They are good backgrounds, not only for the Type II theories, but for the heterotic string theory as well. By running ·our earlier pairing argument in reverse, one can also construct fields in the 27 and 27. This shows that these fields do not gain mass. Note that theory then has N

= 2 supersymmetry.

this has nothing to do, in the case of the heterotic string, with any space-time symmetry. It is entirely a consequence of features of the model on the world sheet.

It is indeed not very difficult to give a microscopic, two dimensional proof of the existance of moduli. 9 This argument can be made particularly easily if one uses vertex operators for the auxiliary F fields,28 along lines used in the arguments of ref. 29. Using these operators, we can compute the superpotential directly in the NS sector. In this discussion we alternate between a sigma model description

339 and the more abstract conformal field theory description. In the -1 ghost number picture, the vertex operator for the radial dilaton is ( 4.32) More generally, the vertex operators for the moduli can be written in the form (4.33) In general b is an operator of dimension (~,~), with left and right U(I) charges (1, -1). For the case of Vr , b = gi;).it/J;. Vm can be written as

-) -.p(w) G--_1 b( W,we

(4.34)

2

(In general, to separate out descendent fields with respect to G from primaries, one writes:

G(z)q'J(w,w) = E(z - wy>-% G-nq'J(w, w)

(4.35)

where n is half integer moded in the NS sector and

1 G_nq'J(w,w) = -2' 7l't

f

dz

1

1

(z - w)n-.

G(z)q'J(w,w)

.

( 4.36)

One makes similar decompositions for other algebras such as the Virasoro and Kac-Moodyalgebras.) For simplicity, we will study the superpotential for R. We will leave it as an exercise to generalize this discussion to all of the moduli in a general eft. Note first that

C+(.z)b(w,w) -

G-(.z)b(w,w)

= 0(1) 1·

•

= -(_ _){)Z'gi;).1 z-w

( 4.37)

The vertex operators for the auxiliary F fields are constructed using the operator, to,

with h

= !, Q = 3 which always exists in (2,2) models (see for example Warner's

340

lectures at this school). In the sigma model or orbifold,

(4.38)

where in the orbifold case,

is simply the antisymmetric symbol, while in the

€ijk

Calabi-Yau case it is the covariantly constant three form. Writing

€(z)(G:::~b)(w,w)

1

-

= (z_w)b(w,w)

(4.39)

the vertex operator for the auxiliary F component of a chiral multiplet is

( 4.40)

which in the sigma model or orbifold case is simply

( 4.41)

Note that the operator product of G- and

b is non-singular, since one can write ( 4.42)

In order to study the superpotential, take two vertex operators for R in the

(-1) picture, VF, and any number of R vertex operators in the zero ghost picture:

V~ = G::: 1 G~! be "'" + ik . ¢G::: 1 be "'" • i

i

2

2

Note that the product Vi1Vi1VF has qR

(4.43)

2

= O.

So the ik·¢ terms cannot contribute

to the amplitude, by U(l) charge conservation. So we need to study correlation functions of the form (using Mobius invariance to fix the locations of VF and the

341

( 4.44) Rewrite the third vertex operator as a contour integral,

( 4.45) Now deform the contour,

50

that it e:ncircles the other vertex operator. In doing

this we encounter no poles, since as we remarked above, the operator product of G- with b is non-singular, while the operator products of G- with itself are nonsingular. Thus the amplitude, and hence the superpotential, vanishes. This argument shows that if one has a (2,2) superconformal field theory, and if it has operators which are highest weight with respect to both the left and right moving N = 2 algebras, then these fields are moduli. One can give them expectation values, obtaining new (2,2) theories. It also proves that one can construct an exact solution of the classical string equations starting from a Riccifiat Kahler manifold at large radius. To see this, one must again think about Leff. At lowest order, there is a massless spin-~ particle, so Leff is supersymmetric. Also, the expectation value of Wand its first derivatives vanishes. So if a superpotential is generated, it can appear first in order g2. But then the potential for the moduli cannot appear before order g4, and the theory is superconformal through order g2. Then by the argument above, no superpotential is generated in order g2. Repeating this argument (including possible non-perturbative contributions to W), we see that no superpotential for the moduli can be generated. 3

Similar arguments can be used to show that, while 27 couplings are renormalized by instantons, 27 3 couplings are not. 29 The trick of using the conformal field theory as a background for the Type II theories can also be used to obtain more information. For example, one can learn much about the local structure of the moduli space by examining the restrictions imposed on the space-time lagrangian by this symmetry.27

342

5. String Perturbation Theory and Beyond It is natural to ask whether the sorts of non-renormalization theorems that

hold in ordinary field theory hold for the string loop expansion as well.

It is

surpringly easy to prove that they do; the proof is quite similar to the argument given above for the sigma model perturbation expansion. 14 In all compactifications to four dimensions, there is a field, called the "space-time axion." It can be thought of as arising from the antisymmetric tensor field, with indices in M4, B Il V ' Its vertex operator is

(5.1) In compactifications preserving N

= 1 supersymmetry, this field lies in a multiplet

with the dilaton field, D, whose vertex operator is

VD = Tfllv82)11({}2)v

+ ik· VnV)eik .., + curvature

terms.

(5.2)

The couplings of these fields include a coupling of D to F;v and of a to FllvFllv (as an exercise, check that the operators Va and VD appear in the OPE of two gauge bosons.) Giving an expectation value to < D

> changes the gauge coupling

constant; indeed, it changes the value of the string coupling constant (string loop expansion parameter). Like the operator

Vb we discussed in the context of the

sigma model, the operator Va is a total derivative at zero momentum:

(5.3) This vanishes on a world sheet of any genus. Thus we learn that the axion field decouples at zero momentum, to all orders of perturbation theory. Now we recall again that the effective action must be supersymmetric, and that the superpotential must be an analytic function of D

+ ia.

But this means that it is independent of

the string coupling, and thus there is no renormalization of the superpotential to any order of the string loop expansion. A number of features of this proof should be noted. any compactification to four dimensions which preserves N

First, it is true for

=

1 supersymmetry

(i.e. which has at least a right moving N = 2 supersymmetry). It relies only

343 on the assumption that the theory is consistent, i.e. that supersymmetry, like other gauge symmetries, is free of anomalies and so the low energy theory must be supersymmetric. A breakdown of this theorem would be an indication that string theory did not make sense. What are the implications of this theorem? In theories in which all U(l) generators are traceless, it implies that supersymmetry is unbroken and that there are no tadpoles for massless fields to all orders of perturbation theory. In particular, it means that the theory is finite. It also shows that massless particles remain massless to all orders of perturbation theory, even if they are not protected by symmetries. This may be desirable in order to understand the presence of light Higgs in nature. In cases where there is a U(l) generator with a non-vanishing trace, Fayet-Iliopoulos terms are generated; some fields do gain mass at one loop and a dilaton tadpole appears at two loops. In all known instances, however, one can shift the ground state (give masses to certain light fields) in such a way as to obtain a new solution to the (quantum) string equations. Moreover, using the sort of reasoning we have used here, one can show that there are no further corrections, i.e. that there are no tadpoles for massless fields in the new vacuum, etc. 30 Does this result extend beyond perturbation theory? Of course, we hope not; supersymmetry must be broken if string theory is to describe the real world. On the other hand, while supersymmetry breaking is not too difficult to achieve in field theory, we have no understanding of how supersymmetry might be broken while at the same time the cosmological constant would vanish. Neither question can be addressed in string perturbation theory, and, lacking as yet any manageable non-perturbative formulation of the theory, we might despair of making any statements about this problem. However, as we will now show, there is an exact,

non-perturbative statement which can be made about this question, based only on things we know and the assumed consistency of the exact theory. The argument has two inputs. The first is the general form of a supersymmetric effective action. The second is that the couplings of Y = eD + ia, at tree level, are independent of any details of compactification, so these are known as well. For simplicity, consider the field Y only (the generalization to models with other fields is left as an exercise). In a theory with N

= 1 supersymmetry, the general potential can be expressed in

terms of two functions: the superpotential, Wand the Kahler potential, K. We

344

have seen that W(Y) = 0 to all orders of perturbation theory. At tree level, K is easily calculated (for example, one can simply look at the dimensional reduction of the ten dimensional effective action;31 since Va and Vd involve only free fields, their couplings are universal):

K(Y)

=

-In(Y

+ yt) .

(5.4)

At weak coupling, the corrections to K are necessarily small. Suppose, now, that non-perturbatively a superpotential is generated. We can ask what form for

W gives broken supersymmetry and vanishing cosmological constant. In an N

=1

supergravity theory, the general form of the scalar potential is

oWV oc ( o'" and H is then of the form (for simplicity we suppose that the symmetry is Z N) .I.'" .I.'" 'I' -+'1'

27riq'" +-N

353

H

-t

H

. H 27rZq

+-N

(7.1 )

The corresponding twist field contains a term proportional to

(7.2) Thus if the A""S have electric charge Q'" (and supposing for simplicity that the electric charge does not involve H) the vacuum has charge

Q"'q'" N

(7.3)

In general, this is fractional. For Calabi-Yau spaces, we have noted that the states in the twisted sector, at large radius, are very massive. Thus the new fractionally charged states, if any, would be extremely heavy, and if any exist today they would have been produced cosmologically. On the other hand, it could well be that in the real world, the radius of the internal space is of order one (in string units). In this case, it is possible to obtain massless, fractionally charged states. Indeed, in orbifold models and in the models of Gepner, one does obtain such states. 38 ,39 It is natural to ask if such a possibility is reasonable. Suppose, in particular, that these particles gained mass of order a TeV or so. Then they could be produced at the SSC, and, being stable, detected. It appears difficult, however, to reconcile such particles with anything like the standard cosmology and the stringent searches which have been made for fractional charges. A simple estimate shows that in the big bang, one would produce at least 10- 10 such particles per baryon, if the particles were strongly interacting. If strongly interacting, all of these particles would end up bound in positively charged nuclei, so there would be no subsequent annihilation in stars. On the other hand, the present limits on fractional charge in matter are about 10 orders of magnitude stronger. If these particles are not strongly interacting, the situation is not much better. While significant annihilation does occur in stars, in the interstellar medium there is still too large a component of these fractionally charged objects; enough of them collect on the surface of the earth to be easily detected.

354

A third question, rather easily answered, is what is the criterion for a discrete symmetry to survive modding out. This question can be answered with or without Wilson lines; we have already discussed it briefly above. Consider first the untwisted sector. Suppose the original symmetry group is G and that one mods out by H. We have already seen that if 9 E G and h E H, the condition that 9 have a well defined action on states in the untwisted sector is g-lhg = h' E H. However, what about twisted sectors? Here the symmetry g, in general, takes states in one twist sector to states in another. To see this, consider the path integral on the torus in the sector twisted in the 0"1 direction by h:

z=

J

[d¢>] e- s

q,(tTl+I)=hq,(tTl)

Then the action of 9 is to change ¢> _ g¢>, which alters the boundary condition to

¢>( 0"1 +1) = g-1 hg¢>( 0"1) = h' ¢>( 0"1). Of course, a similar result holds for twists in the 0"2 direction, so we can in fact easily discuss both (and indeed modular invariance forces us to discuss both) at the same time. The effect of 9 transformations is thus to map the various sectors into one another. Since modular invariance forces us to include all of the sectors, 9 will be a symmetry provided that it maps the sectors into one another with the same phases as implied by modular transformations. It is easy to show that this is the case, at least for Abelian Wilson lines. This can be done in two stages. First, note that before including Wilson lines, there are no extra phases, essentially due to the left-right symmetry. With Wilson lines, there are extra phases, because of the shift of Lo. But for abelian Wilson lines, the sector twisted by h and that twisted by g-1 hg always have the same shift. To see this, diagonalize the Wilson line, Uk; denote its eigenvalues by

,Us. But then Uk' = Ug UkUg has the same eigenvalues (though they may be permuted). So the additional shift in Lo is the same in both states, and no new phases arise due to 9 transformations. As a result, the condition g-1 hg = h' E H is sufficient for 9 to be a symmetry for the full theory. 1

Ul, . ..

355

8. Conclusions Clearly superstring theory remains a tantalizing subject. It is quite striking how easily one comes close to describing the real world. The previous chapters have raised both phenomenological and theoretical questions. Some of these problems are reasonably straightforward. Others are difficult but still look tractable. Finally, there is a class of problems for which some new insights will be required if we are to make further progress. It is perhaps best to conclude by suggesting some problems and directions for future work. Among the relatively easy problems is the search for vacua of string theory with a reasonable phenomenology. For example, perhaps one can find (0,2) vacua, using the construction above, or that of ref. 41, or some other approach (perhaps involving exactly solvable models), which are close to the standard model, and which possesses discrete symmetries which can forbid proton decay, while still allowing suitable fermion mass matrices and the like. Ideally, such models will lead to new and interesting predictions of a qualitative nature (e.g. new light particles, gauge interactions, etc). There are a class of harder problems, on which might still hope to make some progress. One example, which we have not discussed above, has to do with flavor changing neutral currents (FCNC). Even supposing superymmetry is broken in some reasonable way with vanishing cosmological constant, it is necessary that the scalar quark and lepton mass matrix possess a high degree of degeneracy if unacceptable FCNC's are to be avoided. Is there some reasonable scheme which achieves this? Does it lead to other predictions? One possibility is that supersymmetry is broken in a "hidden sector," with supersymmetry breaking fed to ordinary fields by gauge interactions (so, in first approximation, the masses depend only on gauge quantum numbers). Can such a scheme really be viable? Other problems in this hard but not impossible level deal with more fundamental questions. How does one treat time dependent configurations in string theory? Some interesting examples of this type have been discussed recently.42 In a similar vein, there are other types of classical solutions than those discussed here. For example, string theory has monopole solutions, some of which can be written down rather explicitly.43 These solutions raise a number of questions, which are not

356

so easily answered in the conventional first quantized approach. For example, how does one treat collective modes? These questions are interesting, for they take us just beyond our present understanding of the theory, without forcing us to face a full, non-perturbative formulation; they will almost certainly provide clues to such a formulation. Finally, there are the truly difficult problems, such as the problem of vacuum degeneracy and the cosmological constant. We have seen that these problems cannot be solved in any simple way; new concepts are needed if we are to make progress on these questions. Acknowledgements: This work was supported in part by DOE contract DE-AC0283ER40107.

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KS. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B279 (1987) 369. P. Ginsparg, Phys. Rev. D35 (1987) 648. KS. Narain, Phys. Lett. 169B (1986) 41. A. Shapere and F. Wilczek, Institute for Advanced Study preprint (1988). L. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678;

Nucl. Phys. B274 (1986) 285. 8. N. Warner, these proceedings. 9. L. Dixon, in Proceedings of ICTP Summer Workshop in High Energy Physics and Cosmology, World Scientific, Singapore (1987). 10. D. Friedan, E. Martinec and S. Shenker, Nucl. Phys. B271 (1986) 93. 11. D. Friedan, A. Kent, S. Shenker and E. Witten, unpublished; A. Sen, Nucl. Phys. B278 (1986) 289; T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B299 (1988) 601; T. Banks and L. Dixon, Nucl. Phys. B307 (1988) 93.

357 12. E. Witten, in Anomalies, Geometry, Topology, W. A. Bardeen and A.R. White, eds., World Scientific, Singapore (1985). 13. M.J. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B159 (1979) 429. 14. M. Dine and N. Seiberg, Phys. Rev. Lett. 57 (1986) 2625; E. Martinec, Phys. Lett. 171 B (1986) 189. 15. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46. 16. M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Cambridge University Press, New York (1987). 17. C. Lovelace, Phys. Lett. 135B (1984) 75; Nucl. Phys. B273 (1986) 413; A. M. Polyakov, address to the International Mathematics Union, Berkeley (1986); A.B. Zamalodchikov, JETP Lett. 43 (1986) 731. 18. C. Callan, D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B262 (1985) 593. 19. M.T. Grisaru, A.E.M. Van de Ven and D. Zanon, Phys. Lett. B173 (1986) 423; D.J. Gross and E. Witten, Nucl. Phys. B277 (1986) 1. 20. E. Witten, Nucl. Phys. B268 (1986) 79. 21. M. Dine and C. Lee, Phys. Lett. 203B 371 (1988); M. Cvetic, Phys. Rev. Lett. 59 (1987) 2829. 22. M. Dine and N. Seiberg, Nucl. Phys. B306 (1988) 137. 23. D. Nemeschansky and A. Sen, Phys. Lett. B178 (1986) 365. 24. R. Rohm and E. Witten, Ann. Phys. (NY) 170 (1986) 454. 25. D. Gepner, Nucl. Phys. B296 (1988) 757. 26. M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nucl. Phys. B278 (1986) 769; Nucl. Phys. B289 (1987) 319. 27. N. Seiberg, Nucl.Phys. B303 (1988) 286. 28. M. Dine, I. Ichinose and N. Seiberg, Nucl. Phys. B293 (1987) 253; J. Atick, L. Dixon and A. Sen, Nucl. Phys. B292 (1987) 109. 29. J. Distler and B. Greene, Cornell preprint (1988). 30. M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B289 (1987) 589. 31. E. Witten, Phys. Lett. B155 (1985) 151. 32. K. Wilson and J. Kogut, Phys. Reports 12C (1974) 75; G. 't Hooft, Lecture given at Cargese Summer lnst. (1979).

358

33. E. Witten, Nucl. Phys. B258 (1985) 75. 34. J. Distler and B. Greene, Nucl. Phys. B304 (1988) 1. 35. M. Cvetic, Phys. Rev. Lett. 59 (1987) 2829; A. Font, L.E. Ibanez, H.P. Nilles and F. Quevedo, Phys. Lett. B210 (1988) 101. 36. C. Vafa, Harvard preprint (1989). 37. B.R. Green, K.H. Kirkline, P.J. Miron and G.G. Ross, Nucl. Phys. B278

38. 39. 40. 41. 42.

(1986) 667; Nucl. Phys. B292 (1987) 602; D. Gepner, Princeton University preprint (1987). G.G. Athanasiu, J. Atick, M. Dine and W. Fischler, Phys. Lett. 55B (1988) 55. M. Dine and P. Huet, to appear S. DeAlwis, University of Texas preprint (1988). J. Distler and B. Greene, Nucl. Phys. B304 (1988) 1. I. Antoniades, C. Bachas, J. Ellis and D. Nanopoulos, Phys. Lett. 211 (1988) 393; S.P. de Alwis, J. Polchinski and R. Schimmrigk, University of Texas

preprint (1988). 43. T. Banks, M. Dine, H. Dykstra, W. Fischler, Phys. Lett. 212B (1988) 45.

359

(8)

(b)

Fig. 1 Discrete symmetries in ordinary field theory.

a. Two vacua related by a discrete symmetry in an ordinary scalar field theory.

b. In the case of a global symmetry, a symmetry violating perturbation can lift the degeneracy between the vacua.

360

x x x______________

~

___________

Fig.2 Torus for Z orbifold is product of three copies of this two dimensional torus. Rotation by 120 degrees is a symmetry of the lattice. The x's denote the three fixed points under this symmetry.

361

Fig.3 Partition function for the heterotic string in ten fiat dimensions is a product of sums of left and right moving spin structures. P and A denote periodic and antiperiodic boundary conditions. The first factor two factors are for the two groups of sixteen left moving fermionsj the third factor is for the right moving fermions.

362

hlAD +P:J+A~+P D ] A

X

A

P

P

h[AD-p D-AD-AD] A

A

P

P

(8)

h2lAD +P:J+A~+P D ] A

x

A

P

P

h2[AD -p D-AD-ADJl A

A

P

P

(b)

Fig.4 Adding these terms implements the projection on invariant states in the twisted sector.

363

l

AD -P:J+ hA

x

J

hA

[ AD+pD +. . .. hA

hA

J

Fig.5 Additional terms required by modular invariance give rise to twisted sectors.

364

hAD A Fig. 6 A term in the partition function and its image under

T -+

-~

365

V(V)

y

Fig.7 Possible behavior of the potential for Y exhibiting a point with broken supersymmetry and vanishing cosmological constant.

366

Rolling Among Calabi-Yau Vacua

Philip Candelast

Paul S. Green

Tristan Hiibsch+

Theory Group Department of Physics University of Texas Austin, TX 78712, USA CandelasCDUTAPHY

Department of Mathematics University of Maryland College Park, MD 20742, USA PSGCDjulia .umd.edu

Theory Group Department of Physics University of Texas Austin, TX 78712, USA HubschCDUTAPHY

ABSTRACT For a very large number of Calabi-¥au manifolds of many different numerical invariants and hence distinct homotopy types. the relevant moduli spaces can be assembled into a connected web. Here we study the geometry of these moduli spaces. especially near the interfacing regions which correspond to conifolds. certain rather mild singularizations of the manifolds in question. In the natu~al metric. which we show coincides also with the point field limit of the Zamolodchikov metric. all the distances in this web are finite.

tSupported in part by the Robert A. Welch Foundation and NSF Grants PHY-8806377, PHY-8605978 and a fellowship to the Jane and Roland Blumberg Professorship in Physics. ISupported by the Robert A. Welch Foundation and the NSF Grant PHY 8605978. On leave of absence from the Institute "Ruder Boskovic", Bijenicka 54, 41000 Zagreb, Croatia, Yugoslavia.

367

CONTENTS O. PREAMBLE

1. SPLITTING: NEW MANIFOLDS FOR OLD 1.1 A Simple Example in Detail 1.2 Topology in the Neighborhood of the Nodes 1.3 Removing Nodes Deformations of a Node Small Resolutions of a Node 1.4 Cycles Near the Nodes 1.5 Flips and Flops 1.6 Another (Flippable) Example 1.7 A View into Moduli Spaces Generaiities Complex Structure Moduli Kahler Class Moduli 1.8 A Third Example 2. MANY CALABI-YAU MANIFOLDS ARE CONNECTED 2.1 Configurations, ClCYs and Determinantal Splitting 2.2 Double Solids, Generalizations and Discriminantal Splitting 2.3 Contractions Among Some Families of Fibred Products 3. GEOMETRY OF THE MODULI SPACES 3.1 Effective Kinetic Terms and Yukawa Couplings 3.2 Calabi-Yau q-models and the Zamolodchikov Metric 3.3 The Natural Line Element 3.4 The (2, I)-forms 3.5 The (1, I)-forms 3.6 Coordinate-Free Description of the Metric 3.7 Finiteness of Distances Between Distinct Manifolds Finiteness in the Spaces of Complex Structure Finiteness in the Kahler Cone Finiteness in the Complex+Kahler Structure Moduli Spaces 4. AN ORBIFOLD EXAMPLE 5. FURTHER REMARKS ON Calabi-Yau q-MODELS A. Finiteness in the Space of Complex Structures by Direct Computation B. Yukawa Couplings and The Distance Between Manifolds

368

O.

PREAMBLE

This is an attempt to present in accessible form a number of results pertaining to the parameter space of Calabi-Yau manifolds which we believe may bear heavily on the understanding of vacua appropriate to superstring theory. Of course, conformal field theory provides a more general context for compactification than the original point-field limit analysis on Calabi-Yau manifolds [1]. However, this point-field limit analysis has so far provided us with a number of quite robust results [2,3,4] and some very interesting relations to conformal field theory have been recently argued to exist [5,6]. Moreover, the techniques adapted to the study of complex manifolds have been well developed and one has recourse to geometrical intuition which is not yet the case in the more general context. By uncovering some of the intriguing structure of the parameter space of Calabi.¥au manifolds, we hope to provide motivation and some groundwork towards a related study in full-fledged superstring theory. A striking result is that the parameter spaces of distinct Calabi.¥au manifolds touch along certain regions in their boundaries. Such 'interface' regions correspond to certain singular limits of the respective Calabi-Yau manifolds. By including these limit points, the parameter spaces of a large number (perhaps all) of simply connected Calabi.¥au manifolds join together to form a connected 'web' [7]. Calabi-Yau manifolds may be constructed in a variety of ways. We review a number of these ways here though we shall consider in greatest detail the largest known class, the (ICY manifolds, which are defined by transverse polynomial constraints in products of projective spaces [8,9,10]. The moduli spaces of such manifolds are linked utilizing the process of splitting [10,7]. Having connected the parameter spaces of many topologically distinct Calabi.¥au manifolds into a web, it is natural to inquire how distant two topologically distinct m'anifolds are. To this end we study the metric properties of these parameter spaces. There are three a priori different "natural" metrics one could assign and which we describe in § 3 : (1) the metric occurring in the kinetic terms in an effective low-energy action, (2) the Zamolodchikov metric and (3) the Weil-Petersson metric. It is gratifying to find that these metrics actually coincide in the (carefully taken) point field limitd1 . After the complexification of the space of Kahler forms, this metric is Kiihlerian and of a block-diagonal form, with a block pertaining to deformations of the complex structure and another block pertaining to deformations of the Kahler class. We shall describe II Evidence in support of the identification of (1) and (2) was also found well beyond point field limit, considering ~n Coxeter orbifolds [11].

369 in greatest detail the Weil-Petersson metric by employing the techniques of complex geometry. Evaluation of higher corrections towards the complete Zamolodchikov metric is clearly desirable and an intriguing project in its own right but we leave this for future work. We then use this metric to show that any two points in this web may be joined by curves of finite length even though the end-points of the curve may correspond to Calabi¥au manifolds of different topological type [12]. The analysis presented here follows in part our earlier work as reported in Ref. [13]. The present article is organized as follows: In § 1 we study, by means of a few simple examples the process of passing continuously from the parameter space of one Calabi-Yau manifold to another. We also study the local topology of the singular varieties corresponding to the common points of the parameter spaces, their resolutions and the geometry of the relevant moduli spaces. In § 2 we show by means of further examples and general arguments that the parameter spaces of a great many Calabi-Yau manifolds are connected together in this way. With these results in mind, we return to study metric structures for moduli spaces for Calabi-Yau manifolds with a distinguished Kahler class. We determine the relevant metric on this moduli space as an appropriate limit of Zamolodchikov's metric [14] and address the question of finiteness of the path swept out in this moduli space during splitting (§ 3). For reasons of comparison, we discuss qualitative similarities and differences between blowing up the singularities of an orbifold and small resolutions (§ 4). Finally, we propose a framework for au-model generalization of our results and discuss the connection to some recent work on similar ideas (§ 5). 1.

SPLITTING: NEW MANIFOLDS FOR OLD

Here we introduce several examples to illustrate how it is possible to move continuously from the moduli space of one Calabi-Yau manifold to the moduli space of another. Here and elsewhere, except as explicitly noted, our moduli spaces are parametrised by the moduli of the complex structure and those of the Kahler class. 1.1

A Simple Example in Detail

We begin with perhaps the simplest example that concerns the non-singular complete intersection manifold (in the notation of Ref. [8,9], see also § 2.1)

Nt

E

[4114 1

1 ] bll =2

1 1

(1.1)

XE=-168

for which the defining equations, corresponding to the columns of the matrix (1.1), may

370 be written as pI

~ X(X)YI

+ V(X)Y2 =

0,

(1.2)

P = V(X)YI + Y(X)Y2 = 0, with X, V and Y, V any sufficiently general quartic and linear polynomials respectively since a theorem [9] guarantees that almost all polynomials are transverse, i.e, that dpI 1\ dp2 does not vanish on Nt. The fact that the row sums of the matrix add up to one more then the dimension of the respective projective spaces ensures that Nt is Calabi.:vau. A relatively simple choice of transverse polynomials is 2

def

(x~ + x1 - X:)YI

+ (x1 + x~ + X:)Y2 = 0,

(1.3)

Regard now the x's as given and note that we have a system of two equations in the two y's. Since the y's, being the homogeneous coordinates of a ]pI cannot both vanish it must be the case that the determinant of the coefficients vanishes.

C(X) ~r XY - VV

=0

(1.4)

This defines a quintic hypersurface in ]p4. We may regard this hypersurface as either the result of projecting Nt along the ]pI in (1.2) or as the limit of Nt as the radius of the ]pI shrinks to zero. The quintic hypersurface MU defined in Eq. (1.4) is singular and has in fact a number of isolated nodes, i.e., double points, where both C(x) and dC(x) vanish but the matrix of second derivatives is non-singular. This occurs when all four of X(x),Y(x),V(x), Vex) vanish simultaneously. For our particular choice of polynomials this happens when

None of X3, X4, Xs can vanish since then all the x's would be zero. We take therefore Xs = 1 and hence find 16 nodes. Generic X, Y, V, V will of course have these same properties. Note that these singularities occur at points. This is a special feature of three dimensional manifolds. We now perturb Eq. (1.4), to obtain

C(X)

= £(x)

,

(1.5)

with £(x) a generic quintic perturbation, e.g., £(x) = t(L:r=IX;s) which for t i- 0, however small, defines a non-singular quintic M bt C ]p4. The conifold u2 MU corresponds I2Nomenclature : Spaces which fail to be manifolds at only a finite number of nodes are called nodal varieties (see, for instance [15]). In this paper we shall adopt the term "conifold" because it is shorter than "nodal variety" and more euphonious than "nodifold" .

371

to. a common point in the boundary of the relevant moduli space of non-singular quintics in IP4 and of manifolds such as M. The essential point is that the conifold M~ can be either resolved to M or deformed to M· yielding smooth Calabi-Yau manifolds of two distinct topological types

The example we ~ave chosen may appear, on first acquaintance, to be somewhat involved but it typifies a very general process [7] which connects the moduli spaces of a great many Calabi-Yau manifolds and which may in fact link the moduli spaces of all simply connected Calabi-Yau manifolds. In each case there are conifolds that can be interpreted as limit points of both moduli spaces and have a number of isolated nodes described locally by an equation similar to Eq. (1.4) XY-UV=o.

(1.6)

We may regard X, Y, U, V as coordinates in (4 in which case they describe a cone with apex at the origin. The natural question whether the conifolds are at a finite or infinite distance from the smooth manifolds in the moduli space is the subject of § 3.7. We first turn to a local study of these nodes. 1.2

Topology in the Neighborhood ofthe Nodes

M~ is embedded in IP4 by equations (1.4). Around each node p~ E M~, we consider an open neighborhood of p~ in IP4 that is locally like (4 where we can use the four polynomials X, Y, U, V as local coordinates. The cone is described by Eq. (1.6) as a non-degenerate quadric in (4. By a linear change of variables this equation is brought to the form L:!=1((A)2 = O. We may think of as a four-vector and write this equation as (2 = O. The space described by this equation is called a cone because (i) it is smooth apart from the point ( = 0 and (ii) if ( satisfies the equation then so does >.( for any>. so the space is made up of complex lines through the origin.

e

To determine the base of the cone it is helpful to separate ( into its real and imaginary parts then Eq. (1.6) becomes the pair

e - 17

2

= 0,

~ . '7

=0 .

372 The base is the intersection of the cone with a sphere centered at the apex. We take the sphere to have radius r so that

Thus the base is described by the equations

(1. 7) The space of fs is an S3. For each point ofthis S3, say ~ = (r/V2,O,O,O), ~. '" = 0 restricts", to be the 3-vector '" = (0, ",2, ",3, ",4), so that ",2 = r2/2 defines an 8 2• Thus the base ofthe cone a is fiber bundle with base 8 3 and fiber 8 2 • All S2 bundles over 8 3 are trivial so the base must in fact be the product 8 2 x S3. Since the base is connected the cone is a single rather than a double cone. We shall denote this (real) cone by 0, where the quadratic expression on the left is the ratio between the polynomial defining the conifold and a perturbing polynomial which does not vanish at any of the nodes. 53 is then given by x,y real and v = U, while the cap B3 is given by x > 0, y imaginary and v = -u. In these coordinates we have for the holomorphic three-form

n = fdx /\ dy /\ du

,

(1.10)

u

where f depends on the perturbing polynomial but neither vanishes nor becomes infinite at any of the nodes. For future convenience we note that

1n =1 53

z2+y2~
[(O')if> J(O») where the set {if> I(O')} parametrizes the deformations of the background action density. From the point field limit we know that the deformations induced by inclusion of Al and of A2 admit a finite basis and we introduce external world sheet currents WA(O'), WA(O'), T"(O') and Tiit(O') to probe for these deformations: QI'V QI'V

= =

A W (O')QAl'v(Z,Z) ,

WA(O') QAl'v(Z, Z) ,

=

T"(O')Q"l'v(Z,Z) ,

Tiit(O') Qiitiiv(Z, Z) .

It is customary to collectively denote such currents by J/(0') and the corresponding operators if> I( z, Z). Since we are evaluating the metric at the point in the moduli space specified by A o, the expectation values are computed from

(3.7)

where we have indicated a separation Z(O')

= Z + Z(O') ,

into the center of mass coordinate and a remainder. If we now work to one loop order, i.e., quadratic order in Z(O') and the fermionic fields then we may replace GI'V(Z, Z) -> gl'ii(Z, z) in Ao since Gl'ii multiplies terms that are quadratic in the Z(O'). It is now straightforward that

and all other correlation functions between derivatives of Z's and Z's vanish. We shall similarly approximate the functionals Q with their limit that is independent of the nonconstant modes ZI'(O') and ZV(O'). Given the above currents J/(0'), the standard definition of the irreducible twopoint functions in terms of a second functional derivative of the logarithm of the partition

393 function and the expectation values (3.8), it is immediate that the only non-vanishing entries of the Zamolodchikov metric in this limit are GAB

= ~ iMgtd6x [QA"ii(Z,Z)QBii"(Z,Z))

Gail

= ~ iMgtd6x [Qa"v(z,Z)QP""(z,z)).

,

(3.9) (3.10)

We shall obtain this very same metric also from purely point field limit arguments. Higher order corrections to these expressions would, of course, be of great interest. 3.3

The Natural Line Element

As remarked earlier we may take the moduli space to be the space of Ricci-flat metrics of a deformation class of Calabi-Yau manifolds. Let gmn be a Ricci-flat metric for M and 6g mn the variation of the metric under a variation of the complex structure of M. Then with the coordinate condition "'ir6g mn = 0 we have that 6g mn satisfies the Lichnerowicz equation

(3.11) Owing to the special properties of Kahler manifolds the zero modes of this equation of mixed type, 6g"fi, and those of pure type, 6g"v,6g Pfi ' separately satisfy the equation. With a variation of the metric of mixed type we can associate the (l,l)-form

which is harmonic if and only if the variation satisfies (3.11). With a variation of the metric of pure type we may associate the (2, I)-form t!l".xii6gPii dx" A dx.x A dxl' this also is harmonic if and only if the variation of the metric satisfies (3.11). In other words the zero modes of the Lichnerowicz equation are in one--one correspondence with the elements of H(l,l)(M) and H(2,l)(M). The modes of mixed type are therefore variations of the Kahler class. Those of pure type correspond to variations of the complex structure. To see this note that gmn + 6g mn is a Kahler metric on a manifold close to the original one. There must therefore exist a coordinate system in which the pure parts of the metric vanish. Under a change of coordinates xm -+ xm + fm(x) the metric variation transforms according to the familiar rule 6g mn

---+

6gmn

or

or

+ oxmgrn + ox ngmr

.

If f"(x) is holomorphic then 6g"v is invariant. Thus the pure part of the variation can be removed by a transformation of coordinates ·but it cannot be removed by a

394 holomorphic coordinate transformation. In other words the pure part of the metric variation correspond precisely to changes of complex structure. Along with gpr; and its variations. in the o--model corresponding to compactification of superstrings there occurs a two-form B on the internal space which is related to the metric by supersymmetry. The action functional (3.4-3.6) suggests a complexification of the Kahler cone through inclusion of B. One is however always free to set B = 0 in order to recover the moduli space of Ricci-flat Calabi-Yau manifolds. In virtue of the B field equation its variations are required to be harmonic. We take b20 = O. which is certainly the case if the Euler number of M is nonzero. so that oB is of type (1,1). We shall include the B field into the line element on the moduli space but will neglect: (1) derivative terms (these are systematically introduced by o--model corrections). (2) a possible term (g mn ogmn)2. since gmnoBmn vanishes. this term would spoil the symmetry between gmn and Bmn. and (3) terms involving the Es x Es Yang-Mills sector (for the inclusion of this see Ref. [27]). Up to these terms. the most general metric is then

ds 2

=

t iMlmgln(og"IOgmn+OB"'OBmn)gt~x,

= ~ iMg"Pg).r; (og,,).ogpr; + (og"r;og).p + oB"r;oB).p» gt~x.

(3.12)

Upon the identifications

ogpv +-+ T"(o-)Q"pv(Z, Z) ,

(3.13)

ogpr; +-+ Tii(o-)Q/jpr;(Z, Z) ,

(3.14)

6Bpr; - 6gpr; +-+ WA(o-)gAp,,(Z,Z) ,

(3.15)

A oBI''' + ogp;; +-+ W (o-)QAp;;(Z,Z) ,

(3.16)

the metric (3.12) is the same as the Zamolodchikov metric when evaluated in the li~it described in § 3.2 in Eqs. (3.9) and (3.10). It is. somewhat surprisingly. block-diagonal. with a separate block corresponding to variations of the complex structure and of the Kahler class. respectively. We note here that this structure agrees with the result found well beyond the point-field limit. for the case of blown-up toroidal ZN-orbifolds [27] (see also Ref. [31])Ull. As the following IllThe apparent discrepancy with the results of Ref. [31] was caused by a mislabeling of moduli on part of these authors. Upon a complicated transformation to appropriate parameters, which they include, the direct product form and agreement with our results is indeed achieved.

395

study will show, this metric on the moduli space respects the continuity of the volume, the complex structure and the Kahler class of the Calabi¥au manifold all of which are essential from both the physical and the mathematical point of view. We wish now to study the metric on the moduli space of Calabi-Yau manifolds in its own right and in its own terms. It transpires that the metric (3.12) enjoys a number of surprising properties among these being the high degree of symmetry between the roles played by the (2,l)-forms and the (1 ,I)-forms. Since the metric separates into a direct product at least locally, we shall study these two parts in turn. The (2, I)-forms

3.4

We turn now to the deformations corresponding to (2, I)-forms largely following

G. Tian [32]. We set

where the t"', a = 1, ... , b21 are the parameters for the complex structure, i.e., local coordinates in an open neighborhood of B. Each CP'" is a harmonic (2, I)-form. We have also the inverse relation

ogjjv where

1IS1112 ~ ttS1,wpIT'WP and

1 ITPO

J negative V0,

va

> 0,

a

= 1, ... ,27 ,

(4.2)

These restrictions have·a clear interpretation in terms of degenerations of the torus and the orbifold limits of the blown up points. Note that the orbifold limit va -+ 0 has a metric whose (a, b)-components all vanish. The metric is therefore singular since it has no inverse. Nevertheless the distance of these orbifold points from the interior points of the moduli space is finite. On the other hand the distance from the interior to points for which detx - !L::~~1(Va)3 0 is infinite.

=

The fact that the (a, b)-components of the metric vanish implies, in view of Eq. (3.1), that the so-called "twisted" modes do not propagate. This is consistent with the fact that our analysis relies on the classical limit in which the "twisted" modes are indeed confined at the singular points of the orbifold. (Recall that the world-sheet action measures the surface area of the world-sheet which is minimized when the spatial extension of the string vanishes.) q-model corrections then introduce these kinetic terms, corresponding to the propagation of the fluctuations. As discussed previously the boundary of the Kahler cone is actually determined by the condition that the Kahler form be positive and that imposes tighter bounds than (4.2). The ei, are not everywhere positive for all values of the parameters since in the orbifold limit it has components ei,kl

= iOikO,f·

Demanding that the combination xi'eij be positive requires the matrix Xi, to be a positive matrix. This is, of course, a tighter bound than simply demanding that det x > 0 since positivity of x requires all three eigenvalues to be positive while det x > 0 also permits two eigenvalues to be negative and one positive. The e a also have negative eigenvalues. We believe that this leads to extra conditions that require that the va be not too large relative to the xi, and that this requires a boundary V tt to the parameter space which is at a finite distance and which corresponds to parameter values for which Z can be contracted. We remark that it was found in the orbifold limit [31,27] that the complexified Kahler cone W, when restricted to the 'untwisted' sector, is S(J;~f!~)(3)) and has (real) 18 dimensions. The corresponding 'untwisted' sector of the Kahler cone is then only a (real) 9-dimensional subcone of W.

408

Note also that the orbifold singularities can only be blown up. Thus orbifolds are singularizations of only one topological type of Calabi-Yau manifolds and correspond to genuine boundary points in their moduli space. This is in contradistinction to nodes which can be smoothed in two radically different ways (by deformations and by small resolutions) so that nodal varieties correspond to points in the interface regions between moduli spaces of topologically distinct Calabi¥au manifolds. This is to be compared with the fact that the low-energy effective models suffer no change in the gauge symmetries and drastic changes in the number of light matter fields during a passage through a nodal phase, opposite to what happens in the orbifold limit. 5.

FURTHER REMARKS ON. Calabi-Yau O'-MODELS

It is standard that the propagation of strings in a Calabi-Yau manifold may be described by considering the O'-model action (3.4)

where ZI'(O') are the world-sheet complex scalar fields which map the world-sheet into the Calabi¥au manifold. In this point of view, all information on the Calabi-Yau manifold is contained in the unknown Gmn[Z(O'), Z(O')]. Our present analysis pertains to the "point-field limit" of this action, in which Gmn(Z, Z) = gmn(z, z) is taken to depend only on constant modes of ZI'(O') and gmn(Z,z) is the metric of the Calabi¥au manifold selected. As we have shown, large variations of gmn(z, z) that interpolate between topologically distinct Calabi¥au manifolds happen along paths, in the relevant moduli space, of finite length in a metric that coincides with the one motivated from low-energy effective actions [25] and superconformal moduli spaces [14]. Associated to ogmn' there exist massless particles in the low energy effective theory implying that the vacua described by gmn and gmn +ogmn respectively are degenerate. Shifting gmn along the path in the moduli space described above thus seems to correspond to shifting through a number of degenerate vacua labelled by gmn' During the 'horizontal' and later the 'vertical' part of such paths only continuous numerical characteristics, such as the Yukawa couplings, vary. At the points of concatenation, however, the discrete numerical characteristics such as bl l counting the 27*'s and b21 counting the 27's of the gauge Es symmetry jump,- which we understand is a signal of a phase transition. Our result that all the distances in such paths are finite in a physically natural metric we cannot help but interpret as a signal that,- if these transitions are triggered to occur by an as yet unknown dynamical mechanism,- they are completed in finite time.

409 To study this phenomenon beyond the point field limit, we may need a more specific form of the above action. By the very construction of (ICY manifolds, the obvious suggestion for the action is given in superspace notation, in the form A Akin. + Aeon. (see also Ref. [39]), with

=

(5.1 )

Aeon.

1 K = 47rQ:,E

Jd IJ"d c;Aapa(Z(r);ta)+ 2

2

h.c . .

(5.2)

Z(';) are complex (2, 2)-superfields on the l+l-dimensional world sheet, the scalar zeromodes of which are identified as the homogeneous coordinates of the 1P~r factor in the embedding space. Akin. is chosen so that, upon the fermionic integration, we have a standard action describing the tensor product of {pn IJ"-models. The second part, Aeon., then imposes the (superfield) constraints by means of standard Lagrange multiplier technique. This choice of the action has the following favorable properties: 1. The summands in Akin. give (upon fermionic integration) the IJ"-dependent generalizations of the Fubini-Study metrics on the lP n spaces, which also correspond to generators of the (l,l)-cohomology of the Calabi-Yau manifold. Moreover, whenever bn of the Calabi¥au manifold equals the number of lP n factors in the embedding space, this yields a complete basis. The coefficients 9r in Eq. (5.1) are then coordinates in the Kahler cone. 2. Take pa(Z(r); ta) to be a specific choice of polynomials described by the specific choice of coefficients t a in the most general such polynomials. Changes in the choice of t a describes polynomial deformations, which (when taken modulo certain equivalences) correspond to a parametrization the (2,l)-cohomology of the Calabi¥au manifold, yielding often a complete basis (see Ref. [18]). The coefficients t a are therefore coordinates in the space of complex structure. Having explicit terms in a IJ"-model action which appropriately generalize the elements of corresponding cohomology groups, corrections to our present results can be obtained by standard methods of field theory. Computing analogous corrections to other point field limit results, such as Yukawa couplings, also seem now to be straightforward. We note that actions very similar to Eq. (5.1) and (5.2) have been used in the recent literature to exhibit a connection between conformal field theory and Calabi¥au manifolds [6]. To compare Eq. (5.1) and (5.2) with the actions used in Ref. [6]. note that each summand of the kinetic term (5.1) has a separate local scale invariance that allows us to fix e.g. Z(r) I, for each lP~r separately, which corresponds to chosing a

=

410

coordinate patch in every 1P~r. The kinetic term can now be expanded in the vicinity of Zr:i = 0, for /-Lr =f 0 and the leading term is the standard (flat) (Zr:i) t Zr:i kinetic term. Also, expanding the Lagrange multiplier field about its vacuum expectation value, Aa(O') = (Aa) + Aa(O'), the leading term matches the potentials considered in Ref. [6], with (A,.) being the coupling constants. Given these identifications and the action in Eq. (5.1), (5.2), we hope to have provided an avenue connecting our geometrical descriptions of the moduli space for a given Calabi¥au manifold and the connected web of these to field-theoretical models.

ACKNOWLEDGEMENTS

Fruitful conversations regarding the Zamolodchikov metric with Jan Louis and Andrew Strominger are gratefully acknowledged. P. Candelas wishes to thank the physics Department of the University of California at Santa Barbara for hospitality while part of this work was carried out.

411

APPENDICES For the benefit of the reader who prefers explicit computations, we present in the appendix A a direct computation of the asymptotic behavior of i fM!lt 1\ ?it as t -+ 0, largely following Ref. [13]. The relation of this behavior to certain relevant Yukawa couplings is worked out for two examples in the appendix B. In § 3.7, we have derived this behavior by somewhat abstract methods and the exact result is given by Eq. (3.35). A.

FINITENESS IN THE SPACE OF COMPLEX STRUCTURES BY DIRECT COMPUTATION

Using that each node is described locally by an equation XY - UV = 0 in (4, we find that near each node the holomorphic 3-form !It is of the form wI! with f a holomorphic function that does not vanish at the node and deC

w =

dY 1\ dU 1\ dV Y

w has an apparent singularity when Y fact

dY 1\ dU 1\ dV

Y

=

dU 1\ dV 1\ dX

X

= 0, =

but dY 1\ dU 1\ dV vanishes there also. In dV 1\ dX I\dY

V

=

dX 1\ dY 1\ dU U

which shows that w is singular only at the node, when all four of X, Y, U and V vanish. We wish to show by direct computation that the quantity (A.1)

and its first two derivatives are sufficiently well behaved in the limit t -+ 0 for the distance betweenM"t~O and MU to converge. To do so, let us divide the range of integration in (A.1) into a number of neighborhoods in each of which a node will develop when t -+ 0 and the 'rest' of M"t. This last contribution is manifestly well behaved for our purposes and we only need to focus on the integration ranging over the neighborhoods in which the nodes develop. The function f in !It = w / f does not vanish at the nodes, is bounded and holomorphic. Therefore, may be expanded in a power series about each node in an appropriate neighborhood U;, the lowest order being a constant, K i . We shall then approximate f !It 1\ ?it (A.2)

t

}UinM't

with the lowest order term in this series, K;

f

}u.nM't

w I\'W.

(A.3)

412

The contributions from higher terms are even less singular in t. We may choose Ui so that Eq. (A.3) is a constant multiple of

r

r

2

r

J(R R') d2U d2V d y , - JIUI~RI JIVI~lul Jl,Ujrl~IYI~R 1V12

.

This follows from the following observations: 1. We may rewrite the defining equation (1.2) of M~t as XY - UV + t = 0, since the generic perturbation c(x) may, near a node, be identified with the local coordinate t on [3' U [3i.

2. Consider U = {lXI, lVI, lUI, IVI ~ R} and choose Y, U and V to be along U n M~t. The upper limit on the modulus of X = U~-t implies the lower limit, IUV - tl / R ~ lVI, on the modulus of Y. 3. The symmetry of the integral with respect to the interchange of U and V allows us to restrict IVI ~ lUI and double the result.

4. IUV - tl / R ~ IVI ~ R implies IUV - tl ~ R2, which is not satisfied automatically by lUI ~ Rand IVI ~ lUI. It is necessary to further restrict lUI ~ R', where we choose R' sufficiently smaller then R.

Straightforward integration over Y gives J(R,R')

= 21!"

r

JIUI~RI

d2U

r

JIVI~1U1

d2Vlo ( R2 g IUV -

tl

)

To proceed we need to evaluate the following integral U4 :

F(a, A)

= 2JI%I~A r d2 z10g (_Iz _1_1) - a

Clearly, F( a, A) is rotationally symmetric and continuous as a function of a. Moreover, by differentiating twice under the integral sign, one can see that F is harmonic in a and the integrand is harmonic in z provided lal > A. The mean value property for integrals of harmonic functions gives us :

lal >

(A.4)

A.

The other observation tells us that for lal < A, F differs from ! lal 2 by a function which is rotationally symmetric and harmonic in the disk. Combining this with the overall continuity of F, we obtain:

lal < 11 4 The

analysis of this integral is due to Scott Wolpert.

A.

(A.5)

413

In order to apply this to J(R', R), we substitute

UV

z=-

R2 '

whereby

J d1U12U F(.l!LR2' 1U12) R2 . 2

J(R' R) = 1["R4

,

IUI:SR'

In view of the two cases, (A.4) and (A.5), we break up this integral in two parts. The first one is : 1[" R

J

4

d2 U

1U12' 211"

1U1 4

R2

R4 log (itT)

2

R2

R4 log

(1U12)

3

= It I log (itT) .

(A.6)

1["

IUI:S,fii

The second part is : 4

J

d2U

IUI2

(It12

1U1

4

R4 - R4

+8

1U1

4

R2)

,

,fii:Slul:SR'

which we evaluate term by term. Together with (A.6), this sums up to

and is indeed of the form (3.35). To estimate the derivatives of i fM n /\ IT in the same fashion, we would need to include second order corrections to Eq. (A.3) and this manifestly would not affect the asymptotic behavior significantly. However, to complete this direct computation, we would now have to prove that the entire series expansion of 1/ J, upon integration term by term, again converges and yields an expression of the form (3.35). Fortunately, the derivation in § 3.7 guarantees that this is indeed the case. A.

YUKAWA COUPLINGS AND THE DISTANCE BETWEEN MANIFOLDS

n

1. We return to our first example relating to the splitting [: II : of [4115] using the particular choice of defining equations (1.3). Recall that the polynomial resulting from the elimination of the coordinates of the ]PI

C(x)

= Xl(X: + x~ + x:) -

X2(X~

+ x! - x:)

has 16 double points. We take a curve in the moduli space by choosing a family of quintics q(x, t) = C(x) + tp(x)

414

with p = t L:i=o xi as before. then there is an annular region 0 < It I < 8 for which q(x, t) corresponds to a smooth manifold and we wish to examine the form of the Yukawa coupling and of the function {(z(t)) as t -+ o. The procedure is the following: there is an algorithm for calculating the Yukawa coupling associated with variations of t. By Eq. (3.24) this yields information about the third derivative of { along the curve and hence about {(t) itself. Given {(t). the metric along the curve follows from Eqs. (3.19) and (3.22). The algorithm for calculating the Yukawa coupling is explained in Ref. [34]. It derives from a correspondence between differential forms and polynomials and is valid whenever this correspondence is [18). If so. and certainly for our examples. an infinitesimal deformation of the complex structure corresponds to a (2,1 )-form and also to a deformation of the defining polynomial. A deformation of the defining polynomial may correspond merely to a change of coordinate in the ambient manifold

in which case the defining polynomial q changes according to the rule

q(x)

>--->

q(x) + Cj

8q(x)

k

Xk-

. 8 Xj

Thus quintics in the ideal generated by the partial derivatives of q

~=

(88xj(x)) q

do not deform the complex structure and correspond. in fact. to exact forms. The unique (0,3) cohomology class represented by

n (the inverse factor of -i fM n 1\ Ii) is included because the combined quantity depends holomorphically on the parameters t a ) and is represented by the fifteenth order polynomial

Considered modulo the ideaIV(x) is the only fifteenth order polynomial not in the ideal. For the example to hand the third derivative of the complex stucture with respect to t is represented by p3(X) and with the help of Eq. (3.24) we have that K

(t)

3

·a·b·c

=Z

Z Z

8{ 8z a 8z b8z c

'

415

where za ~ (fJza/fJt) and the relation ~ denotes equality modulo~. Both the determinant

2q

""'( ) _ d et (fJfJ (x,t)) vX,t fJ Xj

and the basis for the ideal

Xk

~

(5 + t)xt

+ x~ + x: ~ 0

(5 - t)x~ + x! - x: ~ 0 4XIX~ + tx~ ~ 0 4X2X~

4(XI

-

tx: ~ 0

+ X2)X~ + tx: ~ 0

are quite complicated. However on discarding terms Ul5 in

f(t)

~ 1+

G)

4

~

and setting

(5 + t)

we find the simple relation

It may be shown that the polynomial in x on the r.h.s. of this relation does not vanish as t -+ O. We find that V has a simple zero at t = 0 so we expect

K(t) '"

tC

for some constant C. We shall take the vanishing cycles to vanish linearly with t and suppose that zazbzc(fJ3effJzafJzbfJzc) is asymptotically proportional to (fJ3ef{)t3). By integration we have W) '" Ao + Alt + tt 2 10g t where a possible term of order t 2 is neglected in comparison with t 2 10g t and the scaling freedom inherent in is used to fix the coefficient of the t 2 log t term. It is now immediate from Eq. (3.22) that

e

i

1M n 1\ IT '" a(t) + It l log It I 2

Provided only we can show that a(O) #- 0 we find that we have yet a third demonstration of the fact that the length of the curve converges as t -+ O. So we shall 115This may be conveniently accomplished by means of the MACSYMA proceedure

fullratsubst.

416

now show that a(O) cannot vanish and we shall leave the explicit calculation of I\':(t) to the enterprising reader since this turns out to be surprisingly complicated U16. To show that a(O) -:f. 0 we observe that fM n 1\ is bounded away from zero as t ~ o. To see this surround the nodes of the variety described by the equation C = 0 in IP4 by disjoint balls Bk, k = 1, ... ,16, then in virtue of the fact that the integrand IS manifestly positive we have

n

i

r

nl\n>i

1M

r

nl\n >0

1{q=O}\{UBk }

and to see that the integrand has a well defined nonzero limit as t ~ 0 it is necessary only to observe that n may be explicitly represented in terms of the defining polynomial. In the coordinate patch {Xl -:f. OJ, for example, we have ("\ _ Xldx2 H-

1\

dX3

.!l.9...

1\

dX4

aX5

and this clearly has a well defined nonzero limit outside the B k . 2. Let

As a second example we briefly consider a worse singularisation of the manifold.

and consider the curve

q(X, t)

= r(x) + tp(x) .

11 6 /1': is a rational function regular at t = 0 and it is straightforward to show that /I': '" 3/4t 11 as t -- 00. There are many values of t, however, for which the manifold develops double points. The denominator of /I': is the polynomial which has these values of t as its roots and hence is of high degree. The diligent reader may verify that the manifold develops double points when t = 0, t = ±5,f(±t) = 0 and whenever t satisfies any of the following four equations for any choices of branches of the fourth roots

+ (f( -t))t + (- f(t)f(-t))t (1 - f(t))t + (f( -t))t + ((1 - f(t))f( -t))t (-f(t))t + (f(-t) -l)t + (f(t)(f(-t) -l))t f(t))t + (f( -t) - l)t + ((1 - f(t))(f( -t) - l))t (- f(t))t

(1 -

=0, = 0, = 0, = 0.

The first three of these equations lead to polynomials of degree 40 while the fourth leads to a polynomial of degree 24. Taken together with the factors (t 2 - 25)f(t)f(-t) we expect the denominator of /I': to have degree 156 and hence the numerator to have degree 145.

417

The hypersurface described by the vanishing of r is not smooth since r has zeroes of fourth order (for example when Xl = 1 and X2 = X3 = X4 = Xs = 0). For this curve we propose to calculate !t(t). In the course of the calculation we shall need to evaluate a determinant. So we first note that the 5 x 5 determinant 1 a2

1 1 1 s

= IT ak -

a3

1 1

~

ajakal

+ 2 ~ ajak -

j') and in

particular the so-called coherent states U(g) I>') form a (over- ) complete basis of 7t. The wave function tP(g) corresponding to the state

ItP) E 7t is now defined by (2.13)

The group G acts on these wave functions tP(g') simply as [U(g)tPl(g')

--+

tP(g-lg'), i.e. by

multiplication from the left. From (2.12) we deduce that when g is multiplied from the

right by an element h of the Cartan subgroup tP(g) transforms as (2.14) This shows that

tP is a section of a line bundle over the coset space G IT, which is therefore

the obvious candidate for the phase space r. Indeed, the fact that I>') is annihilated by the positive roots implies that tP depends on only half of the phase space variables. This can be made more precise by using the fact that the representation U can be extended to a holomorphic representation of the complexified group Gc. Further, instead of the wave functions (2.13), let us consider the ratio

tPlwl(g) = tP(g) I >.(g)

(2.15)

I>'). It is clear that tPlwl(g) is holomorphic I>') is the highest weight state, tPlwl(g) is invariant under

where >.(g) is the wave-function corresponding to on Gc. Furthermore, because

right multiplication of g with the Borel subgroup B, generated by the Cartan and the positive roots, i.e. tPho/(gb) = tP(g) for b E B. Therefore, the wave-functions tPhol(g) are

427 holomorphic functions on the coadjoint orbit space f=G/T=GC/ B

(2.16)

This fact is known as the Borel-Weil theorem. By considering the inner product on 'It one can obtain the Kahler potential on f, [18J, and from this one can deduce the Kahler form w. Representing the tangent vectors to f by elements

E

of the Lie algebra, one finds

W(Eb E2) where

).9

= ).9 ([Eb E2J)

(2.17)

= g-l ).g. This is the so-called Kostant-Kirillov form [19J.

Thus, through the use of coherent states, one can extract directly from a representation space 'It the corresponding phase space (f, w). The reverse construction of 'It by applying the rules of geometric quantization to (f, w) is referred to as the coadjoint orbit method.

3

Current blocks and Chern-Simons theory.

In this section we describe some aspects of the space of current blocks of the WZNW theory, described by the action [20J

(3.1) where g is a map from E into a compact Lie group G, and BB = E. Our goal here will be to discuss how, using the operator formalism and some elements of geometric quantization, the WZNW-blocks can be recognized as the states arising from quantizing the space of fiat gauge fields on the surface. As the latter problem is isomorphic to the canonical quantization of 2+1-dimensional Chern-Simons gauge theory, the following discussion may be considered to be the reverse of Witten's derivation of WZNW-theory from CS-theory. Our presentation closely follows that of [11,12J, see also [6,7,8]. This section will also serve as a guide line for how to generalize Witten's result to the case of the Virasoro algebra.

3.1

Description of the current blocks.

We will now give a definition of the space of current blocks using the operator formalism [21J. For the case of U(l)-current algebra this was first discussed in [3].

428 Associated with the invariance of the action (3.1) under g(z,z)

----t

h(z)g(z,z), the

WZNW-theory contains an infinite set of conserved charges

(3.2) which generate the Kac-Moody algebra (3.3) where k is the central extension. The complete Hilbert space of the WZNW-model is composed of integrable representations of the left and right Kac-Moody algebras. These representations are labelled by positive weights A E P+ satisfying the inequality where

t/l. A ~

k,

t/l denotes the longest root [23]. We have [24] ?-l

=

E9 [4>,,] ® [4),,]

(3.4)

"EPt Note that this is a finite set: the WZNW-models are the basic examples of rational conformal field theories. The presence of the infinite Kac-Moody symmetry gives strong restrictions, in the form of Ward identities, on its correlation functions [22]. These Ward identities are conveniently described via the operator formalism of CFT on Riemann surfaces. Let us consider a genus g surface :E and a collection of punctures {Pn , n = 1, ... , N} on it, and attach to each puncture Pn a copy ?-lIn) of the CFT Hilbert space (3.4). The operator formalism now formally associates to these data a unique state

(3.5) defined by the property that when we saturate 1:E,{Pn }} by N 'incoming' states (?-l(n))", one obtains the correlator on :E of the corresponding operators

(t/lnl

E

(3.6) The Kac-Moody Ward identity is now stated as follows. For all Lie-algebra valued functions €(z), holomorphic on:E except for possible singularities at the punctures Pn , we have that

LP !Pn 1 trE(z)J(z) I:E, {P

n }}

n

=0

(3.7)

429 This equation follows from a simple contour deformation argument. Together with some additional physical requirements, such as modular invariance, it uniquely characterizes the state I~, {Pn }). It follows from the decomposition (3.4) of the WZNW Hilbert space, that the corre-

lation functions in (3.6), and hence the state 1~,{Pn}), can be decomposed into chiral components, i.e. the current blocks [22]. In order to explain this, let us for simplicity restrict ourselves to the case of surfaces

~

without operator insertions. To set up the

operator formalism in this case, we choose some arbitrary point P on

~,

and a coordinate

region (z, z) around it such that P = (0,0). To these data one can associate the state which is related to the partition function by Z1: = (Ol~). It satisfies I~) E

I~),

[cpol ® [4'ol and

should therefore have a decomposition

(3.8) i,;

where liJiI:)i E blocks on

~

[CPol

and hij is some matrix. The chiral components liJi1:)j are the current

and span a linear subspace 'H.I: C

[CPol.

All current blocks liJiI:) E 'H.I: obey

the Kac-Moody Ward identity (3.9) for all Lie-algebra valued functions E(Z) that extend holomorphically to

~o

= ~ - D.

Here

D denotes the disk {(z, z)j Izl ~ I} and C = aD. This Ward identity (3.9) projects the infinite dimensional representation space [CPol onto the finite dimensional space 'H.1:. We will give a formula for dim'H.1: later.

3.2

Current blocks from geometric quantization.

Before discussing the quantization problem leading directly to the space 'H.I:, let us first briefly explain how the vacuum representation [cPo] is obtained from geometric quantization of the appropriate coadjoint orbit of the Kac-Moody group. Let us for the moment ignore the central extension k. Then, the group generated by the charges J(f') is the loopgroup

LG, defined as the set of maps g : 51

->

G with the natural multiplication. Let U(g)

denote the unitary representation of g E LG in [cPo] as well as its holomorphic extension to the complexification LGe. Then, from the fact that the vacuum state 10) is annihilated by all positive modes of the Kac-Moody currents J, we deduce that

U(g)IO) = 10)

g E LDGe

(3.10)

430 where LDGC is the subgroup of LGc ofloops extending holomorphically to the disk. So, as done for the finite dimensional groups in section 2.2, we may now associate to a state

IIJI) E [4>oJ a holomorphic function

(3.11) By (3.10), this function is defined on the coadjoint orbit space

fo = LGcl LDGC

(3.12)

This also turns out to be the correct phase space when we take into account the central extension k, except that, instead of functions, the states will be holomorphic sections of some complex line-bundle £/e over the orbit space fo. This line-bundle £/e can most easily be described using the WZNW-action (3.1). (Here we follow [11J.) The relation between £/e and S[gJ is that the curvature of the first is the Kahler form associated with the latter. The WZNW-action is a well-known example of a multi-valued action, of the type discussed in section 2.1. Due to the topological term in

(3.1), SI;[gJ is not a simple number if the surface!; has a boundary, but becomes a section of a line-bundle over the loopgroup(s) associated with the boundary component(s) of!;. Concretely, for each different way of extending the values of g at a boundary component of!; to a disk D we get a different value for SI;[gJ. The variational formula is 6SD[gJ =

~ f

471" 152

tr(6gg- 1 )8(8gg- 1 )

(3.13)

The wave functions of the WZNW-theory have to be defined such that they compensate for this multi-valuedness of the action. Therefore, instead of on the loopgroup LGc, the wave functions have to be defined on the space of maps from the disk D to G e , and transform in the same way as exp S D [g J under variations of g in the interior of the disk

6IJ1D[gJ = 6SD[gJ IJID[gJ

(3.14)

where we have applied the general formula (2.10). This equation defines the states in [4>oJ as sections of the line-bundle £Ie over the coadjoint orbit LGcl ~DGe. Let us now return to our main objects of interest, the current blocks IIJII;) on a general surface !;. In this case we have the additional information ofthe Kac-Moody Ward identity (3.9). As we will now see, this Ward identity is equivalent to the restriction of the infinite dimensional orbit space (3.12) to a finite dimensional compact subspace, which will be the classical phase space of the quantum system we are looking for. Again, first ignoring k,

431 we deduce from (3.9) that (3.15) where LEoGe denotes the subspace of loops in Ge, holomorphically extending to :Eo =

:E-D. Hence, we can identify the current block IIliE} with a holomorphic function (3.16) defined on the double coset (3.17) This space ME is thus the (candidate) phase space, whose quantization will lead to the current blocks. It is a useful mathematical fact [10] that ME can alternatively be represented as: (i) the space of fiat G-connections on the surface :E

ME where

'V

= {A;

F(A)

= O}/

(3.18)

'V

denotes the group of G gauge transformations.

(ii) the moduli space of holomorphic Gc-bundles on:E. That is, ME can be described as the space of multi-valued maps 9 from :E into Ge, with constant transition functions g(z, z) --> gcg(z, z) around the cycles, divided out by the group of single-valued maps into Gc. Since a map into Ge defines a one-component gauge field Az via

(3.19) we thus also may identify ME with the space of chiral Ge gauge fields Az modulo Ge gauge transformations Az

-->

h(8z+ Az)h- 1 •

Again, due to the presence of the central extension, the current blocks IIliE} are not functions but sections of a holomorphic line-bundle .c~ over the phase space ME. The same reasoning as in the case of the disk implies that the sections of .c~ are functionals of multi-valued maps 9 from :E in Ge, which under local variations 9

-->

9

+ 6g transform as (3.20)

Alternatively, by using (3.19), we may take IliE to be a functional of the chiral gauge field

432 A:z, in which case the above condition becomes (3.21) This equation can be shown to be equivalent to the Kac·Moody Ward identity (3.9). Furthermore, it prescribes the behaviour of the functional IJI[A:z] under complex gauge transformations, and thus IJI[A:z] defines a section over Ml;. The above description can be taken as an alternative definition of the space of current blocks. It turns out that the equation (3.21) precisely characterizes the physical wave·functions in the Kahler quantization of 2+ 1 dimensional Chern-Simons theory [5,8,9]. To derive this one first rewrites the Chern-Simons action in the canonical form

CS(A) =

!:. r tr(AdA + ~A3) 4?r Jl;xR 3

(3.22)

!:. Jdt rtr(A A atA + AtF(A)) 4?r Jl;

(3.23)

The time-component At is a Lagrange multiplier imposing the condition F(A) = 0 for the space-components of the gauge field. Therefore, the phase space of (3.22) is given by Ml;, as in (3.18). The kinetic term determines the Kiihler form on Ml; w=

!:. r trc5Az 1\ c5A:z 4?r Jl;

(3.24)

where (c5 A., c5 A:z) is a variation of the fiat gauge field. The most convenient way to perform the quantization of (Ml;, w) is first to quantize the space of all gauge fields (A., A:z) and then impose the constraint F(A) = 0 as an operator constraint on the wave-functionals

[8,9]. The gauge field components satisfy the canonical commutation relation (3.25) Hence, by choosing

A~

as the coordinates in the Chern· Simons wave·functionals, it is

readily verified that (3.21) is indeed equal to the operator constraint F(A) = 0 on the wave function IJI[A:z]. Note that this description of the Hilbert space 'Hl; depends on the choice of complex structure on E.

To show that physical quantities do not depend on this choice, one

needs to construct a projectively fiat connection on moduli space, identifying the Hilbert spaces corresponding to different complex structures. In fact, due to the work of Friedan and Shenker [25], in conformal field theory we already familiar with this projectively fiat connection, namely in the form of the stress-tensor. Its derivation from the 2+1dimensional view point has recently been done in [9].

433

3.3

The physical operators.

Let us now discuss the physical operators acting on 'HE. First we will use the definition of the current blocks in terms of the operator formalism. Because 'HE is then described as a subspace of the basic Kac-Moody representation [3-diagram). The colors of the propagators are the weights J.' of the integrable representations. The possible couplings at the vertices are counted by N IJ11J2 ,W The dimension of 1tr. is found by just counting the number of colored trivalent graphs. Using the result (3.29-3.30) and Weyl's character formula it is a straightforward calculation [4) to show that dim1t'£ = (C(k

+ hn g - 1

L II (1- e '11";a< ml 2

~EP!aE~

r-

g

(3.31)

where r is the rank of the group G and C the order of its center. The sum is over all positive weights A which lead to integrable representations. It would be nice if one could understand this result more directly from the description of 1t'£ given in the previous section. The form of the expression actually suggests that it should be possible to derive it from a fixed-point formula.

3.4

Relation with two-dimensional induced gauge theory

The description of the current blocks of WZNW-models as wave functionals of a onecomponent gauge field has a direct interpretation in terms of the WZNW-model in the presence of a background gauge field. In this section we explain how this leads to new insights about two-dimensional induced gauge theory. Let us consider the partition function (3.32) where S,£(g,A) is the action of the gauged WZNW model [12]. Its precise form will not be

435 important for

US; the only property we will use is that the gauge-field components Az and A z couple to the Kac-Moody currents Jz and Jz respectively. Because of the argument mentioned in section 3.1, we know that ZE[A) can be decomposed into current blocks. Concretely, we have

ZE[A)

= e-~ JA:A~ L

hiiW;[Az) 'lii [Az]

(3.33)

i,j

wher·e i, j = 1, ... , dim?tE and hii is some hermitian metric. The prefactor is dictated by the requirement of gauge invariance, [26). In this decomposition the current blocks 'lii[Az] have the interpretation of the chiral expectation values (3.34) where the subscript denotes the formal projection on the chiral sector labelled by i. The physical constraint (3.21) of the Chern-Simons wave-functions reduces in this situation to the two-dimensional gauge anomaly equation [26) (3.35) which in turn is equivalent to the Kac-Moody operator product of the J:'s. The matrix h ii in (3.33) has to be determined via the condition that the partition function ZE[A) is invariant under the modular (or mapping class) group. At first this seems to require that we explicitly know how the modular transformations act on the conformal blocks, but in fact Chern-Simons theory gives a very direct and elegant way to uniquely characterize hii. In the Hilbert space ?tE of current blocks one can define a natural Kihler inner product, given by the functional integral [8,12) (3.36) which, using the physical constraints (3.21), can formally reduced to a finite dimensional integral over the moduli space of fiat gauge fields. (See [12,8) for details.) The key point is now that, since modular transformations are identified with canonical transformations, they are implemented by unitary operators with respect to the inner product (3.36) on ?tE. Thus there always exists a modular invariant, for which

(3.37) This is called the diagonal modular invariant. Similar formulas can be written for the correlation functions [12).

436 Next let us see what happens when we quantize the gauge field, i.e. functionally integrate over (Az, A z ). So we consider the gauged partition function (3.38) When we insert the expression (3.33) for ZI:[A) we recognize the foimula (3.36) for the inner product of the conformal blocks. IT we in addition use (3.37) we obtain the following surprisingly simple result for the gauged partition function z~aug.d =

2: hij (lJIil lJl j) = dim7-1I:

(3.39)

i,;

The intuitive interpretation of this result is that in the gauged partition function only the physical states of the theory should contribute. Since these are just the Kac-Moody primary states, we expect that the partition functions just counts the number of primary states that can contribute in the different channels, which is equal to the number of current blocks. Note further that the result (3.39) is independent of the moduli parameters of the surface. The gauged WZNW-model therefore defines a topological two-dimensional QFT. This observation that the Hilbert state interpretation of the current blocks essentially trivializes the computation of the partition function of the gauged WZNW-model, motivates the interesting question whether there exists a similar result for the Virasoro case. The natural gauge theory associated with the Virasoro algebra is two-dimensional gravity, so one might hope that if the Virasoro conformal blocks can indeed be identified as Hilbert states of a 2+ I-dimensional quantum theory that similar considerations as above will lead to new insight about two-dimensional induced quantum gravity. We will study this problem in the following section.

4

Conformal blocks and SL(2,R) Chern-Simons theory.

We will now apply the formalism of the previous section to study the space of conformal blocks of the Virasoro algebra

[L(6),L(6)]

LW

L([6, 6])

+ 2~1r

f

3

68 6

( 4.1)

f ~(z)T(z)

The Virasoro representation theory is quite different than for the Kac-Moody algebra, since for the range c ;::: 1 there exists a continuous spectrum of representations. As a consequence we should expect that the phase space associated with the conformal blocks

437 of the Virasoro algebra is non-compact. In this section we will argue that the appropriate phase space is Teichmiiller space, which is indeed non-compact.

4.1

Geometrical Virasoro action.

Let us first discuss the geometrical formulation of the basic Virasoro representation, which will be related to the Teichmiiller space Tv of the disk. To describe Tv we consider an annular region A = {z; a ::; D = {w;

f'

Izl ::;

b} of the complex plane which we attach to a disk

Iwl ::;

I} via the identification map (w, to) = (I( z ),f(z». Two maps I and are called equivalent if f' 0 1-1 extends to a holomorphic function on the disk. The

Teichmiiller space Tv is defined as the set of equivalence classes of the maps (I, /). We can define a transitive action of the group DillS 1 on Tv via

(I(z),f(z»

->

(F 0 I(z), F 0 l(z»

(4.2)

where (F, F) are such that the circle 1/12 = 1 is mapped onto itself. It is easily checked that in (4.2) the S L(2, R)-subgroup of Dill Sl acts trivially. Therefore, we can identify the Teichmiiller space of the disk with the coadjoint orbit

DillS 1 Tv = SL(2,R)

(4.3)

Now let us see what this implies for the basic Virasoro representation [4>0]. We identify the Virasoro algebra (4.1) with the generators of the group of maps F acting on the analytic patching functions vial

I(z)

-+

F

0

I(z)

(4.4)

Again, if we first ignore the central extension c, we can represent F by an operator U(F) obtained by exponentiating the Virasoro generators. The vacuum state (01 is by definition invariant under the group elements F which extend holomorphically to the disk. This implies that the wave functions

'!i[F] = (OIU(F)I'!i)

(4.5)

are in fact holomorphic functions on Tv. This statement again has to be modified to account for the central charge c. For this we will need to know the geometrical action of the Virasoro algebra. This action can be determined in a similar way as before, via the condition that the corresponding Kihler 'in eFT we have two Virasoro algebras Vir ® Vir, which suggests that we can treat the complex coordinates z and % as independent variables.

438 form is that associated with the Virasoro algebra (4.1). This construction has been done in [15,16], with the following result 3

S[F] = _c jaF (8 F _ (82F)2) 2411' 8F 8F 2 8F

(4.6)

This action was first obtained by Polyakov in the context of two-dimensional induced gravity [29]. Applying the general construction, described in section 2.1, of the Hilbert space associated with a geometrical action, we now find the following description of the states in [¢>o]. Instead of holomorphic functions on 72), they are holomorphic functionals

wD[F] on the space of diffeomorphisms w -+ F(w, w) of the disk D, transforming under variations F -+ F + 6F in the interior of the disk as 6WD[F] = 6SD[F] w[F]

(4.7)

The variation 6SD[F] of the Virasoro action is [14]

(4_8) where the (-I,I)-differential Jl.F is defined by Jl.F =

al

81

F(f(z,z),z) = z

(4.9)

These equations provide the geometrical definition of the states IWI:) E [¢>o] as sections of a holomorphic line-bundle £C over 7 D •

4.2

Description of the Virasoro conformal blocks.

Let us now consider a general closed surface E_ For this surface we can describe the Teichmiiller space {Wj

Iwl

~

11:

in a similar way as for the disk. We attach the unit disk D

I} to an annular region A on E - D as depicted in the following figure.

The identification map (w, w)

= (f(z), l(z»

can be used to parametrize Teichmiiller space

439 TE' Two maps I and /' correspond to the same point on TE when /' a holomorphic function on D or when

1-1 0 /,(z)

0

I-I(w) extends to

extends holomorphically to 1: - D.

Now let us discuss the Virasoro conformal blocks on 1:. We again use the operator formalism to associate to the data described above a state 11:) E [IPo) ® [ifio). It can be decomposed into conformal blocks I.pE) E [IPo), satisfying the Virasoro Ward identity2 (4.10) for

e I( z) a holomorphic vector field on 1: - D. This description of the conformal blocks 0

thus depends on the holomorphic identification map

I.

The Virasoro group acts on this

identification map via the transformations (4.4). Modulo the effect of the central charge, we can therefore associate to the conformal block I.pE) the function

(4.11) Combining (4.10) with the description we have given of Teichmiiller space, one can convince oneself that .p[F) represents a holomorphic function on T1:To complete this description of the conformal blocks for the case c

i-

0 we will use

the well-known characterization of Teichmiiller space TE in terms of quasi-conformal maps and Beltrami differentials. Given a fixed Riemann surface 1: and a coordinatization (z, z) on it, we can parametrize the space of all other Riemann surface (1:', (w, w)) in terms of the quasi-conformal transition map z = F( w, w). This therefore describes Teichmiiller space as the space of quasi-conformal maps, modulo local diffeomorphisms. Alternatively, by using (4.9) we can trade F for a Beltrami-differential JI., and describe TE as the space of gauge inequivalent Beltrami-differentials. Collecting all the ingredients, we finally arrive at the following geometrical definition of the Virasoro conformal blocks on 1:. In analogy with the disk, and also with the current case, we define them as functionals .pE[F) on the space of quasi-conformal maps F on 1:, transforming under local variations of F as

c5.pE[F) = _c_ 1(c5F 8 3 J1.F).pE[F) 24?r

1E

( 4.12)

Equivalently, we may take the conformal blocks to be functionals .p[JI.) on the space of Beltrami differentials, in which case the above condition reads (4.13) 'This equation has to be slightly modified if the identification map is not compatible with the projective structure. In general we must add a schwarzian derivative term to the stress-tensor T(w) in (4.10).

440 This condition determines the transformation of the functional ~[Jll under diffeomorphisms, and therefore implies that

~l:[Jll

is defined on Tl:. Equation (4.13) can be shown

to be equivalent to the Virasoro Ward identity (4.10). So far, the discussion of the Virasoro conformal blocks has been completely parallel to that of the current blocks in the previous section. In that case, however, we were at this stage able to point out that the Kac-Moody Ward identity (3.21) was precisely the equation characterizing the physical states of the 2+1-dimensional Chern-Simons theory. We would now like to give the above Virasoro Ward identity (4.13) a similar interpretation.

4.3

Quantization of Teichmiiller space

Motivated by the previous characterization of the Virasoro conformal blocks, we will now investigate the quantization of Teichmiiller space Tl:, and see whether we can make contact with the previous discussion. For this purpose, a very convenient description of Tl: is in terms of the zwei-bein (e+, e-) and the SO(2) spin connection w, [27]. We have

(4.14) where the constraints ga are

(4.15)

The first two are the familiar torsion constraints, which can be used to solve the spinconnection in terms of the zwei-bein, whereas the third constraint gO is the constant curvature condition R = _1. 3 Further, Ditto denotes the group of two-dimensional diffeomorphisms connected to the identity, and LL that of local Lorentz rotations. This identification of Teichmiiller space suggests that the 2+ I-dimensional field theory naturally associated with Tl: is described by the action

~

r

4~

f h,[ ~w

411" JRxl:

dt

[~wdw+

e+de-+w/\e+/\e-]

(4.16)

/\ 8tw + e+ /\ 8t e- +Wt gO + +etg+ + etg-]

which in fact can be recognized as the SL(2, R) Chern-Simons action. It is not difficult to show that, if we restrict the zwei-bein to be invertible, the physical phase space of lWe restrict ourselves here to surfaces of genus 9 ~ 2.

441 this action is indeed given by Teichmiiller space (4.14). The key-point here is that, on solutions to the constraints (4.15), the groups Dillo x LL and that of SL(2,R) gauge transformations are equivalent. Besides reproducing TE as its phase space, the S L(2, R) CS-action (4.16) also provides a natural Kahler form on it, given by

o = 47r ~ 1Ef (~cw " CW + ce+ " ce-)

( 4.17)

which can in fact be shown to be proportional to the Weil-Peterson Killer form, [28J. To quantize (TE'O) we will now follow a similar strategy as that used in [8J in the case of CS-theory with compact gauge group, namely we first quantize the space of all (e+, e- , w)fields, and then impose the constraints afterwards. For our polarization, however, we make a different choice: we take as coordinates the components (et , et, w.), and let their

6!± '

canonical conjugates act as functional derivatives, i.e. e; = 1[etc. The motivation for this choice is that it will lead to wave-functionals which are hol~morphic in the natural complex structure on TE. The physical subspace of the large Hilbert space of all wave functionals lJ! [et , et , wzJ is selected by the operator constraints ( 4.18)

We will now show that these constraints are in fact equivalent to the Virasoro Ward identity (4.13). To this end, let us parametrize the rechte·bein e+ as

e+ = e'P(dz + /-uu)

(4.19)

The two constraints g+ and go then attain the following simple form (here w -k (81{' - ,.,,81{' - 8,." 47r

k C -8w-847r CW

+ /lW )

= wz )

C --

cw

C +CI{'

Using these two constraints, it is easy to explicitly solve for the

(4.20) I{'

and w dependence of

the physical wave functionals. One finds the result ( 4.21) where ( 4.22)

442 and ti[lLl = tirO, 0, ILl. A straightforward calculation now shows that the remaining constraint g- reduces to the Virasoro Ward identity (4.13) on ti[lLl

g-

6 c 3 (8 - 1L8 - 281L)- - - 8 IL hlL

2411"

( 4.23)

where the central charge is given by c = 6k. Thus we have indeed identified the Virasoro conformal blocks as the states of the S L(2, R) Chern-Simons theory (4.16) and thus restored the parallel with the discussion for the Kac-Moody current blocks. In the remaining part of this section we will consider the physical operators and the inner product on the space of conformal blocks.

4.4

Geodesic lengths as physical operators

The operators acting on the space of Virasoro conformal blocks are the Wilson lines of the SL(2,R)-gauge field A = (e+,e-,w): OJ(C) = trj(Pexp

fc Aara)

(4.24)

measuring the conjugacy class of the holonomy of SL(2,R) gauge field A around C. In Teichmiiller theory these conjugacy classes are all hyperbolic and are labelled by a real number 1, called the length of C, defined via

O~(C) = 2COSh'(~)

(4.25)

and equals the geodesic length of the shortest curve on :E with the topology of C. It can be shown that the S L(2, R) Wilson line OJ( C) can be expressed in terms of the stress-tensor T of the CFT as ( 4.26) We would again like to use these operators to organize the Hilbert space 'HI;. To this end let us choose some basis {Ci} of 3g - 3 different non-intersecting cycles on :E, and decompose the Hilbert space HI; into eigen spaces of the corresponding length operators. We will now argue that, perhaps up to renormalization effects, the eigenvalues are ( 4.27) where hi is the weight of the fields passing through Ci. The argumentation leading to this identification makes use of the well-known result that in the length-twist coordinate

443 system on 71; the Well-Peterson Kiihler form is given by ( 4.28) Hence, quantization in these coordinates gives [Ii, OJ]

= k6ij.

This implies that the gener-

ator of the Dehn twist around a cycle Ci, which sends Oi to 0i

+ 211", is ( 4.29)

On the other, it is known that under the same Dehn twist, the Virasoro conformal blocks with representation label hi at

Ci

acquire a phase factor exp(211"hi ). Comparing this with

(4.29) yields the conclusion (4.27). Clearly, there are some assumptions made in this reasoning. In particular there might be some subtleties in the correspondence between the above quantization of 71; using (1,0) coordinates and the procedure, described in the previous section, of first quantizing and then constraining. In that case the formula (4.27) is probably only exact in the classical limit, k -+ 00. Finally, notice that, since modular transformations are symmetries of Teichmiiller space and thus leave (4.28) invariant, they are again identified with canonical transformations.

4.5

The inner product and two-dimensional quantum gravity.

The quantization of Teicluniiller space, as described in section 4.3, also leads to a natural scalar product on the conformal blocks. In this section we will show that this scalar product is given by two-dimensional quantum gravity. To this end we will first describe the action of gravity, induced by a two-dimensional conformal field theory. We consider the partition function Z[g] of a CFT with central charge c in a background metric ds 2 = gabd~ad~b. For the case of the sphere it is well-known that the effective action r[g] = -log Z[g] can be solved from the trace-anomaly equation and is given by [29] ( 4.30) where R is the scalar curvature and

,:l

the scalar Laplacian. This action has been well

studied in the conformal gauge as well as in the light-cone gauge. We will not pick a gauge, however, but work out r[g] in the general parametrization (4.31 )

444 We find that the covariant action r[g] has the following decomposition

( 4.32) where the first term is the well-known Liouville action

SL[

f

R(2)

~ -I R(2)

+ .. ,

(2.9)

j34> is the basic Weyl anomaly coefficient related to the ,B-functions by

p =,8'" - ~KI'U,8;u,

Kl'u

= GI'U + ... ,

(2.10) (2.11)

,8i are the Weyl anomaly coefficients

which appear in the operator expression for the trace of the 2-d energy momentum tensor 36). Power counting and the proper choice of the measure guarantees the renorrnalizability of Z within the loop (0/) expansion (2.12)

dZ

8Z i8Z dA = 8A -,B 8tpi =0,

Direct computation shows that 22)

(2.13)

••

W = Wo -

1,

2

Tinea (R + xV ¢» + ... ,

(2.14)

• See also Refs. 33, 34 and 35 for other discussions of the measure in the a-model. To make contact with string theory we include the standard "-26" contribution of ghosts 37).

493

Wo =

-~(D- 26)(X€71£ + 1~7r

f

R(2) 11- 1R(2)+const.

Hence (see (2.7» (2.15) The coefficients of the €71£ - tenns in (2.14) are consistent with the renonnalizability of Z (Eq.(2.13» as one can check using the known expressions for the ,8-functions 33),36),38) (2.16) I - 26) - .!.a V 2'I' .J. + ~a'2 R~ + 0(a'3) ,89 = .!.(D 6 2 16 AJjUP

•

(2.17)

Substituting the expressions for the bare fields 4> =

1

4>R - '6(D - 26)€71£ + ...

(2.18)

into Z (2.7) we indeed find the cancellation at the leading €71£ tenns. It is possible to prove that the ,a-functions can be expressed in tenns of the derivatives of some functional S 39),40),41),7) ,ai =

ij K

(2.19)

::

where the matrix K is non-degenerate (within the a'-expansion) K

=

KO

') + O( a,

KOGG JjUOl{3 -

kG- 1/ 2 GJjOl G u{3, (2.20)

Under a particular choice of the couplings (Le. in a particular renonnalization scheme) Scan be represented in the fonn 40) (2.21)

(2.22) The relations (2.19), (2.21) were checked by the explicit perturbative computations up to rather high order in the expansion in a ' (for a discussion and references see Ref. 7). The leading a 'independent tenn in K is essentially "one"; the non-diagonal tenns in KO are due to the "mixing"

494 between the graviton and the dilaton in the kinetic term in S (2.21). If we redefine the couplings to diagonalize the kinetic term 2) 1 . r;::;--;:, , cf>= 4vD-2cf> ,

(2.23)

(2.24) we find

- •• - ij' as f3 - It a + .... Thus ItOI is proportional to

2. A relation to string theory is established through the observation 2) that Z (2.1) is the generating functional for the correlators of the massless vertex operators (we consider the tree approximation and hence assume the spherical topology, X = 2)

=L 00

Z

N=O

(_l)N

N

--,-TN'P, N.

TN

=


(2.31)

The on-shell string tree amplitudes AN differ from TN by the infinite factor of the 8L(2, C) Mobius group volume, AN ~ n -I TN. Introducing a short distance cutoff e, one finds !hat (up to power divergent terms) n ~ i ne 20) ,21) • Hence naively AN ~ (ine) -ITN. However, this is not a correct prescription since it does not treat all the short distance divergences in TN on an equal footing. TN contains the following types of logarithmic infinities: (1) Singularities corresponding to the region of integration where N or N - 1 integration points Zi are close to each other. These are "momentum independent" (i.e. present for arbitrary on-shell external momenta) Mobius infinites, which are absent in AN computed with 3 Koba-Nielsen (KN) points fixed. (2)"Momentum dependent" massless pole singularities corresponding to the regions of

495 integration where M = 2, ... , N - 2 points are close to each other. As discussed in Ref. 21, the correct RG invariant relation between the regulari~d expressions for AN and TN is •

8 AN ~ 8R)nETN

(2.32)

Hence the generating functional for the massless string tree amplitudes Z ~ I:N ANcpN (computed with a 2-d curoff e and using the expansion in o/) can be represented as

a

A

(2.33)

Z = 8R)nEZ .

Since the partition function Z is renormalizable, i.e. satisfies (2.13), the same is true for Z (2.34) We would like to stress that Z is the basic object which defines the first-quantized string theory. In particular, it is Z and not the amplitudes which satisfies the renormalizability property. Since a renormalization of logarithmic divergences in Z corresponds to a subtraction of the massless poles in the amplitudes 1),42), the renormalized value ZR of Z should coincide 40) with the effective action (EA) S which reproduces the massless sector of the string S-matrix. Let jJ. be a renormalization parameter, Z = ZR( CPR(jJ.} , jJ.}. Then

dZR=O dt '

8

A

ZR =

t

n

a

.

at ZR = -{3'( CPR} acph ZR( CPR},

== "'TiJ.l,

f./i(} _ dCPR CPR - dt

I-'

(2.35)

.

Using (2.7) (for X = 2) and redefining the fields to absorb the finite part of W, we get Z R J dDyVGR e- 2 Substituting this into ZR (2.35) we finish with 21)

~

"1I.

Z(cp}R = S(cp} = a

f

d DyVGe- 2 "/3,

/3" = /3tP - ~/3~PI'''

(2.36)

(we used cP instead of CPR to simplify the notation). Comparing (2.21) and (2.36) we conclude that EA which reproduces the massless string amplitudes coincides with the functional appearing in the relation (2.19) for the Weyl anomaly coefficients. Since "i is nondegenerate (within the ai-expansion) the string effective low energy equations of motion = 0 are thus equivalent to the conditions of the Weyl invariance of the corresponding renormalizable sigma model.

g;.

In view of (2.35) S

= ZR can be represented in terms of the scale anomaly 21) (2.37)

• The difference between a/8R.ne and (R.ne}-1 prescriptions becomes important at the R.n2 elevel. For example, the 4-point correlator contains the R.n2 e-piece in which one R.ne-factor is the Mobius divergence and the other corresponds to the massless pole.

496

< 9(z) >=

2 baZR _. vgga agab ~ (3'Vi,

Vi =

f

2

d ZvgVi .

While a straightforward generalization of the scale anomaly action (2.37) may not reproduce the full string equations with all massive modes included (see Ref043) it certainly gives the correct effective equations of motion in the massless sector. Let us note also that ~ = -I>, < Vi > is proportional to the massless tadpole computed in a background, i.e. (2.38) Hence the effective equations of motion are indeed equivalent to the vacuum stability condition, i.e. to the vanishing of the massless tadpoles in a background. Eq. (2.37) gives the explicit representation for the effective action in terms of the partition function of the a-model. It is easy to derive, for example, the Einstein term in the EA. According to (2.7), (2.15) (X = 2; we set D = 26 for simplicity) Z

~

f dD!lv'Ge-2~(1 +~aim1lE+

...).

(2.39)

Since the order R term corresponds to the (order k 2 -term in) 3-graviton correlator on the sphere, T3 =< VI V2 V3 >0, the £1lE-divergence in (2.39) can be interpreted as the Mobius divergence present in the regularized expression for T3. Differentiating over £1lE we thus obtain the (k 2_ piece of) usual 3-graviton (Mobius gauge fixed) amplitude which corresponds to the order R term in the EA * (2040) To summarize, there is a deep connection between the low energy (massless sector) string dynamics and the renormalizable a-model at the string tree level which is expressed by the following relations

as. = 0 -a cp'

- = 0 ..... Ali = 0 ,

..... f3'

(2.41)

where S is the effective action reconstructed from the massless string amplitudes, 13i are the Weyl anomaly coefficients «(3-functions) of the 0' model and Ali are the massless I-point amplitudes computed in a massless background. Moreover, S can be explicitly represented in terms of the Weyl anomaly coefficients (see (2.36), (2.37».** * Had we not set D = 26 we would obtain (see (2.15» the order D-26 "cosmological" term in (2040), in agreement with (2.21), (2.36). ** Let us stress that though to compute EA one uses the string S-matrix corresponding to a particular (flat space) string vacuum, being extended off-shell the EA "forgets" about that particular vacuum and, in fact, contains an information about all ("nearby") vacua. For example, it "interpolates" between the flat space, group manifold vacua (assuming that the D-26-term is included as in (2.22» and, in fact, all other "weak coupling" conformal field theories.

497 It would be very interesting to generalize the above string theory -(1 model correspondence to the string loop level. One of the consequences of the equivalence between the string equations of motion and conditions of confonnal invariance of the a-model is that it makes possible to avoid the difficult problem of detennining the stationary points of the EA ( reconstructed order by order from string amplitudes), replacing it by the problem of classification of 2-d confonnal field theories. Similar equivalence at the loop level would provide a way of studying exact string vacua solutions by solving some generalized "confonnal invariance" conditions. 3. The basic property of the string generating functional Z which makes possible to establish the connection with .B-functions is its renormalizability with respect to the local world sheet infinities. To be able to extend the string-a model relation to string loop level it is necessary to have the corresponding renonnalizability property of the loop-corrected generating functional. Before turning to the analysis of string loop corrections it is instructive to add some remarks about renonnalization of Z at the string tree level. Consider the partition function Z =< e-I";V. >0 expanded in powers of I{J, see (2.31). The regularized correlators TN contain divergences proportional (in view of the factorization property) to Tu with M < N. In fact, the divergences come from the regions where 2 or more integration points Zi coincide; if we consider only the low momentum massless particles as the external ones, the operator product V, Vi closes on dimension 2 massless vertex operators • and we get the correlators already present in the sum (2.31), Z ~ 1 + TtI{J + T21{J2 + ... (in more detail, Vi(Zi)Vj(Zj) ~ lulolk;k,-2C;jkVk(Z), u = 2 Zi - Zj, f. ~Iul""k;k, ~ tre + fkikjtn g + ... where we expand in 0/ kikj, etc.). Hence we should be able to cancel the divergences by redefining l{Ji order by order, l{Ji

= I{Jk + tre(B1jk~RI{J~ + B~jkll{J~I{J~I{J~ + ...) + O(tn2g) .

(2.42)

The usual argument for perturbative renonnalizability of Z is based essentially on dimensional (or power counting) considerations. The coefficients which appear in the countertenns (2.42) are in fact proportional to the string scattering amplitudes (see Refs. 1,44,8) and references therein). To see this one should introduce a background x and compute the countertenns extracting the local order V,[x]tre tenn in W[X] = -tnZ[x]. Expanding in power of l{Ji it is possible to check that the "overall" tre singularity in < VI [x + x] ... VN [x + x] > comes from the region where all N points are close to each other. Factorizing on the pole one finds that the coefficient is the usual N + I-point amplitude AN+ I (computed with the Mobius gauge fixed, e.g. ZI = O,Z2 = I,ZN+I = 00). Thusthe.B-functionis.B i ~ EN~2 BNI{JN,BN = (AN+I).ubtr. (to obtain the ,8-function we should subtract all subleading singularities). This result is obviously consistent with the.B ~ ~ relation (2.19) since the EA is S ~ EN(A N ).ubtr.I{JN. Let us now give another argument for the renonnalizability of the string generating functional Z which is based on the usual assumption that the combinatorics of string amplitudes is such that the massless sector of the string S-matrix can be reproduced by an effective • A"physical" explanation of the perturbative renonnalizatibility of Z is that if the external momenta (or derivatives of cp) are small (compared to 0/.,-1/2), we get only the singularities corresponding to massless poles (there is no energy enough to produce massive poles).

498 field theory. By this one implies that Z should be equal to the generating functional s for the field theory S-matrix. The idea is to establish the renormalizability property of s introducing a cutoff through the spac(}-tirne propagator and assuming the proportionality (2.19) between the ,8-functions and the derivative of the EA S. Let (2.43) Consider the functional (2.44) where W is the generating functional for connected Green functions. In the tree approximation

as

WU) = S(tP) - tPJ,

(2.45)

atP = J .

The on-shell S-matrix generating functional is (2.46) tpet

= tPll"'. = tp." + ... ,

~ tp."

= 0,

It is convenient not to specify tp to be equal to tp." (and hence tP to be equal to tpcl) at the intermediate stages (see also Ref. 45). Then

s( tp)

1 = -2tp~ tp + U(tP( tp»

1

U'= 1

,

s(tp.,,) = U(tpcl(tp.,,» + 2(U ~

-I

,

I,

au atp

,

+ 2(U ~ - U )4>(1")

,

(2.47)

,

(2.48) 1

,

U )1"l

£'T/£

< VkVj >......t

2

(3.3)

+o(£n e)

t,k

it is natural to anticipate the exponentiation of the "rudimental" self--energy corrections, z!y = exp(E~~£'T/£) ,

(3.4)

.

~~j

'" (

< V; Vj >.. ).ubtr.

,

where ~ .. is the massless 2-point amplitude on genus n with all self--energy subdivergences subtracted out. This exponentiation was checked on the example of the one loop tachyon leg correction in Ref. 18. Note that (3.4) can be true only if the relative weights of string loop corrections take particular values. As we shall see, the exponentiation of divergences is crucial in order for the RG to act in string loops (see also Ref. 14). Let us now consider the logarithmic tadpole divergences. A careful analysis of factorization of the closed string amplitudes in the "tadpole" limit gives the following result (see Ref. 15 and refs. therein)* ...... 1

L...J ..-4(N......l) £'T/£a.. ,

(3.5)

A(") ' " "

N

t=1

..-4j.;-ll "'< VI

0.

=

... VNOt > ......l,

at "'
l,

4~ct! d2zv'9T2n: 8a xl'8 a xl': +al(1-n)R(2)]

(3.6)

,

(3.7)

where normal ordering is with respect to the propagator on the sphere, (3.8)

We assume that a regular metric is introduced on the world surface, so that the Euler number is

x.. =

1 4 7f

!

d2 zygR(2) = 2( 1 - n) .

(3.9)

at is proportional to a massless tadpole or, equivalently, to the vacuum amplitude. It is possible to rewrite (3.5) in the following way

* In the subsequent sections we shall check this result on several particular examples (see also Refs. 13, 14 and 16).

503

(3.10) where

Vg

= 4~ct

f

2

(3.11)

d z: aoxl'aoxl':

is the (trace of) soft graviton vertex operator, and (see Refs. 14-16) 1

t= 4"D(Xl- 2) < 1 >l .

(3.12)

The £TI£ which appears in (3.5) and (3.10) originates from the graviton and dilaton propagators at zero momentum. In fact, one can represent 0" (3.7) as the combination of the vertex operators for the trace of the soft graviton and the soft dilaton (3.13)

Va =

== 4~0I1

4~ct

f

f

2 d zyfg(: 8 a xl'8°xl' :

d2Z.,jg(aoxl'aOxl' +

-}a/ R(2»

,

(3.14)

~0I1(D-2)R(2».

Note that the dilaton couples to the Euler number (cf. (2.3». We shall also use another equivalent representation for 0" (3.15)

/J,,=2n,

1

12" = 1 + 2"n( D -

2) .

(3.16)

Observing that (3.15) looks similar to the string action (2.3) in a trivial vacuum GI'V

= 61'v,

r/J

= £rIg = CO'II8t,

9

= string coupling

(3.17)

one may try to cancel the tadpole divergences (3.5) in the total amplitude AN = L" A~fi) by adding the counterterms to the vacuum string action 9). To illustrate the idea of renormalization consider, e.g., the genus 3 example. Different tadpole factorizations of 3-loop amplitude give the following divergeces

+ < ... (d304£TI£) >0 + < ... (d I 0 1£TI£)2 >1 + + < ... (d2 02£TI£) >1 + < ... dl 01 £TI£ >2 , where ... stands for VI ... VN' subdivergences,

a.. ~
" are, in general, divergent, containing tadpole (3.19)

504

Suppose now that the correlators are computed with the "bare" string action I = 1R + 61, where the counterterm is a power series in the (renormalized, see below) string coupling andl1l£. Expanding in the (renormalized) string coupling we then get insertions of fJI in the correlators and hence may hope to cancel the tadpole divergences between different terms in AN (for example,

< ... e-6I >0 + < ... >1 + ... = ... + < ... (-fJI) >0 + < ... 01 dll1l£ >0 + ... ~

finite). The resulting expression for the counterterm is 00

fJI

= 2:bn(g)ONl1l£ + 0(ln2g)

(3.20)

n=1 00

= 2:bnOnl1l£ + 0(ln2g), bn ~ a,.,

bn ~ dn .

(3.21)

n=1

Though it may seem that the factorization suggests that the lnmg, m ~ 2 - terms are absent in (3.21), the consistency of the RG suggests that they should be present (and have coefficients related to (dn) fin). Using the expression for On (3.15) one can rewrite fJI as (3.22) (3.23)

al

k

where ) are power series in the (renormalized) string coupling constant. Comparing (3.22) with the vacuum string action (2.3), (3.17) we conclude that the tadpole divergences may be absorbed into a renormalization of the constant vacuum values of the metric and the dilaton, or, equivalently, into a renormalization of the two basic constants: ol and the string coupling 9 (3.24) The consistency of this renormalization procedure depends crucially on whether the relative weights of string diagrams are such that the tadpole divergences actually exponentiate, One may consider the exponentiation to be a consequence of the condition of factorization which together with unitarity 52) fixes the weights of string amplitudes. Alternat~vely, one may impose the requirement ofrenormalizability (which implies exponentiation) and as a consequence, fix the relative weights of string diagrams.

505

4.

GENERATING FUNCTIONAL (OR (J MODEL) APPROACH TO RENORMALIZATION OF STRING LOOPS

The results of the previous section about the elimination of the tadpole and external leg divergences through the renormalization of ol and g and the vertex operators can be reinterpreted in a more fundamental way as the renormalizations of the couplings of the (J-model which appears in the generating functional for string amplitudes Z .Z is the basic object which defines the theory

Z = 0-\ EenjdfJn(m) n=O

1= 10 + 'P'V;,

r Dxexp(-I),

1M"

(4.1)

Z ~ EAN'PN . N

In general, 10 is the string action in a particular vacuum and V; are .the corresponding "massless" vertex operators. We assume that "extended" sets of moduli {m} are used so that the Mobius group volume factor 0 -\ is present for all genera. en are normalization factors (weights) which we prefer to indicate explicitly. Expanding Z in powers of 'P' and putting them on shell (0 'P' = 0) we get the on-shell amplitudes as the coefficients.

Z is formally defined for arbitrary ("off-shell") values of the fields 'P" In (4.1) we do not integrate over the conformal factor of 2-metric, fixing a Weyl gauge. The basic consistency requirement is that Z evaluated for the "true" vacuum values of couplings 'P' (which solve the string equations of motion or generalized Weyl invariance conditions "{3 = 0 ") should be Weyl gauge independent (Le. Weyl invariant)'. Suppose now that a short distance cutoff is introduced in (4.1), which regularizes, in particular the "local", "tadpole" and "external leg" divergences. We can cancel the local divergences separately for each genus by choosing the bare fields to correspond to the "local" (J-model counterterms (see Sec. 2). To cancel the "modular" divergences we are to combine the contributions of different genera. Since it is the product 'P'V; that appears in (4.1) we trade the renormalization of the vertex operators(3.2) for the multiplicative renormalization of the couplings. The counterterm (3.20), (3.22) corresponds directly to the renormalization of the metric and the dilaton in the (J model action (2.3). The corresponding bare couplings are (4.2)

Otad'P' =

O•. I.'P'

al inc: + a~ £n2 e + ... ,

= z;(e)'P~ = 'Pk + Mtj'P~£nc: + ....

( 4.3)

• There is, of course, an ambiguity in splitting the metric on a conformal factor and moduli for each particular 11. It would be nice to have a universal prescription, which relates the splittings for different 11. We anticipate that this ambiguity can be "absorbed" into redefinitions of the couplings 'P"

506 The coefficients which appear here are power series in the string coupling (see also ref. 14). The crucial point is that if Z is renormalizable ( 4.4) it is possible to introduce the p-functions (see (2.34), (2.35»

az

, az

aR:nE -

P (IP) alP'

(4.5)

= 0,

In general, the relation between the bare and renormalized couplings is

( 4.6) The basic restriction imposed by the RG is that P' explicitly on e. This implies that

==

-1i; should depend only on IP' but not (4.7)

We get from (4.2) ( 4.8) Thus the tadpole counterterm corresponds to the "inhomogeneous" term in the ,8-function while the external leg one- to the term linear in the fields. The coefficients in (4.8) have the following structure (see (3.4), (3.21» 00

a'1 = "'g2"b' L...J

00

,p

Mt j

.... 1

= Eln!L~

,

( 4.9)

... 1

where b~ and!L~ are proportional to the finite parts of the massless tadpoles and 2-point functions on genus n (they are both proportional to the genus n vacuum amplitude). Recalling the structure of the counterterm (3.22) we can write the metric and the dilaton p-functions in the following symbolic form (Gl'v = 0I'V + hl'v)

rf;v = A P~

=

G

0I'V

+ Bfl'v4> + Bg:t haP,

A~+ B:4>+ Btrhl'v,

A~

aI,

(4.10)

B ~ MI.

If the theory is defined (regularized) in a way consistent with general covariance in D dimensions, one should be able to rewrite (4.10) as (4.11) (note that A and B should be constants; the terms in (4.11) are the only covariant terms to the linear order in the fields hl'v and 4». Eq.(4.1O) can reduce to (4.11) only if there are relations between the values of the massless tadpoles and the 2-point amplitudes.

507 In (4.11) GJjV and tP are the full non--constant fields (constant vacuum values plus fluctuations). An important observation is that (3's are non-trivial functions of the constant part of tP since it is directly related (see (2.3) and Refs. 53, 2, 54) to the string coupling constant which appears in (4.9). Since we renormalize tP we also should renormalize 9 and it is, in fact, the renormalized values of 9 (or tP) which appears in the coefficients in (4.3). If we split tP on the constant and non--constant parts

tP = tPc +~, tPc = R.ng = con.st

( 4.12)

we get the following expressions for the (3-functions 00

rf;v = LA"e 2 fl\6'GJjv,

(4.13)

... 1 00

(3~ =

L

00

v"e 2 fl\6, +

L

/J."e 2 fl\6, 4> .

(4.14)

... 1

They look unnatural because of the different dependence on tP and 4> and cry for a generalization. The complicated dependence on tPc suggests that the modular counterterms (4.3) and the (3functions (4.8) should, in fact, contain all powers of the fields (in particular, of the dilaton). Then, e.g., the O(~)-term in (4.14) should correspond to the linearization of the exact expression for (3~ which is non-linear in tP. Thus the consistent dependence on tP implies that (3i should have the form * (4.15) (4.16) where dots in (4.15) stand for other possible terms depending on derivatives of the fields. Ifwe further use the explicit expression for the tadpole counterterm (3.20), (3.15) we get 00

FJ = -

L 2 nb"e

2

(4.17)

fl\6 ,

".,1

1

L[1 + 2n(D 00

F2 = -

2

2)]b"e fl\6 .

(4.18)

n=1

To get a deeper understanding of the structure ofthe (3-functions (4.15), (4.17), (4.18) let us consider the "zero-momentum" part ofthe generating functional Z (4.1) or the string partition function computed for the constant vacuum values of GJjv and tP. Assuming that the string

* Note that to be able to rewrite (4.14) in the form (4.15), one should have a relation between the dilaton tadpole and self--energy coefficients v" and j.£" in (4.14). The existence of such a relation .follows from the "exponential" dependence of Z on the dilaton.

508 path integral is defined in a way preserving D-dimensional general covariance (the functional measure is consistent with a regularization, etc., see Refs. 16 and 22) we find (cf. (2.7»

2:=2'.0+2:1 +2:2 + ... = =do

f dDyVGe-2~(olR+ f .f + ...) + f ... )+al

+ d2

. rr; d D yv Ge 2~ (1

= So +

f dDyVGw(t/J)

d DyVG(I+ ... )

. rr; d D yv Ge 4~ (1

• d3

+ ... ) + ...

(4.19) (4.20)

+ ... ,

00

w=

2: d,.e(2f>-2)~ ,

d,. ~ (ol) -D/2 ,

(4.21)

... 1

where d,,( d,.) are (the finite parts of) the higher genus contributions to the vacuum amplitude (see (3.19»*. In (4.19) we have included the tree level contribution (see (2.36), (2.21); So denotes the tree level piece of the effective action). Consider now the divergent parts of 2: (4.22) The renormalizability implies that (see (4.5» A

Z

(I) _

-

.

a

A

f3'-a .ZR' tp'

(4.23)

The renormalized part of 2:R should coincide with the effective action (since renormalization is simply a subtraction of massless exchanges). The operator f3i -> FI Gpl/ . + F2 ~ (see (4.15» represents the insertion of the tadpole counterterm, so that (4.23) is in correspondence with the factorization property of the amplitudes. Let us now show that the expressions for F I , F2 (4.17), (4.18) which follow from the factorization imply that the f3-functions (4.15) are

fv;

aJ••

equivalent to the equations of motion following from the "cosmological tenn "in loop--corrected

effective action (4.20). Namely, let us assume that the tree level relation (2.15) between the f3functions and the effective action holds also at the loop level

. = ",,"··as

f3'

atpi

'

00

S= 2:S.. ,

00

""ij

= 'L...J " ""ij ...

(4.24)

!F O

The loop corrections to ""ij are not important to the leading order (recall that ""ij (2.20) is necessary in order to account for the mixing between the metric and the dilaton in the kinetic term of

* Let us note that d,. are complex because we assume that an analytic continuation prescription 28)

is used in order to get rid of the tachyonic "non-dividing" singularities (see Sec. 3).

509 the tree level EA). Applying (4.24), (2.20) to the case of S given by (4.20) one finds 13),14) (cf.

(2.16» {3i

= {3b

+ {3;ad + ... ,

{3;v

= [a/(Rl'v + 2VI'Vvr/I)

+ ...] + FI(r/I)Gl'v + ... ,

4>_ 1 1 12 I 2 {3 -["6(D-26)-IO!V r/I+0!(8r/1) + ... ]+F2(r/I)+ ... ,

(4.25)

F = -.!.e 2 4>,./":-1(2w

(4.26)

I

4

"1l

+ w')

w' = dJ.,;

'-dr/l'

1

F2 = -16eZ4>dQI(2Dw + (D - 2)w' )

(4.27)

Substituting here the expression (4.21) for the dilaton potential w(r/I), one finds that (4.26), (4.27) coincide with (4.17) and (4.18) if

(4.28)

*.

To find the bare couplings corresponding to the {3-functions (4.15) we are to solve the if there were no higher order tenus in {3i, i.e. if (4.8) were true, we equations Ii ( tp) = diagonal for simplicity would get (we take

MJ

tp(e) = eM1lflE(tpR + Mllal) -

MI-Ial

(4.29) Thus the RG implies the exponentiation of the divergences (cf. (3.4» and, in particular, the presence of higher order ink e countertenus with the coefficients fixed in tenus of the coefficients appearing in the {3-function. The bare fields corresponding to (4.15) are

(4.30) where the in2 e-tenus can be found by applying the general relations (4.7), (4.6). It would be a non-trivial check of the consitency of the RG approach to find that the string loops contain the iri- e - divergences which can be cancelled by inserting the counterterms (4.30) (or, equivalently, that the in2 e - part of Z satisfies (4.23»'. Let us now discuss the origin of the higher order 0 (tpm), m ~ 2, tenus in the {3functions (4.15): Obviously they correspond to the divergences which appear when an amplitude factorizes into two parts (with the numbers of external particles NI and Nz) and all external • It is interesting to observe that the in2e-tenus in (4.30) are different from those which follow from a naive expectation that the. full countertenu 6I can be represented as the sum of the full times Onine. Thus there should be 0 (iri- e) tenus in (3.20). (unrenormalized) tadpoles

a..

510

particles on one part (say the second) have zero momentum (the tadpole divergence thus corresponds to the case when N2 = 0). Clearly, this is a particular case of "momentum dependent" singularity due to massless pole in (kJ + ... + kN. ) 2) which one may try to eliminate by using analytic continuation in external momenta. However, as we have argued above, it is necessary to account for (regularize) such divergences in order to get covariant expressions for the ,B-functions. The continuity in external momenta then suggests that we should regularize the massless pole momentum dependent singularity also for non-vanishing external momenta. This is consistent with the fact that the pole singularities contribute to the Weyl (or BRST) anomaly 49) and is obviously necessary for the correspondence between the ,B--functions and the effective action.

S.

REMARKS ON CORRESPONDENCE WITH EFFECTIVE ACTION

We have already discussed the role played by the effective action in the RG approach at the tree level (see Sec. 2). Most of this discussion generalizes to loops. To construct the EA for the massless fields we start with the massless string scattering amplitudes (with loop corrections included) and subtract all massless poles (see also Ref. 14) *. To subtract the massless exchanges we are to consider all possible "dividing" factorizations of a string diagram. At the same time the corrections to the ,B-functions we discussed in Secs. 3 and 4 come precisely from the massless pole singularities which appear in these factorizations. Hence, qualitatively, we should have

S

rv

E(AN).ubtr.cpN,

,B( '1')

rv

E(AN+d.ubtr.cpN,

(5.1)

N

N

as alP

rv,B

.

In general, we expect to find that: (1) the generating functional Z (4.1) is renormalizable with respect to the infinites associated with all "dividing" singularities of massless string amplitudes ("momentum dependent" as well as "momentum independent" ones). As at the tree level, it is necessary to combine particular amplitudes into Z in order to obtain an object which is renonnalizable with respect to "momentum - dependent" infinites. (2) The EA S( '1') is a renonnalized (finite) value of Z (since all the divergences in Z are due to the massless exchanges, a subtraction of the latter should be equivalent to a renonnalization procedure). (3) The resulting

* Note that we are not assuming an expansion in external momenta since at the loop level this expansion may lead to additional IR singularities (due to massless loops). The EA constructed by expanding in powers of fields will not be manifestly covariant. It may be rewritten in a covariant form by using appropriate non-local structures (like R"f:nf.. 0/1)2 + ol R + ... ), etc.) which are non-analytic in 0/ and momenta.

511 effective equations of motion are equivalent to a generalized conformal invariance conditions f3i = 0 and also to generalized vacuum stability conditions Ali = 0 (the vanishing of tadpoles in a background, see (2.41), (2.38». As we have shown in Sec. 2 the renormalizability of Z is closely connected with the existence of an effective action. The important point is that the existence of the EA (and hence renormalizability of Z) depends on the values of relative weights of string diagrams. These weights are usually assumed to be fixed by the condition of factorization (and unitarity) 52),15),55) which in turn should guarantee the existence of the EA. Alternatively, it is interesting to observe that the assumption of existence of the EA leads to constraints on the weights. Consider first the ordinary field theory, e.g. 80 = tcp.1. cP + t,gcp3. It is possible to represent its quantum effective action aw = cp, e-W(J) = 'Dcpe-so+. f dJ.!..(r) 1'Dx(t)e-

10 =

f

I

,

1= 10 + lint,

(5.3)

'Y.

"I.

dte-1!i?,

Ii" =cpV=g

f

dtecp(x(t» ,

where '1.. are 1 - PI graphs and { r} are the proper time parameters ("moduli" of a I-d metric e( t». To arrive at this expression one should represent 8 in terms of the background dependent propagators and to use the path integral representation for the latter. The "weights" e.. or the relative normalizations of the modular measures are fixed by the functional integral definition of 8 (5.2). The generating functional for the quantum 8-matrix is simply the tree generating functional for 8 (see (2.41), (2.46». Thus the relative weights of the terms in the EA automatically determine the weights of various amplitudes in agreement with factorization property. The situation in string theory is different since if we do not start with a string field theory functional integral we do not know a priori the relative weights in the "first-quantized" path integral (4.1)*. Suppose now that the massless sector at the string 8-matrix can be reproduced by an effective field theory, i.e. that Z(CPi .. ) can be represented as the generating functional for the tree level 8-matrix for some action 8( cp) (see (2.43), (2.46), (2.48» **

Z(CPi.. ) = s( CPi .. ) , S

s( CPi.. ) = 8( cpcl( CPi.. » - CPi".1. cpcl( cpi,,) ,

1 I -I I = U(CPi") - 2"(U.1. U )n ,

(6.10)

517

< ... >,,=

j[d:X] exp(- 4~a' lXV-IX},


,,=

1,

(6.11)

where the propagator is given by (6.7) and the modular measure is

00

(6.12) Oi!

m=l

a

where the product rr~ goes over all primitive elements of SO. The factor in power D is the contribution of the integral over xl', while the contribution of the 2-d reparametrization ghosts is rr~

rr:=2 11 -

k:!'I-

4. The combination d'~~~'fk Ilkk 14 has a natural interpretation of the

measure for the 8L(2, C} group of transformations T = (~~), d' A~Fd'Q, with A, B, C, D expressed in terms of €, 11, k, see Eq. (6.1) (note that the transformation €, 11, k ---+ A, B, C, D is not, of course, one to one). The integration over the moduli ( k, €,11) should be restricted ("by hand") to go over the fundamental region of the modular group (which is not explicitly known in the SP). One may formally not restrict the region of integration assuming that the overall normalization constants can be finally fixed, e.g., from unitarity considerations. The measure (6.12) is clearly projective invariant. Recalling the remark that the SL(2,C) transformation property of V( Z , w} is the same as of the propagator on C we conclude that the Mobius volume factor n in (6.10) cancels out if the external momenta satisfy the tree level on-shell conditions. One can fix the M5bius symmetry by fixing the positions of the three 11. and the KN points Zi. The usual expressions for the loop amplitudes parameters among are obtained by fixing 6 ,1jJ and ZI for n = 1 and, e.g., 6 , 6 ,1jJ for n ~ 2 (note again that n = 1 is a special case) '.

e.,

It is, in principle, straightforward to study the factorization of string amplitudes in SP. One draws a cycle 'Y on C which one wants to pinch and considers the limit in which the points Zi, 116 which lie inside 'Y uniformly come close to each other (e.g. 6 = 1jJ + U, ZI = 1jJ + UWI, U ---+ 0, etc.) A discussion of factorization in SP has already appeared in Ref. 18. We would like to point out, however, that the general analysis of Ref. 15 implies that to get the consistent results, e.g., for the tadpole factorization one should properly account for the additional ghost or 2-d curvature insertions. The example of the factorization on the disc (see

e.,

• Two Riemann surfaces are conform ally equivalent iff their SO's are related by a similarity transformation, i.e. by an overall projective transformation. The remaining 3 n - 3 (n ~ 2) fixed points and multipliers parametrize conform ally inequivalent Riemann surfaces. The projective invariance of the multiloop integrand thus implies that one is integrating over the conformally inequivalent surfaces.

518 Refs. 11, 13, 14, 15, 19 and Sec. 7) suggests that when the string amplitudes diverge one may get different results by using different representations for the world sheet (which are apparently conformally equivalent on shell). A consistent approach is to use a compact curved 2- space representation for the world sheet and to account for the curvature dependence in the modular measure originating from the frame dependence of the coordinates and radii of the holes on the surface. In general, if we start with C with a topological fixture of size a --+ 0 at point w we should go to a curved space representation and define the center of the fixture by 15) w'

=

r

JI'I~.

d2zv'g(Z+W)(Z+W)/

Then da d 2 a3 w

--+

r

JI'I~.

d2zv'g(z+w)~w+O(a2).

l 2 da d wl..;g(iil)(1.!.. 12R(2)( ') O( 4» a'3 +4a w + a .

(6.13)

(6.14)

We shall return to the discussion of this point on the example of the disc in Sec. 7 below. Eq.(6.1O) suggests the following expression for the generating functional for string amplitudes in SP (cf. (4.1» 00

(6.15) ... 0

z" = c,. j

dJ1." L.[dXle-I,

1=10 + I,ft,

Z" = j[dXle-I,

I,ft = 'TJ implies rt -> 0, r2 -> 0 for arbitrary value of k. (so that the boundaries of the holes are not isometric circles).

522 (and dropping f( Zl) + f( Z2) - terms) we can also rewrite the Green function (7.9) in the following way (7.11) (.enlA I> 2

47Tenl k l It is easy to see that only the last non-holomorphic term in V (7.11) gives the leading nontrivial (non-holomorphic) contribution in the € --+ 11 (or A --+ 1) limit (we again drop the tenns depending only on Zl or Z2)

1(1

2

1

2 -2

VI~---8{Vo(€,Z2) + c.c]

=

(7.12)

(7.13)

(=€-11.

+0«(2,(2),

Consider now the expression for the generating functional for the amplitudes (7.7) (see (6.16)-(6.19»

(7.14) The tadpole factorization corresponds to the limit € --+ 11 in which the handle degenerates. Note that in contrast with the general case of the Schottky parametrization the parameters € and 11 defined by (7.6) are formally independent of the mod~lus k (their integration region, in fact, depends on k) and hence the limit € --+ 11 can be taken for arbitrary k. Employing the expansion (7.13) we find in this limit ,

ZI=C1n-

1


0 ,

1 I H = 2"C!co 7rAlineOI .

(7.21) (7.22)

This result is consistent with the general expressions (3.21) and (4.28) if bl

1

= 2"ClcQ

I

7rAI

1

= 4dlciQ

I

,

(7.23)

where d". are the constants which appear in 2: (4.19). If we fix dl by comparing 2:1 with the usual field theory result log det (0/ D) + ... then dl is given by (5.10) (see Ref. 56), i.e.

t

dl = .!..(47r2ai)-D/2 AI 2

(7.24)

«27rol)-D/2 comes from the constant in the string action and (27r)-D/2-from the Gaussian integral nortnalization; the total factor (4 7r 2 aO -D/2 v'G in (4.19) is the contribution of the zero mode of xl'). The choice of do ~ (4 7r2 ci) -D/2 is a matter of convention, being related to the choice of the string coupling. 2. Let us now analyze how the combined "local" plus "modular" renortnalization can be carried out in 2:1. Let us first consider the generating functional for the massless amplitudes in the "parallelogram" representation of the torus * (7.25)

* Note that it is not necessary to fix explicitly the compact U( 1) Mobius group.

524 where (d'T) = [d'T] / I m'T. 2:1 is thus simply the integral over 'T of the partition function of the a model (2.3) defined on a compact torus represented by the parallelogram on a. As we have already discussed in Sec. 2, the partition function of the a-model on a compact space of arbitrary topology is given by (see (2.7), (2.15) and Ref. 22) Zn = an

J

dD yv'Ge 2(r>-I) if> 0 +

~aieneR+ ...)

(7.26)

(we put)( = 2 - 2nand D = 26 in (2.15». For all n Zn is renormalizable with respect to "local" infinities, i.e. satisfies (2.12). This leads to the following apparent paradox: if we renormalize all the "local" infinities in Zn, ZI in (7.25) will be finite and hence 2:\ will also be finite (since, as we have already argued, the integral over 'T does not give new divergences of the type we are interested in). At the same time, 2:1 should definitely contain the "modular" divergences discussed above. To resolve this paradox one should observe that what appears as a "local" infinity in (7.25) is, in fact, a "modular" infinity from the point of view of the string amplitudes defined on the parallelogram. Consider, for example, the 3--graviton amplitude on the torus in the parallelogram representation. The (momentum)2-term in the amplitude corresponds to the h!v-piece of the R-term in 2:1 or in ZI (7.26). This amplitude contains the infinities corresponding to the limits in which two or all three KN points corne close to each other *. These infinites are "local" if considered from the point of view of the a-model defined on the parallelogram but are modular external leg and tapole infinites from the standard string theory point of view-. It is instructive to compare this with what happens on the sphere. The correlator of the three graviton vertex operators on a is again divergent in precisely the same limits (Zi --+ Zj) and Z\ --+ Z2 --+ Z3) reproducing the Rene - term in (7.26) (for n = 0). From the string theory point of view one interprets these divergences as Mobius divergences (see Sec. 2 and Refs. 20 and 21) and subtracts them (e.g. by fixing the positions of the vertex operators or applying the a / aene - prescription (2.33» obtaining thus the finite expression for the 3--graviton amplitude or the R-term in the generating functional (see (2.39) and (2.40».

Zo

Let us now summarize and generalize the above discussion. Consider the standard "Mobius gauge fixed" approach in which one uses the "restricted" sets of moduli {'T} and hence

n2l.

(7.27)

Since Zn is renormalizable with respect to the local infinities, we can rewrite the 0 (ene) term in it as f3A ~ene (see (2.13». Hence the pait of the O(ene)-term in Zn corresponding to the

* By using the expression for the propagator on the torus one can check directly that the result is in agreement with (7.26). - Let us recall that the computation of the a-model (3-function on the torus 34),60) gives the same result as on the sphere. To find the "modular" corrections to f3 one is to give up the parallelogram representation, introducing additional modular parameters (cf. [9]) and defining the "stringy" a-model as the ordinary one integrated over the extended set of moduli.

525 factorizations into the genus 0 and genus n parts can be represented as (cf. (4.23»

az.. z" = z"a + !3b -a. £1IE + . .. . cp' A

Recalling that (3& =

A

•

~~ ~ (see (2.19»

we can absorb the divergence (7.28) into the renonnal-

ization of cps in the tree tenn in 2; = 2::0 order modular correction to the (3-function

Q' _

'j 1-'" - ~o

(7.28)

z.. ,o"cp' = ~'j ~£1IE + ... , obtaining the leading

a'Z.vt acpi ,

(7.29)

Using the expression for ~~. (2.20) we find that (7.29) is, in fact, in agreement with the previous result for the "momentum independent" contribution to the "modular" (3-functions (see (4.25».

Eq. (7.28) is not however, what we would expect to find if 2; were renonnalizable with respect to the sum of the local and modular infinities (7.30)

A

z"

A

=

•

az.. cp'

. aZo cp'

ZnR + !3b -a. £1IE + f3'.. -a. £1IE + ....

(7.31)

In particular, A

A

•

a2;1

2

ZI =ZIR+2(3~-a.£1IE+O(£n e)

cp'

(7.32)

where we have used that to the leading order

(3' az.. o acp'

a:2:0 " acp'

= (3'

=

a:2:0 ~'j az.. acp' 0 acpj

(7.33)

The interpretation of (7.31) and (7.32) is clear: .. one half" of the £1IE-divergence in 2;1 should be "local" and "one-half" - "modular". The "local" divergence is to be cancelled by inserting the local (f30) countertenns directly into 2;1 while the "modular" should be cancelled by inserting the modular «(31) countertenns into

Zo .

We conclude that the generating functional defined according to the standard Mobius gauge fixed prescription does not satisfy the RG equation (7.30) with the full (3-function (which is proportional to the full effective equations of motion). This suggests that we need to modify the definition of 2; in order to ensure its renonnalizability. The qualitiative reason for the absence of "doubling" of the £1IE - divergence in 2;1 is that we have used the parametrization in which the Mobius infinities (which, in fact, are Ii subclass of local infinities, see Sec. 2) are already dropped out. However, the tree level experience suggests that one should first regularize both the Mobius and modular infinities and then use a special prescription of how to "divide"

526 by the Ml>bius volume in order to preserve correspondence with the usual results for on-shell amplitudes. Consider once again the 3-graviton correlator on the sphere and the torus, using the Schottky-type (sphere with two holes) representation for the latter. Introducing a short distance cutoff we find (here < ... >1 includes integration over the moduli)

(7.34) The t'llE-term in < VI V2 V3 >0 and one of the t'llE factors in < VI V2 V3 >1 can be interpreted as Mobius (or local) infinities originating from the limits in which the vertex operators "collide". The second t'llE factor in < VI V2 V3 > I is the modular infinity corresponding to the limit -+ 1'/) in which the holes collide (the handle disappears). If we use the standard prescription (0- 1 '" t'llE) for subtracting of the Mobius infinity we get A~O) '" 0- 1 < VI V2 V3 >0= finite,A~I) '" 0- 1 < Vi V2 V3 >1'" t'llE+ finite, where the t'llE-term in A~I) corresponds to the modular divergence in ZI discussed above. If instead we use the 0- 1 -+ 8/8t'llEprescription 21),22), we get precisely the desired doubling of the coefficient of the t'llE-term in A~ I). What happens is that the derivative 8/ 8t'llE counts the "local" and "modular" t'llE-factors in the e-term is A~I) in an independent way, in agreement with (7.31) and (7.32).

(e

trr

Thus we suggest to use the definition of Z already given in (6.20) and expect to find Z = 8;'IlE (Zo + ZI + ...) = 8;'IlE {2le- ,.. >0

(7.37)

e-+ 1'/ limit (7.38)

* Such counting is also necessary in order to have the sum of the local and modular pieces in the total ,B-function.

527 Next we note that the insertion of 01 (7.18) into the partition function on the sphere can be represented in the following way • (7.39) ~

.",I, f d !lvG( D

'20/ Rf:nE +

...) ,

where in the last line we have used (7.26) (the dilaton factor cancels out since this contribution originates from the torus). Substituting (7.39) into (7.38) we indeed find the lif e R-tenn. The problem which still remains is an apparent absence ofthelnE I dD!lv'G-tenn in ZI. This tenn should be present for a consitency of the prescription (7.35), (7.36)". This suggests that we are still to improve the above qualitative picture. One subtle point is related to the replacement of C by a compact 2-sphere for which we can use the expression (7.26) for the partition function. It would be better to start directly with a curved compact 2-sphere with two holes (with identified boundaries). In this case, however, the expression for the modular measure may change, 3. To get a deeper understanding of related issues let us now analyze the case of the disc correction to the generating functional for closed string amplitudes in the theory of open and closed strings. As in the case of the torus, let us first consider the "MObius gauge fixed" representation of the disc in tenns of the interior of the unit circle on the complex plane. It is not necessary to fix the 8L(2, R) Mobius symmetry on the disc explicitly since the corresponding divergences are power-like (not logarithmic), i.e. the SL(2,R) Mobius group volume is finite if we drop power infinities 61) ,20). Then the momentum independent logarithmic modular divergences in the closed string correlators on the disc come from the limits in which all N or N - 1 vertex operators collide 11) (see also Refs. 14 and 19). They are simply the local divergences when considered from the point of view of the a-model defined on the unit disc. The corresponding generating functional is thus proportional to the partition function (7.26) (we fonnally take n = for the disc since this is in agreement with X = 1)

t

.

ZI/2 =

dl / 2

f

D ."'4J I ,lnER + ...) d!lv Ge- (1 + '20/

(7.40)

and so is renonnalizable with respect to the local divergences. Hence just as in the case of the torus in the parallelogram representation (see (7.27}-(7.30» we find that these local divergences (i) can be interpreted as "modular" from string theory point of view (ii) Can be absorbed into a renonnalization of the couplings in Zo with the proper "modular" countertenn (corresponding • Note that up to nonnal ordering 01 is twice the free string action. The nonnal ordering implies does not act on the overall zero mode factor (0/') -D12 in Zo (see Ref. 14). In fact, that I 8x8x ~ 0/' £.; but the insertion of the D I R(2) -tenn in (3.8) cancels against the derivative of the (ci)-DI2-factor. .. Such tenn is to be expected since the soft graviton amplitudes on the torus are finite only in the standard (Mobius gauge fixed) parallelogram representation.

-k

528 to (4.15), (4.17) and (4.18) with n = tor (4.26), (4.27) with w ~ e-4»; (iii) are only "one half" of the divergences that should be present in ZI/2 in order for Z = 7.0 + ZI/2 + ... to be renormalizable with respect to both local and modular infinities 19). In order to resolve the latter problem let us go to the Schottky-type "plane with a hole" representation of the disc 13). Conform ally transforming (inverting) the unit disc into the plane with a hole one finds the following expression for the generating functional for the closed string amplitudes on the disc 13)

ZI/2 =CI/2 n - Izl/2,

ZI/2 =

f dJ.'l/dexp(~(271"al)V. c5~2)e-I, da a

2

dJ.'I/2 = 3d

471"

(7.41) (7.42)

w,

1 2 1 V(ZI,Z2)=--£njzl- z21 --£nll-

471"

.. ["'1},;=0,

a2 __ (ZI - W)(Z2 - w)

= 1'0 + i>,

I2 (7.43)

where a and w are the radius of the hole and the position of its center on C n is the volume of the 8L(2, C) Mobius group on C In the derivation of (7.41) one assumes that the Mobius gauge is fixed by fixing the positions of the KN points (e.g. ZI = 0, Z2 = 1, Z3 = 00). According to (7.41) we are first to compute the partition function on the plane with a hole, then average over the "moduli" a and w and finally to subtract the Mobius infinities. It is possible to return to the unit disc picture by fixing the formal on-shell L( 2, C) symmetry by the condition a = 1, w = 0, ZI = 0 and then making the inversion u = - ~. The result will be the unit disc representation for the amplitudes in the SL(2,C) Mobius gauge "I = 0 (the Neumann function (7.43) then reduces to the standard expression on the unit disc V = - i,;£nlul - "21211 - "I ;121 2 ). As in the case of the Schottky-type parametrization of the torus, the representation (7.41) makes it possible to clearly isolate the "modular" divergences from the "local" ones. The tadpole modular infinity corresponds to the small hole limit, a --> O. In this limit (see (7.43), cf. (7.13» (7.44) and hence (cf. (7.15), (7.16» • 1/2 = C 1/2 U",-I Z


0 ", 3 (v,t + a2"" vl/2 + "O( a4» e'"

• a

n Z• 1/200 = -CI/2 u","-I (.'Tl£

-I,.. < ",I vl/2 e >0 ,

(7.45) (7.46)

(7.47) The operator 0;/2 is different however, from the correct counterterm 0 1/ 2 (see (3.7» needed in order to renormalize the "modular" divergence in the amplitudes on the disc in a way consistent

529 with the effective action. This problem was pointed out by Fischler, Klebanov and Susskind (FKS) 13). As we have mentioned above (see also Ref. 19) this problem is absent in the unit disc parametrization if one carefully accounts for all local infinities present in the corresponding partition function, or, equivalently, in the correlators of the vertex operators. Let us now discuss how the FKS paradox is resolved in the plane with a hole parametrization. The basic point is that to define off-shell or divergent quantities like Zone shoud use a compact curved 2-space representation for the world surface. This is necessary in order to account for the topology of the world surface in a systematic way and in particular, in order to have a correspondence with the sigma-model approach in which the topology is reflected in the dilaton factor e-x~. The problem is then to find the measure dJ1.1/2 (7.41) in the case of a curved compact sphere with a hole. The approach suggested by Polchinski IS) is to consider a general curved surface with a hole with the metric gab = e2Pliab and to compute the measure by expanding in a --+ O. He pointed out that the 2-" ?TO!

7T0I w-+z

-

-

= 2{j1'" lim 8.8w V( z, w) w-+z

I b = - -l 8 2 {jl'" lim (wO(z)t;;-b w (ill) 7f W-foZ

= __I_{jI'"wO(z)r1wb(z) 871'2

ob

where we have used that 8.8w (enE(z, w) + enE(z, ill»

= O.

It can be proved in general that

(7.56) Hence

71'~,
g

2

"

1- < 4wO!'

'

f

d2 z !;;g8 xl'8·xl' Vli •

1 >" = --D 2'

(7.59)

As was discussed in Ref. 16 (see also Ref. 15)) this universal result is consistent with the covariant expression for the "constant" part of the string partition function •

~< e- I,..

z"

f

dDy( 1 +

>,,=

~h~ + ...) =

f

f d y(l- < Vr >" hl'v + ...) = D

dDyVGO + ...),

GI'V

= 0I'V + hl'v .

(7.60)

In the case of torus X = 0 and hence the subtlety related to the use of (3.8), (7.58) is irrelevant (we, of course, drop the quadratic divergence ~ aaVo). In the parallelogram representation only the last term in (7.4) contributes to (7.55) and we find again (cf. (7.19))

~ < '7t'Q

fd

2

U:

1 axl'ax V :>1= --4 ol'V lim Ul---+U1

fd

2

U

8t8z(U12 -U12)2

(7.61) (we have used that the area of the parallelogram is I roT). In the case of the Schottky-type parametrization V (7.9) belongs to the general class (6.7) and hence we get (7.55) with w = (z

~)(~ 0 (see (7.10)) (7.62)

e

If we fonnally put = 0, 'Tj = 00 and map back to the parallelogram (z e21f;U) we get ~ = 4w 2 d 2 u = 4w 2 ImT, in agreement with (7.61). If instead we substitute (7.62) into

J

J

the modular integral (7.7), (7.8), we get

A

A-I - C\u.

(I) -

1

ol'V = - 8w 2 Ct).ln

-1

f[d ]Ie~ed21] . -1]1 < Vg

I'V

T

f

4

>1

d ed 1]c£2 z 2

2

Ie -1]1 2 Ie - zl21z -1]1 2

'

(7.63)

where ). I was defined in (7.17). Thus we find the agreement with the result in the parallelogram representation if

(7.64)

* Note that Eq.(3.8) is true in dimensional regularization in which one can ignore the contribution of the measure (0(2) (0) = 0). In other regularizations one is to account for the contribution of the measure in order to reproduce the covariant result (7-60) 16);22). In general, to derive (7.59) and (7.60) one is only to use that AV = 0(2) (1,2) - ~ and to set 0(2)( 1,1) = O.

533 Eq.(7.64) looks like the standard expression for the Mobius group volume. However, in contrast to the tree level case, here the integration region is not the full a'and in fact, depends on k (hence the n -I factor should be better placed under the integral over ,,). This suggests that a possible modification of the ansatz for Z which should apply to the case of the vacuum amplitude may look like • - Z

J[d ]J I" _ f.121" d"d_ f.£fJ fJ121f. _ 1]1 2 < e 2

A-I

- Ci"

2

T

-I,..

(7.65)

>1

where we have introduced one extra moduli parameter" which appears only in the measure (which is formally projective invariant) but not in the Green function. Now the measure gives the .ellE-factor already for the vacuum amplitude and hence (7.65) gives the finite expression for the "constant" part of Z after we differentiate over .e1lE.

8.

GENUS TWO EXAMPLE OF RENORMALIZATION

Below we are going to explain how the renormalization of the string generating functional can be carried in the two-loop approximation. We shall demonstrate that the condition of renormalizability of Z (or, equivalently, the condition that the massless sector of the string S-matrix can be reproduced from an effective action) fixes the overall coefficients (or relative weights) of string loop corrections to Z. Let us first consider the disc and annulus corrections to Z in the open-closed Bose string theory. Since the annulus factorizes on two discs we expect to find the logarithmic divergence in its contribution to Z (which we shall denote as Zf) (8.1)

(8.2) Here r/> and G are the bare values of the fields which we know already from the study of the tadpole renormalization on the disc. We shall consider only the renormalization of the zero momentum part of Z and hence will ignore the "local" counter terms. According to the general expression (4.15)-(4.18), (4.26), (4.27) (8.3)

, For example, z belongs to the exterior of the two circles and hence the limits z -+ f., excluded from the integration region (unless -+ 1]).

e

Z

-+

1] are

534 where for the disc w = dl/le-~R. Substituting (8.3) into the disc contribution to (8.1) we find that the divergence in the annulus contribution cancels out if 16) '(I)

dl

1 l -I = -16dl/ldO (D - 2) .

(8.4)

If we start directly with the well-known expression for the string partition function on the annulus we get

1 = ci[ 2e l

(D - 2)£11E: + finite] .

-

(8.5)

Dropping the quadratic divergence and comparing with (8.2) and (8.4) we find the agreement if the overall constant dl takes the following value J

CI

1 l -I = -d1/ldO 16

(8.6)

(note that do, dI/2 , ri!1 and dl are proportional to o/-DI2). Equivalently, one may say that it is the factorization condition that fixes dl • Let us now repeat the same analysis in the case of the torus and genus 2 contributions to Z. Since the genus 2 surface may factorize on the two tori we should find

Z=ZI+Zz+ ... =dl

!

dDy../G+dl

dl=d2+4,I)£I1E:.

! dDy../Ge2~+

... ,

(8.7) (8.8)

The renormalizability of Z implies that it should be possible to cancel the divergence in Zz by substituting the expression for the bare metric (8.3) in the torus term in (8.7). Using that for the torus w = dl, we find the following conditio~ on 1)

4

(8.9)

Let us now check this prediction by directly computing the divergent part of the genus 2 partition function using the Schottky parametrization. According to the general expression for the modular measure in SP (6.12) we have

(8.10)

535

a

m-=2

This expression for the genus 2 string partition function was studied in Ref. 66 where the equivalence between the loop measure in SP and in the parametrization in terms of the period matrix 67) (in which the modular invariance is explicit) was checked by expanding the SP measure in powers of the modular parameters *. If we formally fix the Mobius group by choosing 6 = 00, 7)1 = 1,7)2 = 0, 6 == E, (8.10) reduces to

. _f

~ - C2

2

d kl ci? k2ci?E 2 -D/2 Ikd4Ik214IEI4If(kl,k2,E)1 (dett) .

(8.11)

The "dividing" factorization limit corresponds to E -+ O. To find the expansions for f and t = I mTab in powers of E we apply the results of Ref. 66. The generators of the Schottky group (8.12) (8.13)

Analyzing the E-dependence of the multipliers of the elements of SG which are products of N factors TaN = T:: ... T::; , Ta = {n, T2} one finds that kaN ~ EN, N ~ 2. Since we are interested in the logarithmic divergence in (8.11) we are to expand to the order IEI2. We have k"k.,m Tal = {TI,T2}, T"" = {Tj"T2", n, m = 1,2, ... }. kal = {k l , kd, k"" = (I k~)l(i k;)Z, etc. Therefore (see (6.5), (6.12))

2e

00

f

=

II 1(1 - kr)( 1 -

krw 2(D-2) {I + O(e

,e2)},

(8.14)

m=1

2'ITi7'JI

= .enkl + O(e), 2'ITi7'J2 =

27Ti'li2

E+ O(e),

tab

= .enk2 + O(e)

,

(8.15)

= ImTab .

The non-trivial 0 ( lEI) 2) correction factor thus comes only from the expansion of (det t) -D/2_ factor in (8.10), (det t)-D/2 = [tllt22 -

(2~)2 (ReE)2r D/ 2 =

(tllt22)-D/2[1+

+iD(27Ttll)-1(27Tt22)-IIEI2 + O(e ,e)] ,

(8.16)

* Let us note that while the formal agreement of the SP measure with the general Belavin-Knizhnik expression 25) dp." = Id3 n-3 yf(y) 12 (det t)-D/2 (where f is the holomorphic (3n-3, 0) form without zeroes or poles in the interior of the moduli space but with double poles at the parts of the boundary corresponding to surface degenerations) is obvious, the modular invariance of it remains to be proved.

536

A

_

C

d2 -

2

f

2

2

2

d kl d k2 d f- II""

I

m

m

Ik141k2141€14 m=l (1-kl)(1-k2)

1-2(D-2)

,

(8.17)

(enlk11 enlk 2'1 )-D/2[1 + _1_D(enlkd enlk21 )-11f-1 2 + O(e,e)]. 271" 271" 4(271")2 271" 271" Thus finally (cf. (8.5»

1

2

d € 4 2 1 2 2 2 d2 = C2 • W{(271") tl + 4"D(271") )'lI€1 + A

+O(e,e)} 71" 421 32 ., =C2[2g2 (271") t l -4"D(271") ),Iene+fm~te],

(8.18)

where the zero momentum tachyon and massless scalar tadpoles on the torus tl and),1 were defined in (7.17)', and hence (see (8.8» ,./(1) "'2

1 C2( 2 71" )3\2D = -4" Al .

(8.19)

Comparing this with (8.9) we get (8.20) This finally determines C2 if we use the relation between the coefficients dl (the vacuum amplitude) and),1 (the tadpole) for the torus (7.24) (in general dl = q),I)' Let us emphasize that having fixed the overall coefficient in 2:2 we automatically determine the normalization of the 2-loop amplitudes generated by

Zz .

The result that the logarithmically divergent part of 3,2 is proportional to D is consistent with the general factorization formula (3.10) (note that XI = 0 and < Vg >1 ~ D). To explain why massless propagator contributes effectively ene let us rederive the restriction (8.9) starting from the assumption that 2: can be reproduced by an effective field theory (for similar analysis in the case of the annulus see Refs. 14 and 16). According to the discussion in Sec. 5 (see 5.6) we should have

i

A

Zz

=

82

= (U2 -

1 aUI

-I

aUI

2' atp' !J.'j atpi )'1';. + ...

,

(8.21)

• We are assuming, of course, that the integrals in (8.11) are restricted to the fundamental region and hence the integrals over kl and k2 in the factorization limit appear restricted to the fundamental region of the modular group of the torus.

537 If we use the a-model parametrization of the fields G/l V = 8/l v + h/l v and , i.e. use the tree action (2.21) (2.22) then llifl is non-diagonal and is expressed in terms of I';~ (see (2.29), (2.30), (2.20». We need only the "GG" element of this matrix since in the a-model parametrization UI = dl J dDyVG + .... Hence we find 9#tAj l ~ ~ df8/l V (8/l c,.lJv(3) 80/(3ll-1 (0) ~ df Df.'fIE (where we have used that (0/ ll) -I (0) ~ -f.'fIE, see (2.50». Equivalently, we can use the "Smatrix parametrization" of the fields (2.23}-(2.27) in which the propagator diagonalizes. The sphere plus torus contributions to the EA are then given by (D=26)

=

o/do

! dDY(-~h'llhh'- ~'flo' 1 '/l

+2' h/l

+

1

D

2' v'D _

llh~~v = flo-I (8f0/8p) -

D

+ ... ) + dl

!

dDy(l+

,

2 + ... ) ,

~ 2 8/l

V

8",(3), flo

=-

0

(8.23)

(we use the harmonic gauge for the graviton). Using (8.21) (or directly integrating over hi and

') we get for the O( df f.'fIE) - term in 82' S2

liD = -d2l dO_ f.'fIE( - - - - + 2 D - 2

D2 ) 4( D - 2)

1 2 -I = --d dO f.'fIED . 4 l

(8.24) This result is in agreement with (8.9). The two terms in the brackets are the contributions of the graviton and the dilaton exchanges.

9.

APPROACH BASED ON OPERATORS OF INSERTION OF TOPOLOGICAL FIXTURES

In this section we shall rederive some of the results of the previous sections by using the representation of the loop part of the string generating functional in terms of the operators insertion of holes, handies, etc. Similar approach was originally suggested in Ref. 18 and recently discussed also in Ref. 32 (our formulation is closer to that of Ref. 32).

t

1. Let Io = J ~ zv'§xll x ,ll = - V2 be the free string action on an arbitrary curved 2-surface (for notational simplicity we shall often set 2 no/ = 1 and do not explicitly indicate the index of x/l). Consider the expectation value


= ![dX]e-rOF[x], < 1 >= 1

(9.1)

• We again replace ( 1/ k2 hz --->0 by -0/ f.'fIE. This normalization is consistent with the 2-n= 1 h n =2'

1,

n = 0,1 ... ,

J -

-

~x,Vn'~ax,Va=O

(9.19)

(the subscript n indicates that Vn = Vn - Va corresponds to a genus n surface). The representation (9.17')-(9.19) may be a bit misleading: it may look as if it is possible to integrate first over the moduli and then to compute the expectation value on the plane. However, this is not possible to do in general since the a-model action Ja + Jim and the expectation values depend implicitly on moduli through the integration region for the 2-d coordinates (for n ~ 1 the amodel is, in fact, defined not on a sphere but on a higher genus surface). Still (9.17)-(9.19) may

540 be useful in the case when we study particular regions of the moduli spaces corresponding to surface degenerations (or when we consider the vacuum partition function, putting Jint = 0). Let us now demonstrate how the representation (9.18) works on the examples of the disc and the torus. Expanding the non-trivial part fJ of the Neumann function on the disc in the "plane with a hole" representation (7.43) in powers of radius of the hole, using 8(z - w)-I = niP) (z - w) in order to reduce 8(z - w)-n to an-I 6(2)(z - w) and integrating by parts to get rid of the 6-functions, one finds the following expression for the functional h (9.19) (see also Refs. 18 and 32) (2no/ = 1) (9.20) If we now consider the limit a -> 0 (: eh ,/> := 1 + 4na2 : ax ax : + ...) and substitute (9.20) together with the correct modular measure given by (7.50) into (9.17), (9.18) we easily reproduce the con.sistent result (7.51) and (7.52) for the logarithmically divergent part of the disc contribution to the generating functional Z1/2. Note that the normal ordering in (9.18) leads to the normal ordering in the operator 01/2 in (7.52).

Using the Schottky-type (plane with 2 holes) representation for the torus, substituting the non-trivial part fJ of the corresponding Green function (7.11) into (9.16) and going through the same steps as in the derivation of (9.20) we get the following expression for the corresponding functional hI (9.19) 18).

hI

-f1

m!(

~-

I)! 1

= 4n {

2 I 4£nlkl [xW - x(Tj)]

~mkm [arl«e +c.c] }

Tj)ma{x)](€)[a;--I«e - Tj)ma"x)](Tj)

== hI (e, Tj, k, [x])

.

(9.21)

The first term here originates from the last non-holomorphic term in (7.11). Like the second term in (9.21) it can be represented (by using Taylor expansion) as an infinite series in powers of Tj and derivatives of x (cf. (9.13))

e-

• It is useful to note that

541

:e

h1

:= 1 +

t:~~II€ -

7)1 2 : 8x8x : W + ...

(9.22)

Considering the tadpole factorization limit € -+ 7) and integrating over,€ and 7) (see (7.8), (7.15» we reproduce the result (7.20) for the divergent part of the generating functional ZI . It is possible also to derive the general expression for h" and hence for the operator of insertion of a topological fixture Q" (9.14) for arbitrary genus n representing E and w in the Green function (6.7}-(6.9) in terms of the Laurent series in z and w 18). As we discussed in the previous sections, to have the correct factorization 15) and hence renormalization properties of Z it is necessary also to go from the flat plane with holes to a curved 2-space. Then the curvature dependent correction terms appear in the normal ordering relations and in the modular measure. 2. In order for the renormalization group to operate at string loop level the tadpole divergences found above should, in fact, exponentiate to become counterterms for the string = c.,O-1 J dJ.L" < e- I - 61 >", 8I = 2:::1 b"C)"t:11E + .... action (see Secs.3 4, Eq.(3.20» Higher powers of t:11E appearing from the expansion of e- 61 in the string coupling should cancel higher order tadpole infinities in higher genus contributions to Z.

z..

For this to happen the relative coefficients of the loop contributions to Z should take particular values. Let us assume that the exponentiation does indeed take place, for example, the tadpole divergence on the torus exponentiates. Consider the regions of the moduli space (for each genus n) corresponding to surfaces with small and far separated handles. It is then natural to expect that the exponentiation of the I-loop tadpole divergence implies that the contribution to Z coming from the specified regions of moduli spaces can be effectively represented (9.23) where (see also Refs. 18 and 32) (9.24)

I

~2PI>'17I't:'IlEC)I[xl+ ... ,

-I

PI=CICO

(9.25)

(cf. (7.15), (7.20) and (7.38». Eq. (9.23) corresponds to the "renormalization group improved" perturbation theory or gives a "dilute handle gas" approximation for Z •. The representation (9.23) may be true only for small handles since it is only in this case that we can ignore the dependence of Jint on the moduli through the integration region (for small handles we may approximately consider the a-model as being defined on a sphere).

• Let us emphasize that the possibility to "resum" Z in the form (9.23) depends crucially on the use of the extended parametrization of moduli with the 0 -I appearing as a universal factor for all contributions to Z.

542

In Eq. (9.23) we have explicitly indicated the dependence on the (bare) string coupling constant. Since g is related to the constant part of the dilaton it should renonnalize together with cP (see Secs. 3 and 4). This observation suggests that there should be higher order tenns in (9.23) which should generate the renonnalization of g = e~' in the exponent. A natural guess is

z = a-I Eg2(t>-I)c" f djj." < e-I,..+g'H, >" + ... 00

(9.26)

... 0

where the wave in djj." indicates that the corresponding integration regions over moduli do not contain configurations in which all handles are small and a separated (it is assumed that we have already summed the contributions of such configurations). The consistency of the RG demands that we should be able to resum (9.26) further, generating higher order tenns in the exponent (which produce higher order tadpole renonnalizations) (9.27)

if" = p"

f d[J,,,: e

h

.:,

p" = c,.cQI.

(9.28)

The hat in the modular measure indicates that the measure and the integration region are chosen in a way that avoids overcounting equivalent configurations after we expand (9.27) in powers of g and compare with the usual expansion in genus. if" contain the lowest order 0(£71£) singularities corresponding to the factorization on afinite part of genus n tadpole and a correlator on the sphere. In this way we may reproduce the expression (3.21) for the tadpole countertenn. As we already noted, a representation based on (9.23), (9.27) can be valid only approximately because of the dependence of Iifll. on moduli through the integration region. However, Iifll. is absent in the case of the vacuum partition function. In this case we may expect that Z can be exactly represented as a correlator on the sphere (9.29) where F is given by (9.27) and n -I should be defined using some regularization prescritpion (e.g. a -I -+ ala£7I£). F can be considered as non-local (depending on all powers of derivatives of xl' integrated over moduli) addition to the free string action. Only the leading 0 (£71£) infinite part of F should be local, since it should coincide with the tadpole countertenn (3.21) Foo = - 2::1 b"O,,£7I£ + O(£n2 g). Eq.(9.29) represents a resummation of the usual perturbation theory *

* There may be two possible strategies of how to try to "resum" the usual perturbative expansion for the string partition function. The first is to represent Z as a path integral for a simple free string action over a complicated "universal moduli space" 69) , which includes all moduli spaces for finite genera as subspaces. The second is to represent Z as a path integral simply over a sphere but for a complicated 2 - d action. According to some previous suggestions 70) this action may depend non-trivially on some additional to xl' variables (ghost fields). According to our ansatz (9.29), this action may depend only on xl' but in a complicated non-local way.

543 3. To illustrate how the functional F in (9.27), (9.29) is constructed let us consider the Q(g4)-tenn in F (9.27). Expanding e:F in powers of g we should reproduce the ordinary perturbation expansion in genus n. Namely, we should have A

Z2(0) =

1

1

A

A

2

con- < H2 + "2(HJ) >0

(9.30)

where Z2 (0) is the usual genus 2 contribution to the string partition function, given e.g. in the Schottky parametrization, by Eq.(S.lO). Thus we get the following equation for the modular measure in 1I2 (9.2S)· (9.31) where we have noted that 0= 1 and used the superscripts (1,2) to denote the two different sets of genus one moduli. dJ1.1 is given by (9.24), (7.S) and hi - by (9.21). The expectation value of the product of the two operators Q =: e h easily computed in general using (9.7), (9.12), (9.4)

= exp{ -Qtr£n( 1 +

2

60 . V(I)

(l +

~

:

(9.14) can be

. V(2)

-Qtr£n[ 1 + (~(I) + ~ (~) . Vol} 2 = exp[ -~tr€.n( 1 -

~o

. V(I)

.

~

. V(2)]

(9.32) ==

==M2. Substituting here the expression for VI (7.9) or (7.11) we detennine M2 (e(l) ,7)(1) , k(l); e(2) , 7)(2) , k(2). Note that M2 depends non-trivially on the both sets of moduli (the expectation value < ... >0 "couples" them). The expression in the exponent can be simplified by using (once) the relation ~V = -~wa(Z)t;;-b\Wb(Z) (see (6.S». This relation can be used twice in the first tenn of the expanion of M2 in powers of V (cf. (7.62»

(9.33)

1+

DI(6-7)\)(6-7)2)lj2 21' ( )( C) 2(2'11-)2£nlk\1 £nlk21 d z\d Z2 (Z\ - 6) z\ -7)1 Z2 - ... 2 X(Z2 - 7)2)

1-2 + ...

• Note once again the crucial role of the use of a parametrization in which the Mobius volume factorizes.

544 where we have applied (7.10) and changed the superscripts (1,2) for the subscripts for correspondence with (S.lO). Using the fonnal projective invariance of M2 and the measures dJJ" we may fix it in the same way as we did in Sec. S: 6 = 00, 1)1 = 1, 1)2 = 0, 6 = E. The limit E-+ 0 then corresponds to the factorization of the genus 2 surface on two tori (see (S.11)-{S.,1S». According to (7.33) the second tenn in (9.31) contains the logarithmic singularity which appears to cancel the corresponding singularity (S.lS) in dJJ2 if tq2 = (271")4, cf. (S.19), (S.20) (the same condition guarantees the cancellation of the quadratic tachyonic infinites originating from the leading order tenns in the expansions of the measures) '.

J

The cancellation of the tadpole divergence in

J dJJ2 against the singularity in the ex-

pectation value of the square of the one-handle operator in (9.31) does not imply that all leading multiple divergences in the generating functional Z may be represented by multiple insertions of iIi. In the case of a non-trivial background (lint =I 0) one is to consider factorizations with external "legs" on one partofadiagram. However, if a generalization of (9.29) with:F -+ :F -lint is true at least as some approximation (cf. (9.23), (9.26), (9.27» (which should be sufficient for the analysis of the RO), the singularities which appear in :F and hence the corresponding countertenns should be universal. Treating H" as a kind of vertex operators, we expect to find singularities when the integrands of (arbitrary number of) H" and (or) of the ordinary vertex operators in lint "collide". This easily explains the exponentiation of the external leg divergences 18) and also implies that the loop-corrected ,B-functions should contain tenns of all powers in the massless fields ". The question of renonnalizability of Z is thus reduced to that of renormalizability of the theory 1 I - :F on the sphere.

=

There are some subtleties in the renonnalization procedure at the ink e, k ~ 2, level. Consider, for example, the genus 2 contributions. Splitting HI, H2 into a finite piece and a logarithmic divergence (proportinal to 0,,), H" = H", + HfIIX), we find for the part of Z2 which is divergent to the tadpole singularities (9.34)

o

The first two tenns here are (i'1lE:) while the third is O(in2 e). From the analysis offactorization one, however, expects to find an additional 0 (in2 e) tenn corresponding to the 2-point function on the torus connected with the tadpole on the torus and the sphere with external leg insertions. One possibility could be that the combinatorics of string diagrams is such that this extra tenn combines together with the tenn corresponding to the two separate tadpole insertions on the sphere to be in agreement with the last term in (9.34). We expect, however, that there is an additional O(g4in2 e) term present in the exponent which produces an additional in2 e • Note the presence in (9.33) of the same factor D which we have found in Sec. S from the expansion of the determinant of the period matrix factor in dJJ2. •• At this point we disagree with Refs 17 and 32 where no higher order tenns in ,B are found (these authors do not account for the contributions of momentum dependent singularities of the amplitudes (see Sec. 3).

545 term in (9.34). This is suggested by the observation that we are also to renormalize the string coupling which appears in:F. To check this it is desirable to study the OPE relations for the operators Nfl. We finish with the remark, that the approach discussed in this section seems to be potentially very powerful. Combined with the requirement of renormalizability it may give a clue to a more fundamental formulation of the theory. To see if this is actually so one is still to get a deeper understanding of how the RG is actually realized at higher ( n = 2,3 ... ) loop orders and to resolve the problem of proper counting of higher f.nk£, k ~ 2 divergences. There also remain some open questions concerning a prescription for subtraction of the Mobius volume divergence.

Acknowledgments The author is grateful to Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International ~entre for Theoretical Physics, Trieste, where this work was completed. He is also most grateful to D. Amati, E. Sezgin and S. RandjbarDaemi for the hospitality extended to him at ICTP and SISSA and for useful conversations. Finally, he would like to thank D. Amati and J. Russo for helpful discussions of some issues related to the topic of this paper.

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550

BABY UNIVERSES Andrew Strominger Department of Physics University of California Santa Barbara, CA 93106

551 Table of Contents

I. Introduction II. Topology Change and Third Quantization in 0+1 Dimensions 2.1 Third Quantization of Free One-Dimensional Universes 2.2 Third Quantization of Interacting One-Dimensional Universes 2.3 The Single-Universe Approximation and Dynamical Determination of Coupling Constants 2.4 The Third Quantized Uncertainty Principle III. Third Quantization in 3+1 Dimensions 3.1 The Gauge Invariant Action 3.2 Relation to Other Formalisms IV. Parent and Baby Universes 4.1 The Hybrid Action 4.2 Baby Universe Field Operators and Spacetime Couplings V. Instantons-From Quantum Mechanics to Quantum Gravity 5.1 Quantum Mechanics 5.2 Quantum Field Theory 5.3 Quantum Gravity 5.4 Axionic Instantons 5.5 The Small Expansion Parameter VI. The Axion Model and the Instanton Approximation VII. The Cosmological Constant 7.1 The Hawking-Baum Argument

552 7.2 Baby Universes and Coleman's Argument Acknowledgements References Figures

553 I. INTRODUCTION

The subject of baby universes and their effects on spacetime coupling constants [1,2,3,4,5,6,7] is in its infancy and rapidly developing. The subject is based on the non-existent (even by physicists' standards) Euclidean formulation of quantum gravity, and it is therefore necessary to make a number of assumptions in order to proceed. Nevertheless, the picture which has emerged is quite appealing: al\ spacetime coupling constants become dynamical variables when the effects of baby universes are taken into account. This fact might even solve the puzzle of the cosmological constant [8,9,10]. The subject therefore seems worth further investigation. Several important, as yet incompletely answered, questions are 1. How does one describe, even formally, a system of interacting universes? An ordinary

quantum mechanical system is described by a set of initial data along with laws governing time evolution. Since each universe has its own time, such a description is not directly applicable here. Consistent laws of physics and interpretational rules for the many-universe system must be found.

2. How can the dynamics 01 the many-universe system be approximated? The description of the many-universe system will involve a formal sum-over-four-geometries and topologies which is intractable for a variety of reasons. In order to better understand its properties, and certainly to make any physical predictions, approximation methods must be developed. One aspect of this is that, since we appear to live in a single universe, it 'is important to understand when the single universe approximation is valid.

3. To what eztent can low-energy couplings be predicted independently 01 the precise details 01 Planck-scale physics? It has been argued that one effect of baby universes is to shift the low-energy cosmological constant to zero, and that this result does not depend on details of Planck scale physics. Why is this possible? Can other low-energy couplings be determined as well?

554 In this article we will address all three of these questions. We will have something

fairly definite to say concerning the first two. The last question appears unresolved at present [11,12,13,14,15,16J, so our comments will be incomplete. The organization is as follows. In Section II we consider a toy model of topology change in a no-space one-time dimensional universe. This model will be used to illustrate, in a simple context, the proposal of "third quantization" as a means of defining a system of interacting universes. We show how the third quantized equation of motion becomes a dynamical equation for the second quantized coupling constants in the one-dimensional universe. In Section III it is explained how third quantization is applied by analogy to 3+ 1 dimensional universes. A new feature, third quantized gauge invariance, arises and is discussed. In Sections IV, V and VI small expansion parameters and approximation methods for describing this third quantized field theory are discussed. IV describes the parent-baby universe approximation, which involves an expansion in the ratio of low-energy scales to the Planck or baby universe scale. V contains a mini-review of instantons, beginning with quantum mechanics and discussing in some detail the subject of gravitational instantons. In VI we discuss the axion model, for which the third quantized system can be quite ex-

plicitly described in the instanton approximation. In VII we discuss the possibility that baby universes are responsible for the vanishing of the cosmological constant. I would like to thank my collaborator, Steve Giddings, for many discussions on baby universes. Much of the material herein is part of the collective folklore, and should not be attributed to Giddings and me. I have otherwise attempted to give proper references. Sidney Coleman and Stephen Hawking have in particular had a deep influence on many aspects of the subject.

555 II. TOPOLOGY CHANGE AND THmD QUANTIZATION IN 0+1 DIMENSIONS

In this section we are going to consider, as a toy model, a quantum field theory in a universe (or universes) with zero space and one time dimension. In the absence of topology change, this is mathematically equivalent to a quantum mechanical particle(s) moving in a potential. IT topology change is allowed, space is not described as a point, but as a set of points. The interacting many-universe system is no longer equivalent to a quantum mechanics model. We shall see that it becomes equivalent to a 'third quantized' manyparticle quantum field theory on a larger space, and that the third quantized equation of motion is an equation for effective coupling constants in the one dimensional universe. Mathematically, this section contains nothing new. We are merely repeating the wellknown steps leading from first quantized, single-particle quantum mechanics to a second quantized, many-particle quantum field theory. However, we shall say different words as we take these steps, in order to explain their relevance to the problem of topology change and interacting universes. The reader should be forewarned that, while there are many features common to the problems of topology change in one and four (the real case of interest) dimensions, there are also important qualitative differences. For example, in four dimensions the spatial topology is described by some (possibly disconnected) three-manifold. It has been shown [17] that all three-manifolds are cobordant. This means that given any two three-manifolds, there always exists a smooth interpolating four-manifold whose boundary is the two given three-manifolds. As we shall see in the next section, this leads to a very natural expression for the quantum transition between the two topologies in terms of a functional integral on the interpolating four-manifolds. In contrast, in one spacetime dimension topology change never occurs smoothly in this sense. The spatial topology is just a set of disconnected

556 points, and n points is cobordant only to n points. There is no smooth one-manifold which interpolates between one point and two points. It is therefore much more natural to consider topology change in four (in fact two or higher) dimensions than in one dimension. In one dimension the topology-changing interaction must be introduced in a rather artifical way, and its form is rather unconstrained. Nevertheless, it does provide an instructive example. *

2.1 Third Quantization of Free One-dimensional Universes Consider a one dimensional universe described by the second quantized action (see e.g. [19]) S=

iT,

T/

1··

2

d.,.(e- XJlXll gllJl - em)

(2.1)

where the summation convention is implied for the index II- = 1,2, .. D. The second quantized fields are the einbein e and the D "matter" fields XIl, although e turns out to be non-dynamical. We refer to the fields XIl and e and the action S as 'second quantized' in keeping with our aim to stress the analogy with four dimensions. e is related to the one-dimensional (timelike) line element by ,u2 = _e 2d.,.2. m 2 is the one-dimensional cosmological constant. gllJl is a constant Lorentzian metric. (We are interested in the case gllJl = '1IlJl but we consider this more general action because we will later need to analyti-

cally continue'1IlJl to 61l Jl to define a path integral.) They are one-dimensional versions of the usual second quantized objects in a four dimensional spacetime. S is invariant under local diffeomorphisms of the world line

*

6.,. = E("')

(2.2)

6XIl(.,.) = E(.,.)XIl(.,.)

(2.3)

In some ways a better example is provided by string theory, which was in fact a guiding example for third quantization of four dimensional universes [10,9,18]. However the analogy is somewhat obscured by the important role played by Weyl invariance in string theory, which does not have an obvious four dimensional analog.

557 oe = e(r)e(r)

+ E(r)e(r)

(2.4) (2.5)

oS =0

The diffeomorphism invariance must be fixed in order to define the quantum theory. One choice is synchronous gauge

e(r) = 0

(2.6)

e(r) = N

(2.7)

which implies

for some constant N. N is not a gauge degree of freedom since the proper length of the world line is JNJ(rf - ri). Since the metric involves the square of e, there are two choices of N (plus or minus) which describe the same geometry. We restrict N to be non-negative in order to avoid double-counting (Le. to give a single cover of moduli space). This gauge choice leaves unfixed the global translation r-+r

+ constant.

This symmetry is generated

by the Hamiltonian

(2.8) where P" =

IrXl-'gl-''' or, in the quantum theory, pI-' =

-i~. Invariance of the quantum

theory under the (unfixed) global translation symmetry is then obtained by the constraint H,p(XI-') = 0

(2.9)

which must be satisfied by physical second quantized states,p. (2.9) is the famous WheelerDeWitt equation [201. In the absence of topology change this equation encodes all the dynamics of the theory.

(Mathematically, (2.9) is equivalent to the time-independent

Schroedinger equation for a free particle wave function in D dimensions with energy E =

_m 2 .),

558

In this simple model, tP(XI') is the second quantized "wave function of the universe." Since the metric is trivial in zero space dimensions, tP is a function only of the D matter fields XI'. It gives the probability amplitude for observing the field configuration of the universe to be XI'. To third quantize this free second quantized field theory, we write an action which is a functional of the second quantized wave function tP(XI'):

s =! = !

J J

dDXHtPHtP dD X H(gl'lI\1I't/>\1I1t/> + m 2t/>2)

(2.10)

Variation of this S with respect to tP leads directly to the Wheeler-DeWitt equation. This equation contains all the information of the second quantized theory, so the two formulations are equivalent. At this level, third quantization is rather trivial. We shall see later on that allowing topology change in the second quantized theory leads to an interacting third quantized theory. To facilitate the discussion of topology change, we would like to describe this system in terms of a path integral "sum-over-one-geometries." Such expressions are well defined

only in Euclidean space. In order to obtain a convergent path integral, we must Wick rotate both the one dimensional and ten dimensional metrics in (2.1), i.e. e-> - ie

(2.11)

'11'11->01'11

(2.12)

T-> - iT

(2.13)

xO->ixO

(2.14)

or, equivalently

559 The Lorentzian path integral is then obtained by analytic continuation (in both variables) from the Euclidean expression. The path integral "sum-over-one-geometries" is defined as the weighted sum over all gauge inequivalent field configurations on the line interval with initial (final) value

Xf(xj).

We choose the coordinate system to run from T, = 0 to Tf = 1. One then has in synchronous gauge, (2.15) The integration over the "modular" parameter N is the one remnant of the integration over the einbein which survives the gauge fixing. The motivation for the notation "G E" will be evident shortly. In synchronous gauge, the Euclidean action is given by (2.16) where gl''' here is a Euclidean metric. It is then a standard problem in quantum mechanics to show that, [21J

(2.17a)

where (2.17b) K is the kernel of a quantum mechanical system with Hamiltonian H.

It obeys the

(Euclidean) Schroedinger equation

(2.18) This in turn implies the desired result that GE is a Green function for H: (2.19)

560 The fact that the Euclidean sum-over-geometries with fixed boundaries gives a Green function for the Wheeler-DeWitt operator is also true for higher dimensional universes. It was verified for the two dimensional universes described by string theory in [22], for four dimensional minisuperspace models by Halliwell in [23] and was argued more generally to be the correct interpretation of the sum-over-four-geometries in [24,10]. It is also possible to obtain a third quantized formula for the quantity G E. The answer is (2.20) where SE is the Euclidean version of (2.10). Normalization of the right hand side to divide out disconnected vacuum diagrams is implicit. This formula is obvious since (2.19) states that GE is the inverse of the kinetic operator H appearing in SE. Hence the notation GE. G E is a Euclidean Green function of the third quantized field theory. The real time Lorentzian Green function is then obtained by Wick rotation either of (2.20) or (2.15). This one dimensional problem is actually simple enough that the entire analysis can be done without Wick rotation of either Xo or ".. This is also possible for the two dimensional universes desribed by string theory, where the real time (light-cone) and imaginary time (Polyakov) methods give the same results. However in four dimensions only the Euclidean methods are understood. For that reason it is useful to discuss the Euclidean formalism in one dimension. In particular, our treatment of the indefiniteness of the four dimensional action (due to the conformal factor) will be analogous to the treatment given here (and in string theory) of the indefinite action of the field Xo.

2.2 Third Quantization of Interacting One-Dimensional Universes Now we are going to introduce topology changing interactions. For the moment, we will just discuss the construction and properties of the sum-over-one geometries describing topology change without ascribing any physical interpretation to it. After the construction

561 has been understood, we will be in a better position to state our interpretation. Topology change is introduced in the path integral by allowing the basic process illustrated in Figure 1 and all its iterations to occur with an arbitrary weighting A. This process clearly represents one universe splitting into two, or two joining into one. We impose the natural boundary condition that the values of Xl'(r) on each of the three world lines are equal to

xC where they meet, and then integrate over all values of XC. *

The three point function of Figure 1 can be easily expressed in terms of products of two point functions. The result is

GE(Xr,X~,X~) =

-A

J

dDXCygGE(Xr, XC)

GE(X~,Xtj)GE(X~,Xtj) .

(2.21)

Iterating this process leads to arbitrarily complicated diagrams, as illustrated in Figure 2. The rule for computing the n-point function is to include every diagram with n external lines, and to count every inequivalent geometry on that diagram once and only once. A third quantized formula can be obtained for these interacting Green functions just

as was done for the free Green function in (2.20). We require.an action SE[cf>] with the property that the correlator of n field operators weighted by SE exactly reproduces the second quantized Euclidean sum-over-one-geometries with n boundary points. Such an action is given by (2.22) It is then a matter of combinatorics to verify that (2.23)

*

While this is a natural boundary condition, many others are possible. We could multiply by functions of Xl' or its r derivatives before integrating, have four or more world lines meeting, or attach group theory factors. This would lead to inequivalent third quantized field theories, in the latter case, gauge theories. This ambiguity is eliminated in higher dimensions by requiring smooth geometries.

562 Green functions for scattering Lorentzian universes governed by the free action (2.1) are then obtained by Wick rotation. We stress that the third quantized action is defined by the requirement that it reproduce the sum-over-one-geometries. This definition carries over to higher dimensions. Let us now turn to the physical interpretation of these diagrams. This is a highly non-trivial issue, and one on which there is not general agreement. As far as I can tell, the physical interpretation of the sum-over-geometries including topology change cannot be derived from the theory without topology change. Rather, one must simply postulate the interpretation, and then check to see if it is consistent or sensible. It is not even ruled out (though it seems unlikely) that there is more than one sensible interpretation. Interpretations which are apparently inequivalent to that presented here can be found in [25,8,26J. Clearly a larger Hilbert space is needed to describe a theory with topology change. The states must have support on configurations with all possible numbers of universes. How can such a state be extracted from the sum-over-one-geometries? We postulate:

The man!l universe s!lstem is described b!l a Schroedinger state 'II[4>(xi), XOJ

0/ the

third quantized Hilbert space. Here i = 1,···D - 1, and

XO

is a second quantized field operator which has been

chosen to serve as a third quantized "time" coordinate. The state

I'll'

> obeys the third

quantized Schroedinger equation: (2.24) where Jl is the Hamiltonian of the third quantized action (not to be confused with the Wheeler-DeWitt operator H). What is the interpretation of the state

I'll >? I'll > can be decomposed into components

with definite universe number in the following sense. Let JI be the universe number

563 operator defined in the standard manner from the action S. (JI of course does not commute with the full )/.) We may then define orthonormal eigenspaces

(2.25)

Jlln >= nln > and decompose

IIIf > at some moment Xo: (2.26) n

IIfn(XO) is then the "probability amplitude for n universes at time XO" or the "probability amplitude for n universes with field values XO". More generally, eigenstates of some complete set of observables, such as tJ>(x'), can be constructed. These represent coherent states (of indefinite universe number) of universes with wave functions tJ>(x'). The state

11If>

may then be decomposed at time

XO

in terms

of these eigenstates, and describes a general many-universe state. Note that the specification of an auxilliary variable playing the role of time (in this case

XO)

is necessary to make sense of the question "How many universes are there?n. Since

each universe has its own intrinsic time r, and these times are unrelated to one another, the intrinsic time r is not a suitable auxilliary variable. Instead, we use the second quantized field

XO as a

"time" variable. We can then ask the sensible question "How many universes

are there on which the matter field variable

XO

takes the specified value?n

In more general models, there may not be a variable such as

XO

which can play the

role of time, or a corresponding Hamiltonian formulation of the third quantized theory. In this case other descriptions of the theory might be developed, as for example briefly described in [10,91. However, in many cases of interest a time variable is available, and for simplicity we restrict our attention to these.

564 2.3 The Single-Universe Approximation and Dynamical Determination of Coupling Constants The third quantized state

lilt >

in general describes a system in which interactions

between universes are not small and the single-universe approximation is not valid. It is certainly true that the single-universe approximation is at some level valid for the universe we inhabit (these investigations were not motivated by any experimental observation). One therefore is especially interested in when the single-universe approximation is valid. This is a dynamical issue. In four dimensions, we shall see that the dynamics associated to the Einstein action

(plus axions) leads to universes at two widely separated scales. These are small (roughly Planck-scale) baby universes and large (roughly Hubble-scale) parent universes. One can then compute the effect of baby universes on parent universe dynamics, and ask when the latter is well approximated by single-universe dynamics. In one dimension, all universes have the same size-they are spatially just points. There-

fore there can be no separation of scales as in four dimensions. It is nevertheless possible to construct a model which mimics some (but not all) of the features encountered in four dimensions, as follows. Consider two species of one dimensional universes described by the actions

(2.27a)

and (2.27b)

where mp#mB. Sp describes a 'parent' and SB a baby universe. Note that we have set D = 1. Now include topology changing interactions of the form parent-parent-baby and

565 baby-baby-babyas illustrated in Figure 3. As before the values of X(r) must all equal Xo at the junction of the world lines, and all values of Xo are integrated over. The resultant sum-over-one-geometries is generated by a third quantized action, but now there are two fields (tPP and tPB) which create and annihilate the two species of universes. Since there is only one X, the third quantized field theory is one dimensional. It is described by the action

S[tP] = -;jdX(-(\7tPp)2 + m~tP~ 2g

- (\7tPB)2

+ m~tP~ + IttP~tPB + ~tP~)

(2.28)

A factor of g2 has been scaled out of the action to facilitate discussion of the (third quantized) semi-classical limit. Let us now consider the limit of very large mp. There is then a clear energy gap between the sectors with zero and one parent universe. The couplings conserve parent universe number mod 2, so the one-parent universe state cannot decay into a state with no parent universes. The large value of mp suppresses pair production of parent universes. It is therefore consistent to restrict attention to the one parent universe sector.

The single parent universe propagates in a plasma of baby universes. A typical process in its evolution is depicted in Figure 4. We wish to determine to what extent the parent universe dynamics, including baby universe effects, can be described by an ordinary second quantized effective action on the parent universe. To this end, we introduce a hybrid description which treats the baby universes in a third quantized manner, the single parent universe in a second quantized manner and includes parent-baby interactions by means of a mixed interaction Lagrangian S I. The utility of this alternate description will be evident shortly. S I will take the general form SI = j

dreL£i(r)tP~ •

(2.29)

566 where l,(r) are as yet unspecified local second quantized operators on the parent universe and with no parent universe and baby universes in the state

Iw B >, given by the third quantized expression

G(X"X,) =

J

f)= ,8la(X} > +,8'la'(X} >

(2.37)

where 1,81 2 + 1,8'1 2 = 1. To understand the behavior of a parent universe in this state, it is useful to insert an ideal clock into the parent universe. One may then discuss correlation functions of n field operators at times rh·· ·rn. These are given by (2.38) Since the a-states are orthogonal, this separates into two pieces:

J J

< X(Td·· ·X(Tn} >a,a' = 1,81 2 DX(r}iSp+iS11a]X(rd·· ·X(Tn} + 1,8'1 2

DX(r}iSp+iS11a']X(rd·· .X(rn } .

(2.39)

Each of these pieces looks like an ordinary correlation function, but in universes with different coupling constants a and

a'.

This separation occurs because second quantized

operators in a single universe (correponding to physical observables) do not affect the baby universe state and so cannot connect- the states la > and la' >. Thus an observer

569 who measures the value a can never talk to one who measures

a'.

Given that the result

of some measurements which indicate that the coupling constants are a (a') all future measurements will agree that the coupling constants are a (a'), as argued by Coleman [6]. This result may be rephrased using the Copenhagen interpretation of quantum mechanics [7]. Initially the coupling constants of the universe are not well defined, rather they are governed by a probability distribution. However performing a measurement collapses the wave function into the state la > (la'

» with probability 1.81 2 (1.8'1 2). (This was

shown explicitly in [7]). All future measurements are then consistent with some definite value of the coupling constants. An important feature which naturally emerges here is the idea of probability distributions for coupling constants. While all coupling constants are subject to shifts by baby universe effects, given some initial conditions or other criteria for choosing a baby universe state, it may be possible to determine their most probable values. States of the form (2.37) are still far from the most general baby universe state. In general, away from the semiclassical limit g2 = 0, one has

(2.40) Since

tPB

onalize

does not commute with itself for different values of X, we cannot possibly diag-

tPB

for all X, and the a-states do not exist.

How then, can one interpret such a system? The answer to this question is not obvious, and is not generally agreed upon. We advance here the following interpretation of the formulae for nonzero g2, which will reduce to the preceding interpretation in the limit g2 -> O.

In the limit g2->0, the value of the parent universe potential a(X). and its first derivative at one value of X uniquely determine, along with the third quantized equation of

570 motion, the values of a(X) at all other values of X. In principle this could lead to definite predictions for values of coupling constants. If g2#O, the baby universe state is subject to quantum mechanical fluctuations, and definite predictions for values of unmeasured coupling constants are no longer possible. Instead, one must speak of conditional probability amplitudes for the results of various measurements. For example, given that the potential at Xl and X2 have been measured to be (2.41)

and (2.42)

we may then ask for the conditional probability amplitude that the potential at an intermediate point Xs is measured to have the value as This is given by (2.43)

where SB is the third quantized baby universe action and the path integral is over all paths obeying al =

(Xl)

a2 =

(X2)

as = 4>(Xs )

(2.44)

and is suitably normalized so that the probabilities sum to one. It should be stressed that this does not mean that coupling constants do not take definite values. Once a coupling constant has been measured to lie in a certain range, the wave function is collapsed and it retains that value regardless of what other measurements are made. What is becoming "uncertain" here is the predictions implied by the third quantized equation of motion for relations among coupling constants.

571 Our interpretation does, however, imply that in practice it will be difficult or impossible to actually obtain precise measurements of all couplings. For example, if the field X runs over an infinite range, the probability amplitude A(o. Diffeomorphism invaria.nce of matrix elements between two physical states 14> > a.nd 14>' > is then a consequence of

< 4>' 1Hp. (x) 14> >= 0

(3.4)

which follows directly from (3.2) a.nd (3.3). Note that (3.4) is weaker tha.n the usual Wheeler-DeWitt [20] physical state condition

Hp.(x)l4> >= O. However, only the weaker matrix element condition ca.n be dema.nded on physical grounds, a.nd in string theory imposing the stronger Wheeler-DeWitt condition would eliminate all states. As for third qua.ntization of one-dimensional universes, the equation of motion of the free third quantized action should reproduce the second quantized physical state condition. The action which accomplishes this is·

(3.5) • This may suffer from the "doubling problem" of closed string field theory in that there may be multiple copies of the physical spectrum at differing ghost numbers.

575 This action has a gauge invariance generated by

614> >= QIE >

(3.6)

so we see quite generally that the third quantized field theory is a gauge theory. This gauge symmetry must be fixed in order to compute correlation functions. Presumably by doing so one can reproduce the second quantized sum-over-four-geometries, although this has not been done explicitly. The third quantized gauge invariance is preserved by interactions between universes. This can be deduced (as in string theory) from the second quantized sum-over-fourgeometries from which the third quantized interactions are derived. Consider the sumover-four geometries with n fixed three-boundaries. The sum of the BRST charges on the n boundaries vanishes by BRST conservation. This implies that the coupling of a BRST exact or "pure-gauge" state to n - 1 physical states vanishes. The diagrams thus obey Ward identities which are equivalent to gauge invariance of the interacting third quantized field theory. The nature and consequences of this third quantized gauge invariance remain to be understood.

3.2 Relation to Other Formalisms Third quantization is a specific form of the non-linear generalizations of quantum mechanics which have been discussed in the literature. Before the incorporation of topology change, the quantum mechanical wave function of the universe obeys a linear equation such as the Wheeler-DeWitt equation or (3.3). Including the effects of topology change amounts to adding non-linear terms to this equation [32]. In [40], Weinberg has discussed experimental signals of and constraints on non-linear quantum mechanics models. Part of that discussion may be relevant here. An apparently different way of defining a many-universe system has been proposed by Hartle and Hawking [25,41]. They compute the n-universe wave function as the sum-

576 over-four-geometries with n fixed boundary components, and demand that each universe separately obeys the linear Wheeler-DeWitt equation. This contrasts with third quantization, in which this same sum-over-four-geometries is an off-shell Green function and accordingly obeys a non-linear Schwinger-Dyson equation. In the Hartle-Hawking program, it is hoped that an appropriate complex contour for integration over the conformal part of the metric will insure that the Wheeler-DeWitt equation is obeyed [42]. The discussions of this and the previous sections are illustrations of the apparently general phenomona that generalizing a quantum field theory by including topology change leads to a quantum field theory on a diferent (usually bigger) space. This notion was pursued in the direction opposite to that taken here by Green [43] who suggested that the two dimensional quantum field theory of the string world sheet itself arises in this manner. The idea was taken to its logical extreme by Srednicki [37] who suggested that topology change occurs at all levels, and the universe is actuall described by an infinite sequence of quantum field theories.

5n IV. PARENT AND BABY UNIVERSES

A third quantized field theory in general allows the joining and splitting of universes of all sizes. However, the joining or splitting of a macroscopic universe from our own would lead to rather dramatic effects which have not been observed. We are, therefore, particularly interested in theories where such processes are dynamically suppressed. This suppression indeed occurs in the semiclassical approximation to theories governed by the Einstein action at long distances. By dimensional analysis, the action associated to nucleation of a universe of radius R is R2 M;. The nucleation of 'baby' universes large relative to the Planck length is, therefore, exponentially suppressed.t The third quantized description of baby universe phenomona is discussed by Banks [9] and in reference [10]. We then have two widely separated scales: the baby universe scale, denoted J.l., and the parent universe scale,

tt. The construction of the third quantized theory simplifies p

in an expansion in the small parameter instanton approximation also requires

lM'.

p.

p

iT.: « p

{Later it will be seen that validity of the 1}. To leading order in this approximation

we have only the diagrams of Figure 5 representing wormholes connecting parent universes to themselves and to one another. There are no diagrams representing bifurcation of parent universes, this is assumed to be exponentially small in

~.

Bifurcation of baby universes

is also negligible relative to the interaction depicted of three baby universes coupling via a parent universe, since the latter process is enhanced by a phase space factor of the cube of the parent universe volume in Planck units. The processes in Figure 5 resemble Feyman diagrams with the wormholes representing propagators and the parent universe vertices. Thus the baby universes couple to one another via interaction with the parent universes, whose second quantized couplings are t This assumes the sign of the action is positive. In fact for the nucleation of one universe from nothing, the sign is negative and production of large universes may be enhanced. Further discussion of this important sign issue can be found in Sections V and VII.

578 in turn determined by the state of the baby universes. This provides an unusual feedback mechanism between long and short distance physics-the baby universe system knows about the long distance couplings. This circumstance forces us to reexamine the usual lore that values of low energy couplings - such as the cosmological constant - can be understood independently of short distance physics. This will be further discussed in Sections VI and VII. 4.1 The Hybrid Action

To analyze the parent-baby universe interactions, it is convenient to construct a hybrid representation of these diagrams using second quantized parent universe variables and third quantized baby universe variables as was done for the one dimensional parent-baby universe model in Section II. This hybrid formula will reproduce the Euclidean sum-overfour-geometries. We simply state the answer, various pieces of which are derived in [6,7,10,9,13,44] and leave it to the reader to verify the combinatorics. Let baby universe field operator

tPi be a mode of the third quantized

tP which creates or annihilates a

baby universe of type i from

a parent universe (4.1) where the Ii are a set of orthonormal functions on the space of (small) three metrices on S3. The effect of nucleation of a small baby universe (of type i) on an observer in the parent universe are equivalent to the insertion of a local operator, denoted 'ci(Z), at the nucleation event

Z

[3,4,5,6,7]. Let Gij be the propagator defined by the sum-over-four-geometries

from a baby universe of type i to type j, i.e.,

(4.2) The second quantized sum-over-four-geometries is then reproduced by the third quantized functional integral [10,13]

579

(4.3) where

(4.4) and (4.5) Summation over the (potentially continuous) indices i,i is implied. "'I here is the coefficient of the Euler character which appears in the second quantized action. For the topologies we consider, this counts the number of closed loops of universes. e- 2-y therefore plays the role of a third quantized Planck's constant. The functional integral

J[)3 g•

denotes

an integration over small three geometries (corresponding to baby universes) as well as possible additional matter fields.

q,

is a function on this space.

J[)491

is an integration

over large four geometries (plus possibly other matter fields) on the parent universe. Ai are the fundamental coupling constants. We see from this formula that the effective parentuniverse coupling constants are Ai -

q,i

and are shifted by the baby universes.

Without going through all the details, it can be seen qualitatively how (4.3)-{4.5) reproduces the sum-over-four-geometries. The parent universes act as vertices, and the integral over large four geometries in the interaction term of S corresponds to the fact that there is one vertex for each configuration of the parent universe. The use of G- 1 as the kinetic operator then produces the desired factor of G with each wormhole propagator. An important issue, which awaits further clarification, is the relation between the action (4.4) for Euclidean universes and the action for Lorentzian universes. Formally the two are related by factors of i (arising from the rotation of the lapse function) exactly as

580 in the case of one dimensional universes discussed in Section II. However this issue cannot be entangled from the problem of the conformal factor, which renders the action indefinite in Euclidean space. A proper understanding of this issue is essential to further progress in the subject.

4.2 Baby Universe Field Operators and Spacetime Couplings To understand the relation between the third quantized baby universe operators

tP. and

second quantized coupling constants, it is convenient (but probably not essential) to assume the existence of a Hilbert space representation of the third quantized field theory. The are then operators on this Hilbert space, and the state I'"

tP.

> of the baby universe system

is described as a state in this Hilbert space. Measurements on a single parent universe are then represented as correlation functions of second quantized operators O.(X.) in this state

Let us now suppose that the third quantized theory is classical in the sense that the loop expansion parameter e- 2, is taken to zero. In that case, the tP's all commute and may be diagonalized in terms of coherent a-states I{ad > obeying tPil{ad > = ail{ao} >

< {ao}l{aH > = IIo6(ao - aD.

(4.7)

If the third quantized system is in an a-state, the correlation function (4.7) becomes, after

normalization,

We see from this expression that the second quantized parent universe couplings (below the wormhole scale) are shifted by the eigenvalues ao of the third quantized field operators

tP,

[6,3,7,4,5,451·

581 In general, there is at least one species of baby universe for every local operator ..ci(X) so, in the absence of a symmetry forbidding the coupling of the baby universe, one expects that all low energy couplings will be shifted by the a parameters. The field ';S; 0 (X) .

(4.11)

topologies

Coleman attempted to derive this formula from the Hartle-Hawking wave function of the universe. We see here that it arises in computing expectation values in the third quantized baby universe ground state. In general the state

1tJ. > of the baby universes may not

be one of the

I{a;} >

eigen-

states, as discussed in Section 2.4. For example suppose it is in a linear superposition of two eigenstates

1tJ. > = .B1{ai} > +.B/I{a~} >

(4.12)

582 where 1.81 2 + 1.8'1 2 = 1. The correlator of n parent universe operators then becomes

2 = 1.81 < Ol(Xl)" ·On(Xn) > {ai}

+ 1.8'1 2 < Ol(Xd·· ·On(Xn) >{a:}

.

(4.13)

The correlator is split into two separate pieces because neither second quantized operators nor the full third quantized Hamiltonian connect different a-states. There is a superselection rule which prevents us from interfering different a-states. This means that an observer who measures a will never know about the one who measures d. Given that the results of a set of measurements indicate a set of couplings {a;}, all future measurements will be governed by dynamics with the couplings {a;} [6]. This generalizes to states involving superpositions of all possible eigenvalues of the form

II >= II;

i:

da;/( {a;}) I{ a;} >

(4.14)

where

(4.15)

I({a;})f*({a;}) is then the probability that the universe is governed by the set of couplings {a;}. Allowed values of {a;} are constrained by the third quantized equation of motion. It is natural to consider general states of the form (4.15) because there is no particular

reason that the baby universes are in an a-state. For example the third quantized ground state is in general not an a-state. Predictions for physical couplings are possible if the baby universe wave function is highly peaked on a subspace of the {a;}. The preceeding discussion has assumed the existence of a set of coherent states obeying 4>il{aj} >= a;1{aj} >. Such states exist only in the semiclassical limit for which

583

[.,;J

= O. Away from that limit

[.,;J

is in general order e- 27 , and the a-states

do not exist as states in the third quantized Hilbert space. The eigenvalues of is also imaginary. This imaginary

part is directly related to the decay rate of the particle from XF into the true vacuum [46J. The validity of this derivation, and the instanton approximation in general, depends on the smallness of ee- S /

r

s/r , which in turn follows from small g2.

can be seen by noting that the instanton density,

N T

dF dB

The relevance of small

< t} >, is given by

--2

< _ >= g2-= = Ke-S/9 < 1.

(5.15)

When this parameter is not small, the instantons are close together and interactions between instantons are important. Ignoring these interactions, as we have done here, is known as the dilute gas or dilute instanton approximation. The form of the bounce solution also tells us what IXF > decays into. There is a moment "s of time symmetry where

X ("s) = 0 and X("s) =

XT. Cutting the instanton

in half at this moment, one obtains a saddle point which contributes to the matrix element

(5.16) After the particle tunnels to XT, it then oscillates around the (true) bottom of the potential. The instanton thus describes tunnelling between two classical solutions of the same energy. Note that the entire history of the particle can be described semiclassically. Before tunneling it is approximated by a classical particle at the stable minimum. It then undergoes semiclassical tunneling. After tunneling its evolution is again classically described as oscillating motion. The initial data determining the post tunnelling behavior is obtained from the instanton solution. Such a semiclassical description of instanton processes in quantum field theory and quantum gravity is also possible. The second type of instanton which arises in quantum mechanics exists for the potential of Figure 8. This double well potential has two degenerate minima at X+ and X _. To find

589 the instanton, the potential should be turned upside down. It is then evident that there is a kink solution X(T) which begins at X_ for T = X+ at

T

=

+00.

-00

and asymptotically approaches

Unlike the previous example, it does not bounce back. Because of the

boundary conditions X(±oo) = X±, this solution can not be continuously deformed to either of the trivial solutions X(T) = X±. Correspondingly, there is no negative mode. The instanton gives a real contribution to the matrix elements In < X+le-HTIX_ > = In < X_le-HTIX+ > = KTe-S/r/

(5.17)

as computed in the dilute instanton approximation. This implies that the states IX± > do not diagonalize the Hamiltonian. Rather H is diagonalized by the coherent states

IX+ > ±IX_ >, and the energy splitting between these states can be computed using (5.17). The lesson to learn here is that the existence of an instanton with no negative modes in general signifies that the quantum vacuum is constructed as a coherent superpostion of classical vacua.

0.2 Quantum Field Theory

We now turn to some examples in four dimensional quantum field theory. Consider the action (5.18) where the potential V is the same function appearing in the first example above with vacuum decay. Vacuum decay occurs here as well.

There is a Euclidean solution of

the form illustrated in Figure 9, with a round bubble of the true vacuum inside a sea of the false vacuum. This solution can be continuously deformed to the false vacuum, and correspondingly has one negative eigenvalue. It therefore represents a decay process.

590 The state to which the false vacuum decays can be found as in the quantum mechanics example, by cutting the instanton in half at the moment of time symmetry. This reveals a bubble of true vacuum in the sea of false vacuum, and all time derivatives of fields vanish. Despite the lower energy of the true vacuum, this tunneling process conserves energy exactly because there is energy in the bubble wall. The nucleated bubble then begins exponential acceleration, as governed by the classical equation of motion, and grows indefinitely. Now suppose V is as in Figure 8, corresponding to degenerate vacua. Tunneling from one minima (4)+) to another (4)-) can not proceed via a bubble as in the preceeding example because energy could not be conserved. Instead all of space must tunnel at the same time. But the action of such an instanton diverges like the volume of space. Tunnelling is therefore exponentially suppressed and vanishes in the infinite volume limit. In Yang-Mills theory, instantons mediate vacuum topology change. Classical vacua of Yang-Mills theory with Fpov = 0 are labelled by integers n=trJAAAAA

(5.19)

R3

and may be denoted

In> . A Yang-Mills instanton with non-zero integral tr J F

(5.20) A F =

1 tunnels between two such

classical vacua. There is then a non-zero matrix element

(5.21) so that the

In > vacua do not diagonalize the Hamiltonian.

ized instead by the

(J

The Hamiltonian is diagonal-

vacua

(5.22)

591 Thus, as in the degenerate double well, the quantum vacua are coherent superpositions of classical vacua.

5.3 Quantum Gravity

Classically, topology change is forbidden in general relativity. If we pick initial data on a space with arbitrary topology and evolve using Einstein's equations, the topology does not change (unless there is a singularity, in which case general relatively itself breaks down). Quantum mechanically topology change might occur in principle, but there is no real argument that it must or must not occur. Our attitude is simply to assume that it does occur, try to find models where the effects are calculable, and then to see whether these effects are interesting or even measurable. We shall also assume that the Euclidean path integral and instanton techniques are applicable for computation of topology change. That is the transition amplitude between two manifolds M[ and MF is given by

< MFle-HTIM[ >=

L:

J MF

[)4ge- S

(5.23)

topologies MI

for some action S. There are several immediate problems with this formula: 1) In general, there is no well defined time 'T' to use on the right hand side. This formula only makes sense when M[ and MF are three manifolds which bound a four-manifold that has an asymptotically flat region. T is then the Euclidean time as measured in this asymptotic region. This will not work if M[ and MF are compact. In that case, the Euclidean sum over-four-geometries instead gives a Green function of the third quantized field theory, as discussed in Sections II and III, and instantons provide an approximation to this Green function.

592 2) The sum-over-four-topologies in (5.23) is problematic because four-manifolds are unclassifiable. Demonstrating the equivalence of two four-manifolds is a Goedel unsolvable problem [47). In practice we simply restrict ourselves to some subset. H this subset is closed under composition of the functional integral, the theory thereby obtained is at least naively self-consistent. 3) The Einstein action is unrenormalizable. This problem can be remedied by simply imposing a cutoff IS on loop integrations. The semiclassical loop expansion is then an expansion in requires

iT.;, p

iT.;p «

where M" is the Planck mass. Validity of this expansion therefore

1. The physical idea behind this cutoff is that some new physics (such

as string theory) which does not have the divergence problems of the Einstein action is relevant above the scale IS. We assume here (as suggested by string theory) that a generally covariant cutoff procedure exists. 4) The most serious problem, in my view, is that the Einstein action is unbounded from above and below, so expression (5.23) is not well defined even with a cutoff. To see this let us fix the sign of the action so that a transverse traceless graviton around flat Euclidean space (hTT) has positive action (5.24) where () is the flat metric. With this sign, a conformal transformation of the metric decreases the action (5.25) So for example a sphere of volume V, which is related to flat space by a (singular) conformal transformation, has action

(5.26)

593 which grows in magnitude with the size of the sphere. This bizarre fact is ultimately the origin of the Hawking factor [48] in his analysis of the cosmological constant, as discussed in Section VII. However, there do not seem to be many examples in quantum field theory with indefinite actions, and we don't really understand yet how to deal with such systems. My approach in these lectures will be to treat the indefinite modes in much the same way that the indefinite mode Xo is treated in the one and two dimensional cases, as mentioned at the end of Section (2.1). In the following this amounts to simply ignoring the fact that the functional integral is unbounded, and obtaining results by formal manipulations. There is a good physical reason why the functional integral for gravity is unbounded, and understanding this may ultimately lead to a resolution of the indefiniteness problem. The Euclidean path integral with periodic boundary conditions is equal to the canonical partition function at a temperature related to the periodicity. However the canonical partition function is, and should be, divergent for gravity because of negative specific heat associated with the attractive force. Even setting all these problems aside, it is exceedingly difficult to find gravitational instantons of the Einstein action which contribute to any physical process. Let us first consider tunneling from flat R3 to N where N is a connected but topologically nontrivial three manifold. The tunneling process always conserves energy, so N must have zero energy just as does flat R3. We now encounter the following theorem of Schoen and Yau

[49]: Theorem: There are no asllmptoticalill flat solutions 01 Einsteins equations with zero energll except flat space.

This rules out any such tunneling processes. We might then consider tunneling

(5.27)

594 which might conceivably be accompanied by topology change of a spin structure. However, we are foiled by another theorem of Schoen and Yau [49]:

Theorem: There are no as!lmptoticall!l fiat lour geometries which are Ricci fiat except fiat R4.. Now we might try to make M3 disconnected e.g., M3 = R3 ED S3. The extrinsic curvature induced on MS from the interpolating four-manifold must vanish by the analog of the boundary condition (5.8). The interpolating manifold could then be as depicted in Figure 10. Then we encounter the theorem of Cheeger and Grommol (restated) [50]:

Theorem: Given an as!lmptoticall!l fiat lour geometr!l with n > 1 compact interior boundaries with vanishing extrinsic curvature, the Ricci tensor alwal/s has a negative eigenvalue somewhere. This rules out such instantons for pure' gravity, or for gravity coupled to a scalar field for which the Ricci tensor obeys R",v = V ",,S, then becomes

L:

dq

e,qa("')~(q) =

4>(a(x))

(6.5)

where 4> is the Fourier transform of~. Putting this all together, redefining 4> by a minus sign and Fourier transforming with respect to q, we obtain

(6.6) where G-1 is the Fourier transform of

0- 1 and

is non-local in 1'-space.

part of the axion field a orthogonal to the zero mode

l'

a=

a-

l'

is the

on S4. In rewriting the integration

over the axion field in terms of a (rather than B) this zero mode is omitted. However the

va integration combines with the overall

l'

integration to give an integration over the full

axion field.

a can be evaluated in the saddle point approxand a = O. One then obtains our final expression

The integral over large four metrics and imation, for which

RI'll

=

gI'll (A

+ 4>(1'))

for the third quantized action

(6.7) If the charge q is quantized,

l'

becomes a periodic variable.

In the semiclassical limit e- 21

eigenstates of 4>(1') for all

-->

0, coherent states can be constructed which are

l'

4>(1')10:(1') >= 0:(1')10:(1') >

(6.8)

0:(1') is constrained to be a solution of the classical equation of motion of S. It represents 3M'

the trajectory of a particle in the potential e 8 (A.{4».

Comparing with (4.8) and using

equation (6.4), we then see that the effective second quantized action for a parent universe interacting with baby universes in the state 10:(1') > is Self =

M2 81rM2 d"xy'9(---E..R + - _ P \ll'a\ll'a + A + o:(a)) 161r 3 .

J

(6.9)

606 In conclusion: The spacetime arion potential is given by the classical trajectory of the

particle governed by the third quantized action (6.7). One interesting consequence is that, because of the singular nature of the third quantized potential at A + t/> = 0, the axion potential appears to always have a minimum where the bare cosmological constant A is just cancelled. To the low energy observer, this would appear to be a fine tuning of coupling constants. Here it is a conseqence of the third quantized equation of motion. This vanishing of the cosmological constant is more general than our specific model, and is further discussed in a more general context in the next section.

607 VII. THE COSMOLOGICAL CONSTANT.

Recently there has been much discussion of the possibility that non-perturbative effects in quantum gravity might account for the observed vanishing of the cosmological constant. The basic argument appears in a 1984 paper of Hawking's

[48] which received little

attention at the time. This paper was slightly preceded by Baum [68] which contains some of the important ideas. We begin by restating their argument in a slightly more modern form.

7.1 The Hawking-Baum Argument. Consider the following formula for the correlation function of n operators

(7.1) where 8 is the Einstein action plus matter

(7.2) and the functional integral is over matter fields A and metrics g on 8 4 • Formulae of the type (7.1) have been discussed by Hartle and Hawking [25,26,41] in the context of the wave function of the universe. (7.1) may also arise in an approximation to spacetime correlation functions in a third quantized theory; alternately it may simply be postulated. However the relation between the Euclidean formulae (7.1) and real time Lorentzian expectation values is poorly understood at present. Hawking's mechanism exists whenever the cosmological constant becomes a dynamical variable. To take a familiar case, let us suppose that A is a Yang-Mills field and there is a corresponding fJ angle. The vacuum state is then in general given by

If >=

J

dfJf(fJ)lfJ > .

(7.3)

608 In a fI eigenstate, the effective cosmological constant is well known to be

(7.4) for an appropriate mass M of order the Yang-Mills confinement scale. Correlation functions in the state

II > involve an integral over fl. This is equivalent to integrating over the

cosmological constant. The formula for correlation functions in the state

II> is

< 01(Xd .. ·On(Xn) >/ =

f f dfl

S. DgDAI/(fI) 12

e- S (9)(Ol(Xd" ·On(Xn))

(7.5)

where

8(fI) =

f

M2

~xyg(-16~R + A(fI)

+ C(A))

.

(7.6)

Let us now approximate the integral over the metric by its saddle point:

(7.7) We assume that A