Quantum field theory, statistical mechanics, quantum groups and topology : proceedings of the NATO advanced research workshop, University of Miami, 7-12 January 1991 9789814537605, 9814537608

Table of contents :
On looking for solutions of the star-triangle relation, R. Baxter
level crossing and the chiral potts model, B. McCoy
a quantum-generated symmetry - group-level duality and conformal and topological field theory, H. Schnitzer
differential equations for quantum correlation functions, V. Korepin
distinguishable particle in delta interaction, J. McGuire
subcritical strings in 1

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Quantum Field Theory, Statistical Mechanics, Quantum Groups, and Topology

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Proceedings of the NATO Advanced Research Workshop

QUANTUM FIELD THEORY, STATISTICAL MECHANICS, QUANTUM GROUPS AND TOPOLOGY University of Miami

7 - 12 January 1991

edited by

Thomas Curtright, Luea Mezincescu &.. Rafael Nepomechie Department of Physics, University of Miami

.~ World Scientific

. . . Singapore· New Jersey· London· Hong Kong

hblUJuJby

Worid SciUtiflC Publishill, Co. Pte. LId. POB01128, Fanu Road, Sill&~9128 USA cJfiu; SIIile IB, 1060 MaiD Street, RiyetEdce.. Nl 07661 UK olJiu: 13 L)1IIOII M~ Toaeridsc.l.ocIdoII N20 SOH

Librvy oC COIII,rtSI CataJoai.·u,· Publicatioo d.II:a i$ available.

QUAN'Il1M nELD THEORY. Sl'A11S11CAL MECHANICS, QUANnIM GRoups, AND TOPOLOOY Copyriahl e 1992 by World Scieillific PublWUlI1 Co. Pte. LId.

All ri,lW ru~nwL TIIU boaI:. or pam /Ju'~4

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k np~ed ill QIIJ 10"" orlly_JIrWilIU, ~~,rottic orm«;IuutiCDl. includin,pI!otoc.,u.,. r«orditf,ortIIfJ inlOf7PllJtioft Jt(Jtrlg~ attd uttWN2l spklff _ btoMft or t(J k itrvotl«l, ....uilotlr

wrillDl pumiDiMl/rc- /Ju hblish~r.

ISBN 9789814537605 ISBN 911.o:z.196G-6 (pbk)

Printed in Sinppore by Utopia Preu.

v

CONTENTS

Preface Integrability in Random Matrix Models L. Alvarez-Gaume

ix 1

On Looking for Solutions of the Star-triangle Relations R. J. Baxter

11

Casimir Effect in Conformal Theories in Higher Dimensions John L. Cardy

17

Quantum Cosmology on the Worldsheet A. R. Cooper, L. Susskind, and L. Thorlacius

26

A Note on the Branching Rule for Cyclic Representations of Uq{gln) Etsuro Date, Michio Jimbo, and Kei Miki

46

Exact S-matrices for Toda Theories based on Lie Superalgebras G. W. Delius, M. T. Grisaru, S. Penati, and D. Zanon

55

Superconformal Algebras as Hidden Symmetries in Topological Conformal Field Theories Tohru Eguchi, Shinobu Hosono, and Sung-kil Yang

68

Quantized Planes and Multiparameter Deformations of Heisenberg and GL{N) Algebras D. B. Fairlie and Cosmas Zachos

81

p-Adics and q- Deformations

93

Peter G. O. Freund

The Quantum Group Structure of 2D Gravity Jean-Loup GenJais A Note on Liouville Theory and the Uniformization of Riemann Surfaces Cesar Gomez and German Sierra

97

115

vi

Gauss Codes and Quantum Groups Louis H. Kauffman Finite-size Effects as a Probe of Non-perturbative Physics: Perturbed CFTs, Factorizable Scattering, the Thermodynamic Bethe Ansatz, and All That Timothy R. Klassen and Ezer Melzer Differential Equations for Quantum Correlation Functions V. Korepin Correlation Functions of Sine-Gordon Model at Free Fermion Point as Fredholm Determinants H. [toyama, V. E. Korepin, and H. B. Thacker

123

155 175

176

Level Crossing and the Chiral Potts Model Barry M. McCoy

184

Distinguishable Particles in Delta Interaction J. B. McGuire

193

Exactly Solved Models with Quantum Algebra Symmetry Luca Mezincescu and Rafael!. Nepomechie

200

A Quantum Generated Symmetry of Conformal and Topological Field Theory S. G. Naculich, H. A. Riggs, H. J. Schnitzer, and E. J. Mlawer

218

Noncritical Strings Beyond c = 1 Joseph Polchinski

233

Sub critical Strings in 1 < D ~ 25 Christian R. Preitschopf and Charles B. Thorn

241

Recent Results on Two-dimensional Supergravity z. Qiu

255

Quantum Supergroups N. Reshetikhin

264

vii

Loop Calculations in BRST Quantized Chiral W3 Gravity K. Schoutens, A. Sevrin, and P. van Nieuwenhuizen Solvable Lattice Models, Corner Transfer Matrices, and Infinite Dimensional Lie Algebras H. B. Thacker

283

302

Null Strings, Twistors, and Higher-Spin Algebras P. K. Townsend

313

Integrable Field Theory of Self-avoiding Polymers in 2D A. B. Zamolodchikov

324

Using Functional Methods to compute Quantum Effects in the Liouville Model T. Curtright and G. Ghandour

333

List of Participants

347

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ix

PREFACE

In recent years, much progress has been made on integrable two-dimensional systems and conformal field theory, and on related topological quantum field theories in three dimensions. Quantum groups and knot theory provide important mathematical links between these subjects. Because of the close interrelation of all these disciplines, new discoveries in one of them have quickly led to progress in the others. This volume contains the proceedings of a NATO Advanced Research Workshop on these interrelated disciplines. The workshop took place 7-12 January 1991 in the new James L. Knight Physics Building at the University of Miami. The principal goal of the workshop was to advance the pace of discoveries in theoretical research on the chosen topics by providing an opportunity for presentations of new results, and by providing a forum to discuss likely directions for future work. The workshop brought together a unique combination of leading researchers in integrable systems, conformal field theory, quantum groups, and knot theory, thereby providing an opportunity for a cross-fertilization of ideas. Hopefully, the experience has led to further progress in all ofthese areas. A large portion of the workshop dealt with so-called "integrable models." Such models provide useful paradigms in quantum field theory and statistical mechanics. They serve as foundations for constructing many of the most important theoretical structures iIi physics, including string theory-the most acceptable candidate for a unified theory of all interactions which incorporates the standard model of particle physics. The proceedings contain a number of new results in two-dimensional physics. Among the topics that are discussed: integrable lattice models, including the chiral Potts model and its generalizations; quantum 2D gravity and Liouville theory, and their relation to string theory; and various aspects of integrable perturbations of conformal field theories. We thank all the participants for making the workshop an enjoyable and informative gathering, and we thank the funding agency, NATO, for providing a generous and realistic level of financial support. We especially thank Giovanni Venturi, former Director of NATO Advanced Research Workshops Program, for his strong encouragement. Finally, we thank George Alexandrakis, Chairman of the Department of Physics, and David Wilson, former Dean of the College of Arts and Sciences, University of Miami, for their enthusiastic sanctioning of the workshop. The Editors

1) E. MelJer 8) H. Thacker 2) V. Korepin 9) O. Alvares 3) J. Curlright 10) J. Schwall M. Grisaru 11) L. Susskind 5 L. Alvares-Gaume 12) J. McGuire 13) P. Townsend 6 M. Jimbo 1) C. Zachos 14) H. Schniber

4l

15) 16) 11) 18)

P. Freund C. Gomes D. Fairlie C. PreHschopf 19) T. Curtright 20) R. Nepomechie 21) J-L. Gervais

22) D. Caldi 23) A. Sudbery 24) G. Sierra 25) T. Eguchi 26) Z. Qiu 21) T. McCarty 28) C. Thio

29) 30) 31) 32)

P. van Nieuwenhuisen J. Polchinski A. Zamolodchikov L. Mesincescu

Participant. not shown: R. Baxter, J. Cardy, W. Fishier, G. Ghandor, L. Kauffman, H. Kleinert, B. McCoy, A. Perlmutter, N. Reshetikhin, A. Tsve1ik, H. Verlinde.

J. McGuire (I.) and R. Baxter

D. Fairlie (I.) and B. McCoy

P. Townsend (1.), T . Curtright, and A. Zamolodchikov (immediately before experimental work on "superimmersioDs")

Quantum Field Theory, Statistical Mechanics, Quantum Groups, and Topology

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INTEGRABILITY IN RANDOM MATRIX MODELS

l. Alvarez-Gaume Theory Division, CERN CH-1211 Geneva 23, Switzerland alvarez~cernVTn

INTRODUCTION Some aspects of the recently discovered non-perturbative solutions to non-critical strings [1] can be better understood and clarified directly in terms of the integrability properties of the random matrix model used to define the coupling of (p, q)-conformal matter to 2D-gravity. Soon after the appearance of the original papers on the subject , Douglas [2] showed that the continuum limit of these models can be described in terms of special representations of the Heisenberg algebra. He also exhibited some interesting connections with the KdV hierarchy [3] which strongly indicate a deep interplay still to be elucidated between string theory and integrable models. In [4] we have made a modest attempt to clarify these connections. This lecture will be based on this work. * BASIC DEFINITION OF INTEGRABILITY The notion of integrability for a classical mechanical system of n degrees of freedom goes back to the nineteenth century and is due to J. Liouville. A system with n degrees of freedom described by the Hamiltonian H(p, q) and the Poisson Bracket {,} is said to be integrable if one can find n conserved quantities Ii(P, q), i = 1 ... n in involution, i.e. with vanishing Poisson brackets {Ii,Ij} = O,i,j = 1, .. . n. When these conditions are met, the general solution to Hamilton's equations can in principle b.e reduced to quadratures. An efficient procedure to constuct integrable systems is to write them in terms of Lax pairs [7]. If the dynamical equations of motion can be written in terms of two matrices L, B with L symmetric and B anti-symmetric as dL = [B L] dt '

(1)

Ie 1 = -trL k

(2)

then the quantities lie

*

The use of discrete integrable systems to clarify some olthe properties of matrix models is abo described in [5, 61 where use is made ofthe properties of the Toda lattice.

2 are automatically conserved. This is proved by showing that the eigenvalues of L are time independent. Defining the matrix U according to dU =BU dt

(3)

the time evolution of L implied by (l)is L(t) = UL(O)U-l and therefore the eigenvalues of L, L(t)~(t) = ~,.~,.(t) are time independent. This way one relates the theory of Integrable Systems with isospectral problems. In general we have to check whether the eigenvalues of L Poisson-commute before we can claim that Liouville integrability is satisfied. One of the basic examples of an integrable system is provided by the Toda lattice [8]. The Hamiltonian is given by

H=

41:p! + 1:(e-('l·+l-V.) - 1) ,. ,.

(4)

with equations of motion dq,.

Tt=P,. (5) The Lax pair form of these equations was found by Flaschka (for references see for example [8]). Defining the variables

(6) (5) becomes Ii,. = a,.(h,. - 6,.+1)

b,. = 2(a!_1 - a!)

(7)

the equations (7) can be written in matrix form (1) by introducing the matrices L

= 1: , a,E~+1 + 1: h,E" (8)

With free boundary conditions the L-eigenvalues Poisson-commute and we have an integrable system. In the usual Toda lattice the index n runs from -00 to +00, but in the applications to the matrix models we are interested in, n runs from 0 to 00. The Lax pair [(1)] can also be written as the compatibility condition for an auxiliary linear problem L~=~~

~

-=B~

(9)

dt These two equations are compatible as long as the Lax equation is satisfied. Notice that if we think of the ~ variables as the amplitudes of oscillators at sites n of the

3 lattice, the eigenvalue equation (9) takes the form of a three-step recursion relation (10) reminiscent of the recursion relation for orthogonal polynomials [9]. We will see that this analogy is not a mere coincidence. Although we have written the simplest time evolution equation generated by the Hamiltonian H = H2 = tr L2 /2, we could instead define the time evolution equation in terms of the higher conservation laws H~ obtained by taking higher order traces of the operator L. These evolution equations are known as the higher Toda flows

(11) where the Poisson brackets are computed with the standard Poisson bracket between p and q. In the original Toda equation we cannot consistently set b = OJ however, if we

look at the higher flows, it is possible to construct another integrable hierarchy known as the Volterra hierarchy defined in terms of a Lax pair

L=

L

.,[il;,Et,k+l

k~1

satisfying tr L 2.. +1 = 0 and

B

= L yl R k R k+I E k,k+1

(12)

k

yielding the equations of motion

(13)

In the continuum limit this equation reproduces the Korteweg-DeVries equation. Some of the conserved charges in the Volterra hierarchy are

H3 =

L GR~ + RnR~+1 + RnRn+IRn+2)

(14)

n

The Hamiltonian structure which produces (13) out of HI is

(15) It is a relatively tedious exercise to verify the Jacobi identity for this Poisson bracket. The explicit form of the Hamiltonians (14) can be obtained using the staircase procedure explained in [10]. The basic properties of the Volterra hierarchy can be summarized in the following results:

4 Theorem 1. The k-th :6.ow of the Volterra hierarchy can be written as a Lax pair

aL =

at.

[(L~k,L)

(16)

where L~ is the antisymmetric matrix whose upper triangular part coincides with L21o. The second important result is Theorem 2. There is a second Hamiltonian structure for the Volterra hierarchy compatible with the first (15) one [11) {R,.,Rmh

= R"Rm(R,. + R m)(5n,m+l -

5",m-d + R"Rm(5",m+2 - 5",m-2)

(17)

It is an unpleasant exercise to verify the Jacobi identity for [(17)]. The compatibility between the two Poisson structures is equivalent to saying that any linear combination of the two is again a Poisson structure. It satisfies the Jacobi identity. When this happens, this is a 'very strong signal about the Liouville integrability of the theory. Finally Theorem 3. The following identity (analogous to the Gel'fand-Dickii [12] relation in KdV) is satisfied,

(18) These three theorems guarantee the complete integrability of the Volterra hierarchy. Notice that if we assume some homogeneity properties of the Hamiltonians H" with respect to their R,. dependence, we can use (18) to compute all Hamiltonians starting from Hi' These properties play an important role in the analysis of the double scaling limit for this hierarchy. We point out that the second Poisson structure produces in the continuum limit the Virasoro algebra [13), and, as stressed by the authors of reference [13] it can be taken as the starting point in the lattice quantization of Liouville theory. RELATION WITH ONE-MATRIX MODELS Given some even potential VeAl = ~gpA2P and a basis of orthonormal polynomials P"(A) with respect to the measure e- V (),), the operation of multiplication by A is represented by a Jacobi matrix L of the same form as in the previous section. As we change the couplings g;, the polynomials P,,(A) also change:

ap" - = (Mi )nlPl 8g;

Since A is independent of the couplings, differentiating AP"

aL ag;

= [M;,L]

= L"kPk we obtain (19)

The couplings g; are all independent and therefore we can vary them independently, implying that the g;-:6.ows commute. In terms of the matrices Mi this implies

a

a

-M' - -M; - [M;,M'] = 0 ag;

J

agj

J

(20)

In the space of couplings A = M;dg; represents a :6.at O(n) gauge field and (19) means that L changes by parallel transport in the presence of the gauge connection A(g).

5 The matrices M, are anti symmetric and they should be local to have a continuum limit. In this context, locality means that (M,)mn = 0 for 1m - nl > p finite. We could modify M, by adding a symmetric matrix X. However, the symmetry of 8L189, requires [X,L] = o. If L is generic (all its eigenvalues are distinct) then X has to be a polynomial in L and therefore it is irrelevant in the construction of the flows. The matrices M, are determined by the conditions (1) M, is antisymmetric, (2) [M" L] is a Jacobi matrix of locality one, and (3) M, is homogeneous of degree 2i in the matrix elements of L. This is 50 because V = L: 9p>.2p is unchanged under >. .... et>. accompanied by 9p .... 9p et- 2P • These three conditions fix M, to be given oy L~, the matrices defined in the previous section:

(21) The flows are generated only by the even powers L~. We know that the only nonvanishing matrix elements of [L~, L] are (m,m±l). If instead we considered [L~+1 ,L], only the diagonal matrix elements would be different from zero. Consequently, L~ generates flows and L~+1 provides initial conditions. This cannot be seen so clearly in the continuum limit, where the operators generated by [L~n, L] and [L~n+1, L] differ only by a total derivative. Summarizing, L(9,) is an orbit of the Volterra flows. What is left to do is to determine the initial conditions for the flows. If V(>.) is a polynomial, then the operator did>' is also local (in the Jacobi sense), and it can be chosen antisymmetric. Write (22) Then

J d~ = - Jd>'e-V(~)V'(>')PkPI +

o=

(e- VPkPI)

d>'

(23)

Nkl

+ NI"

and hence N - V'(L)/2 is antisymmetric, and if n =degree(V), then N"r = 0 unless Ik - rl :5 n - 1. We can choose N to be antisymmetric. Therefore, we can expand N as a linear combination of L~. Since [N, L] = 1, this means that only odd powers of p appear, and therefore [N, L] = 1 becomes a set of initial conditions for the Voiterrra hierarchy n

L 2j9}O)[L~-1 ,L] = 1

(24)

j=1

In conclusion, the generic one-matrix model is equivalent to a Volterra hierarchy with a particular initial condition . It is possible to show that this initial condition does not intersect any of the multi-soliton sectors of the Volterra equation. To study the mechanical properties of this system, we may define a Volterra equation with a finite number of points, instead of considering the equation on the semi-infinite line. For a

6 finite number of particles the initial condition becomes 1 0 0

0

010

[N,L)=

(25)

0 0 1

o

-n+ 1

The converse of this result can also be proven. Starting with the Volterra hierarchy and taking into account that R" > 0, we can use Favard's theorem [9] which guarantees that, given a set {rl) then dictates[7] the most singular term in its operator product expansion (OPEj~ith the stress-energy tensor (taking rl = 0 for simplicity):

J

TI'I/(r) 25. The action is taken to be local and coordinate invariant, but can otherwise be quite general. It is also assumed that some covariant non-perturbative method of regularizing the theory exists. Unfortunately no such method is known at present for carrying out the continuum theory path integral in a manifestly covariant manner. Instead we shall have to rely on a prescription which is the analogue of old-fashioned methods of regularization and renormalization in gauge theories. Before the invention of gauge-invariant regulators, a procedure which worked was to regularize the theory in a non-covariant way and then compensate for the resulting non-invariance by allowing the Lagrangian to contain non-gaugeinvariant terms, such as a photon mass. At the end of the day, the gauge symmetry is re-imposed through Ward identities, which place constraints on the values of the added terms. A particular version of this method for two-dimensional gravity follows: • Gauge fixing: The first step is to remove the over-counting of metrics due to general coordinate invariance. For each worldsheet topology introduce some fixed fiducial, metric -Yab. Then choose coordinates such that the physical metric is conformal to the fiducial metric,

'" .

'Yab = e 'Yab·

(2.1)

The original path integral is replaced by an integral over the matter fields Xi (0-) and the one remaining degree of freedom of the metric ¢>( 0- ), which is called the Liouville field . • Regularization: In order to define the gauge-fixed path integral the ultraviolet divergences of the theory need to be regularized. For example, a nonperturbative regulator can be introduced by discretizing the worldsheet. This discretization is to remain fixed and not be summed over as in matrix models. The regulator involves a shortest fiducial length defined by, (2.2) where fa is the line element connecting the nearest lattice points and Ii tends to zero as the cut-off is removed. A more covariant definition would refer the cutoff scale to the physical metric 'Yab, but then the regularization would depend on the Liouville field which is being integrated over. Thus we are obliged to use a non-covariant regularization procedure in order to have a concrete definition of the continuum theory path integral.

29

• Renormalization: Performing the path integral over short distance fluctuations of both the matter and gravitational fields generates various interaction terms, involving

, the role of time in target space is played by the two-dimensional scale. The tachyon field, T(X), is of primary interest because it controls the two-dimensional cosmological constant. A cosmological term in the original classical action corresponds to a tachyon background which grows exponentially with increasing two-dimensional scale,

(2.4) As we shall see, this remains qualitatively true in the quantum theory, as long as the tachyon background remains weak, but the rate of the exponential growth is modified by quantum fluctuations. The exponentially growing background will eventually become strong and then non-linear effects in the target space theory can no longer be ignored. The two-dimensional cosmological constant will still be governed by the behavior of the tachyon background in the non-linear regime, but the connection between the two is more subtle. The string theory equations of motion, obtained by setting beta-functions to zero, are derivable from a target-space action. For simplicity, we will work within a truncated theory, containing only the lowest order couplings, T(X), ibeX) and apl/(X)' To leading order in derivatives, the target-space action for these fields is

[10] 1= -

2:3 JdD+l XVGe-2~ C5;D + R+4(vib)2 - (VT)2 - 2V(T) + ... }, (2.5)

where VeT) = _T2 + ... is the tachyon effective potential. Since renormalization group beta-functions are not universal, the detailed form of VeT) will depend on the regularization and renormalization prescription used. This is believed to correspond to field redefinition ambiguities in the target space equations. In fact all higher order terms in the tachyon beta-function can be arranged to involve target space derivatives, and therefore be removed from the potential leaving only _T2 [11,12]. It should be 1)

We will use the string theory names for the target-space fields, but the reader should keep in mind their cosmological interpretation.

31

stressed that using such a prescription in no way alters the fact that the target space equations are non-linear and are in general not exactly satisfied by a simple exponential tachyon background. The important question to ask is whether there exists a renormalization scheme in which the target space fields can be identified with Wheeler-DeWitt amplitudes of a one-dimensional universe. We will return to this point in section 5, where we propose an appropriate scheme and present a calculation of V(T) to all orders in T.

3. Quantum Cosmology in Two Dimensions The equations of motion which follow from the target space action (2.5) are

v 2T - 2vc)· vT = V'(T), 2

2

25-D

v ~ - 2(v~) = - - 6 - + V(T), 1

(3.1) 2

1

2

R,.." - 2G,.."R = - 2v,..v"c) + G,.."v ~ + v,..Tv"T - 2G,..,,(VT) . These equations have a simple solution, the so called linear dilaton background [13], which for D > 25 is given by T=O,

G,.." = 17,..",

(3.2)

c) = - ~xo. 2

The target space is Lorentzian and it is the conformal mode, Xo, which is timelike. This means that the kinetic term of XO in (2.3) has the "wrong" sign and the Euclidean action of the two-dimensional theory is unbounded from below. This is analogous to the instability of the Euclidean path integral in four-dimensional gravity, which lies at the heart of Coleman's argument for the vanishing of the cosmological constant. On the other hand, it means that Euclidean two-dimensional gravity coupled to D > 25 matter is ill-defined and the renormalization group computation, which led to the target space equations (3.1), can only be viewed as a formal argument. Ideally the theory should be reformulated on a worldsheet of Lorentzian signature but it is unclear at present how to perform the steps involved in the quantization of such a theory (regularization, renormalization, etc.). In four-dimensional gravity people have sought to circumvent this problem by formally rotating the contour of path integration over the conformal factor into the complex plane to obtain a well defined integral [14]. While this formal procedure can also be applied in the two-dimensional theory, its validity has been called into question [15].

32

We will use the target space picture to define the two-dimensional theory for D > 25. The equations of motion (3.1), which were arrived at via a formal derivation based on a Euclidean worldsheet, lend themselves to an interpretation as a Lorentzian field theory of strings. Our assumption, which maybe unwarranted, is that a consistent Lorentzian worldsheet formulation would lead to the same target space field theory. Since the equations are non-linear, singular geometries which describe splitting and joining strings will have to be included in the Lorentzian two-dimensional path integral. In addition, the path integral will receive contributions from universes being absorbed or emitted from the background, which also involves two-dimensional singularities. By contrast, in Euclidean space the metric can be chosen with no singularities. We only use Euclidean methods to compute renormalization group betafunctions, but our subsequent discussion of the two-dimensional cosmology takes place with Lorentzian signature. An important difference between the worldsheet theory and four-dimensional gravity is that the gravitational coupling in two dimensions is dimensionless, so there is no proper Planck scale. However, as is well known, the strength of the string coupling depends on the dilaton field in target space. A key feature of the linear dilaton background (3.2) is that the string loop coupling constant is related to the two-dimensional scale,

g

= go e~ = gO /jx

O

(3.3)

We can only expect the effective field theory to be simple where this coupling is weak. For D > 25 the theory is strongly coupled for sufficiently small strings and target space quantum mechanics (string loops) are important in the ultraviolet on the worldsheet 6 • One can say that a Planck scale is spontaneously induced, and define it by the point at which goe~ = 1. The factor of gO can be absorbed by a constant shift of the dilaton. The effective Planck-scale is then set by q-l. It depends on the number of scalar fields in the theory, and in particular, D -+ 00 is a semi-classical limit for gravitational fluctuations 7. Another way to see that q-l defines the Planck scale in this theory is to consider the relation between the classical conformal mode and the quantum variable, !4> = Xo. A particularly interesting cosmological system is given by an expanding universe which starts out at small scale. The question of initial conditions is complicated, just as it is in four-dimensional quantum cosmology, because the theory is strongly coupled early on. We will assume that the short distance physics can be summarized by some unknown initial state at the Planck-time, which then evolves in the weakly 6 In contrast with the D :5 1 case, where the strength of quantum corrections tends to zero in the limit of metrically small strings. 7 For D :5 1 a corresponding semi-classical limit is reached as the number of scalar fields is formally taken to D ..... -00.

33 coupled theory. In a classical theory this means initial conditions on the target-space fields and in a quantum theory it corresponds to a wave-function in target-space. A background tachyon field can be added to the linear dilaton solution (3.2). Its beta-function equation depends on the shape of the effective potential, V(T), and is non-linear. For the moment we will assume that the background field is weak. The tachyon equation can then be linearized as follows,

-a~T + a?T - q80T + 2T = 0,

(3.4)

and if we further assume that the tachyon background only depends on XO, we find solutions (3.5) Such a homogeneous background configuration is the D > 25 analog of the D < 1 two-dimensional field theories discussed by David [16] and by Distler and Kawai [17]. One of the solutions decays in the weak coupling regime X O -+ 00 but the other one grows exponentially with scale. The system is unstable and is likely to form a condensate of background tachyons. In the D -+ 00 semi-classical limit we recover the classical cosmological term (2.4) from the exponentially growing solution. At this point the quantum behavior is qualitatively the same as in the classical theory, but the classical scale factor, et = eixo, has been renormalized to efXo, with a =

-J + V'f +2.

In order to make contact with a more conventional Wheeler-DeWitt description of cosmology, let us consider fluctuations of some target space field in the exponentially O growing tachyon background, TB(X o) = A e(-2!l.+V i!..+2)X , • Take, for example, a tachyon with some non-zero space-like momentum k. This corresponds to a onedimensional universe with some matter excitation. The target-space action (2.5) is not time-translation invariant. In order to describe physical fluctuations it is convenient to absorb the e-2~ pre-factor by a field redefinition, which has the form

U(X) = e-~(X) T(X)

(3.6)

for tachyons. A fluctuation Ur.(X) = Ur.(Xo) eir.;x; satisfies a linear equation,

(3.7) Near the top of the potential the tachyon background is well approximated by the exponential form (3.5) and we can drop the contribution of all but the leading terms

34

of the potential V(T) in the fluctuation equation, whereupon (3.7) becomes

(3.8) where ~ = AV"'(O). If we change variables from XO to the scale factor a takes a more conventional form,

= e~.xo this (3.9)

Up to factor-ordering ambiguities, this is the Wheeler-DeWitt equation derived from the mini-superspace Lagrangian of two-dimensional gravity,

a 2 --[k 1 2 q2 - 2 -(2+-)+Aa].

L=-(-) a

0: 2

4

(3.10)

The three terms in square brackets are the matter, curvature and cosmological constant energy densities. It seems that we have recovered a more or less conventional Wheeler-DeWitt description of large scale cosmology. In particular, the problem of the cosmological constant is the usual one. In order to obtain vanishing cosmological constant, the exponentially increasing solution for T(XO) must be fine-tuned to zero. In other words, the tachyon must be delicately balanced at the top of the potential. We are ignorant about the short distance physics, which is supposed to determine the initial state, so we have no way of gauging how likely it is to find the system balanced at the top of this potential. At any rate, such a fine-tuned initial state is not allowed in a quantum theory, because of the uncertainty principle. However, this is not the whole story. Even if the tachyon background starts out near the top of the potential it will eventually roll into the region where the higherorder non-linear terms in the tachyon beta-function cannot be ignored. As we have already mentioned, different renormalization prescriptions in the two-dimensional theory will lead to different evolutions for the tachyon background. Since the - T2 term in V(T) is universal the different schemes will all agree near T = 0, but away from the origin they can present very different pictures. For example, the question of whether V(T) has a minimum is scheme-dependent. The key issue here is to identify the definition of the tachyon field most closely corresponding to Wheeler-DeWitt amplitudes in the two-dimensional cosmology. In section 5 we propose a candidate scheme for calculations and obtain the tachyon potential to all orders in T within that framework.

35 It should be emphasized that the non-linear effects that we are talking about do not disappear in the semi-classical limit D -+ 00. In particular, the splitting and joining events described by the non-linear terms of the target-space equations are unsuppressed even at large scales. This may seem surprising because the string coupling is becoming weak, with e+ = e-f xo . Indeed, the canonical tachyon field U(XO) defined in (3.6) satisfies

[v2 + (2 + ~)]U = ~e-fXOU2.

(3.11)

As we move toward the semi-classical limit q -+ 00, though, the tachyon mass squared 2 increases as so that the unstable exponential growth of U compensates the decreasing coupling strength. The existence of string interactions, along with the tachyon instability, shows that the usual Liouville model described by an exponentially growing tachyon background is not the complete theory in the D > 25 case.

T'

4. The Running of Coupling Constants Before delving further into the two-dimensional cosmology, we would like to clarify the connection between the target space equations of motion and the renormalization group flow of couplings in the two-dimensional field theory. The equations of motion for the target-space fields are that the beta-functions of all two-dimensional couplings vanish. From this one might conclude that the couplings seen by a two-dimensional observer would not run. This, however, is not the correct interpretation. We can think of the equations for the target-space fields as renormalization group equations with I XO identified with the logarithm of the renormalization scale. The XO dependence of the coupling functions T,~, G"II ... hence determines their evolution with scale. This connection may appear unfamiliar because the equations of motion (3.1) are second-order in XO derivatives, whereas the usual renormalization flows are controlled by first-order equations~ The higher-order nature of the flows is a special feature of theories containing gravity, where the scale itself is a dynamical variable. The situation. is siinilar to the issue of time evolution in the Wheeler-DeWitt formulation of quantum gravity. We begin with an equation HWD Iw) = 0 which seems to imply that no time evolution occurs. Reinterpreted, though, the equation tells us how the wave function of matter evolves with the expansion of the universe. The Wheeler-DeWitt equation, like the equations of motion for T,~, GplI , is secondorder. The first-order SchrOdinger equation is only recovered in a semi-classical limit 8 The equations of motion (3.1) are of course only the leading order approximation to the exact beta-function equations, which include terms with an arbitrary number of derivatives.

36

in which gravitational fluctuations become unimportant [18]. In our two-dimensional theory, the semi-classical limit corresponds to taking D -+ 00 or equivalently q -+ 00.• In this limit we will see how the target-space field equations reduce to the familiar renormalization group equations, and how gravitational corrections to the ordinary renormalization group beta-functions can be obtained systematically in a "large q" expansion. We consider, as a simple example, the case of fluctuations about the linear dilaton background at the top of the tachyon potential with a flat target-space metric. A field An at the nth mass level in string theory will contribute to the effective action a term

2~~

f v'Ge- +

2 {(VAn)2 - 2(I-n)A! + ... } .

(4.1)

Its equation of motion in a linear dilaton background is (4.2) As we saw previously, this equation has unstable solutions for n = 0, which describe the tachyon rolling off the top of its potential. Note, however, that the solutions for n ~ 1 are stable for all values of q. In other words, the dilaton, graviton and higher couplings do not become "tachyonic" for large D. Now, recalling that the scale factor is a = ef Xo , we can rewrite (4.2) as a2 {} 2 aq {} 2 ] [-(a-) +-a-+k -2(1-n) An=O. 4

Thus when

~ -+ 00

{}a

2{}a

(4.3)

we find a first-order equation, (4.4)

where we have used that a -+ ~ + O(!:r) for large q. This is the usual lowest order Callan-Symanzik equation for a coupling of bare dimension 2n. In particular the field hpil has anomalous dimension _k2 as expected. The exact target space equations of motion will include complicated higherderivative terms, which are difficult to compute explicitly, but in the semi-classical limit they will all be suppressed by powers of q-2 in the same way as the secondorder term in (4.3). To see that, note first of all that the only effect of the linear dilaton background (3.2) on beta-function calculations is to shift the anomalous dimensions of vertices. For example, a tachyon with target space momentum kp has its dimension shifted from dl: = 2 - k2 to dl: = 2 + iqko - k2 , but this is the only place

37 where an explicit factor of q enters into the tachyon beta-function. One can easily convince oneself of this by considering sigma model Feynman graphs [2]. As a result, all terms in the equations of motion with higher-order derivatives, with respect to the conformal mode, pick up factors of q-2, when we express the equations in terms of the scale factor a. These terms will therefore all vanish in the q -+ 00 semi-classical limit and should be viewed as gravitational corrections to the renormalization group beta-functions computed on a flat worldsheet. In fact, by using simple manipulations, we can rewrite the target space equations of motion as conventional renormalization group equations with gravitational corrections. This is easily illustrated for the example considered above. The right-hand side of (4.4) is the leading contribution to ,8~(Ai)' the beta-function of An on a flat worldsheet. To obtain the leading order gravitational correction to ,8~ we write the second-order equation (4.3) as

a

] a

1 [2(a-a ) + 1 a-a An q a a

= .8n0 (Ai) ,

(4.5)

and solve for the corrected beta-function to the next order in ~,

(4.6)

We can use this trick to rewrite any higher-derivative term in the target space equations as a contribution to the renormalization group beta-functions, suppressed by some powers of q-2. In this way a systematic large q expansion can be developed to compute gravitational corrections to beta-functions. The higher-order equation of motion for a given target space field has a number of solutions. For example, we have a choice of sign in the exponential tachyon background (3.5). Only one of these solutions reduces to the expected classical behavior in the limit of large q, and it is not hard to see that this semi-classical branch also provides a solution to the corresponding first-order renormalization group equation with gravitational corrections. We have so far been considering two-dimensional cosmology with a trivial matter sector, consisting of several free fields. A more complicated theory, involving an interacting matter sector coupled to the conformal mode, provides more stringent tests of the above ideas. One can, for instance, study an asymptotically free sigmamodel coupled to gravity. This case was considered in reference [2], taking three of the target-space dimensions compactified to a sphere of time-dependent radius r(XO), but

-38

leaving the remaining D-3 spatial coordinates flat. We will not go into the details here, but only discuss the qualitative behavior. The sigma-model coupling strength is l/r, and the XO dependence of r(XO) gives the running of the coupling with scale. This is easily checked by inserting a metric of the above form into the target spate equations of motion (3.1). In the semi-classical limit the standard renormalization group flow,

(4.7) is reproduced. This calculation is valid for large r, where the sigma-model is weakly coupled, but breaks down as the system passes into the strongly coupled regime. However, since we know that the flat space sigma-model contains only massive particles [19], we may speculate that well below the induced mass scale, the sigma-model degrees of freedom decouple. This would correspond to the effective central charge of the matter becoming smaller at some point in the evolution of the universe. All this has important consequences for the cosmological constant. The non-trivial sigma-model dynamics generates a two-dimensional vacuum energy, which manifests itself as a source term in the tachyon equation of motion. As explained before, it is the exponential growth of the tachyon field as it rolls off the top of the hill that gives rise to the cosmological constant term in the Wheeler-DeWitt equation (3.9). We might imagine that it would be possible to "fine tune" the initial conditions so that the tachyon stays balanced at the top, and the cosmological constant would thus vanish. In our simpler examples in which the target-space was flat, we saw that this could indeed be done. Now, however, the coupling of the sigma-model to the two-dimensional gravity will make it impossible. There are terms in the target space effective action which couple T and Gpv • They can be determined explicitly by beta-function calculations, but for our argument it will suffice to note that there must be some such term because string theory has a non-zero graviton-graviton-tachyon vertex. There will thus be an extra source term involving some power of the target space curvature in the tachyon equation of motion. As the three-sphere contracts, this will knock the tachyon from the top of the potential. We would therefore have to search for new fine-tuned initial conditions to make the tachyon end up balanced at the top of the potential at large scales. This need to account for the matter vacuum energy is just the familiar cosmological constant problem.

39

5. The Tachyon Beta-Function and the Cosmological Constant In this section we return to the issue of a. la.rge-scale cosmological constant. We will follow the system as the ta.chyon ba.ckground rolls off the top of its potential into the non-linear region a.nd investigate whether observers in a two-dimensional universe, interacting with the background, would register a non-zero cosmological constant. In order to discuss the evolution of the ta.chyon ba.ckground at la.rge scales we need to compute its beta-function. A more or less standard perturbative approach is described in reference [2] and the leading terms are obtained there. Such calculations rapidly get quite involved and it is not tra.ctable to compute the complete beta-function to all orders. In addition, the cosmological interpretation of the results is sensitive to the choice of perturbative renormalization prescription. In the semi-classical limit the problem simplifies enormously. The general a.rgument given in the previous section can be applied to the equation of motion for a homogeneous ta.chyon ba.ckground. As q - t 00 all terms with higher derivatives, with respect to the conformal mode, will be suppressed. Since all space-derivatives vanish for homogeneous backgrounds the equation becomes first-order in the semi-classical limit,

a:a T = -V'(T) ,

(5.1)

and the dynamics is completely determined by the shape of the effective potential. The problem is reduced to finding the beta-function for tachyons with vanishing ta.rget space momentum, i.e. for a constant tachyon field. This might appear to be a trivial task since, according to (2.3), a constant ta.chyon ba.ckground only contributes a cnumber, [,. J d2 a .,ff, to the two-dimensional action. This is too simple a view to take, for it does not take into a.ccount the effect of the regularization which is required to define the quantum theory. H, for example, the ultra-violet divergences a.re cut off using a hard sphere regulator, then the excluded volume introduces non-trivial effects even when T is constant. The ta.chyon potential V(T) can be obtained using a lattice method introduced in reference [2]. This regularization scheme is pa.rticularly suitable for cosmological applications because the renormalized couplings a.re directly identified with WheelerDeWitt amplitudes. Begin by introducing a squa.re lattice on the fiducial coordinate spa.ce with lattice spacing f. In ea.ch cell we define an amplitude W on the boundary by integrating over the two-dimensional fields in the interior of that cell, fixing the values on the bounda.ry. This defines an effective theory that lives on the lattice edges. The remaining integration over the bounda.ry values of the fields yields the full path integral. The integrand of the effective theory is given by the product over

40

all cells of the cell amplitudes. Schematica.lly, Z=

J

I I 'DcPboundary \l1( cPboundary) • cells

(5.2)

Renormalization can be carried out by fixing the field values on some sub-lattice and integrating over the field on the remaining lattice edges. The amplitudes \l1 are by construction Wheeler-DeWitt amplitudes. To introduce target-space fields we can expand \l1( cPboundary) in terms of string modes. Let X be the zero-mode part of X on the boundary. Then

(5.3) where \l10 is the free theory amplitude. By requiring the long wavelength behavior to be independent of the cutoff we can define beta-functions for the target space fields T, GplI ••• • This is certainly not a convenient scheme for beta-function calculations in the presence of general couplings, but in the special case of a constant tachyon field, we can obtain the full answer. Then the partition function is simply

Z=

J

II'DcPboundary

(1- T) WO(cPboundary)

cells

= (1 -

T)N Z(T

~~

= 0),

:r.

where the total number of cells N is proportional to The free energy is therefore F = f 2 10g Z = log(l- T). The running of the coupling T with f is defined by requiring the partition function to be independent of the cutoff scale. This implies

o= ~~F(T) au;2 = -!F(T) + ~F'(T)aT. f3

f2

(5.5)

af

We define the beta-function in the usual way,

j3(T) =

aT f af

F(T)

= 2 F'(T)

(5.6)

= - 2(I-T) log(I-T).

This zero-momentum tachyon beta-function corresponds to the following potential, 1

V(T) = -T + "2T2 - (I-T)2Iog(I-T) ,

(5.7)

which has the form of an unstable tachyon potential near T = 0 and has a stationary point at T = 1, which is singular (V" '" 00). The potential cannot be continued past

41

the singularity but, as we shall see, the tachyon field never rolls beyond T = 1. To see how this works, insert (5.7) into the first-order equation of motion,

o

a-T= -2(1-T)log(1-T). oa

(5.8)

This is easily solved by writing (l-T) = eS , so that

(5.9) There is one integration constant which is determined by initial conditions on T, (5.10) For small T this solution reduces to the classical cosmological term (2.4) with cosmological constant A. The equation for S is linear so we see that this example provides a realization of the fact that tachyon field redefinition can eliminate the non-linear terms in the equation of motion. However, the resulting field S is no longer proportional to the Wheeler-DeWitt amplitude in equation (3.6). Now consider a two-dimensional universe containing some matter excitation interacting with the tachyon background (5.10). The fluctuation equation it satisfies is linear,

o

o=a oa T + k 2 T + V"(T) T =(a!

(5.11)

+ k2 - 2 + Xa 2 ) T.

In the semi-classical limit this is equivalent to the Wheeler-DeWitt equation (3.9) with non-vanishing cosmological constant. It is very important for the large-scale cosmology that the tachyon potential we obtained takes precisely the form (5.7). It is the singular behavior at T = 1, due to the logarithm, which allows a non-vanishing cosmological constant at large scale. It is quite striking that in spite of the apparently complicated non-linear evolution of the tachyon background T given by (5.10), the linear Wheeler-DeWitt equation obtained from (5.1) is precisely that of mini-superspace Liouville theory in the q - t 00 limit [3,15,20,21]. It is interesting in this context to note that the tachyon background also satisfies a linear equation,

0-2 a oa (l-T)+Aa (l-T)=O.

(5.12)

42 This suggests the alternate definition for the canonical tachyon field (3.6),

(5.13) In the large

q

limit the following linear second-order equation for

8) 2U- -

-1 ( aq2 8a

q2 - 2-U +'xa U = 0,

4

fJ, (5.14)

is equivalent to (5.12). Equation (5.14) differs from the k = 0 Wheeler-DeWitt equation (3.9) by the term due to the bare dimension of the tachyon. This is the equation that the SL(2, C) vacuum of string theory satisfies [3], and it is natural to identify that state with the most symmetric or Hartle-Hawking state of a onedimensional universe [3,22]. For any non-zero cosmological constant the tachyon background (5.10) approaches the minimum of its potential at T = 1 as the two-dimensional universe evolves to ever larger scale. It is unclear what conformal field theory, if any, corresponds to a tachyon field sitting at rest at T = 1, but we suspect it to be a rather trivial one. By the arguments of Kutasov and Seiberg [20,23] it cannot be a standard matter theory coupled to gravity. Apparently the T = 1 fixed point describes the asymptotic behavior of an expanding universe long after all relevant scales (e.g. the cosmological constant scale) have been passed. The only remaining degrees of freedom are conformal matter fields from which the scale of the metric decouples. The situation is analogous to that in QeD at very large distance scales where the only degrees of freedom are massless pions. Another closer example is provided by the D = 0 onematrix model. In this case a non-zero cosmological constant corresponds to a matrix potential slightly off criticality. The model flows to the trivial Gaussian matrix model at large scales and the random surface interpretation breaks down.

6. Conclusions Our results can be summarized as follows. Two-dimensional quantum cosmology can be formulated as a string theory with background fields. The dynamics of fields in the string theory target-space determines the values of coupling constants in the two-dimensional universe. In particular, the cosmological constant in two dimensions is governed by the background tachyon field, which satisfies a non-linear equation of motion. Nevertheless, the non-linear dynamics is such that, for generic initial conditions, a two-dimensional universe interacting with the background obeys a standard linear Wheeler-DeWitt equation with a non-zero cosmological constant.

43

If this picture is correct then there is nothing in the classical target space dynamics which favors a vanishing cosmological constant at large scales. The question remaining is whether the effects of wormhole topologies can change this conclusion. Including arbitrary worldsheet topologies in string theory turns the classical target-space field theory into a quantum field theory. As emphasized by Coleman, the couplings of the worldsheet field theory become quantum variables. However, the effective value of Planck's constant for the target-space theory is itself a field, and is given by nef[ ex e2+ = .e-qx•. Thus quantum corrections are expected to become negligible for large XO, and hence only to influence small scales. Now consider the quantum mechanics of a tachyon field depending only on Xo. Its Lagrangian is

v qX.·2 2 2'2e (-T +2T + ... ), 90

(6.1)

where V is the volume of (Xl, ... ,XD)-space. If this volume is infinite, then quantum fluctuations are negligible and the tachyon evolves classically. If the volume is finite, then the tachyon, and therefore the two-dimensional cosmological constant, is a true quantum variable. To describe it, a quantum wave-function for the target space fields must be introduced. The form of this wave-function at some value of XO for which e- qX• is already small summarizes the effect of small wormholes. If we assume that this wave-function is of some generic form at XO '" 0, then since the subsequent behaviour rapidly becomes classical, the only effect of wormholes is to provide a generic probability distribution for the initial conditions. Thus, we see no way in which target-space quantization can force the large-scale cosmological constant to zero.

44

REFERENCES [1] S. Coleman, Nucl. Phys. B307 (1988), 867" Nucl. Phys. B310 (1988), 643. [2] A. Cooper, L. Susskind and L. Thorlacius, Two-Dimensional Quantum Cosmology,SLAC-PUB-5536, April 1991, to a.ppear in Nuclear Physics B. [3] J. Polchinski, Nucl. Phys. B324 (1989), 123. [4] E. Baum, Phys. Lett. 133B (1983), 185. [5] S. W. Hawking, Phys. Lett. 134B (1984), 403. [6] T. Banks, Physicalia Magazine, vol 12, Special Issue in Honor of the 60th Birthday of R. Brout,(Gent 1990). [7] S. R. Das, S. Naik and S. R. Wadi a, Mod. Phys. Lett. A4 (1989), 1033; S. R. Das, A. Dhar and S. R. Wadi a, Mod. Phys. Lett. AS (1990), 799. [8] T. Banks and J. Lykken, Nucl. Phys. B331 (1990), 173. [9] A. A. Tseytlin, Int. Jour. Mod. Phys. AS (1990), 1833. [10] C. G. Callan, D. Friedan; E. J. Martinec and M. J. Perry, Nucl. Phys. B262 (1985), 593; E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B271 (1986), 561; C. G. Callan and Z. Gan, Nucl. Phys. B272 (1986), 647. [11] T. Banks The Tachyon Potential in String Theory, Rutgers preprint, RU-91-08, 1991. [12] A.A. Tseytlin On The Tachyonic Terms in the String Effective Action, Johns Hopkins preprint, JHU-TIPAC-91004, 1991. [13] A. Chodos and C. B. Thorn, Nucl. Phys. B72 (1974), 509; T. 1. Curtright and C. B. Thorn, Phys. Rev. Lett. 48 (1982), 1309; R. C. Myers, Phys. Lett. 199B (1988), 371; I. Antoniades, C. Bachas, J. Ellis and D. Nanopoulos, Phys. Lett. 211B (1988), 393; S. P. deAlwis, J. Polchinski and R. Schimmrigk, Phys. Lett. 218B (1989),449. [14] G. W. Gibbons, S. W. Hawking and M. J. Perry, Nucl. Phys. B138 (1978), 141. [15] J. Polchinski in Strings 90, Proceedings of the Superstring Workshop, Texas A&M University, March 12-17, R. Arnowitt et al. eds., (World Scientific 1991). [16] F. David, Mod. Phys. Lett. A3 (1988), 1651. [17] J. Distler and H. Kawai, Nucl. Phys. B321 (1989), 509. [18] T. Banks, Nucl. Phys. B249 (1985), 332. [19] A.B. Zamolodchikovand A.B. Zamolodchikov, Annal.Phys. 120 (253), 1979.

45

[20] N. Seiberg, Notes on Quantum Liouville Theory and Quantum Gravity, Rutgers preprint, RU-90-29, 1990. . [21] G. Moore, N. Seiberg and M. Staudacher, From Loops to States in 2-D Quantum Gravity, Rutgers preprint, RU-91-11, 1991. [22] D. Birmingham and C.G. Torre" Phys.Lett. B 194 (1987), 49 [23] D. Kutasov and N. Seiberg, Number of Degrees of Freedom, Density of States and Tachyons in String Theory and eFT, Rutgers preprint, RU-90-60, 1990.

46 A NOTE ON THE BRANCHING RULE FOR CYCLIC REPRESENTATIONS OF Uq(g[n)

Etsuro Datef, Michio Jimbot and Kei Mikif

t Department of Mathematical Science Faculty of Engineering Science Osaka University, Toyonaka, Osaka 560, Japan t Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606, Japan ABSTRACT

For Uq (g[,,) at qN

= 1 we introduce a class of cyclic representations

labeled by highest weight representations of the algebra of lower rank Uq(gln-1)' Assuming N is large, we give decomposition rules for tensor product of these modules with the vector representation of Uq (g[,,). Tensor product with general highest weight representations is also studied. O. Representations of the quantized enveloping algebra Uq(g) are of particular interest at qN = 11 )-3). They are called cyclic when the N-th power of Chevalley

generators ef', fiN act as nonzero scalars. Finite dimensional cyclic representations for affine type g are relevant to a new family of solvable lattice models known as chiral Potts models and their generalizations 4)-15). In this paper we introduce for Uq(gl,.) a class of cyclic representations p(;r, e) labeled by a representation ;r of Uq (g[n-1) and a parameter E (C X)2"-1. We shall consider the decomposition of Uq(gl,.)

e

modules of the type 7r ® p(;r, e), where 7r (resp. ;r) signifies an irreducible highest weight representation of Uq (g[,,) (resp. Uq (g["-l))' The results are described in section 2-3 below. The construction in the present note arose through the effort to understand the functional relations of transfer matrices 16 )-19) for the generalized chiral Potts models ll ),14). These models are associated with the minimal cyclic representations of Uq(gl,,), which in the above notation is p(triv, e) with triv being the trivial representation of Uq(g[,,_l)' Starting from p(triv,e) and repeating the decomposition of tensor products with the vector representation, one arrives at the family p(;r, e). The results of this note are still of preliminary nature, in that we have not yet considered the subtleties about the appearance of indecomposable modules 20 ) and also

47 the extension to the affine case. In section 1-3 we prepare the notations and state the results. The rest of the paper is devoted to outline the method. Our point is to consider an auxiliary algebra A defined in section 2. This has some similarity in spirit to the auxiliary algebra exploited by Arnaudon21 ), but seems to be different. In particular our A has Uq (gln-1) as a quotierit algebra. In section 4 we introduce a special basis of 00 r ® p(;r, e) (0 = the vector representation) with respect to which the action of Uq (gIn) looks almost like p(;r', e') with some ;r', e'. Section 6 deals with the case of r = 1. The general case is discussed in section 7. 1.

Throughout this note we let q denote a primitive N-th root of 1 with N odd

11 = Uq(gln-1)' By definition U is the C-algebra presented by the generators ei, I; (1 ~ i ~ n - 1) and q±" (1 ~ i ~ n) through the following defining relations. ~ 3. We set U= Uq(gln),

q~'qej

=

qCj qCi,

q£iq-Ci

q"ejq-"

= l,;-6,;+lej,

[ei, f j 1=

0ij - - - - 1

f;l q-q

c ti -

(

=1=

q-CiqCi,

q. = AB - ABA. Let now A be the C-algebra defined on the generators Ui, Ti (1 ::; i ::; n '- 1) through the defining relations (5). For a representation (1i', V) of A we let p(1i',e) denote the representation of U defined by the formula (4). Note that each weight space of p(1i', e) has the form V ® Cw m . When 1i' is the trivial representation 1i'(Ui) 1i'(Ti) = 1, p(1i',e) reduces to the minimal cyclic representation 22 ),14). Define uijE A (1::; i,i::; n) by Uii = 1, Ui;+! = Ui, Ui+!i [Uik, Ukj]q-' [iTik, Ukj]q'

= (1 -

q-2)Uij

= (1 -l)Uij

= Ti

=

and

(i < k < i),

(i>k>i)·

It can be shown that [Uij,

ulozl = [Uji, ukzl = [Ui;, Ulk] = [Uji, UII,] = 0

[Uij,Uik]

= 0,

[Uji,Uki] = 0

(i

(k < i < i < l),

< j,k or i > j,k).

In particular {uljh::;j::; .. (resp. {u ..jh::;j::; .. ) are mutually commutative. The following shows the precise relation between A and U. Proposition 1. (i) There exists an algebra homomorphism '{J: A ~ U given by

= q-U, + (1 - q-2)q-I'ei_l, '{J(Ti) = q2e, + (1 -l)q 2 below, best described in the oscillator notation introduced.

DEFORMATIONS TO q-HEISENBERG ALGEBRAS Now generalize the results of the previous section in oscillator instead of coordinate language, to the Heisenberg algebra of N classical oscillators Ai, for Iii = 1, ... , N and A_Iii == Alii: (6) [Ai,Aj] = sgn(i) Oi+j,O , where sgn(i) = i/lil. Thus, indices i < 0 describe creation operators, while i > o ones describe annihilation operators. Without loss of generality in what follows, we introduce the real antisymmetric matrix Cij which, in general, depends on N(N - 1)/2 parameters; and we further introduce the N real parameters ri == rlil' For qt = l/q as above, define Nk == A_kAk • Consider then the statistics-altering invertible deformations r!V·+l - 1

•

(ri - 1)(1

+ Ni )

A.

(7)

84 for i

> 0; consequently, again for i > 0, hermitean conjugation yields: r!'l· -1

(8)

(ri•-1 )N.i A_i , and hence the inverse: N·

(ri - I)Ni

rf· -

where

1

r;' -1 --1rj -

= a_jaj

.

(9)

These explicit deformations obey the following quommutator algebra of qdependent classical oscillators a.;aj - q( i, j) ajai

where q(i,j) =

l

= sgn( i)

ogn(ij) Cljllil

(1

r~sgn(i)-1)/2 Oi+j,O

+ (rj;fn(i) -1)

(10)

OHj,O) .

(11)

These oscillators obey anyonic statistics and comprise our central result. Viewed as coordinates, they roughly parallel Manin's multiparameter quantized phase-space as extended by Sudbery[31. Observe that this particular multiparameter q-grading satisfies, for i ±j =I- 0, the general symmetry conditions imposed on the quommutator algebra (10) by index relabelling covariance, herlniticity, and braid associativity: q(j, i)

= q(i,j)-l = q( -i, _j)-l = q(i, -j) = q( -i,j)

.

(12)

More specifically, the relations (10) are but a special case of the following generalization, ajak = R(j, k, s, t) a,a, + p(j) 0iH,O , (13) where repeated indices are summed over. To test associativity, compare the following two braid transposition schemes. On the one hand, ai(ajak)

=

aia,a,R(j,k,s,t)

+ aiP(j) OJ+k,O

a,.a"a,R( i, s, 'U, v )R(j, k, s, t) + p( i) a,R(j, k, -i, t) + aiP(j) 0i+/.,o a ..apaqR(v, t,p, q)R(i, s, 'U, v )R(j, k, s, t) + p( -t) a,.R(i, s, u, -t)R(j, k, s, t) +p(i) a,R(j, k, -i, t)

+ aiP(j) 0iH,O .

(14)

On the other hand, (a.;aj)a/c =

a,a,a/cR(i,j,s,t) + a/cp(i) 0Hj.O a,a.aqR(t, k, v, q)R(i,j, s, t)

+ p( -k) a,R(i,j, s, -k) + alcp(i) OHi,o + p( -v) aqR(t, k, v,q)R(i,j, -v,t)

a.. apaqR(s, v,'U,p)R(t, k, v, q)R(i,j, 8, t)

+p( -k) a,R(i,j, s, -k)

+ akP(i) OHi,o .

(15)

Hence associaiivity ("consistency") is ensured by R(s,v,'U,p)R(t,k,v,q)R(i,j,s,t) = R(v,t,p,q)R(i,s,u,v)R(j,k,s,t} ,

(16)

85 with implied summation over repeated indices; and also

p(-t) a" R(i,s,v,-t)R(j,k,s,t)+p(i) at R(j,k,-i,t)+p(j) a; p(-v) a. R(t,k,v,s)R(i,j,-v,t)+p(-k) a" R(i,j,v,-k)+p(i)

b;+Io,o=

a~ bH;,O'

(17)

To find the above particular (10), we postulate

R(j,k,s,t) =

bj,t bk,.

q(j,k) ,

(18)

(no summation, in contrast to the Wess-Zumino alternative.) This satisfies the trilinear consistency equation identically, and gives for (17):

p( i) q( i, k )q(j, k) ak bj+i,O + p( i) q(j, k) aj bi+k,O + p(j) ai bj+Ie,o = p(j) q(i,k)q(i,j) as b;+k,o + p(i)q(i,j) aj bHle,O + p(i) a. bj+i,O •

(19)

This is satisfied if q(i,j) = q(j,-i), q(i,-j)q(i,j) = 1 for Iii =f Iii, (and hence q(-j,k)q(j,k) = 1 for Ijl =f Ikl.) Moreover, for Iii = Iii, it is satisfied if q(j,j) = 1 and (q(j, -j) -1 )(q(j, -j) +p(i)/ p( -i») = 0, in agreement with (10),(11):

p(i)/p(-i) = _r:gn(i). The N(N + 1)/2-parameter algebra (10-11) stands by itself, i.e. it emerges as a solution of the braid relations even in the absence of the deforming functionals (7); historically, we only found these functionals after discovering the algebra. Further note, with T. Curtright [11] , the alternate solution q(j,i) =f 1, which then amounts to no more than q(j,j) = -1, yielding Manin's fermionic plane[2,5]. It follows from (19) that q(j, -j) = -1 and the p(j) = p( -i)'s are unconstrained, hence absorbable into the oscillator normalizations. The salient feature of this solution is nilpotence, i.e. self-exclusion. Self-exclusion may thus be pitted against statistics, by judicious choices of parameters. E.g. in (3), the choice q2 = -1 makes different oscillators anticommute with each other, so symmetric wavefunctions such as (XY + Y X) 10) vanish, but does not impose self-exclusion. On the other hand, it makes different oscillators of the above "fermionic" solution commute with each other, albeit still excluding themselves. This is a striking feature of Klein transformations[4]. The coproduct of the q-Heisenberg algebra (10) induced by the map (7) is cocommutative: rt(N,) -1

(ri ~(Nk)

(20)

-1)~(Ni)

= 1 ® A-kAk + A_k ® Ale + Ale ® A_Ie + A-leAIe ® 1

,

where the inverse map (9) must be employed to express the answer in terms of the aiS after reduction. By virtue of the above procedure, you may also verify that the one-parameter multioscillator algebra of ref.[12] fails the test for associativity if its parameter is real; or, if it is complex, and the creation operators are the usual hermitean conjugates of the destruction operators for the complex conjugate parameter. The only consistent value for that parameter then is 1.

86 When (18) is relaxed, a different solution to (17) is provided by the oneparameter generalization of the covariant quantum plane to N dimensions[I), with deforming maps given in ref.(10), further generalized to several parameters in refs.16]. An even broader Ansatz which admits yet more general solutions of (17), and is related to an R matrix of Cremmer and Gervais!131, maintains an additive grading, in the sense that the sum of the indices of each pair of oscillators is the same within a quommutator: [(;-lc)/21

a;alc

=

L

q(j, k, 8) a/c+.a;-.

j >k .

(21)

• =0

The general associative solution is!14]

q(j,k,O)

=

q(j,j - 3,1) q(j,k,l)

>.a-b-1 p.a-b-2 q(j,j - 2, l)q(j - l,j - 3,1)(>. + p.2) for all j > 3 (_>. + p.2) q(j -1,k,l)q(j,j - 2,1)>' for j > k + 3 (22) (>. _ p.2) r-1 . p.(r-1)(i-Ic-r+2) q(J - 8, k + r - 8 - 1,1) (>. + p.2)r-1(p.2 _ >.)r-1 x

!!

q(j,k,r)

x (5(j - k - 2r) >.r(r-2)(>. + p.2) + (J(j - k - 2r - 1) >.(J-Io-r-1Xr-1»),

r ~ 2,

where >',p., q(j,j - 1,1), and q(j,j - 2,1) are free parameters. However, it is not known whether this can be made to also include negative values of the index.

GL,(N) ALGEBRAS The oscillators of the previous section may be arrayed in bilinears['I] to construct generators

i,; > 0:

E i; = ala;

(23)

with GL(N), quommutation relations

Ei;EImo - Q(i,j,k,m)EImoEi; = 5;."Eim - Q(i,j,k,m)5i.mEIc; , Q(i,j,k,m)

= q(-i,-k)q(j,-k)fq(m,-i)q(m,j).

(24) (25)

Note the index-relabelling symmetry Q(i,;,k,m) = Q-1(k,m,i,;), and the hermiticity condition Q(i,j,k,m)1 = Q(m, k,j,i), given E!; = E;i' This then is a deformation of the conventional "physicists' basis" of GL(N).2 It may be noted that the N-dimensional representation of the classical GL(N):

(26) 2 A. Sudbery (unpublished) has anticipated some parts oHhis structure. One may wish to contrast this to a broad class of Jordan-Schwinger realizations, such as ref.[1S) which construct q-algebru in terms ofindependent (COj 0) q-oscillators, unlike the multiparameter q-dependent q-oscilJatoD considered here.

=

87 auo serves as a representation of the quantum algebra (24). This is a not uncommon feature of the defining representation of quantum algebras.

Our particular deformation (11),

Q(i,j,k,m) =

l

(C"+Ci~+C~;+C;. ) 1 + (r; - 1 ).5;,Ie

1 + (Tm - l).5m ,i

(27)

is left unchanged by the transformation eij --> C;j + bi - bj • As a result, the number of independent parameters in Q of our deformation is (N - 1)N/2 - (N - 1) = (N - l)(N - 2)/2 parameters of q-type, in addition to the N parameters Ti, so (N - 1)N/2 + 1 in all. For N > 2, however, if we wish to abstract the algebra (24) away from the particular realization (23), we must take all the ris to be equal to one: The reason is that only then is braid associativity satisfied without the imposition of quadratic eon8traint& on the generators. (To be sure, these constraints are satisfied by the realization (23).) The total number of parameters remaining is thus (N - l)(N - 2)/2 for N > 2, and 2 for N = 2. In the Ti --> 1 limit, it is straightforward to check that the Casimir invariants of this deformation are not q-dependent:

(28) (summation over repeated indices). In fact, they are just the classical ones, as is evident from the invertible map of the classical generators eij = A! Aj to the quantum generators

This simple map actually transcends the q-oscillator realization and maps all representations of GL(N) to their correspondents in GLq(N). As explained in ref.[16J, such deforming maps completely refer the representation theory of GLq(N) to that of its classical limit, automatically inducing a consistent coproduct. As an illustration, consider e.g. the GLq(2) given directly by N formulas:

[En, E22J = 0 , En E 12 - r,E'2En = E'2 , E22E12 - Ti' E12E22 = -Ti' E12 ,

= 2 in the above

E'2E21 - Tl'T2E2,E12 = En - Tl'T2E22 , E2lEl1 - TlEnE21 = En , E21E22 - Ti' E22E2l = -ri' E21 . (30)

The following invariant commutes with all four generators, and reduces to the linear Casimir in the classical limit:

For the special values

1/T2

= Tl == T2,

this Q grading for N

= 2 in (27) reduces to (32)

88 which amounts to no more than an "r-central" extension of the SU.(2) deformation of Witten[S]: r2 En - r- 2E22

Wo

+ 1/r E 12 /Jr + 11r E 21 /Jr + 11r Ell + E22 r

W+

-

W_

-

T

-

(33)

In the classical limit, this quantum algebra reduces to a mere SU(2)xU(1) as T collapses to a center. The invariant (1 - T)/(1 + (r - llr)Wo) commutes with all four generators and yields the linear Casimir in the classical limit. As a result, the generator T is representable as T = (1 - c)1 - c(r - 1lr)Wo for an arbitrary real parameter c. This indicates that the representation theory of the extended algebra is no different than that of the original 8U.(2) of ref.[8], a fact easily confirmed by inspection of the deforming functionals of ref.[16]. That reference provides the deformation of the quadratic Casimir. Loosely speaking then, T "mods out" of the algebra. For contrast to the above, also consider the Wess-Zumino oscillators (5) which directly yield a different[6] GL q (2) via (23):

[En, En]

=0

[Eu

+ E 22 ,E21 ] = 0 (35)

The proper deformation of the quadratic Casimir invariant, up to functions of the above linear invariant, is C == CI/C 2 , where C2 = 1 + (r' -1)E22 , (36) so that [C1,2,Ell ] = [C1,2,E22 ] = 0, hence

r 2E21 C1,2 = C1,2E21,

r 2 C 1 ,2E12 = E 12 C 1 ,2,

(37) It is normalized to reduce to the conventional operator in the classical limit. DISCUSSION AND APPLICATIONS The q-Heisenberg algebras introduced and deformed here, beyond the relaxation of nonlocal commutativity discussed in the introduction, may also serve as building blocks for various hamiltonian constructions and, since multioscillator wavefunctions

89 have unconventional symmetries and wavefunctions such as (a1 a~ - q( 1, 2 )~al) 10) are absent, anyonic physics[I7]. It is also inter~;ting to ask whether the structure of the SU(2)®U(I) electroweak theory could be illuminated through embedding in GLq(N) as realized above. Here, we merely illustrate in passing an application to the definition of q-exponentials whose arguments are defined with power series coefficients lying in a quantized space, and a construction of Hamiltonians with simple quommutation rules which could bear on the time-evolution problem in deformed quantization. Recall that Jackson's q-derivative with respect to a variable z is defined by

D j( ) q z

= j(qz) -

j(z) z(q _ 1) ,

(38)

i.e. the slope of the chord to the graph between qz and z. Jackson's conventional q-exponential is 00

eXPq(z)

z"

== L: -[J' o n.

[nJI == [n][n - IJI, (39)

where

specified by the eigenfunction property

(40) The analog 'eigenfunction' for a power zn in the argument of the q-exponential then is[18] eXPqn(zn), so that Dq eXPqn(zn) = Dq(zn) eXPqn(zn). But how can a corresponding 'eigenfunction' be defined for a general power series 00

j(z)

= L: ajz j

(41)

?

j=O

Here is a possible answer: The function[19]

Eq(z) == eXPq(alz)eXpq2(a2z 2)expq3(a3 z3 ) ...

(42)

obeys the eigenvalue condition

(43)

=

provided the coefficients aj in j are not c-numbers, but, instead, N 00 q-annihilators of the type covered above, obeying ajak = lakaj for j > k > OJ this is to say that 2Cllolljl in (11) is chosen to be k (}(j - k) - j (}(k - j). As a further exercise, finally consider building Hamiltonians which possess simple quommutation relations with oscillators. Take N q-oscillators of the type introduced here, and q2 = W to be an Nth root of unity, w N = 1. Moreover, for positive indices, choose Cji = i - j and Ti = 1. Thus, q(i,j) = wsgn(ij)(lil-Iji) so that

aiaj -

wsgn(ij)(lil-lji)

Consequently, for j > 0, q(i,j)q(i, 1

_

ajai

j) =

= sgn( i)

bi+j,o .

(44)

w-sgn(i).

The nonhermitean Hamiltonian N

H(l) =

L: j=1

al_jaj ,

(45)

90 (where the index 1 - 1 in fact denotes -N), quommutes simply with the oscillators:

(46) (where the index 1 +N denotes 1). Given these quommutation rules, the Hamiltonian may be chain-advanced past any function of oscillators; but simple conventional quommutators with the Hamiltonian only result out of homogeneous functions of such q-oscillators. In the classical limit and in the phase-space language, this Hamiltonian, H{I) = :Z:NP1 + :Z:lP2 + ... + :Z:N-1PN, ultimately describes the classical mechanics of dN:z:i/dt N

= :Z:i·

There is, in fact, an entire family of similar Hamiltonians, N

H{k)

=L

(47)

aHaj ,

j=l

with k = 0,1, ... , N - 1, and the index of the first oscillator is k - j - N whenever necessary so as to be always negative. These all commute with each other, and are pairwise connected by hermitean/canonical conjugation, except for Ho (and H N / 2 for even N) which are self-conjugate. They are the Jordan-Schwinger realizations of the circle-shift matrix h and its powers[20.21):

(48) so that Ho is a mere number operator with canonical commutators. The quommutation eigenstates of these shift-Hamiltonians are[21J N

a(k)

= Lwk(i-l)ai

N

a(-k)

= Lw-Io(i-l)a_i

(49)

.=1

i=1

where k = 0,1, ... , N - 1, so that a(n)H(k) - w- Io H(k)a(n)

= w-nloa(n)

,

(50)

w-Io a ( -n)H(k) - H(k)a( -n) = -w-nIoa( -n) ,

(51)

related to each-other by hermitean conjugation. As a consequence, all powers of (a{n»N are null commutation eigenstates. Note the new grading a(n)a(k)

= a(k -1)a(n + 1)

a(n)a( -k)

= a( -k + l)a(n -1) + bn,h ,

(52)

and their hermitean conjugates. Hamiltonians such as H(I) may serve as toy-models for addressing the problem of time-evolution of q-deformed systems. In standard quantum mechanics, timeevolution issues from a similarity transformation involving the Hamiltonian, and, as a consequence, commutation relations hold for all times if only they are valid at t = O. Quommutation relations (44), however, are not preserved for the transforms

(53) which are required by (44) through

db· d;

= wbi H(I) -

H(I)bi

db_ i = w -lb -iH () dt 1 -

H ( 1 )b_ i

.

(54)

91 (This is reminiscent of, but crucially different from (4.18) ofref.[22]). Alternatively, preservation of the quommutation relations via the conventional similarity transformation (55) dictates that the time evolution (54) be bought at the price of the time t not being a mere parameter anymore. Instead, it must quommute with the quantum variables: (56) whence the canonical conjugates (57) which endow d/dt with the same q-grading as H(l). Thus, t commutes with H(l) to yield (54). The q-grading of t then varies not only with the quantum variable, but even with respect to the hamiltonian H(k). At present, this appears problematic. We anticipate further applications in the future.' We wish to record our gratitude to P. Bay ton. T. Curtright. P. Fletcher. J. Nuyts. A. Sudbery. and S. Vokos for their interest and suggestions.

REFERENCES 1. D. Fairlie and C. Zachos, Phys.Lett. 256B (1991) 43. 2. Yu. Manin, Comm.Math.Phys. 123 (1989) 163. 3. A. Sudbery, J. Phys. A23 (1990) L697; "Non-Commuting Coordinates and Differential Operators", in the Proceeding& of the Argonne Worbhop on Quantum Group3, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1991, p. 33; "Matrix-element Bialgebras Determined by Quadratic Coordinate Algebras", August 1990 preprint; and private communications. N. Reshetikhin, Lett.Math.Phys. 20 (1990) 331; M. Rosso, Comm.Math.Phys. 124 (1989) 307. 4. P. Jordan and E. Wigner, ZS.f.Phys. 47 (1928) 631; G. Liiders, Z.Naturforsch. 13a (1958) 254. 5. J. Wess and B. Zumino, Nucl.Phys.(Proc. Suppl.) ISB (1990) 302; B. Zumino, Mod. Phys.Lett. A6 (1991) 1225; A. Schirrmacher, J. Wess, and B. Zumino, Z.Phys. C49 (1991) 317; W. Pusz and S. Woronowicz, Rep.Math.Phys. 27 (1989) 231. 6. A. Schirrmacher, Z.Phys. C50 (1991) 321; S. Vokos, Argonne preprint ANLHEP-PR-91-07, to appear in J.Math.Phys. 7. P. Jordan, ZS.f.Phys. 94 (1935) 531; J. Schwinger, "On Angular Momentum", AEC preprint NYO-3071 (1952).

92 8. E. Witten, Nucl.Phys. B330 (1990) 285; T. Curtright, in the Proaeding, 01 the Argonne Worklhop on Quantum GroupI, T. Curtright, D. Fairlie, and C. Zachos (eds.), World Scientific, 1991, p. 72. 9. V. Bargmann, Rev.Mod.Phys. 34 (1962) 300. 10. P. Kulish and E. Damashinsky, J.Phys. A23 (1990) L415; also see M. Ubriaco, Puerto Rico preprint LTP-017-UPR. For a discrete approach, see E. Floratos, Phys.Lett. 252B (1990) 97. For one oscillator, see J. Cigler, Mh.Math. 88 (1979) 87; V. Kuryshkin, Ann.Fond.L.de-Broglie 5 (1980) 111; A. Jannussis, G. Brodimas, D. Sourlas, and V. Zisis, Lett.Nuov.Cim. 30 (1981) 123. 11. T. Curtright, private communication. 12. D. Coon, S. Yu, and M. Baker, Phys.Rev. D5 (1972) 1429. 13. E. Cremmer and J .-L. Gervais, Comm.Math.Phys. 134 (1990) 619. 14. P. Bayton, unpublished. 15. A. Macfarlane, J.Phys. A22 (1989) 4581; L. Biedenharn, J.Phys. A22 (1989) L873; Cop. Sun and H-C. Fu, J .Phys. A22 (1989) L983; M. Chaichian and P. Kulish, Phys.Lett. 234B (1990) 72. 16. T. Curtright and C. Zachos, Phys.Lett. 243B (1990) 237; T. Curtright, G. Ghandour, and C. Zachos, J.Math.Phys. 32 (1991) 676. 17. Y-H. Chen, F. Wilczek, E. Witten, and B. Halperin, Int.J.Mod.Phys. B3 (1989) 1001. 18. D. Fairlie, in Symmetriel in Science V, B. Gruber et al. (eds.), p. 147-158, Plenum, 1991. 19. C. Devchand, private communication. 20. D. Fairlie, P. Fletcher, and C. Zachos, Phys.Lett. 218B (1989) 203. 21. E. Floratos and T. Tomaras, Phys.Lett. 251B (1990) 163. 22. A. Jannussis et al., Had.Jou.

I)

(1982) 1923.

93

p-ADICS AND q-DEFORMATIONS Peter G.

o. Freundl

The Enrico Fermi Institute and Department of Physics University of Chicago, Chicago, IL 60637 The study of spherical functions on symmetric spaces!l) on the one hand, and the study of p-adic strings(2) on the other hand, have independently led to the suggestion of a connection between quantum groups and p-adic symmetric spaces. In an attempt to clarify the nature of this connection, I have studied!S) the problem of scattering on symmetric spaces. At its simplest one deals!41 with scattering on the real hyperbolic plane H(oo) = SI(2,R)/SO(2). In this case the zonal spherical functions (zsf's) are best described(5) in terlDll of the Poincare unit disk D (SU(I, 1)/S(U(I) x U(I))) of radial coordinate tanh r and angular coordinate o. The zsf's, labeled by a real parameter ~ (corresponding to the continuous spectrum) are then determined by three requirements:

1) A zsf q,(oo)(tanh r) is a complex-valued function on D which depends on the radial coordinate (tanh r), but not on the angular coordinate (0) of D. In other words, it's pull-back to the group SL(2,R) is SO(2) ''''invariant.

2) q,~oo) is well behaved and suitably normalized at the origin (1) 3) q,~oo) is an eigenfunction of the Laplacian (- SchrOdinger operator) on D

(2) and therefore of every SU(I,I) invariant differential operator on D (as all such erators are functions of A).

OJ)-:

At large distances from the origin (r -+ 00), q,~oo) has the simple asymptotic form (up to terlDll suppressed by factors e- 2nr n = 1,2, ...)

(3) 'W...k aupporied ita pan by lJae N.io... SeieDce FODDd.io., PRY 90 00386.

94 involving an incoming and an outgoing wave and the J08t fundion c(oo)(±~). In the theory of symmetric spaces this c(oo)(~) is known"! as the Harish-Chandra c-function and is given by (a special case of) the famous Gindikin-Karpelevil! formula

(4) The corresponding S-matrix is

(5) with ~ playing the role of momentum. All this has a p adic counterpart. Instead of .11(00) = SL(2,R/SO(2) = SL(2, Qoo)/maximal compact subgroup of SL(2,Qoo), consider n"J = SL(2,Q,)/SL(2,Z,) (SL(2,Z,) being the maximal compact Bubgroup of SI(2,Q,». This n") is a discrete space,le) a tree with p+ 1 edges meeting at each vertex. In this tree there is a natural concept of distance between vertices: the number of edges which make up the unique simple path (i.e., no edge is traversed more than once) path between these vortices (see figure I). Now we can choose any vertex as the tree's ·center", e.g. the vertex 0 in the figure. Then each vertex V is separated from 0 by a distance dOyEZ+. For instance, doo

= O,doA = I,do~ = 2,doc = 3, etc ••...

The natural generalization of the zsf on on the tree, whose value at a vertex V depends solely on the := r from V to the origin o. Then we follow (2) and require

n") is a fundion 4Jt distance doy

(6) and to match condition 3) of the archimedean case of 4J~oo), we also require 4J~)(") to be an eigenfunction of the laplacian on the tree. This laplacian (we also call it A) is defined A la Cartierl7J in a manner consistent with the mean value theorem. Given a function I(V) of the vertex V, we define (see figure 2).

, I I

,/ Fig.2 A/(V)

=

E y,:4v,v=1

For spherical functions

I(Vi) - (p + I)/(V)

(7)

95 the exception at r = 0 taking into account that the distance r is positive semidefinite. The final requirement that 4>~'(r) be an eigenfunction of the tree-laplacian with eigenvalue E>. is then

p4>r'(r + 1) + 4>r'(r - 1) = (EA + P + 1)4>r'(r) for r ~ 1

(9a)

and

(9b) remember r takes only integer values. The equatioDB (6) and (9) fully determine 4>r'(r). Write El + P + 1 = p/)./2 + p-/)./2, (10) which amounts to a convenient choice of '\. Thenl7,8)

4>~'(r) =

(1 + ~) {c(p'(,\)p(¥r + c(P'(_,\)p(-i~-l)r}

(l1a)

with the analogue cP (,\) for H(P' of the Harish-Chandra c-function given by a counterpart of the Gindikin-KarpeleviC! formulal·,!!(

c(P'(,\)

= 1-

p-l-il

1 - p-i>.

(l1b)

From these tree Jost-Barish-Chandra functions c(P' we can construct non-c:asual Oust for p-adic strings) S-matrices S(P' for scattering on the p-adic hyperbolic planes H(P'. These yieldlSJ an adelic S-matrix which explicitly involves the Riemann seta function and which is causal on account of the prime number theorem of Hadamard and de la Vall&, Poussinl 88

Returning to the connection with quantum groupe, we notice that the 4>r' are polynomials in the variable Z = !(p'l/I + p-/)./2) (12a) 2

or equivalently Laurent polynomials in the variable

11= p,A/2 , 80

that

1

_

Z = 2(11 + II 1).

(12b)

(12c)

In fact they are a special case of the Rogers-Askey-Ismail (RAI) polynomials C,,(Zj tlq) =

:E 1+1=..

(ti q),(tj q)1 11'-1 (qj q),(qj q)i

(13a)

'J~O

where (aj q). = (1- a)(1- qa) •.. (1- q'- l a).

(13b)

To see this connection, notice that using Eqs. (12) we can convert the ,\ variable to the variable z, and thus define the polynomials

(14)

96 with 4>rl(n) given by the equations (11). Finally, define the Macdonald polynomials!l]

p.. (2:;tlg):=

~:::~:C"(2:;llg)

(15)

Then the desired relation is

"'.. (2:;!) p

= (1 + !6..o)P .. (2:; !IO). p p

(16)

The RAI polynomials arel1) essentially (aee Eq. (IS)) the Macdonald polynomials for the root l)'lltem A1 corresponding to SL(2). No surprise then that they appear in our context. The special case t gl/1 of (15) corresponds essentially to spherical functions on SL(2)f as shown by Koomwinder,(ll) while the ordinary casel1) is obtained by letting t gl/1 -+ 1. The full quantum group with two parameters whose spherical functions yield the fun two parameter (I and g) family of Macdonald polynomials (15) remains to be nailed down.

=

=

I wish to thank Leonid Chekhov for valuable discussions. REFERENCES

1. I. G. Macdonald, in Orthogonal Pol"nomiala: Theor" and Practiee, P. Nevai ed., Kluwer Academic Publ., Dordrecht, 1000, p. 311; I. G. Macdonald, Queen Mary College preprint 19S9. 2. P. G. O. Freund, in Super,tring, and Partiele Theor", L. ClavelIi and B. Harms eds., World Scientific, Singapore, 1000, p. 251; P. G. O. Freund, ·On the Quantum Group - p-adica Connection" , EFI ~90, to appear in the Sakita FutBehrift. 3. P. G. O. Freund, Phya. Lett. 25'1B (1991) 119. 4. R. F. WellrhaIm, Phya. Rev. LeU. 65 (1000) 1294; M. A. Olshanetsky and A. M. Perelomov, Phya. Reports 94 (1983) 313. 5. S. Helgason, Topie, in Harmonie Anal",i, on HomogenoUB Spaee" Birkhiuaer, Basel, 1981. 6. F. Bruhat and J. Tita, Publ. Math. I.H.E.S. 41 (1972) 5; A. V. Zabrodin, Comm. Math. Phya. 123 (19S9) 463; Mod. Phys. Lett. A4 (19S9) 367. 7. P. Cartier, Proe. S"mp. Pure Math. Vol. 26 A.M.S. Providence, 1973, p. 419. S. F. Mautner, Am. J. Math. 80 (195S) 441; ibid. 86 (1964) 171. 9. I. G. Macdonald, Spherieal FunetionB on a Group 0/ p-odie 7\!pe, Ramanujan Institute, Madras, India 1971; R. P. Langlands, Euler Produd" Yale Univ. Press, New Haven, 1971.

10. R. Aakey and M. E. H. Ismail in Studie, in Pure Mathematic" (P. ErdOs, ed.) Birkhiuser, Basel 1983 p. 55. 11. T. Koornwinder, in Orthogonal Pol"nomial, Theor" and Practice, P. Nevai, ed., Kluwer Academic Publ., Dordrecht,l990, p. 257.

97

THE QUANTUM GROUP STRUCTURE OF 2D GRAVITY Jean-Loup GERVAIS Laboratoire de Physique Theorique Ecole Normale Superieure 24, rue Lhomond 75231 PARIS CEDEX 05 - FRANCE.

1. INTRODUCTION.

The algebraic approach to quantum gravity in the conformal gauge - that is to the Liouville field theory - was developed very early by Neveu and myself1,2,3,4,5,6,7. Recently it has been making substantial progressS,9,10,11,12,13. This comes about because there exist decompositions of inverse powers of the metric into operators that precisely transform under irreducible representations of the underlying quantum groupS,s in the standard form. Their non-commutativity as quantum-field operators coincides with the non-commutativity that is induced by the "quantum" deformation of this group in the mathematical sense. Their braiding and fusion properties are known explicitly, since they are given by the universal R matrix and q-ClebschGordan coefficients respectively9,12. Some related progress have also been made for the WZNW models 14 . The most recent advance 10,11,12 concerns the challenging problem of the strongcoupling regime. In our early works 5,6 Neveu and myself had put forward the special cases of central charges C grav = 7, 13, and 19, for which we showed that a particular operator with a real Virasoro-weight is closed under braiding. We were thus led to the idea that the theory should make sense for these special cases. Making use of the properties of the operator-algebra derived from the quantum group structure just mentioned, it has been possible to prove a unitary truncation theorem 11,12 which shows that indeed 2D gravity is a consistent conformal theory at the above central charges. These notes review the main lines of the above-mentioned topics. Some background material scattered in the early papers 1,2,3,4,7,15 are included for completeness. The present discussion has been extended the Toda field theories, leading to new deformations for Lie algebras of rank higher than one 16 . We shall not discuss this topic here. 2. THE CLASSICAL STRUCTURE. First recall some basic points about the weak coupling regime. In the conformal gauge, the classical dynamics is governed by the action: S

= _1_ Jd2z.,fi{~9"~aac)a~c) + e2C) - ~Roe)} 4~~

2

4

(2.1)

98

9..6 is the fixed background metric. We work for fixed genus, and do not integrate over the moduli. As is well known, one can choose a local coordinate system such that g2 is complicated and not very illuminating. We shall come back to it below in a suitable form. It is simple to derive from (2.10) that -

4~ {T;-), ,p;(CT')}P.B. =

[c5(CT - CT')8cr• + ic5'(CT - CT')] ,p;(CT')

which means that ,pi is a primary field with weight A classical primary field A,1 with weight A is such that -

4~ {T;-), A.1(CT')}P.B. = [c5(CT -

CT')8cr•

-

= -1/2.

(2.15)

More generally, a

Ac5'(CT - CT')] A.1(CT').

(2.16)

101 This means that A.1dz.1 is invariant by analytic mapping z -+ F(z). Moreover it also follows from (2.10) that

_~{ T(o-), T(o-')}p B = [6(0- _ 0-')80-' _ 26'(0- _ 411"

'Y

'Y

••

0-')1

T(o-')

+ 6'11(0- -

'Y

0-') (2.17)

'Y

so that ,T satisfy a Poisson-bracket realization of the Virasoro algebra with central charge C = 3h in standard notations. The last formulae were recalled as a preparation for the following theorem. For 0- i- 0-', any primary field A.1 satisfies Poisson-bracket algebras of the form

= 'L Ck(o-,e)1/>k(o-')

{A.1(o-), 1/>j(o-')}P.B.

(2.18)

Ie

where C10 depends upon the primary field A.1. e is the sign of 0- - 0-'. The proof goes as follows. The Poisson bracket (2.16) vanishes for 0- i- 0-', and it follows from the Schrodinger equation (2.5) that

[-a!. + T(o-')] {Aa(o-), 1/>j(o-')}P.B. =

0

(2.19)

so that the left-member is a linear combination of 1/>1 and 1/>2 and (2.18) follows. Since the right-member of (2.17) is a sum of 6-functions at 0- = 0-', the linear combination is different for 0- > 0-' and 0- < 0-'. q.e.d. 0 Applying this result to the fields 1/> themselves, one moreover sees, by reversing the role of 0- and 0-', that they satisfy Poisson bracket algebras of the form

{1/>j(o-), 1/>1o(o-')}P.B. = Ls;':(e)1/>,(o-'),1/>m(o-)

(2.20)

1m

where e is the sign of 0- - 0-'. The relevance of this fact to our discussion is that this last relation is equivalentt to the Poisson-bracket relations which is the limit of the corresponding braiding property of Uq (sl(2» for q -+ 0 . This discussion may be generalized to the higher powers (1/>dP.(1/>2)~ which we rewrite as 1/>~) with 2J = p. + v, and 2m = v - p.. One may verify 7 that they are primary fields with weight ~ = -J/2, and that, for -J S m S J, they satisfy a differential equation of order 2J + 1. As a result their Poisson bracket algebra is closed in a way similar to (2.20) and again corresponds to the Poisson-bracket relations which are the limits ofthe Uq (sl(2» quantum- group braiding-structure. In this limit, one recovers the standard sl(2» algebra, which is precisely the one mentionned above that leaves the =

v'V Of MJ LJ ± M + IJ IJ,M ± 1 >,

J3IJ,M >= M IJ,M >. (3.19)

These operators satisfy the Uq (al(2»-commutation-relations (3.20) In

9,12

1) For

e

the operator algebra of the fields was completely determined. > IT > IT' > 0, these operators obey the exchange algebra

11"

N t(J')( ') t(J)( ) (J, J ')N' MM' "N' IT "N IT,

(3.21)

-J5,N7':,J; -J'5,N'5,J'

(J,J')Z:;~,

=

« J,MI® < J',M'I) R (IJ,N > ®IJ',N' »,

R coincides with the universal R matrix of Uq (sl(2».

(3.22)

106 2) For 0

< < (1'

(1"

e

< 71", the fields obey the exchange algebra (3.24)

(J,J')r:.;=, =

« J,MI® < J',M'I) R (IJ,N > ®IJ',N' », 00

R

= e(2ikJ ®J (1 + ~ 3

(1

3}

(3.25)

-2ik)" -ikn(n-1}/2

-e

Lnj!

e-iknJ'(J_)" ® eiknJ'(J+)n).

(3.26) 3) The two exchange formulae are related by the inverse relation (3.27) -J~N~J; -J'~N'~J'

4) The short-distance operator-product expansion of the

e}.::\(1')e}.::)(u')

=

efields is of the form:

J , +J.

L {(d(u _ u,».o.(J)-.o.(J, }-.o.(J·}

J=IJ,- J ,I (3.28)

=

where d(u - u') 1 - e-i(a-a'}, (JloM1iJ2,M2IJ1,J2iJ,M1 + M 2) denotes the Clebsch-Gordan coefficients of Uq (sl(2» , and ..:l(J) := -hJ(J + 1)/71" - J is the Virasoro-weight of e}.:}(u). 5) Define the quantum group action on the fields by

e

(3.29) Then the operator-product e}.::}(u) e}.::}(u') gives a representation of the quantum group algebra (2.6) with the co-product generators (3.30) where the tensor product is defined so that (3.31) and where each term in the expansion over J transforms according to a representation of spin J. Similar formulae hold in the other half circle. Eq. (3.30) coincides with the standard co-product, and is thus non-symmetric, the two definitions being related by the universal R matrix. The exchange properties of the fields (Eqs (3.21)-(3.26}) show that their quantum mechanical structure precisely matches this asymmetry, so that the transformation law (3.29) is fully consistent with the operator-algebra.

e

107 Obviously the same structure holds for the hatted fields. One replaces h by Moreover the hatted and unhatted fields have simple braiding and

h everywhere.

fusions. 9 The most general (2J,2J) field

e(J!2 ~ e~) 'f!! MM M

has weight

) CLiou- 1 l:J.Kac ( J,JjCLiou = 24

214

((J + J + l).JCLiou -

1 - (J - J).JC Liou _ 25) 2,

(3.32)

in agreement with Kac's formula. 4. THE CASE OF REAL SCREENING CHARGES If CLiou > 25 the screening charges Q± are real. This is the weak coupling regime which is connected with the classical limit ('Y -+ 0). In this region h and h are real and the structure recalled above is directly handy. Let us briefly discuss how the powers of the metric are reconstructed. Consider for instance exp( -J Q_ ~). There are two types of cases one may distinguish. 1) One may consider, as is most usual, closed surfaces without boundary. Then the natural region is the whole circle 0 ::; (J' ::; 21r. We are aiming at the quantum version of Eq. (2.8). It will involve the fields e~, together with their counterparts

~>C:I:+) whose exchange properties are similar. Concerning the latter one should remember that they are functions of z', that is, are anti-analytic functions. so that the orientation of the complex plane is reversed. This may be taken into account simply by replacing i by -i in the above formulae for the e-fields, that is by taking the complex conjugate of all the c-numbers without taking the Hermitian conjugate of the operators. The appropriate definition of ~ «(J') is

(4.1) where (lJ,W)MY is the complex conjugate of (IJ,w)M). In addition to taking the complex conjugate of IJ, w)M' we have introduced an additional phase factor for later convenience. One may see that it does not change the braiding and fusion properties, up to overall normalizations. Concerning braiding, for instance, this is true because (Ill J2)fi/~&, is non-zero only if PI + P2 = MI + M 2. As is usual in conformally invariant field theory we assume that the right- and left- movers commute. Thus we take the e-fields to commute with the I-fields. There are two basic requirements that determine exp( -J Q_ ~). The first one is locality, that is, that it commutes with any other power ofthe metric at equalr. The second one concerns the Hilbert space of states where the physical operator algebra is realized. The point is that, since we took the fields e~) and to commute, the quasi momenta w and w of the left- and right-movers are unrelated, while periodicity in (J' requires that they be equal. This last condition is replaced by the requirement that exp( -JQ_~) leave the subspace of states with w = W invariant. The latter condition defines the physical Hilbert space 1iph l/' where it must be possible to restrict the operator-algebra consistently. At r = 0, the appropriate definition is:

t::)

e-JQ-~«(J')

J

= CJ L M=-J

(_l)J-M

e~)«(J')t'1«(J')

(4.2)

108 where CJ is a normalization constant. It is invariant under the quantum group action (3.29) if the fields transform in the same way as the fields:

e

J

e

-;:(J) _ Mt 17" > O. (4.4)

In checking locality, one encounters the product of two R matrices. It is handled by means of the identities N,)* - «J J )-M,-M2)* - - ( JJ)M'M2 « J 1, J 2 )-N2-M, -M2 2, 1 -N2 -N, 2, 1 N2 N,

(4.5)

that follow from the explicit expressions (3.23), (3.26). In this way one deduces the equation (Jl,J2)fi'~2 «(J1,J2)=~ :::.~.r = bp"N, bP2,N2 (4.6)

L

M,M2

from the inverse relation (3.27), and the desired locality relation follows: e-J1(L'f1(Ul) e-J2(L'f1(U2)

= e- J2cL 'f1(U2) e- J1GL 'f1(Ul)

(4.7)

On the other hand, the precise form of the factor eih(J+M) in (4.1) is dictated by the requirement that 1ip hy. be left invariant, as we show next. This is seen by re-expressing (4.2) in terms of"" fields. One gets, at first e- Ja _'f1(U,T)

J

L

= CJ

(_l)J-M eih(J-M) IJ, w)M (IJ, W):'M)*""~)(U)~~J)(u)

M=-J

(4.8) Using Eq. (3.22) of Ref. 9 one writes J

L

(_l)J-M eih(J-M) IJ, w)M (lJ, w):'M)*

=

M=-J J

L

(_l)J-M eik(J-M) IJ, w)ii{ IJ, W + 2p):~

(4.9)

M=-J

If w = w, this becomes, according to (3.16,3.17), J

L

(_l)J-M eik(J-M) IJ, w)M IJ, w M=-J

+ 2p):~ = bm,p C~)(w).

(4.10)

As a consequence, and when it is restricted to 1ipkp., Eq. (4.2) is equivalent to e- Ja _'f1(u)

= CJ

J

L m=-J

C!.{)(w)""~)(u):t...J)(u)

(4.11)

109 and the condition w = iii is indeed left invariant, according to (3.11). 2) One may also consider gravity with boundary, following Ref. 1,2,13. A typical situation is the half circle 0 ::; u ::; 1T. One may set up boundary conditions such that the system remains conformal, albeit with one type of Virasoro generators only. The left- and right-movers become related as is the case for open strings. The appropriate definition of the metric becomes 13 : e-Ja_~(u) _ c "A(J) -

J ~ M,N

t(J)(u) 'oN t(J)(21T - u)

( 4.12)

M,N '>M

where

=
(413) ,.

where a depends upon the boundary condition chosen. These operators are mutually local and closed by fusion l3 • A similar structure appears for factorizable scattering on the half line18 ,19 • So far the present discussion assumes that q is not a root of unity, that is, deals with irrational theories. The problem of specializing q to a root of unity has, however, been essentially reduced to the equivalent limit in the representation theory of Uq (sl(2» which is a much studied problem. Clearly, the present discussion applies whenever the screening charges are real, so that it also describes the C < 1 models, as a continuation of the Liouville theory. There still remains the difficulty already pointed out in the classical case, that positive powers of the metric are difficult to handle. This is a major difference between the proper region of the Liouville theory ( CLio" > 1) and the region of statistical models (C < 1), since (3.32) gives negative or complex weights for positive J and J in the former case, so that one must deal with negative J or J. The above discussion must be continued to negative spinsll,12. One may show that equation (3.14) is equivalent to !(2.i\2J)eih(7n/2+(W+7n)(J-M+7n» F.q (a , IJ , w)M m - V \J+MJ

b·, c·,

e- 2ih (w+7n»)

,

( 4.14)

where a = M -J (resp. a = -M -J), b = -m-J (resp. b = m-J), c = l+M-m (resp. c = 1 - M + m ) for M > m (resp. M < m) and Fq(a, bj Cj z) is a qdeformed--so called basic- hypergeometric function. The continuation to negative J is a direct consequence of Rodgers identity20

where

rq

denotes the q-deformed gamma-function. Equations (4.14,4.15) give

)M=(2" (h»1+2J( 2J )r q(w+m+J+1)I_J_l )M IJ ,w7n Uln J+M rq(w+m-J) ,w 7n .

(4.16)

This exhibits a symmetry between J and -J - 1 which is also shared by the R matrices and Clebsch-Gordan coeflicientsll,12. It is the basis for defining operators

110 with negative J. The crucial point of (4.16) is to show that ",~J-l) and et"J-l) are to be considered for -1::; m ::; 1, and -1 ::; M ::; 1, respectively. 5. SOLVING THE REALITY PROBLEM OF STRONGLY COUPLED GRAVITY: THE UNITARY TRUNCATION THEOREM Next, we consider the region 1 < C Liou < 25, which is relevant to the strong coupling regime of 2D gravity. In this case, h and Ii are complex and Ii = h*. We choose the imaginary part of h to be negative for definiteness. In the weak coupling regime of gravity, the solution of the conformal booststrap we just outlined arised in a natural way from the chiral decomposition of the 2D metric tensor in the conformal gauge, that is by solving Liouville's equation. It is thus legitimate to study the strong coupling regime by continuing this chiralstructure below CLio ... = 25. Complex numbers appear all over the place. However-in a way that is reminiscent of the truncations that give the minimal unitary models- for C Liou = 7, 13, 19; there is a consistent truncation of the above general family down to a unitary theory involving operators with real Virasoro conformal weights only. The main points of this theoremll ,12 are briefly summarized next. The truncated family is as follows: a) The physical Hilbert space. It is given by6,lO,1l,12: 1-.

1-..

r=O

r=O n=-oo

=EB1i-(w~) =EB EB

1iP"lI'

(5.1)

F(w r ,.,) ,

(5.2) The integer

8

is such that the special values correspond to

CLiou

= 1 + 6(8 + 2),

8 = 0, ±1j

h+

Ii = 811'.

(5.3)

Ll(W r ,.,) is positive and in 1ip h ll a the representation of the Virasoro algebra is unitary. The torus partition function corresponds to compactification on a circle with radius R = J2(2 - 8) (see Ref. 10). b) The restricted set of conformal weights. The truncated family only involves operators of the type (21,21) noted x OJ then there exists a unique (up to conjugation in SL(2,R)) fuchsian group r c SL(2,R) such that: ~=H/r

(4)

The induced metric on ~ is one of constant curvature equal to -1 and it can be written in terms of a Liouville field 4> as e~ldzI2. Next we consider differential equations whose monodromy group is a given fuchsian group r. The simplest examples are the equations

(5) with

1["-1 :

~ ---+

H, the uniformization map of

~.

For ~ = C· /{P1 , ••• ,P,,}, the Riemann sphere with n punctures, the schwartzian derivative {'/I"-1, z} has the following general form:

(6) for some C~ depending on the positions of the punctures {P,,}. The parameters C. are known as uniformization parameters and they can be considered as coordinates on the Teichmiiller space of Riemann spheres with n punctures. Moreover they are constrained to satisfy the relations:

(7) Given the representation (6) of the schwartzian derivative we can translate the problem of finding the uniformization parameters of a given Riemann surface E = H/r to the one of finding the values of Ck such that the equation

(8)

have as monodromy group the fuchsian group r of the Riemann sphere with the n punctures P1 , ••• , P". It is important to point out that we can in principle find some values for the C k parameters such that the monodromy group of (8) is fuchsian but it is not the one that uniformizes ~.

117

UNIFORMIZATION EQUATIONS VS. DECOUPLING EQUATIONS We will consider a standard conformal field theory in its Feigin-Fuchs Coulomb gas representation. The energy momentum tensor is given by:

(9) which correponds to a conformal anomaly c constant" , such that: Q

= 1 + 3Q2.

We introduce a "coupling

, 2

=-+,

(10)

The cases , E [0, V2] correspond to the region c ~ 25 and for imaginary, in the interval [0, iV2] we cover the region c ::; 1. The primary fields are (11)

with

1-n

an,m

1-m

= -2-' + -,-

Their conformal weights are given by the standard formulae:

(12) The BPZ equation for the decoupling of the null vector in the V2,l-Verma module is given by [2]:

(13) We will consider only values of ai's such that we can screen the total charge by including just one screening operator J+(z) = e-YOP('>, which means [3]: N

2,

Lai=-+i=1 , 2

(14)

L(1-ni) = 1

(15)

or, equivalently for ai == an;,m;:

i

~:::tfJ

litLi- .

The upshot of the Whitney trick is that any immersed plane curve is regularly homotopic (Regular homotopy is generated by the projected Reidemeister moves of type II and type III.) to a curve that is in one of the standard forms shown below:

134

···~~ootJC5··· This result is known as the Whitney-Graustein Theorem

[W).

The code for an immersed curve in the plane is a product of elements U and U-1 where these denote elements of the form U-1= Mab Mcb and U = Mab Mac.

b c

b _\G

LV

By convention we take U to have Whitney degree 1 ,and U-1 to have Whitney degree -1. Then the Whitney degree of the plane curve with a given minImax code can be read off from a reduced form of the code word. This completes our description of the minImax coding for immersed plane curves and its relation to the Whitney Graustein Theorem. We next turn to a formalism for regular isotopy invariants of links that involves a mixture of the Gauss code and the minImax code.

135

IV.

The Abstract Tensor Model For Link Invariants

By augmenting the minImax code to include information about the crossings in a link diagram, we obtain a complete symbolic description of the link diagram up to regular isootopy. (Regular isotopy is generated by the Reidemeister moves of type II and 111.) In this section we deal with the abstract form of this symbolism. Along with the cup,cap and identity arc formalisms of the previous section, we now add symbols for the crossings in the diagram and further relations among these symbols that parallel the Reidemeister moves. A ,crossing will be denoted by

ab Rcd

or by

-ab R cd'

according to its

We shall take this type with respect to a chosen direction. direction to be the vertical direction on the page. Thus the R without a bar refers to a crossing where the overcrossing line proceeds from lower right to upper left, while the barred R refers to a crossing where the overcrossing line proceeds from lower left to upper right. This is illustrated below.

Note that in this abstract tensor algebra individual terms commute with one another. The inherent non-commutativity is contained in the index relations, and these involve the distinctions left, right, up and down in relation the the chosen direction. In the tensor notation that chosen direction has become the vertical direction of the written page.

136

Reidemeister moves with respect to a vertical direction take the forms indicated in Figure 2. In this Figure are shown representatives for the moves that generate regular isotopy. Thus the moves are labelled I, II, III, IV, V where II and III are the usual second and third Reidemeister moves in this context. Move I asserts the cancellation of adjacent pairs of maxima and minima. The move IV is an exchange of crossing and maxima or minima that must be articulated when working with respect to a given direction. Move V is a direct consequence of move I and move IV; it expresses the geometric "twist" relationship between one crossing type and the other. Each of these moves has an algebraic counterpart in the form of the abstract tensor algebra. A list of these algebraic moves is given below. (Here we have codified just the abstract tensor algebra for unoriented links for the sake of simplicity. The oriented versions are similar but more complicated to write in this formalism.) The moves are labelled, Cancellation, Inverses, Yang-Baxter The Yang-Baxter Equation is an abstract Equation, Slide and Twist. One method for form of the Yang-Baxter Equation for matrices. representation of the abstract tensor structure is to assume that the indices range over a finite set, and that repeated upper and lower indices connote a summation over this set (the Einstein summation convention). With this convention, the axioms become demands on the properties of the matrices. It is worth noting that the axioms for an abstract tensor algebra and the distinctions between cups, caps and the two forms of interaction are an articulation of the basic distinctions up, down, left and right in an oriented plane.

Definition.

AT A, is the free b ab l>a' Rcd ' Mab, Mab,

The abstract tensor algebra ,

multiplicative abelian group on the symbols

modulo the axioms stated below. The indices on these symbols range over the English alphabet augmented by any extra conventional set of symbols that are appropriate for a given

137

application. Thus elements of ATA are products of symbols modulo commutativity and the relations that are expressed by the axioms. The axioms can be regarded as rules of transformation on expressions, so that two expressions in ATA are equivalent if and only if there is a finite sequence of transformations from one to the other.

Axioms for the Abstract Tensor Algebra

= ~:

I.

Mai Mib

II.

R~b-R I j "1 j cd

(Cancellation)

(Inverses)

III.

(Yang-Baxter Equation)

IV.

V.

(Regular Isotopy)

(Slide Move)

=

ia 'b MCIRdj MJ

(Twist Move)

We translate an (unoriented) link diagram into an expression in the abstract tensor algebra by arranging the diagram for the link so that it decomposes into cups, caps and crossings with respect to the vertical (I shall refer to the special direction as the vertical.). Label the nodes of the decomposition and write the corresponding tensor.

138

b

b

I. )

li.

Y.

c:z.X'" c:

b

"J

~ Figure 2

139

Theorem 4.1. Let K be an (unoriented) link diagram arranged in standard position with respect to the vertical. Let T(K) be an abstract tensor expression (free of deltas) corresponding to K. Then the diagram K can be retrieved from T(K). (Here it is assumed that b the axioms other than the use of the delta (ba) for the tensor algebra are not applied. That is, T(K) is taken simply as a code for the link diagram.) Proof. To retrieve the link diagram from T(K) first place crossings corresponding to the R's in the expression T(K). (I shall say the R's when referring to R's or R-bars.) Note that in T(K) every upper index is matched by a unique lower index. Choose any R, and draw in a plane a glyph corresponding to that R. Now choose an index on the R in question and find the other R (there is a unique other) that contains this index. Place another glyph in position with respect to the first so that their indices are matched. Note that the relative placement is unique. For example, the first R may have a lower left index to be matched with an upper right index of the second R. In this case the second R is placed to the lower left of the first. After all the .R's have been successively interconnected in this manner, place the cups and caps corresponding to the M's. Since the expression is known to come from a link diagram, this is possible, and will give a reconstruction of the diagram. We leave the placemant of the cups and caps for last because the sites for their nodes are then chosen by the placements of the R's. This completes the proof. Remark. This proof can be regarded as a generalization of the usual procedure for writing a braid from an expression in the braid group. It is particularly important for our conSiderations, beacause of the next result. Theorem 4.2. Let K and L be two links arranged in standard position with respect to the vertical. Let T(K) and T(L) be corresponding expressions for K and L in the abstract tensor algebra, ATA. Then K and L a're regularly isotopic as link diagrams

140

if and only if the expressions T(K) and T(L) are equivalent in the abstract tensor algebra. Proof. If K is regularly isotopic to L then there is a sequence of the diagrammatic moves 1,11;111, IV (V) taking K to L (See e.g. [K3J. This is the reason for articulating these particular moves with respect to a direction.) By the translation from moves to abstract tensors, this gives a sequence of axiomatic transformations from T(K) to T(L). Hence T(K) and T(L) are equivalent in ATA. Conversely, if T(K) and T(L) are equivalent in ATA, then there is a sequence of axiomatic transformations from T(K) to T(L). Apply Theorem1 to each expression in this sequence and obtain a regular isotopy from K to L. This completes the proof. Remark. We have shown that the association K -------> T(K) of links to expressions in ATA is a faithful representation of the regular isotopy category of links. When ATA is represented by actual matrices, and the Einstein summation convention, then this association gives invariants of knots and links such as the Jones polynomial and its generalizations. We will give specific examples in the next section. At the abstract level, the correpondence gives a "complete invariant" for knots and links. This suggests that if there are sufficiently many representations, then the matrix invariants derived from ATA may be able to distinguish topologically different links. I conjecture that any given matrix invariant will not separate all distinct knots and links, but that the collection of all representations of ATA does have this property.

v.

From Abstract Tensors to Quantum Groups

Abstract tensors associate matrix-like objects to the crossings, maxima and minima of a link diagram. (The diagram is arranged with respect to a given direction that we take here to be the vertical direction of the page.) The Gauss code (Section 2) suggests considering non-commutative algebra elements arrayed along the edges of the link diagram. The natural one-dimensional ordering imposed by the edges keeps track of the order of multiplication. These two pOints of view come together when we assume that the

141

abstract R-matrix has a tensor product decompositon in terms of matrix elements associated with the two transversal edges at a crossing. In particular let us assume that

=

where the summation runs over a suitable set of elements and

Hiding the indices, we may write

R

= 1:e8e' e

where the tensor product sign insures the convention for assigning the indices as shown above. As in our previous discussion, the individual elements such as

e:

commute with one another, so that

the non-commutative algebra structure that is inherent in the category comes from the pattern of index connections. Note that in an actual matrix algebra representation we expect that the sums are either finite or convergent. It is possible to consider infinite sums such as the formal power series that arise from the exponentiation of matrices. The diagrammatic interpretation of

R

R

is

shown below:

= ~e®e/ e

In this diagrammatic, we can dispense with the direct labelling of the indices, since they correpond to the lines emanating from the algebra elements. Multiplication of algebra elements corresponds to connection of the lines. The convention for multiplication in the

142

diagrammatic algebra will be to read upwards for a left to right ordered' product. Thus AB corresponds to a diagram with the upper leg of A connected to the lower leg of B as shown below. Let n denote the line algebra generated by the cups, caps and diagrammatic algebra elements placed on the lines of the diagram.

AB Link diagrams do not proceed only in the vertical direction. In fact the cups and caps give us a natural anti-morphism s: n ~ n of the diagrammatic algebra n defined by the equation s(X)

where M denotes the algebra the element for the cup, and

=

MXW

e~ent

X

corresponding to the cap, denotes the top to bottom

traverse of a vertical algebra element X. At the matrix level, is the transpose of the matrix for X. We shall call s the antipode of n.

X

t'--->~

S

W

X

143

Lemma A. s: 0 -----> 0 is an a,nti-morphism. That is, s(XY) = s(Y)s(X) for any algebra elements X and Y. Proof. x

Use the fact that

MW=WM=1, plus that fact that

= x, or contemplate the diagram below.

= S(y)S ()().

Remark. In regard to the cups and the caps, it is useful to have the following diagram for these elements and their transposes. This diagram indicates the element or its transpose (indicated with a bar) as it appears in a product composed along the indicated direction.

144

Lemma B.

Proof.

-

~

This lemma shows that we can move algebra elements around the diagram, by regarding the traverse of a maximum or a minimum as composition with the antipode s.

Lemma C.

R-1=

rs(e)8e'

e Proof. This formula for the inverse of R is a direct consequence of the twist relation of the previous section. The diagrammatic proof is given below:

I :. Ft'

Ft4t+») = ~Xe' 5'?. as el±)(r)

= e.(0)+B(±)r2 _.!i lim .!In[(4>.lexP(-AJcYld2e .)] 211' L~oo L

= e'(O) + B(±)r2 + ~ L..J a\±) ',n rny -

I

(4).I4>i)

(20)

,

n=l

e

e

where the integral is over the cylinder W Re E [O,R), 1m E (-t,t)}, and the role of the term B(±)r2 is to cancel any bulk term arising from the last term (or rather its analytic continuation to large r), to implement the normalization condition e\±)(r)/r -> const(±) as r -+ 00. The fact that B(±) is nonzero, generically, can be understood from a CPT point of view as arising from the mixing of

. is primary

165 (in general one has to use the transformation properties of the non-primary question) we have

(2 )1-1/

al~ = --+-('1') n. ~ ~

(=f"±t

,,-1

J},

,,-1

j=1

11' ZJ

;=1

f II (2 IZ~I)!I (ilil>(l,l) II il>(zj,zj)li)conn

4>.

in

(21)

The correlators here are connected (with respect to the "in- and out-states" created by 4>.) critical (n + 2)-point functions on the plane, with (il ... Ii) denoting

(4).( 00,(0) ... 4>j(O, 0»). Actually, the discussion of CPT so far has been formal in that we have ignored This is no problem for the 9-related models (9 f. possible divergencies in the

al:;1.

A~1» discussed in the previous section, since there all al~ are finite. 1t This follows immediately from the fact that d < 1 in these theories (fo; the perturbed non-unitary CFTs leading to the A~~-related PESTs the argument is slightly more involved and can be found in [8]). On the other hand, if d ~ 2(",,-1), then is UV divergent (unless it vanishes due to some symmetry of the theory). Note that for given n the UV divergencies in are the same for all i, since they arise from the region where one or more (all, for the leading divergence) Zj -+ 1. It is commonly assumed that these UV divergencies can be regularized by analytic continuation of the in d. This analytic continuation method is presumably a consistent renormalization scheme (although we are not aware of a proof), as long as d f. 2{t,:1). If d = 2(",;1) the coefficients al~ diverge even after analytic continuation. It is well known that these divergencies'signal the presence of terms non-analytic in I'll in e.( 1'). At present no general systematic method is known to explicitly determine these non-analytic terms within a CPT framework. Within the TBA approach, however, they can often be determined.[28] In the cases studied so far it turns out that the B(±)1' 2 term in (20) has to be replaced by a term proportional to 1'2 In l' •

al:;1

al:;1

al:;1

Even if d f. 2(,,:1) the analytic continuation method is only of limited use in the cases we are interested in, where the perturbing field is not almost marginal, since to analytically continue in d one needs explicit expressions for the Except for

al:;1.

the a\~), which are just proportional to operator product (OPE) coefficients (see (24)

below), they are known generically only for a~~{ and a~~;, see e.g. [27] and references therein. In all other cases evaluating using eq. (21) is not easy, even numerically, and in practice is feasable (numerically2 only if convenient representations for the correlators in (21) exist. In this way ~.1 was calculated[29,8] for several models. In the next subsection we will discuss a general and often the only practical way (found so far) to estimate the al~. Note that even if some al~ diverge, the expansion coefficients al~ of the scaied energy gaps el±)(1') = el±)( ~) - e~±)(1') are finite and can be estimated using the just mentioned method.

al:;1

If We here ignore IR divergencies, which may occur in the aJ~ with i > O. The correct way to reguralile l271 these divergencies is by analytic continuation in the sealing dimension of 4>;.

do,

166 4.2 The Truncated Conformal Space Approach The starting point of the TCSA[30j is a Hamiltonian formulation of perturbed CFTs on the cylinder,[31 j

H;..(R)

=

HCFT(R)

HcFT(R)

=

271"

+).

f dz C)(z) ,

-

c

(22)

1i(Lo + La - 12) ,

where the integral is along a "line of constant time" around the cylinder, and La (La) are Virasoro modes whose eigenvalues ll. (.1.) are left (right) conformal dimensions. As long as the effective coupling )'RY is not too large, the low-lying eigenvalues can be calculated approximately by truncating the Hilbert space of the CFT to a finite-dimensional subspace (by ignoring all states above a certain level', or scaling dimension, say) on which H;..(R) is diagonalized numerically. This method has been discussed and applied extensively,[30,32,18,17,33j one important subtlety was however not noticed until recently.f27] Namely, that the eigenvalues of H;..(R) all diverge as the truncation is removed, if there are UV divergencies in CPT, i.e. if d 2: 1. As remarked earlier, the coefficient a\~ diverges if d 2: 2(n;1), but can be regularized by analytic continuation in d (as long as dol 2(";1»). This analytic continuation has to be done "by hand", however, and the TCSA does not do it automatically. So for d 2: 1 only difference& of eigenvalues, like ei(r), can be calcUlated with the TCSA. Studies[30,32,18,17,33j of many cases where d < 1 have shown that even for relatively low truncation levels, e.g. I = 5, the TCSA gives quite accurate results for the lowlying eigenvalues for small to moderately large volumes. For d 2: 1 we have recently demonstrated[27] that the same is true for energy gaps (although, perhaps not too surprisingly, the TCSA does not seem to be accurate up to as large volumes as it is in the cases without UV divergencies). The Hamiltonian formulation of perturbed CFTs also offers a general method to calculate the aj,,,, or at least the Let e~l](r) denote the scaling functions obtained from the TCSA at level I. For any finite I, at least, they obviously have an expansion of the form

a.,,,.

e!l](r) =

f= a!~ r"Y

.

(23)

11.=0

The "truncated coefficients" a!~ can be calculated in standard Rayleigh-Schrodinger perturbation theory. In terms of the OPE coefficients G}ij == C~i+~i = (ilc)(l,l)li), we find[27] for the first four terms (see [27) for the next one)

(24)

167 where dji = dj - dj, gij = (iii), Clij = L:~ (g-1 )i/,C/cjS';'j (with 8; = ~i - .:5.i ), and the prime on the summation symbols indicates that they ate to be restricted to the truncated Hilbert space. Alternatively, (the I --+ 00 limit of) these expressions can be obtained[27] directly from (21) by radially time-ordering the correlators and inserting decompositions of the identity between neighboring ~ fields. In cases where expressions for the CFT correlators in terms of elementary or familiar special functions are known, one can obtain formulas for the ai,n in terms of more explicit multiple sums (see [29,8]), which are related to the above general formulas by nontrivial sum rules for the OPE coefficients of descendant fields. The simplest example is that of aO,2, where one can even obtain a~!2 for finite 1 (by comparison with (24)), namely[27] all]

=

_(211")2(1-V)

K,2

t (_d)2

± /c=0

0,2

k

_1_ 2k + d

(25)

Here we see the UV divergence of CPT and the TCSA very explicitly; this sum converges as 1--> 00 only if d < 1 (and dolO, -2, -4, ... ).

4.3 Comparison with the TBA Even the few coefficients a~~ calculable (analytically or numerically) in CPT provide nontrivial tests of the TBA results. By "calculable in CPT" we of course mean calculable up to the appropriate power of K,±, see (21), which (given our present knowledge) must be provided as external input to the CPT calculation of the smallvolume spectrum. So if we obtain from the TBA two or more (nonzero) a;,n that are also calculable in CPT, we can extract K, from one of them and the rest are then used as consistency checks. In this subsection we will say a few words about the comparison of TBA and CPT results for the ground state (i = 0) of the perturbed CFTs considered in sect. 2. Consider first the perturbed unitary CFTs. Note that except for the E~l) case all these perturbations are by the leading thermal conformal field, i. e. the one coupled to the temperature (mass term, in QFT language); furthermore, in all the statistical systems corresponding to these perturbed CFTs there is a duality transformation relating perturbations in opposite directions (the perturbing field is Zz-odd w.r.t. the Kramers-Wannier symmetry restricted to the self-dual critical theory). Therefore the ground state energy should be independent of the sign of A in these theories, implying in particular that a~~J = 0 for all odd However, the scattering theories corresponding to .>. positive and negative, describing the phases of unbroken and spontaneously broken symmetry, respectively (by convention), are different. Strictly speaking, the minimal PESTs are conjectured to describe the A > 0 phase of the theories, while the scattering theories describing the A < 0 phase are related to the PESTs by duality.[32] In the E~l) case, on the other hand, the perturbation is "magnetic" and opposite signs of A lead to the same massive QFT. The immediate consequence a~~J = 0 for all odd is the same as in the above mentioned thermal perturbations, although for a different reason. The form of the TBA expansion (17) is therefore exactly as expected from CPT, with iio,n = a~;i" and ii = 2y. For the

n.

n

168

I Theory ~

II

A~2)

(M;,5)

iO.097048456298606( 6)

Ai2)

(M;,7)

M~,5

1 A~l)

Di

l)

D~l)

0.040537955423786(4)

iO.33412100338590(6)

I

TBA

CPT

00,2

1.3587274893(2) .10- 4

1.3587274892973(2) .10 4

00,3

-4.7582753(2) .10- 6

-4.75829(3) . 10-6

00,4

2.130038(6) . 10- 7

2.130039(5) . 10- 7

00,2

1.2819264579(3) . 10- 4

1.281926(2) . 10-4

00,3

-5.1850727(4) .10- 6

-5.185073(4) .10- 6

00,4

2.66176(3) . 10- 7

2.661753(5) . 10- 7

00,2

2.2841438952(4) .10-3

2.2841438951(4) .10-3

aO,3

-5.825723(1) . 10- 5

-5.825(5) . 10- 6

00,4

.10- 6

-2.29(3) .10-6

00,..

-2.26634(4)

"0.178484948224174(8) " aO,4 112.2923742792(4) .10- 4 1 0.133325360490478(5) 0.15797698617775(2)

2.2925( 4) . 10- 4 1

aO,4

5.3139311369( 4) . 10- 5

5.31394(8) . 10- 6

aO,4

1.0052507265(3) .10- 4

1.00524(2) . 10- 4

. 10- 4

1.982005(1) . 10- 4

D~l)

0.18403147021451(3)

00,4

1.982005645(1)

E~l)

0.09283439222673(2)

00,4

2.430562917(2) .10-5

2.4305632(4) .10- 6

00,4

. 10- 5

1.18088962(4) . 10- 5

E~l)

0.06203236135476(2)

1.180889634(2)

Table 21 Comparison of TBA and CPT results for the expansion coefficients ao,.. of eo( r) in several perturbed eFTs with diagonal S-matrix. The first column gives the theory, either in terms of the affine Lie algebra to which it is related, cf. Table I, or directly as a perturbed CFT (the prime denoting the -PI,3-perturbation of the indicated minimal CFT in the first three cases). K, in the second column is obtained, in the first three models, by comparison of aO,1 from the TBA with ao,I! K" which is analytically calculable in CPT, CC. eqs. (21) and (24); in all other cases, where ao,.. = 0 for odd n, it is obtained by comparison of aO,1 with aO,2 / K,2 .

The TBA results for the 00,.. are obtained from the numerical solution of the corresponding TBA equations. The CPT result for 00,2 in the A~2) and M~,6 cases and 00,4 in the A~l) and D\i) cases is obtained (cf. [8]) by a numerical evaluation of the corresponding integrated correlator in eq. (21). In all other cases we obtained the CPT results using eq. (24) to calculate the a~q.. for truncation levels I = 0, I, ... ,5, which were then extrapolated to I =

00.

The model M~,5 was first studied in [9). For the M~,5 case, K, a8 well as CPT and TBA results for 00,2 were first given in [20) (though to JIDlch lower accuracy).

169 perturbed non-unitary CFTs all o$,~J # 0, and 0.0." = 0$,72 (the sign in the superscript depending on one's conventions), ii = y. A comparison of TBA and CPT results for several perturbed CFTs is given in Table 2. One sees that the agreement is excellent. We have now come full circle through a long series of nontrivial arguments/conjectures and calculations: We started from certain perturbed CFTs that were argued to be integrable arid furthermore purely IIiassive. The corresponding scattering theories were conjectured to have certain purely elastic S-matrices, but not without the "minimal" tI•• "non-minimal" ambiguity (d. sect. 2). The TBA was then used to study the infinite-volume thermodynamics of these PESTs, which allows one to obtain the finite-volume ground state energy. The UV limit analysis of the latter, sect. 3, resolved the ambiguity in the S-matrices, showing that the minimal ones describe the corresponding perturbed CFTs. And finally, the UV expansion of the ground state energy obtained from the TBA for the minimal PESTs was tested against predictions of CPT (or the TCSA), which was essentially our starting point.

5.

RECENT DEVELOPMENTS

We now describe recent work extending the discussion of the ground state energy in PESTs in two (overlapping) directions. One is to calculate the ground state energy of theories with massless particles and/or non-diagonal S-matrices using the TBA, the other to establish analytical results for the finite-volume energies of excited states. For these problems there are as yet no exact results as general as those for Eo(R) in PESTs; rather, what is known derives from a study of specific (classes of) examples. There are however analytical large-volume results for the energies of I-particle states at zero momentum (= finite-size masses) which are of more general applicability, and will now be briefly summarized. (For results on 2-particle states, d. [34,30,18].)

5.1 Finite-Size Mass Corrections in Large Volume If a stable particle in a QFT is enclosed in a finite box, its mass changes from its infinite-volume value, since the virtual cloud surrounding the particle is "squeezed" by the box. More precisely, two processes contribute to the leading finite-volume mass shift (corresponding to the two terms in eq. (26) below). In the first the particle itself disintegrates into two virtual constituents, which "travel once around the world" before recombining to give back the original particle. In the second, virtual particles in the cloud travel around the world. Liischerl35] first studied this problem for QFTs with very simple mass spectra. We then generalized[lT] his work to (almost) arbitrary purely massive QFTs (in any dimension). The proof of the formula for the leading mass shifts is given within an effective Lagrangian framework using general "diagrammatics", allowing for arbitrary local interactions. One first isolates the leading Feynman diagrams contributing to the finite-size shift of the self-energy of a particle, and then deforms the integration contours in these diagrams to express them in terms of universal quantities, namely scattering amplitudes. In the second step analyticity properties of the vertex functions and propagators of the theory have to be taken into account, which (generically) allows one to calculate the mass shift only for particles below the 2-particle threshold of the lightest mass. The final result

170 for the mass-shift ~ma of a particle of (infinite-volume) mass ma < 2m1 in a 1+1 dimensional (not necessarily integrable) theory on a cylinder of circumference R, can be written as

b,c

(26)

where S:t( lI) is S-matrix element for the process ab -+ abo Here the primes on the summation symbols indicate that one should only sum over terms larger than the error term O( e- uaR ) (which is discussed in detail in [17], we just mention here that it is substantially smaller in 1+1 than in higher dimensions), P denotes the principal value of the integral, lI(z) = 0, !, 1, respectively, for z < 0, z = 0, z > 0, and finally Rabc = - Mabc i Resll=iu~b S:t( lI) ,

(27)

where u~& was defined after eq. (3) and Mabc = 1 if c (the antiparticle of c, in case c =f c) is a bound state of a and b, Mabc = 0 otherwise. Obviously we have to refer the reader to the original works[35,17] for a proof and a detailed discussion of these results and some of the subtleties involved. Let us just mention that as a first application we compared[l7] our analytical large-volume prediction (26) with numerical results from the TCSA and lattice simulations[36] for theories with exactly known (or conjectured) S-matrix. The agreement is excellent. If one does not know the S-matrix, eqs. (26-27) can be used to extract mass ratios and 3-particle couplings (which is essentially what the Rabc are) from numerical results for the finite-volume energies of (zero momentum) I-particle states. 5.2 Integral Equations for Finite-Volume Energies

Recently the TBA integral equations for the ground state energy in massive PESTs have been extended to various other situations, namely to the ground state energy in factorizable theories which are massless (but not scale invariant) and/or have non-diagonal S-matrices, and also to excited state energies. In all the cases studied so far the scaling functions e(r) = -!E(R) (where r = Rm, m being some mass scale of the theory, see below) on the cylinder are of the form N

e( r)

= - 4: 2 0 a_l

€a(lI)

=

!

00

dll va( lI) 1n(1

+ t ae- '* < sMTlexp(aQMT(z,-z»lsMT >

(7)

179

III.

Determinant Formula

In order to find a determinant formula mentioned, we are going to employ the many-body description of the massive Thirring model built over a reference state. This is a standard procedure for Bethe ansatz soluble field theory. Let the reference state In> be such that 1P,,(x) In >= O. An N-body state built over this reference state labelled by a set of rapidities (31 ... (3N can be written as

where

L crESN

N

(-I),g""

II X"K (XK I (3cr(K»)

(9)

K=1

We have denoted by SN the permutation group of N-th order. Periodic boundary coditions simply mean Lm sinh (3,. = 271'(integer) and the psuedoparticle density is given by D

= !:f; = f!:e A •

We are going to take the infinite-volume limit defined by N -+ 00, L D kept finite. Our goal is to establish the following formula in the limit:

Det (1 m

471' with'Y -

+ 'YK) = exp TrIn (1 + 'YK)

-+ 00

with

,

fA d(3e-im.inh(3(lfl-IIdu", «(3) U"l «(3) -A

e" -1 .

(11) (12) (13)

Here the determinant Det and the trace Tr are with respect to the function space. Explicitly,

TrK n

= IT [: dYK L K=l

-:&

Ott··o(1fN

K"N'" (YN - yt) K"'''' (Yl - Y2)'" K"N-,."N (YN-l - YN) .(14)

180 We begin the proof of eq. (11) by evaluating its denominator: N

II (2Lcosh(:JK).

< fJJ. .•• (:IN I fJJ. ••• (:IN >= (N!)2

(15)

K=l

To evaluate the numerator, let us state the following formula

(16) The normal ordering is with respect to the reference state A simple combinatorics tells us that < (:Jl'" (:IN equals

Ig

I {l >.

Q( -z, z)K

gl fJJ. ••• (:IN >

The integrations of ZK+l ••• ZN are over the full period and force the rapidities of the respective one-particle wave functions composing XN and XN to be equal. We find XK

=< fJJ. '''(:IN Ig Q(-z,z)K gl fJJ. ···(:IN > / < fJI '''(:IN I (:Jl"'(:JN >

= .1

I [ ' dill'" dllK

(N - K). -,.

E (-)..,.P-.gnQ

IT

c5pU ),QU)

j=K+I

PEsfl

QE5N

IT

X"i

(IIi I (:JPUl) X"i (IIi I (:JOU») 2Lcosh(:JpUl

i=l

(18)

It is not difficult to see that the summation 1

I

E (- y.,.P-.g,.q

(N - K). PES Qe'l:,

IT

c5p U),OUl'"

(19)

i=K+I

is equivalent to

(20)

181

Here the first summation means summing over the possibilities to choose K rapidities out of N rapidities without allowing any duplication. The choice from XN and the choice from XN must be the same. So far we have not specified the set of N-rapidities of our N-body states. In order to build the true ground state in the infinite-volume limit, we choose f3 to be i7r+real. To be more specific, f3n

= i7r - f3~, sinhf3~ = (- t+1[(n + 1)/2] ~:

.

(21)

(Here [ ] denotes the integer part.) Let us denote by 'DN a set of rapidities occupied by the N- body states. After some rearrangements, we see that eq. (18) can also be written as

The constraint f3i :f: f3j for i :f: j prevents us from factorizing the expression and carrying out the summation. These terms with f3i = f3i for i :f: j are, however, down at least by a power of and are ignorable in the infinite-volume limit. Observing

t

(23) we finally find limXK

=

L (-Y9""rr[' dy[ L

00.

2In the same paper it is shown that all the kernels are related to the Riemann-Hilbert problem, which helps us to find asymptotics of correlation functions.

183

References [1) A. Lenard, J. Math. Phys. 7(1965)1268. [2) M. Jimbo, T. Miwa, Y. Mori, M. Sato, Physica 1D(1980)80. [3) V.E. Korepin, Comm. Math. Phys. 113(1987)117; V.E. Korepin, N.A. Slavnov, Commun. Math. Phys. 129(1990)103; V.E. Korepin, N.A. Slavnov, ITP-SB-90-72. [4) V.E. Korepin, A.G. Izergin, A.R. Its, N.A. Slavnov, Int. Journ. of Mod. Phys. B(1990)1003. [5] A.R. Its, A.G. Izergin, V.E. Korepin, Commun. Math. Phys. 129(1990)205. [6] V.E. Korepin, A.G. Izergin, A.R. Its, Commun. Math. Phys. 130(1990)471; Its A.R., Izergin, A.G., Korepin, V.E., Phys. Lett. A141(1989)121; ITP-SB-89-92; ITP-SB-90-70. [7] Korepin V.E., Slavnov N.A., Nucl. Phys. B340(1990)759. [8] McCoy B.M., Perk, J.H.H., Shrock, R.E., Nucl. Phys. B220(1983)35; Its A.R., Izergin, A.G., Korepin, V.E., Novokshenov, V.Yu., Nucl. Phys. B340(1990)752. [9) S. Coleman, Phys. Rev. D11(1975)2088; S. Mandelstan, Phys. Rev. D11(1975)3026. [10) H. Bergknoff, H.B. Thacker, Phys. Rev. D19 (1979)3666. [11) V.E. Korepin, Theor. Mat. Fiz. (SOV.) 41 (1979)169.

184

Level Crossing and the Chiral Potts Model Barry M. McCoy

Institute for Theoretical Physics State University of New York Stony Brook, NY 11794-3840

Abstract We review the physics of level crossing in the chiral Potts model. In particular we discuss how the excitation spectrum changes qualitatively as a function of the parameters of the model and discuss the implications of this on perturbative methods of computation.

1

Introduction

The way one thinks about physics is often conditioned by one's past experience. Consider, for example, the solution of eigenvalue problems in quantum mechanics. Usually the first problems one encounters are Hamiltonians which are the sum of a kinetic energy term T and a potential energy term

= - LI

1 (j2 2m, ox2I

v = E ifo(Xk -

XI) •

(la)

(lb)

10,1

The solutions of eigenvalue problems of this form have an important property which follows from the form alone, namely that the ground state is non-degenerate and the ground state eigenvector has all positive entries. For differential operators such as (1) this is a consequence of the Sturm-Liouville theory [1). More generally, considering the Hamiltonian as a matrix, the ground state will be non-degenerate when the off diagonal elements are all nonpositive (and the matrix is irreducible). This follows from the theorem of Perron and Frobenius [21· A particular consequence of these theorems is that, as a function of the diagonal potential V, ground states can never cross. This is of great importance if we wish to study the ground state energy by doing perturbation in the potential V because the

185

ground state vector of the "free" system is smoothly connected to the ground state vector of the interacting system as a function of all the parameters in V. The simplicity of this elementary problem often permeates our thinking about more general situations. It is most common to split any Hamiltonian into 2 pieces, one "free" piece which may be treated exactly and an "interacting" piece which is treated by perturbation theory. This is an especially familiar proceedure in quantum field theory where the perturbation theory is sometimes used as a definition of the theory. Unfortunately, however, in many of these more elaborate examples the positivity properties which guaranteed that the ground state was unique and non-degenerate for the simple example (1) . are not present and the worry arises that the state computed by perturbation may not actually be the true ground state of the system. To gain intuition into this more general situation it is most useful to study solvable many body problems where level crossing occurs. One such model which has proved to have a rich physical content is the integrable chiral Potts model. The integrable chiral Potts model can be defined either as a 1 + 1 quantum spin chain or a 2 dimensional statistical mechanical model. The integrable spin chain has the Hamiltonian [3] L N-l

'It == -

E E {an(Xjt + a n(Zj ZJ+1t}

(2)

j=1 n=1

where in a direct product notation

(3a)

Zj == IN ® ... ® Zjth ® ... ® IN with IN the N x N identity matrix and the N

X

(3b)

N matrices Z and X have elements (4a) (5)

and X/,m == D/,m+1

(mod N) .

(4b)

Moreover an == exp[i(2n - N)if>/N]/sin(1rn/N)

an

= k' exp[i(2n -

N)~/N]/ sin(1rn/N)

(6a) (6b)

with cosif>=k'cos~.

Thus for N ~ 3 the Hamiltonian 'It depends on the 2 parameters if> and k'.

(7)

186 As a 2 dimensional statistical system the model is specified by the horizontal and vertical Boltzmann weights [4,5,6]

W~,(n) = Wt.,(O) W;',(n) W;',(O) where a p , bp , Cp, dp and

IT (d"b, - apCqwi)

(Sa)

bpd, - Cpa,w j

j=1

= IT (wapd, -

dpa,wi) Cpb, - bpCqW1

j=1

(8b)

a" b" c" d, lie on the rapidity curve aN +k/~

= kdN

(9a)

= k~ (9b) 3 2 with k +k'2 = 1. This is a curve of genus N - 2N2 +1 for k ::f 0, 1. From the Boltzmann klaN +~

weights we construct an NL x NL transfer matrix L

T{n},{n'}

= II W;',(nj -

(10)

n/)W;'q(nj - nj+l/) .

j=1

Here n

= 0, 1,··· N -

1 (mod N). When N

= 2 the system reduces to the Ising model.

The connection between T and 1i is that as p -. q.

Tp" with

= 1{1 + u const } + u1i

(11)

and ei~/N = wl/2apdp (12) bpdp bpCp where u measures the difference between p and q. This model is integrable by virture of the commutation relation ei4>/N

= wl/2apCp

(13) It is perhaps useful here to point out that mathematically there is a great deal of difference between k' and~. From (9) we see that k' appears in the definition of the curve. Indeed, k' is not removable from the curve and thus the moduli [and hence the period matrix and the Jacobian) of the curve (9) depends on the one parameter k'. On the other hand if> does not appear in the curve (9) but only in the Boltzmann weights (8) (meromorphic functions). Thus the moduli of (9) do not depend on if>.

We also note that when k'

= 1 that (9) reduces to the Fermat curve (9c)

where eN = k~. This curve has genus !(N - l)(N - 2). We also note that if ~ -+ 0 then a~ = -~. This is only consistent with (9c) if ~ and both -. 00. But from (8)

f

we see that when this happens we must also have

aN

~ ",

bN

= -7!/r e q

-.

00

as well. Thus when

187 ~ --+ 0 the Boltzmann weights degenerate from meromorphic functions on a higher genus Riemann surface to meromorphic functions over the genus 0 Riemann sphere. In other words sending I/J --+ 0 changed the genus of the curve on which the Boltzmann weights are defined even though the modulus is independent of I/J. ThisreHects the mathematical fact that moduli space is not closed.

The curve (9) is a complex manifold and on this complex manifold the Boltzmann weights (8) are in general complex. This general situation does not correspond to real interaction energies. However, there is a real manifold where the Boltzmann weights are positive, specified by (14a) a;c" = W1/ 2 b; = ~ < 11" these level crossings are of the spirit (universality class if you will) of the imposition of a magnetic field on the 6 vertex model. However, as we reach 4> = ~ = 0 the sea of zeroes which provides the fermi sea of the vacuum disappears and is replaced by a fermi sea of complex conjugate pairs (to use the language of Bethe ansatz roots). This is a qualitative change which will change the central charge of the system. Clearly it will be of great interest to study a cross over function to see how one central charge changes into another. But it is quite unlikely that such a drastic change can be understood by a power series expansion of either the ground state energy or dispersion relation in terms of 4> = ~. The mechanism is clearly more complicated than (say) a thermal perturbation of the critical Ising model. This perturbation sends the dispersion relation E = Pinto E = (m 2 +p2 )1/2 but does not change the structure of the vacuum. Indeed, a major feature of level crossing transitions seems to be that if one tries to "perturb" about the "free" vacuum that the "perturbation" causes the bottom to drop out of the spectum and the "free" system becomes unstable. This is clearly related to the fact that the perturbation acting on the ground state vector of the free system does !!Qt give the ground state vector of the perturbed system. The importance of this observation is that it seems to occur many places in physics, of which perhaps the most notable current example is matrix models of quantum gravity [18]. These models are surely not of the simple form of the Hamiltonian of (1) where Perron-Frobenius applies and they are notorious in that when defined as perturbation in terms of a genus expansion about a classical Hat vacuum the perturbation series is badly divergent. Conversely when defined as matrix integrals the expansion about the Hat vacuum will involve potentials which are unbounded below. Similar problems of definition would arise in chiral Potts models if we took continuous litnits too soon and tried to define the model by perturbation because as soon as the perturbation causes the ground states to cross the perturbation has nothing to do with the correct answer. The almost universal potential for level crossing transitions becomes apparent if we consider even Hamiltonian (1) acting on fermions. The positive Perron-Frobenius vector is disallowed by the antisymmetry of fermi statistics. Thus we reach the conclusion that for systems with fermions and massless excitations in the "free" state it is very difficult to rule out the possibility of a level crossing that changes the vacuum and throws the system into a new Hilbert space when a coupling constant is changed. Such effects do not readily admit a study in terms of a semi-classical limit and can perhaps be called "strongly quantum mechanical" problems. There is every indication that in phenomenon from quantum gravity to the quantum Hall effect level crossing plays a most important role.

191

Acknowledgement I am pleased to thank Prof. T. Curtright for the opportunity to participate in the 1991 Coral Gables Conference. I am also pleased to acknowledge extensive discussions with Dr. G. Albertini and S. Dasmahapatra. This work was supported by NSF grant #DMR-91006648.

References 1. C. Sturm, Jour. de Math. Pure et Appl. (1)1(1836) 106; J. Liouville, ibid. (1 )1(1836) 253; (1)2(1837) 16; see also "Methods of Theoretical Physics", P.M. Morse and H. Feshbach, McGraw-Hill, New York, 1953, vol. 1, pp. 719-726. 2. O. Perron, Math Ann. 64 (1907) 248; G. Frobenius, S.B. Deutsch. Akad. Wiss. Berlin, Math-Nat. Kl. (1908) 471, 511; ibid. (1912) 456; see also F.R. Gantmacher, "Matrix Theory", Chelsea Publishing Co. 1959, vol. 2. 3. H. Au-Yang, B.M. McCoy, J.H.H. Perk, S. Tang and M.L. Yan, Phys. Letts. A123 (1987) 219. 4. B.M. McCoy, J.H.H. Perk, S. Tang and C.H. Sah, Phys. Letts. A125 (1987) 9; H. Au-Yang, B.M. McCoy, J.H.H. Perk and S. Tang in "Algebraic Analysis" ed. M. Kashiwara and T. Kawai, Academic Press 1988, p. 138. 5. R.J. Baxter, J.H.H. Perk and H. Au-Yang, Phys. Letts. A128 (1988) 138. 6. H. Au-Yang and J.H.H. Perk, Advanced Studies in Pure Mathematics 19 (1989) 56. 7. R.J. Baxter, Phys. Letts. A133 (1988) 185. 8. G. Albertini, B.M. McCoy and J.H.H. Perk, Advanced Studies in Pure Mathematics 19 (1989) 1. 9. G. Albertini, B.M. McCoy and J.H.H. Perk, Phys. Letts. A135 (1989) 159. 10. G. Albertini and B.M. McCoy, Nucl. Phys. B350 (1991) 745. 11. R.J. Baxter, Phys. Letts. A146 (1990) 110. 12. B.M. McCoy and S-S. Roan, Phys. Letts. A150 (1990) 347. 13. G. Albertini, preprint Nordita-91/43 S. 14. G. Albertini, S. Dasmahapatra and B.M. McCoy (preprint). 15. V.V. Bazhanov and Yu. G. Stroganov, J. Stat. Phys. 59 (1990) 799.

192 16. R. J. Baxter, V.V. Bazhanov and J.H.H. Perk, Int. Jour. (1990) 803.

0/ Mod. PilUS. B4

17. G. von Gehlen and V. Rittenberg, J. Phys. A19 (1986) L625. 18. E. Brezin and V. Kuakov, Phys. Letts. B236 (1990) 144; M.R. Douglas and S.H. Shenker, Nucl. Phys. B335 (1990) 635; D.J. Gross and A.A. Migdal, PhUS. Rev. Letts. 64 (1990) 127.

193

DISTINGUISHABLE PARTICLES IN DELTA INTERACTION

J.B. McGuire Department of Physics Florida Atlantic University, Boca Raton, FL 33431 [email protected] INTRODUCTION The meaning of delta interaction I borrow the phrase "delta interaction" from Gaudin[ll, who uses it as an abbreviation for "particles of equal mass in one dimension interacting with equal strength delta function potentials." It is customary to formulate this problem by specifying the Hamiltonian of the system 1

H

N

d2

= -- E-2 - gEEb(:Il. 2

.=1 d:ll i

:Ili)

(1)

'>i

and to seek a complete set of stationary states satisfying Hi/! = Ei/!. It is not necessary to abide by this custom, for the system is "completely integrable," satisfies the "Bet he ansatz," and is "non-diffractive"; but a few of the phrases of conventional wisdom which are said to account for the confluence of properties which lead to the algebraic factorization of the problem. The Hamiltonian is permutation symmetric and all degrees of freedom are on equal footing; but the full symmetry of delta interaction, the symmetry which allows the algebraic factorization, transcends permutation symmetry. In fact the symmetry of state functions of the stationary states with this factorization symmetry transcend all of the symmetries of the Hamiltonian. One consequence of this factorization symmetry is that it is possible to represent the state function as a linear combination of plane waves in every region of state space. If the state function is expressed in this form the set of "pseudomomenta," the k. in the plane wave factors e"''''i, are integrals of motion. There are N of these integrals of motion because the system is completely integrable. Unfortunately, these integrals of motion are not the eigenvalues of the generators of an operator algebra. To express this transcendent supersymmetry as the symmetry of an operator algebra, note that a linear combination of plane waves is a simultaneous eigenstate of a set of N commuting hermitian operators, denoted SN, where each operator is a

194 symmetric function of the particle coordinate derivatives. That is, an element of SN is

(2) and SN is the set of all Sn. Because the eigenvalues of SN are real the integrals of motion are real or come in complex conjugate pairs. Because the eigenvectors of SN are orthogonal the state functions of states with different integrals of motion are orthogonal.

£:r

Any function of the elements of S N could replace E as the generator of the time dependence, and the state function in a stationary st~te would be unaffected. It is possible, therefore, to define delta interaction in a way which is independent of the Hamiltonian. The state function in a stationary state - a function in the N dimensional state space of the particle coordinate random variables - has imposed upon it the conditions of delta interaction:

(i) The state function is continuous when any two particle coordinates agree. (ii) The normal derivative of the state function is discontinuous at the equal coordinate boundary. This discontinuity is a constant multiple (2g, twice the delta function strength constant) of the state function on the boundary. A system described by a state function which is an eigenstate of any function of the elements of SN and satisfies the conditions (i) and (ii), I will call a system in delta interaction. THE PRlMARY SYMMETRY OF DELTA INTERACTION The only Hamiltonian system in delta interaction is (1). Choosing any operator

1£( SN) to generate the time dependence implies a Schrodinger equation 'H.ll!

.Oll!

= tOt

(3)

(Note that this is not the same as changing the kinetic energy term in the Hamiltonian.) The delta interaction state function will automatically be an eigenstate of the Hamiltonian (1), an eigenstate of every element of SN and and eigenstate of every function of the elements of SN. This is the primary aymmetry of delta interaction. This primary symmetry is established by showing the internal consistency of a set of assumptions with respect to the state function of the delta interaction problem[2,3j. All state functions satisfying primary symmetry are eigenstates of the Hamiltonian (1). The eigenstates of the Hamiltonian may be represented as a linear combination of plane waves in every region of state space; passage from one region to another is effected by repeated application of the two-particle reflection and transmission amplitudes; passage from a region to a non-adjacent region is independent of path. The path integral formulation of this quantum system shows that only finite number of paths contribute to the amplitude dynamics. When this primary symmetry

195 is exact, as it is for delta interaction, the path integrals cancel by destructive interference, except over finite number of distinct paths. This cancellation is independent of the operator chosen to generate the time dependence, i.e. this property does not depend upon what function of the SN modulates the path phases in the action integral. The calculation of amplitudes is reduced to a sum over complementary events. The internal consistency of finite sums over a finite number of competing paths requires satisfaction of an algebraic "condition of no diffraction" or a "star-triangle relation." When such a condition is satisfied the state functions may be classified according to primary symmetry. Further classification requires a study of other symmetries consistent with primary symmetry.

SECONDARY SYMMETRIES OF DELTA INTERACTION Primary symmetry is consistent with both translation and permutation symmetry, but these two symmetries are, in general, only partially consistent with each other. Since both have an important role to play in the physical constraints imposed upon delta interaction, it is necessary to study both to define the limits of their mutual compatability. Translation symmetry The SN are symmetric combinations of derivatives, and the state functions upon which they operate are linear combinations of plane waves. Under these circumstances the translation operators Tk - operators which translate the kth coordinate by a fixed amount - commute with the SN and commute with each other. It is therefore consistent to assume both primary symmetry and eigenstates of Tk which satisfy (4) where the translation eigenvalues >'k are independent of coordinates and real for normalizable I]!. In general >'k will be an analytic functions of the integrals of motion, and different for each coordinate. Finite density state functions In order to examine the properties of a large system with many degrees of freedom it is necessary to compute the state function in the finite density bulk limit, that is the limit N --+ 00, L --+ 00, N / L = p. Finite particle density may be assured by imposing the further requirement that >'k be independent of the integrals of motion, i.e. that the >'k be real constants. In these normalizable eigenstates of Tk the particle probability density is always positive and periodic in every translation symmetric coordinate. The average of the probability density is identified with the particle density. It will develop that maximum compatability with permutation symmetry will require the further restriction that these constant translation eigenvalues have the form \ \ 27rnk (5) Ak

=

A

+ --r;-'

i.e. that the state functions be quasiperiodic. The translation symmetric special case of a state function with constant translation eigenvalues which satisfy the constraint

196 (5), I will call a finite den3ity 3tate function. A method for finding integrals of motion and the explicit determination of finite density state functions will be discussed elsewhere. The result I seek here is a demonstration that a single class of finite density state functions is sufficient to determine the complete set of state functions for states of finite density. Further, the process used to generate the complete set from the single class is independent of the integrals of motion, and is thus independent of delta interaction. Permutation symmetry The generators of the permutation group,

P"", permute particle coordinates (6)

and commute with the SN, because the SN are symmetric in particle coordinates. These generators are self-inverse, or idempotent

(7) The p •• do not commute with the T.,but it is easy to show that

(8) i.e. that translations in any pair of coordinates are similar under permutation of that pair of coordinates. Assuming a simultaneous eigenstate of the SN and translation

(9) which implies

(10) Thus state functions which are simultaneous eigenstates of the SN and translation have the following properties under permutation: 1. Operation with the generators of the permutation group doe, not affect the co-

ordinate translation eigenvalues. State functions which are transformed by the permutation of a pair of coordinates remain eigenstates of the SN and translation with the same eigenvalues.

2. Operation with any generator of the permutation group permutes the translation eigenvalue" among the coordinate, which label the generator. Translation eigenvalues are permuted among the coordinates according to the coordinate pair appearing in the permutation. This is true in general, and in particular for the special case of finite density states. Compatability of permutation and translation can be assured if the eigenvalues of translation are chosen so that any permutation on a finite density state function produces a finite density state function. Suppose a finite density state function with its arguments in standard order such that

(11)

197 Operation with any set of permutations will scramble the order of the coordinates in ip.. but the properties above imply that the translation eigenvalue associated with a coordinate in an assigned position in the standard order will be the same value as that of the coordinate in that position in the standard order, e.g.

-

The particular choice form for translation eigenvalues given in (5) assures that finite density state functions permute into finite density state functions. Young's recipe The permutation generators do not commute among themselves, so it is not possible to simultaneously put them in diagonal form. There are, however, irreducible representations of these generators such that

P",ip: = LA:iip~,

(12)

i

where ~j are d sets of d x d matrices whose elements are independent of coordi. nates. The representations may be chosen so that the matrices are the same in all representations. The basis vectors are orthogonal in different representations, i.e.

L ipiip~ = 0, r

of. t.

(13)

i

Given an arbitrary ip, satisfying no particular symmetry conditions, a recipe, devised by Young[41, exists for projecting onto basis vectors in the d d-dimensional irreducible representations. To describe the recipe I outline the steps required to project an arbitrary state function with N = 9 degrees of freedom,ip(zl'" Z8), into particular irreducible representations of the permutation group. The first step is to fix a Young diagram; a collection of N boxes arranged in rows and columns such that the number of boxes in each row is non-increasing from top to bottom and the number of boxes in each column is non-increasing from left to right. For example, the 9 particle diagram

From the diagram a ,tandard tableau is generated by associating particles with the boxes, e.g., 1 2 61 3 4 81 5 7

"9 -

-

such that the particle numbers decrease along each row and column. This Young tableau implies a three-step schedule for creating a standard set of basis vectors in an irreducible representation.

198 Step 1. Symmetrize with reapect to the entriel in the rowl (e.g !,6). Step 2. Antiaymmetrize with respect to the entriea in the columna (e.g 4,7). The result of these two steps is a linear combination of state functions which is one basis vector of an irreducible representation of the permutation group of 9 objects. Step 3. The remaining balil vectorl of the irreducible reprelentation may be found by operating with the generatora of the permutation group until nothing new re,ult,. Basis vectors in orthogonal irreducible representations associated with the same diagram are found from a different standard tableau generated by permuting particles not in the same row or column (e.g. 4,5) - taking care to satisfy the decreasing particle number rule - and repeating the three steps of the schedule. Application of Young's recipe to a finite density state function Young's recipe involves only repeated application of the permutation generators, and thus, when applied to a finite density state function produces a finite density state function with the same translation eigenvalues. Therefore all of the basis vectors in the d sets of d-dimensional irreducible representations associated with the diagram have the same eigenvalues of translation. Finally, then:

(1) Every finite density state function has an associated Young diagram.

(2) The state function and diagram can be used as input to the Young recipe to generate a complete set of d2 degenerate independent (unfortunately not orthogonal) finite density state functions. (3) The process of generation of the complete set of state functions is independent of delta interaction -

it is Young's permutation recipe.

If finite density state functions can be found for every diagram the state functions for stationary states of distinguishable particles can be derived. This represents a massive savings in effort when compared to the usual method of reduction according to permutation symmetry and subsequent application of periodic boundary conditions. Young's recipe generates state functions of such monumental complexity that, in practical terms, reduction according to permutation symmetry is no reduction at all. For any but the simplest cases involving attractive delta interaction the dimensionality of the irreducible representations is essentially singular in the bulk limit, i.e. the dimensionality increases at least as eN. Because primary symmetry dictates that finite density state functions are degenerate within the Young diagram - as opposed to the degeneracy of the tableau - a simplification of essentially singular proportions obtains. CONCLUSIONS (1) The state function of a system in delta interaction is a normalizable quantum field. Each degree of freedom, or particle, is labelled by a random variable. w·w is

199 the multivariate probability density of these random variables. (2) State functions may be represented as linear combinations of N! plane waves in each region of an N dimensional state space partitioned into N! regions corresponding to one-dimensional particle orderings. This representation of the state functions is the primary symmetry of delta interaction. (3) This symmetry may be expressed as the symmetry of an operator algebra. A complete set of commuting operators are the symmetric functions of coordinate partial derivatives. Because the system is completely integrable there are N of these integrals of motion. (4) Any Hamiltonian consistent with primary symmetry (i.e. a symmetric function of particle coordinate derivatives) may generate the time dependence in a Schrodinger equation. The integrals of motion are independent of the Hamiltonian. (5) Although permutations and translations do not commute, translations are similar under permutation for finite density state functions. This similarity means that the permutations rearrange but do not change the translation eigenvalues. (6) Every finite density state function is labelled with a Young diagram. (7) The schedule of operations of the Young recipe on a standard tableau generates all of the states in the irreducible representation. These state functions comprise the complete set of state functions identified with the diagram. All of the states with all of the diagrams comprise the complete set of finite density state functions for distinguishable particles. Thus the principal conclusion:

All that i, required to completely characterize the ,tationary ,tate, of a 'lIliem of di.dingui,hable particle$ in delta interaction il a lingle let of finite demity .tate function" one for each Young diagram. Thi, characterizaton i, valid for any generator of time dependence con.i,tent with the primary $ymmetry of delta interaction. REFERENCES 1. M. Gaudin La Fonction d'Onde de Bethe, (Masson, Paris, 1983).

2. J.B.McGuire J.Math.Phys. 5 (1964) p.622 3. C.N .Yang and C.P.Yang Phys.Rev.147(1966) p.321 and p.327 151 (1966) p258 4. There is an enormous literature on the representation theory of the permutation group. A personal favorite is H.Boerner Repre$entation$ of Group. (North Holland, Amsterdam, 1963)

200

EXACTLY SOLVED MODELS WITH QUANTUM ALGEBRA SYMMETRY

Luca Mezincescu and Rafael!. Nepomechie Department of Physics University of Miami, Coral Gables, Florida 33124 ABSTRACT We have constructed and solved various one-dimensional quantum mechanical models which have quantum algebra symmetry. Here we summarize this work, and also present new results on graded models, and on the so-called string solutions of the Bethe Ansatz equations for the A~2) model.

1. INTRODUCTION

The concept of symmetry is fundamental for the description of physical systems. In many cases, such symmetry is codified by a Lie (super) algebra. A generalization of this structure, the so-called quantum Lie (super) algebra, has recently emerged l - 7 • Our ultimate goal is to understand how quantum algebra symmetry is implemented in physical systems, and to explore the consequences of this symmetry. Such symmetry may eventually prove to be useful for field theory in 4 spacetime dimensions and for string theory. To date, quantum algebras have been identified as the common mathematical structure linking three types of physical systems: topological (Chern-Simons) field theory in 3 spacetime dimensions 8 , integrable lattice models9 , and rational conformal field theories 10 and their integrable perturbations ll • Over the past two years, we have studied primarily the connection between integrable lattice models and quantum algebras. Among the three connections of quantum algebras to physical systems noted above, this is the most direct. Furthermore, it is within this context that quantum algebras were first discovered. . In the course\of our investigations 12 - 18 , we have constructed and solved various one-dimensional quantum mechanical models which have quantum algebra symmetry. Here we summarize this work, and also present new results on graded models, and on the string solutions of the Bethe Ansatz equations for the A~2) model. The construction of these models requires two main ingredients: the R matrix, which

201 can be interpreted as a two-particle scattering amplitude, and the K matrix, which can be interpreted as the amplitude for a particle to reflect elastically from a wall. The integrability of these models comes from demanding that the scattering be consistent with factorization. In Section 2, we introduce R matrices via the Zamolodchikov algebra, and summarize some of their important properties. In Section 3, we introduce K matrices through an extension of the Zamolodchikov algebra. In particular, we describe the graded case, which we illustrate with an example connected to the superalgebra 81.£(211). In Section 4, we construct open chains of N "spins" (generators of a quantum algebra) with certain nearest-neighbor interactions, which are integrable and which have quantum algebra symmetry. For these models, the transfer matrix (i.e., not just the Hamiltonian) commutes with the generators of a quantum algebra. We also comment on the solution - namely, the eigenvalues of the transfer matrix and the Bethe Ansatz equations - of these models. In order to calculate quantities of physical interest, one must first solve the Bethe Ansatz equations in the N -+ 00 limit. For the A~l) case, these so-called string solutions are well known. In Section 5, which is a result of a collaboration with A.M. Tsvelik, we investigate string solutions for the A~2) model of Izergin and Korepin. We find new types of string solutions, but we are not able to formulate a general string hypothesis. We summarize our results in Section 6. A more detailed account for the simplest case of A~l) can be found in Ref. 13.

2. R MATRICES Yang-Baxter equation We briefly review here how the (graded) Yang-Baxter equation follows from the associativity of the (graded) Zamolodchikov algebra. This algebra is abstracted from studies of scattering in massive relativistic quantum field theories in 1+1 dimensions with an infinite number of conservation laws 19 • The Zamolodchikov algebra has generators Aa(u), where 1.£ is the so-called spectral parameter, and Q = 1 , ... , n. These generators obey the relations

(2.1) The matrix apRa.p.(u - v), which may be interpreted as a two-particle scattering amplitude, is called the R matrix. By setting 1.£ = v in the above relation, and by assuming linear independence of monomials of second degree, we learn that the R matrix is regular,

R(O)

= 1',

(2.2)

where l' is the permutation matrix,

(2.3)

202 Moreover, by interchanging A",(u) A,B(v) twice using (2.1), we obtain the unitarity relation R( u) P R( -u) P = 1. (2.4) Consider now the monomial of third degree A",(u) A,B(v) A.y(O). Associativity of the Zamolodchikov algebra, as well as the assumption of linear independence of monomials of third degree, imply that

",,BR,,,,,,B"(u - v) "''''YR",,'Y''(u) ,B"'Y"R,B''Y'(v) = ,B'YR,B"'Y"(v) "''Y"Ra,,'Y'(u) ",",B" R"",B' (u - v).

(2.5)

This relation is the well-known Yang-Baxter (or factorization) equation. Introducing the notation RI2 = R ® 1, so that

and similarly defining R13 and R 23 , the Yang-Baxter equation can be rewritten in the compact form

R 12 (U

v) RIS'(U) R 2s(v) = R 2S (v) R 13 (u) R 1 2(U - v).

-

(2.6)

We remark that the Yang-Baxter equation transforms covariantly under "gauge" (or "symmetry-breaking") transformations4 ,20,15 of the R matrix I

R 12 (u - v)

-+

2

I

2

B(u) B(v) R12(u - v) B( -u) B( -v),

(2.7)

where B(u) is a diagonal matrix with the properties

B(u) B(v) as well as

1

= B(u + v), I

B(O) = 1,

2

(2.8)

2

B(u) Rdv) B( -u) = B( -u) R12(v) .B(u).

(2.9)

Here we have introduced the notation I

B=.B®1,

2

B=.1®B.

(2.10)

There is a graded version of the Yang-Baxter equation. Following Kulish and Sklyanin21 , we introduce a Z2 grading of the Zamolodchikov algebra, by considering the generators A", to be homogeneous elements with parity pea) =. p(A",) equal to either 0 (even) or 1 (odd). These generators obey the relations (2.11) We assume that ",,BR,,,,,B' are commllting numbers, and that if ",,BR,,,,,B' i= 0, then p(a) +p(,B) +p(a')+p(.B') = 0 mod 2. Associativity of this graded Zamolodchikov algebra leads to the graded Yang-Baxter equation 21 ,

(- )p(,B" >[p(-Y">-p(-Y>] ",{3R"'''{3''(u - v) "''''YR",,'Y''(u) (3"'Y"R,B''Y'(v) = (- )p(,B">[p(-Y">-p('Y')]

(3'Y R,B"'Y"(v) "''Y"R",,,'Y'(u) ",",B"R""{3'(u - v). (2.12)

203 Solutions of the Yang-Baxter equation An R matrix is said to be quasi-classical if it depends on an additional parameter 1] which plays the role of Planck's constant, so that

R(u,1])1 '1=0

= const

(2.13)

1.

There are three known classes of regular quasi-classical solutions of the Yang-Baxter equation: elliptic, trigonometric, and rational (corresponding to the three types of functions of 1.£ that appear in R(u) ). Being interested in quantum algebras, we focus on the trigonometric solutions. Such solutions are associated 22 with affine Lie algebras g{l.), where 9 is a simple Lie algebra (A" = su(n + 1), Bn = o(2n + 1), 0" = sp(2n) , D" = o(2n), etc.) and k( = 1,2,3) is the order of a diagram automorphism (T of g. That is, (Tic = 1. The cases k = 1 and k > 1 are often referred to as "untwisted" and "twisted", respectively. For instance, in the case of A~2) in the fundamental representation, the diagram automorphism is given by the complex conjugation map (T : ,XA -+ _,XA*, where ,XA are the eight Gell-Mann matrices. We shall later make use of the fact that the automorphism (T leaves invariant a sub algebra go of g. (This subalgebra go is in fact the maximal finite-dimensional subalgebra of the affine algebra g(Ic).) In the A~2) example, it is clear that (T leaves invariant the purely imaginary matrices .x 2 , .x s , ). 7 , which generate an 81.£(2) subalgebra of su(3). A table listing every simple Lie algebra 9 which has a nontrivial diagram automorphism, along with the corresponding sub algebra go which is left invariant by this automorphism, is given in Ref. 23, and is reproduced in Ref. 16. The simplest example of an untwisted R matrix is the spin 1/2 Ail) matrix

R(1.1)(u) _

-

Sh(U + 1]) 1 ( JI sh(u + 1])sh(-u +1])1

) shu sh1]

sh1] shu

. sh(u+1])

(2.14) In this gauge, the R matrix is "symmetric"; i.e., it is both P invariant (P12 R12 P12 = R 12 ) and T invariant (R~~t. = R12). The gauge transformation (2.7) with B(u) = diag( cu/2 ,c- u/2 ) yields the symmetry-broken R matrix

e

sh 1.£ u sh1]

sh 1] shu

CU

'h(U+"J· (2.15)

which is only PT invariant,

(2.16) The transposition ti refers to the

it!>

space.

204

The R matrices associated with the nonexceptional affine Lie algebras in the fundamental representation have been given by Bazhanov24 and Jimbo4 • (For the graded case, see Ref. 25.) Although in general these R matrices do not have either P or T symmetry, they do have PT symmetry (2.16). Except for A~) (n > 1), these R matrices have crossing unitarity, 1

1

R~Hu) MR~~(-u - 2p) M- 1 = 1,

(2.17)

where M is a symmetric matrix (Mt = M) which can be deduced from Ref. 24. Moreover, except for D~2), these R matrices (in the so-called homogeneous gauge used by Jimbo) satisfy (2.18) [R(u),R(lI)] =0, where

R(u) == 1'R(u).

(2.19)

Connection with Quantum Algebras The prototype quantum algebra is Uq [su(2)], with generators 8 = {S+, S-, S"} which obey 25' .

-25'

q [S+ , S-] = q q _- q-1

'

(2.20)

where q is a complex parameter. Given two sets of generators 81 , 82 of this algebra (with ,8 = 0), the generators 8 in the tensor product space are given by

[81 2]

(2.21) The generalization to Uq[g] for any simple Lie algebra 9 is discussed in Refs. 3 - 6. Faddeev, et al. 6 emphasize an R-matrix formulation of quantum algebras. Taking Uq [su(2)] again as an example, define

R± =

(2.22)

lim R(u) ,

u--+±oo

where R(u) is the spin 1/2 Ail) R matrix inthenonsymmetric gauge (2.15); and define the upper, lower triangular matrices

-5'-1 ) T_ = ( -q-q ( q -l)S+ q 5'-1. ,(2.23) . with q =

e".

The relations with

f

= {+ , - }

(2.24)

205

hold if and only if the operators § obey the algebra (2.20). Moreover, consider two sets of such matrices Ta, T2± constructed from S10 S2 respectively. The coproduct matrices T± are given by (2.25) where the symbol ®indicates the tensor product of the algebras and the usual product of the matrices. They are expressed in the form (2.23) in terms of the operators § given precisely by the comultiplication rule (2.21). An important identity is

[R(u),Uq [su(2)J] = 0,

(2.26)

where here by Uq [su(2)] we mean coproducts of the generators. For the general case of an R matrix of the type 9(1·), the corresponding result is 4 (2.27) where 90 is the subalgebra of 9 which is left invariant under the diagram automorphism of order k. In particular, for both AP) and A~2), the matrices R( u) commute with Uq [su(2)].

3. K MATRICES Reflection-factorization equation We now extend the Zamolodchikov algebra (2.1), by introducing the additional relation (3.1) The K matrix aKa'(u) can be interpreted as the amplitude for a particle to reflect elastically from a wall 26 • By setting u = 0, we see that

K(O) = 1.

(3.2)

Furthermore, using the relation (3.1) twice, we obtain the unitarity relation K(u) K(-u) = 1.

(3.3)

Consider now the monomial of second degree Aa(u) A,B(v). There are two different ways by which one can apply each of the Zamolodchikov relations (2.1), (3.1) twice to obtain an expression proportional to Aa.(-u) A,B'(-v), Using again the assumption of linear independence, we obtain the relation21 - 28 ,n 1

2

R 12 (U-V) K(u) P12 R 12 (u+v) P 12 K(v)

2

1

= K(v) R 12 (u+v) K(u) P 12 R 12 (u-v) P12 , (3.4)

206 to which we shall refer as the reflection-factorization equation. This equation transforms covariantly under the gauge transformation (2.7), provided that the K matrix transforms as follows, K(u) -+ B(u) K(u) B(u). (3.5) By repeating the above calculation using instead graded Zamolodchikov generators (which obey the relation (2.11)), we obtain the graded reflection-factorization equation,

Here we have assumed that aKa' are commuting numbers, and that if aKa' then p(a) + p(a') = 0 mod 2.

=f. 0,

Solutions of the reflection-factorization equation Given a solution R(u) of the (graded) Yang-Baxter equation, one can solve the (graded) reflection-factorization relation for the corresponding K{u).

,pin 1/! A~l) : For the spin 1/2 A~l) R matrix (2.14), there is a one-parameter family of diagonal K matrices given by 27,28 KW(u,e) =

1 vi sh(u + e)sh(

-'1£

+ e)1

. (Sh(u+ e ) ), - sh(u - e)

(37) .

where e is an arbitrary parameter. spin 1 A~l) : For the spin 1 A~l) matrix

K(l)(u,e)=p(u,e) (

R(l,l)

given in Refs. 29,2, we find 13

sh(u + e)sh(u -1] + e) -sh(u-e)sh(u-1]+e) sh( '1£

-

e) sh( '1£

) ,

+ 1] -

(3.8)

e)

where p(u, e) = [sh(u + e) sh( -'1£ + e) sh(u -1] + e) sh( -'1£

-

'1

+ e»)-i .

Just as there is a fusion procedure 30 by which R(l,l) may be obtained from R( i,i), there is a similar fusion procedure 13 by which K(l) may be obtained from K(i) and

]l'r. = alo + iyr. with i

Yl =

i

2'

and . (1 11") - 4 + 41/ '

Yl = ~

(5.12)

-2'

Y2 =

'C4 -

Y2 = ~

41/ 11") '

(5.13)

respectively. The first string (5.12) is the positive-parity 2-string of Takahashi. Suzuki. The second string, which does not appear for Ail), has been studied numerically in Ref. 45. M = 3: We find the 3-strings of Takahashi-Suzuki, Yl = 1,

and

11"

Yl = 1 + 21/ '

Y2 = 0, 11"

Y2 = - ,

21/

11"

(0 < 1/ < "2)'

Y3 =-1 11"

Y3 = -1+-

21/

11"

(2" < 1/ < 11"),

(5.14)

(5.15)

of positive and negative parity, respectively. We also find the solution 1

Y3 =

2-

11"

21/'

(5.16)

which can be interpreted as a combination of a negative-parity 2-string and a positive-parity I-string. M = 4: We find the positive-parity 4-string of Takahashi-Suzuki, (5.17)

214

In addition, we find four new candidate 4-strings: 5 4

11' 1 11' 1 - - -47]' 4 47]' 4

{y,,} = {- - -

11'

5

11'

+ -47]' --4 + -} 47]

11'

11'

('5 < 7] < '3

and

11'

2'

311'

< 7] < 5'"), (5.18) (5.19)

3

{y,,} = {4" 3

11'

1

+ 47]'4" 11'

1

11' 1 47] '-4" 11'

1

11'

3

+ 47] '-4" 11'

3

11'

47]}'

11'

{y,,} = {4" - 47]'4" + 47] '-4" - 47] '-4" + 47]}'

(5.20) (5.21)

For (5.20), (5.21), the analysis of the BA equations is quite intricate, and we have not been able to confirm that these string configurations are in fact solutions. The group-theoretic significance of these new strings has so far eluded us. The fact that in string configurations there occur steps of both 1 and accompanied by necessary factors of 4~' makes it difficult to formulate a general string hypothesis. This impedes further progress in computing the thermodynamic properties of this model. On the other hand, in the noncritical regime 7] = pure imaginary, the situation is much simpler. Let us make the replacement 7] --+ i7] (with 7] real) in the BA equations (5.1). Evidently, :1:" is determined modulo 11'/7]. Repeating the steps (1)(3) in the above analysis, we find only the positive-parity M-strings of TakahashiSuzuki; i.e., A" = :1:0 + iYk with

!,

{Yk}

M-1 M-3

3-M 1-M

= {-2-'-2-""-2-'-2-}'

(5.22)

We do not expect significant difficulties in calculating thermodynamic properties in this regime.

6. CONCLUSIONS We have obtained a number of results concerning integrable spin chains in connection with quantum algebras. We have presented a generalization of the Zamolodchikov algebra which accommodates reflecting walls, and which reproduces the algebraic relations that are obeyed by the K matrices. By either directly solving these relations or implementing a fusion procedure, we have obtained new K matrices corresponding to the trigonometric R matrices for certain (graded) Lie algebras. We have extended Sklyanin's approach for constructing integrable open quantum spin chains to PT-invariant R matrices, and we have used this formalism to construct and investigate a large class of models with quantum algebra symmetry. These models may be solved by the analytic Bethe Ansatz. Finally, we have exhibited new types of string solutions for the A~2) model of Izergin and Korepin.

215

We are frustrated by the difficulty of solving the Bethe Ansatz equations, even in the N -+ 00 limit. These equations have a "group theoretical" origin, as they implement the construction of irreducible representations of a certain algebraic structure. Therefore, there should be a straightforward algorithm for obtaining their solutions. This, in turn, should enable one to standardize the calculations of thermodynamic properties, such as specific heat and magnetic susceptibility, of the corresponding models.

7. ACKNOWLEDGMENTS We are indebted to T. Curtright, P. Freund, M. Jimbo, E. Kiritsis, P. Kulish, E. Melzer, N. Reshetikhin, V. Rittenberg, A. Tsvelik and A. Zamolodchikov for valuable discussions. This work was supported in part by the National Science Foundation under Grant PRy-gO 07517.

APPENDIX The solution of the graded Yang-Baxter equation corresponding to 8[(211)(2) in the fundamental representation is given by 21.25 fa

\

b

y

r c

r R(u)

y

b

=

(A.l)

a

c

x c

x x

y

c

d/

y

where a

= 1,

b=

sh u ch( u - 1]) , sh( u + 21]) ch( u + 1])

c=

shu , sh(u + 21])

=

1 [Sh u _ .ch 1] sh 21]] sh( u + 21]) ch( u + 1] ) , sh 21] sh u sh 21] x = , y = . sh(u + 21]) sh(u + 21]) ch(u + 1]) d

The parity assignments are given by p(l)

= p(2) = 0 ,p(3) = 1.

(A.2)

216

8. REFERENCES 1. E.K. Sklyanin, Fund. Anal. Appl. 16 (1982) 263; 1'T (1983) 273. 2. P.P. Kulish and N.Yu. Reshetikhin, J. SOy. Math. 23 (1983) 2435. 3. V.G. Drinfel'd, SOy. Math. Dokl. 32 (1985) 254; 36 (1988) 212; J. SOy. Math. 41 (1988) 898. 4. M. Jimbo, Commun. Math. Phys. 102 (1986) 537; Lecture Note, in Ph1l,ic" Vol. 246 (Springer, 1986) 335. 5. M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11 (1986) 247; Int'l J. Mod. Phys. A4 (1989) 3759. 6. L.D. Faddeev, N. Yu. Reshetikhin and L.A. Takhtajan, Algbr. Anal. 1 (1988) 129; Algebra Analysis 1 (1989) 178 (in Russian). 7. S.L. Woronowicz, Commun. Math. Phys. 111 (1987) 613; 122 (1989) 125. 8. E. Witten, Commun. Math. Phys. 121 (1989) 351; Nucl. Phys. B322 (1990) 629; B330 (1990) 285. 9. V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523. 10. See, e.g., A. Tsuchiya and Y. Kanie, Adv. Stud. in Pure Math. 16 (1988) 297; K.-H. Rehren and B. Schroer, Nucl. Phys. 312 (1989) 715; G. Felder, J. FrOhlich and G. Keller, Commun. Math. Phys. 124 (1989) 417; 130 (1990) 1; G. Moore and N. Seiberg, Phys. Lett. B212 (1988) 451; Commun. Math. Phys. 123 (1989) 177; G. Moore and N. Yu Reshetikhin, Nucl. Phys. 328 (1989) 557; L. Alvarez-Gaume, C. Gomez and G. Sierra, Phys. Lett. B220 (1989) 142; Nucl. Phys. B319 (1989) 155; B330 (1990) 347; J.-L. Gervais, Commun. Math. Phys. 130 (1990) 257; Phys. Lett. B243 (1990) 85; L.D. Faddeev, Commun. Math. Phys. 132 (1990) 131. 11. See, e.g., F.A. Smirnov, Int'l J. Mod. Phys. A4 (1989) 4213; T. Eguchi and S.K. Yang, Phys. Lett. B224 (1989) 373; B235 (1990) 282; A. LeClair, Phys. Lett. B230 (1989) 103; D. Bernard and A. LeClair, Nucl. Phys. B340 (1990) 721; C. Ahn, D. Bernard and A. LeClair, Nucl. Phys. B346 (1990) 409; N.Yu. Reshetikhin and F.A. Smirnov, Commun. Math. Phys. 131 (1990) 157; F.A. Smirnov, Commun. Math. Phys. 132 (1990) 415. 12. M.T. Batchelor, L. Mezincescu, R.I. Nepomechie and V. Rittenberg, J. Phys. A23 (1990) L141. 13. L. Mezincescu, R.1. Nepomechie and V. Rittenberg, Phys. Lett. A14'T (1990) 70; R.I. Nepomechie, in Supera.tring8 and Particle Theory, ed. by L. Clavelli and B. Harms (World Scientific, 1990) 319; L. Mezincescu and R.I. Nepomechie, in Aryonne Work.shop on Quantum Groups, ed. by T. Curtright, D. Fairlie and C. Zachos (World Scientific, 1991) 206. 14. L. Mezincescu and R.I. Nepomechie, Phys. Lett. B246 (1990) 412. 15. L. Mezincescu and R.I. Nepomechie, J. Phys. A24 (1991) L17. 16. L. Mezincescu and R.I. Nepomechie, Int'l J. Mod. Phys. A6 (1991) 5231. 17. L. Mezincescu and R.I. Nepomechie, Mod. Phys. Lett. A6 (1991) 2497. 18. L. Mezincescu and R.I. Nepomechie, Nucl. Phys. B, in press. 19. A.B. Zamolodchikov and ALB. Zamolodchikov, Ann. Phys. 120 (1979) 253; A.B. Zamolodchikov, SOy. Sci. Rev. A2 (1980) 1. 20. K. Sogo, Y. Akutsu and T. Abe, Prog. Theor. Phys. 'TO (1983) 730.

217

21. P.P. Kulish and E.K. Sklyanin, J. SOy. Math. 19 (1982) 1596. 22. A.A. Belavin and V.G. Drinfel'd, Funct. Anal. Appl. 16 (1982) 159; SOy. Sci. Rev. C4 (1984) 93; D.A. Leites and V.V. Serganova, Theor. Math. Phys. 58 (1984) 16. 23. V. Kac, Infinite Dimenllional Lie Algebrall (Cambriqge University Press, Cambridge, 1985). 24. V.V. Bazhanov, Phys. Lett. 159B (1985) 321; Commun. Math. Phys. 113 (1987) 471. 25. V.V. Bazhanov and A.G. Shadrikov, Theor. Math. Phys. 73 (1987) 1302. 26. A.B. Zamolodchikov, unpublished. 27. LV. Cherednik, Theor. Math. Phys. 61 (1984) 977. 28. E.K. Sklyanin, J. Phys. A21 (1988) 2375. 29. V.A. Fateev and A.B. Zamolodchikov,Sov. J. Nucl. Phys. 32 (1980) 298. 30. P.P. Kulish and E.K. Sklyanin, Lecture Notes in Physic II 151 (Springer, 1982) 61; P.P. Kulish, N.Yu. Reshetikhin and E.K. Sklyanin, Lett. Math. Phys. 5 (1981) 393. 31. A.G. Izergin and V.E. Korepin, Commun. Math. Phys. 79 (1981) 303. 32. R.J. Baxter, Ezactly Solved Model8 in Stati8tical Mechanics (Academic Press, 1982); L.D. Faddeev and L.A. Takhtajan, Russ. Math. Surv. 34 (1979) 11; J. SOy. Math. 24 (1984) 241; A.A. Vladimirov, JINR preprint P17-85-742; L.D. Faddeev, in Lell Houchell Lecturell 19B! ed. by J.B. Zuber and R. Stora (North-Holland, 1984) 561; H.J. de Vega, Int'l J. Mod. Phys. A4 (1989) 2371. 33. P.P. Kulish and E.K. Sklyanin, J. Phys. A24 (1991) L435; in Proc. Euler Int. Math. Inst., 1st Semester: Quantum Group8, Autumn 1990, ed. by P.P. Kulish, in press. 34. L.C. Biedenharn and M. Tarlini, Lett. Math. Phys. 20 (1990) 271; V. Rittenberg and M. Scheunert, Bonn preprint (1991). 35. A. Connes, Publ. Math. IHES 62 (1985) 257. 36. J. Wess and B. Zumino, CERN preprint (1990). 37. F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel, J. Phys. A20 (1987) 6397. See also M. Gaudin, Phys. Rev. A4 (1971) 386; La fonction d'onde de Bethe (Masson, 1983). 38. A.M. Tsvelick and P.B. Wiegmann, Adv. in Phys. 32 (1983) 453. 39. M. Takahashi and M. Suzuki, Prog. Theor. Phys. 48 (1972) 2187; V.E. Korepin, Theor. Math. Phys. 41 (1979) 953; K. Hida, Phys. Lett. A84 (1981) 338; M. Fowler and X. Zotos, Phys. Rev. B25 (1982) 5806. 40. B.L. Feigin and D.B. Fuchs, unpublished; Vl.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 (1984) 312; B251 (1985) 691; C. Thorn, Nucl. Phys. B248 (1984) 551. 41. E. Date, M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Nucl. Phys. B290 [FS20] (1987) 231; M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Comm. Math. Phys. 119 (1988) 543; V.V. Bazhanov and N. Yu. Reshetikhin, Int'l J. Mod. Phys. A4 (1989) 115. 42. D. Kastor, E. Martinec and Z. Qiu, Phys. Lett. B200 (1988) 434; J. Bagger, D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 60 (1988) 389; F. Ravanini, Mod. Phys. Lett. A3 (1988) 271.

218

A QUANTUM GENERATED SYMMETRY OF CONFORMAL AND TOPOLOGICAL FIELD THEORY' S. G. Naculich Department of Physics, Johns Hopkins University Baltimore, MD 21218, USA and H. A. Riggs, H. J. Schnitzer, and E. J. Mlawer Department of Physics, Brandeis University Waltham, MA 02254, USA

l. INTRODUCTION The research reviewed here 1-4 deals with the remarkable duality of classical affine Lie algebras 6 for the dual pairs {G(N)K , G(K)N}, where G(N) denotes any of the classical Lie algebras SU(N), Sp(N), or SO(N). The effects of this duality appear in conformal field theory,1-4,7,l1 the representation theory of the braid group,1-4,9,lO knot polynomials,2-4,9 Chern-Simons gauge theory,2-4 integrable lattice models,6,8,9 quantum groups, 1,4,9 and the An series Hecke Algebras at roots of unity.10 Recently, we have found a similar duality between the models based on the affine Lie superalgebras SU(n + Nln)K a.nd those based on SUCk + Klk)N. 5 It appears that this group-level duality may well have broad implications for our understanding of twoand three-dimensional field theories. We emphasize that the duality we consider relates observables of different systems: they have different values of the central charge and different conformal weights. Nevertheless, each dual pair of theories is closely related: dual WZW models have conformal blocks which provide 'dual' bases for the braid and fusion matrices, while dual Chern-Simons theories have gauge invariant observables with identical expectation values. 'Supported in part by the US Dept. of Energy under contract no. DE-AC02-76-ER03230.

219 In this report we focus on the {SU(N)K , SU(K)N} duality for simplicity, presenting the analogous results for Sp(N)K and SO(N)K without detailed exposition, for which we refer the reader to references 2-4. We also report recent results on an analogous duality relating SU(n + Nln)K and SU(k + Klk)N. We will begin by describing several consequences of this group-level duality for the Wess-Zumino-Witten (WZW) models based on Lie groups, which are of central importance as building blocks of a wide class of two-dimensional conformal field theories. This duality provides relationships between SU(N)K and SU(K)N WZW models, which on the face of it is unexpected, since group and level play such different roles in WZW models and the associated Kac-Moody algebras. (The parameter N fixes the rank and dimension of the global symmetry group, while the parameter K limits the number of primary states in the spectrum of the quantized theory.) There are some hints of such a duality involving the interchange of Nand K in the normal ordering correction -+ K!N to the Sugawara-Sommerfield stress tensor, T

0

+--+

0

tPA

+--+

~l

SU(3).

+--+

SU(4)a

0

An analogous statement holds for the other groups. It can then be shown1,2 that the conformal weights h and h of the fields t/JA and ~I' respectively, satisfy the dimen,ion formula -~ for SU(N)K (1) h(t/JA) + h(tfJI - j for SO(N)K and Sp(N)K •

--)_{j

Note that the right hand side does not depend on the shape of the tableau, but only on the number of boxes r of the reduced tableau A. The conformal weights are given by

(2) where QA(N) is the quadratic Casimir invariant calculated with the square of the long roots normalized to be two, and 9 is the dual Coxeter number for G(N). A second crucial insight is that although the integrable irreducible representations of SU(N)K and SU(K)N' or those of SO(N)K and SO(K)N, are not in one-ta-one correspondence, there is a one-ta-one correspondence between cominimal equivalence clu,e,1-3 of integrable teRlor representations. (The integrable irreducible representations of Sp(N)K are in one-ta-one correspondence with those of Sp(K)N') These equivalence classes are generated by certain discrete symmetries of the extended Dynkin diagram (a ZN symmetry for SU(N)K, a Z2 symmetry for SO(N)K' and the trivial symmetry Zl for Sp(N)K) which connect the representions belonging to each class. For SU(N)K, the conformal weights of the fields comprising a cominimal equivalence class in which the representation a appears are simply related to the conformal

221 weight of a by

h(um(a»

- m)K = h(a) + m(N2N -

mr(a) . r + mteger

(3)

where u corresponds to the basic, discrete symmetry operation. The tableau of the SU(N) representation u(a) is obtained by adding a row of width K to the top of the reduced tableau of a (followed by reduction of the tableau). This means that a and u(a) transpose, for most a, to the same representation in the dual theory. An example is,

a=w

ii

=ffD

SU(N)6

u(a)=r

>(')~r

-

ffD

SU(6)N

In this example we have assumed that N > 4. The representations a, u(a), ... , u N - 1(a) (not all of which need be distinct) form such an SU(N)K cominimal equivalence class and provide a representation of the ZN symmetry. The primary fields of SU(N)K corresponding to representations in the same class differ only by free fields; thus, the idea of cominimal equivalence identifies a useful isomorphism of the underlying dual Hilbert spaces. Similarly, for SO(N)K, the conformal weights of the (in general two) members of a cominimal equivalence class are related by (4) h( u( a» = h( a) + half integer , while for Sp(N)K each represention by itself forms a cominimal equivalence class. We have shown1,1l that the chiral conformal blocks of SU(N)K and SU(K)N satisfy orthogonality and completeness relations. This is illustrated by the four-point correlator of primary fields of SU(N)K

(5) This correlation function can be decomposed into a sum over chiral contributions from a finite number of intermediate channels. Invariance under SL(2, C) transformations allows one to simplify the functional dependence of these chiral conformal blocks so that

n.

(6) where Z,j = Z, - Zj and Z = Z12Z34/ Z14Z32' as a sum of contributions in terms of a set corresponds to an appearance of the identity Eq.5 GP(Z) = EIA~(z)

Each conformal block can be written of invariant tensors lA, each of which in the operator product of the fields in

; P = 1 to n •.

(7)

A

For large N, the IA are linearly independent, while for N small enough there are nc dependency relations (8) ~IA = 0 ; IS = 1 to nc ,

E A

222 where the c~ are certain constants. 1 In the latter case the conformal blocks are not unique, but subject to a "gauge" change n,

G~(z)

--->

G~(z)

+ E c~ g=(z)

(9)

1'=1

which leaves Eq. 7 invariant for arbitrary functions g:(z). There exists a particular "gauge" choice for which the blocks of Eq. 7 satisfy an orthogonality relation

(10) where a.p (aq ) are the monodromy coefficients of SU(N)K (SU(K)N), and fA(Z) is a kinematical factor. There is also a gauge invariant completeness relation satisfied by the conformal blocks G~(z) and G~(z) of SU(N)K and SU(K)N, respectively, the explicit form of which is given in reference 1. These orthogonality and completeness relations are proved by direct examination of the Knizhnik-Zamolodchikov equation,14 and impose constraints on the braid matrices of SU(N)K and SU(K)N' The braid matrix for Eq. 5 is defined by

GP(Z17 Z3,Z2,Z4) =

EB pI

pp' [ ; 1

;] 4

GP'(Z17 Z2,Za,Z4)

(11)

where the left-hand side of this equation is defined by analytic continuation. More generally we consider the braid matrix B which interchanges two A legs of an arbitrary conformal block in a WZW model based on any classical simple Lie algebra G. We consider segments of the conformal blocks with copies of A primaries on the external legs, and braid matrices which interchange these legs. The fusion rules (12) determine the dimension and basis states of the space of conformal blocks on which B acts. The eigenvalues of any braid matrix correspond to the fields appearing in Eq. 12, and are given in terms of the conformal weights of those fields by ,

(13) where fe = 1 (-1) for c appearing in the symmetrized (anti-symmetrized) product of Aa ® Ab if a = b. With the braid matrix B in G(N)K which interchanges two Aa representations in diagonal form, consider iJ in G(K)N in a dual diagonal basis, specified by transposing all primary fields in external channels and by appropriate cominimal equivalents of transposes of primary fields in internal channels. From Eq. 1 and the property that feEe = e-i"r, one can show that the diagonal entries for a given braid matrix satisfy2 for SU(N)K for SO(N)K and Sp(N)K

(14)

The phase which appears in this equation only depends-remarkably-on the number of boxes r of the tableau of the representation a.

223 The modular transformation matrix Sab = (Sa b)* characterizes the mixing of the affine characters XG(-r) of the integrable highest weight representations of G(N)K under the modular transformation T -+ -l/T :

(15)

Xa(-l/T) = ESab Xb(T) . b

The multiplicities of the primary fields in the operator product expansion of a pair of primary fields are given by the coefficients Nab" of Eq. 12, and they are related to the quantities SGb by Verlinde's formula12 (16) One can establish the dualities 2,3,7,8 for SU(N)K' for SO(2n + IhHI and Sp(N)K ,

(17)

which, when combined with Eq. 16, gives the duality of the SU(N)K, Sp(N)K, and SO(2n + Ihh+l fusion coefficients3 ,8 N ab "-- N db"'(0)

(18)

where w(c) specifies a corninimal equivalent of C. (Specifically, w(c) = uA(c) where tl. is an integer that only depends on the difference of tableau boxes r( a) + r( b) - r( c). ) An example of such a fusion coefficient duality is

SU(3)4

Al

A2

IF' x IF

"

+

!fxlF

!fIl

SU(4)a

u 2 (AJ)

+ rH'

EH

!

!

u(A 2 )

A3

+

2 x IF'

+

2XIF

! U(A3)

As

A4

+ +

Effi'

! ~

A4

+ = +

! IJ:D

u(As)

An analogous, but more complicated fusion coefficient duality holds for SO(N)K if either N or K is even, the details of which are presented in reference 3. Several surprising results involving spinor representations of SO(N)K are also proved in this reference.

3.

CHERN-SIMONS THEORY

Several of the results we have described have natural interpretations as relations between the gauge invariant observables for pairs of Chern-Simons theories based on dual affine Lie algebras. The d = 3, topologically invariant Chern-Simons theory is defined by the action

1= -K

411'

f tr(A

i\

2 i\ A i\ A) dA + -A

(19)

3

where A is the gauge connection of G(N), and quantization forces K to be an integer. We consider a dual pair of such Chern-Simons theories, each defined on the 3-manifold S3. Gauge invariant observables are given by the expectation valu~s of \

224 linked products of Wilson lines and graphs in S3, with each component specified by a representation A of G(N)K. An simple example is that provided by the expectation value of an unknotted Wilson loop,u (WA.(unknot»)

= (unknot

jAa)

= ~:

(20)

where, for a loop C, we have the loop operator

WA.(C)

= Tr A.

Pexp

JcI A

(21)

From the modular transformation matrix duality in Eq. 17 we have2

= (unknotj A;;)

(unknot; AB) O(N)K

O(KIN

.

(22)

To obtain relations between other observables, we now derive a duality of skein relations. Choose a surface S2 that divides S3 so that two untwisted segments of components of a Wilson line observable, with representations a and b, respectively, puncture it at exactly four points. The Hilbert space 1£ associated with this surface, considered as the boundary of the right half of S3, is f = ~(Nab')2 dimensional (see Eq. 12). The path integral on the right half of S3 will produce a state .,po E 1£. Alternately, the internal lines can be braided so that the f states.,pi Bi .,po, i = 1, ... , f, are produced on the corresponding boundaries S2, as shown here:

=

}{ These f

x

+ 1 states are linearly dependent, /

"L,!3;.,pi =

(23)

0

;=0

Insertion of the expression for .,pi yields the characteristic polynomial satisfied by B /

/

i=O

;=1

2:.a: Bi = II(B -

pi) = 0 .

(24)

The superscript 8 signifies standard framing; the eigenvalues of Eq. 13 for a = b correspond instead to vertical framing, and are related to those of standard framing by (25) In the dual situation in the theory based on G(K)N, with A the dual representation for the Wilson line observable, we have a dual characteristic polynomial f

II(B-p'n=o ;=1

(26)

225 From eqs. 1, 13, and 25, we have

pipi = 1

(27)

TI(B- 1 - pi) = 0

(28)

giving f

i=l

so that B-1 in G(K)N satisfies the same characteristic equation as B in G(N)K. 2 Thus, simultaneously interchanging N and K, A and X, as well as B with its inverse leaves the skein relation invariant. Now consider a knot 1C and its mirror image knot iC (obtained by exchanging all under and over crossings in a diagram of the knot). The duality of the skein relation means that the expectation value of a knot with A in G(N)K will satisfy the same skein relation as that of the mirror image knot with X in G(K)N. We will call such pairs of observables dual expectation values. For the knots that can be untied with this skein relation, the associated expectation values are reducible to a polynomial of the unknot expectation value. Since the expectation values of these mirror image knots will have the same pattern of reduction via identical skein relations, they reduce to the same polynomial of the dual unknot expectation values. In these cases 2,3 (8 indicates standard framing)

a

b

The Wilson link (Cab) with writhe four. One can also obtain dual relations between the link observables (Cab) obtained via braid closure of w = 2n twistings of two Wilson lines defined with representations a and b, shown in the figure above for n = 2. The quantity w, the writhe of the link diagram, is the difference in the number of over and under crossings. In order to obtain identities between expectation values that will allow simplification of the crossings in such a link expectation value, we can use the spectral decomposition based on the fusion rule in Eq. 12 to show that (v indicates vertical framing) (30)

The general braid matrix eigenvalue duality

Pab,.

_ _{±ei. . (a)r(bl/NK

P;;b,,,,(~)

-

±l

for SU(N)K , for Sp(N)K and SO(N)K ,

(31)

226

allows one to obtain dual relations between these expectation values. For example, for SU(N)K' we can use Eq. 30 to derive such a dual relation,3 i.e.

N ..,: (unknot;c) (Po.b,ct 2n N_"(c) (unknot;w(c») e-2n1ri.(.. )r(~)/NK (~ _ )2n ..6 o.b,.,( c) e-2n1ri.(.. )r(~)/NK

L

N;;;"(c) (unknot;w(c») (p;;;,.,(~»2n

c

(32) In the last line C denotes the mirror image link of £ (in the sense that it has all under and over crossings interchanged), whose writhe is the negative of that of £. In this derivation we have used the fusion coefficient duality, the quantum dimension duality, and the eigenvalue identity in Eq. 31. In the case of n = 1 this calculation and the identity (C({b) = S;;,/ Soo produces the modular transformation duality of Eq. 17 (since Sao = JK/N Sao). Similarly, the analogous calculations for Sp(N)K and SO(N)K show that S;;; = S:~ = SGb , (since here Sao = Sao). If a = b, these results can be recast into standard framing. The expectation values in vertical and standard framing differ by a phase

Using this a natural normalization is possible in all cases, so that we obtain an exact equality of expectation values

(£ ..6) G(NlK = (C({b) G(KlN An extension of these results to general classes of observables is given in reference 4, the results of which we describe next.

4.

GRAPH OBSERVABLES AND TETRAHEDRA

In the previous section we have reviewed two approaches to a duality qf knot and link observables in Chern-Simons theory. The first, based on the characteristic polynomial of the the braid matrix, applies to all knots, but is only effective for a restricted class of representations. The second, based on the spectral decomposition directly, works for all representations, but only applies to some knots and links. It was shown in ref. 15 that all Chern-Simons observables can be expressed-via a series of spectral decompositions, Hilbert space basis changes, and tetrahedral eXclsionsas sums of products of planar tetrahedra expectation values. We have shown 4 that both this pattern of reduction and the tetrahedral expectation values are the same for dual knot and link observables with arbitrary representations on each component of the observable. It can also be shown that arbitrary (planar and nonplanar) graphs have duals. Here we review the main elements of this argument, which completes the demonstration that the all Chern-Simons observables have duals. The duality of planar tetrahedra is of independent interest since (up to normalization conventions) they have the same values as the q-6-j symbols of the related quantum groups with q equal to the associated root of unity. It is then immediate

227 that such q-6-j symbols are dual. In addition, these same quantities are directly related to certain limiting values of the Boltzmann weights of integrable lattice models, as well as to the braid and fusion matrices of WZW theory, by means of simple phases. One can formulate a set of equations that overconstrain and in general uniquely determine the expectation value of any given of tetrahedra. We have shown that given such a tetrahedra in G(N)K and this set of equations, there exists a dual tetrahedra in the dual theory G(K)N which satisfies exactly the same set of equations. This allows us to conclude that these dual tetrahedra have exactly the same expectation values (given certain natural phase conventions in both theories). The argument goes as follows. The first step is to consider three different bases of the braid matrix Hilbert space of section 3, as shown in the figure below, and their consistency as bases of the same space.

It)

=

lu) =

These bases are related by basis transformation coefficients as follows

Is)

LFotlt)

Is)

~::2

The conformal weights of these SU(n + Nln)K representations reduce to those of

SU(N)K as follows 6 hSU(_+N\a)K( {A, X})

= hSU(N)K(A) + hSU(N)K(A) + T;'TA/2N(N + K)

(36)

One can then use the SU(N)K conformal weight identity in eq. 1 to obtain an analogous result for SU(n+Nln)K' Each supercharaderof an SU(n+Nln) representation in this set equals the character of the SU(N) representation labeled by the same tableau. These results then lead to dual relations between SU(n + Nln)K ChernSimons observables and their duals in SU(k + Klk)N. A complete development of the further consequences of these results for the WZW models, Chern-Simons theories, and quantum supergroups based on SU(mln)K is given in ref. 5.

6.

CONCLUSION

We have found a uniform mapping relating the objects in one WZW model or Chern-Simons theory with those in the dual: object

operation

G(N)K

N ...... K

A-primary field A-fusion rule

transpose A transpose A transpose Nab" A-conformal block transpose A dimension formula h braid eigenvalue product is constant equals SOA/SO,O proportional to Sab (unknot;A) equals under ...... over skein relation equals (knot; A) equals (link; a, b, ... ) (planar graph; a,b, .. . ) equals

dual object

G(K)N A-primary field A-fusion rule

N_"'("C) _ ab A-conformal block

h dual eigenvalue

~OA/SO,O S;;,; (unknot; X) dual skein relation (mirror image knotj X) (mirror image link; ii, b, ... ) (same graphjw(ii),w'(b), ... )

In each case there is a twist that inverts braid group generators, transposes tableaux, or exchanges under and over crossings of Wilson line knot or link diagrams. An appropriate context in which to understand the occurence of these dualities in a natural way is that of conformal embeddings. The infinite series of dual pairs

231 that we have considered also occur in pairs in

U(NKh :::l SU(N)K ® SU(K)N ® U(1h O(NKh :::l SO(N)K ® SO(K)N O(4NKh :::l Sp(N)K ® Sp(K)N

(37)

Similarly for the duality of the Lie superalgebras considered in section 5 we have the conformal embedding5

+ N K jlh :::l SU(n + Njn)K ® SU(k + Kjk)N ® U(1h (38) I = 2nk + kN + nK. There is an analogous conformal embedding for the U(l

where orthosymplectic series as well.5 Since group-level duality is a symmetry of quantum groups,1,4,9 and quantum supergroups,5 it should have manifestations wherever these structures appear. It is clear that group-level duality is a pervasive symmetry that appears in many aspects of two and three dimensional field theories. Since its full scope and deeper significance have yet to be elucidated, further study of this symmetry and its consequences should yield additional surprises. We are happy to thank M. Bourdeau for recent collaborations, the results of which are reported in ref. 5 and reviewed here in section 5.

References [1) S. G. Naculich and H. J. Schnitzer, Phys. Lett. B244 (1990) 235; Nucl. Phys. B347 (1990) 687. [2) S. G. Naculich, H. A. Riggs, and H. J. Schnitzer, Phys. Lett. B246 (1990) 417. [3) E. J. Mlawer, S. G. Naculich, H. A. Riggs, and H. J. Schnitzer, Nucl. Phys. B 352 (1991) 863. [4) S. G. Naculich, H. A. Riggs, and H. J. Schnitzer, "Symmetries of Chern-Simons Tetrahedra and Quantum 6-j-Symbols under Group-Level Duality," Brandeis preprint BRX-302. [5) M. Bourdeau, E. J. Mlawer, H. A. Riggs, and H. J. Schnitzer, "The QuasiRational Fusion Structure of SU(mjn) Chern-Simons and WZW Theories," Brandeis preprint BRX-319, in preparation. [6) 1. Frenkel, in Lie Algebras and Related Topics, Lecture Notes in Mathematics, no. 933, D. Winter, ed. (Springer-Verlag, Berlin, 1982), 71; M. Jimbo and T. Miwa, in Adv. Stud. Pure Math 4 (1984) 97; 6 (1985) 17; M. Jimbo, T. Miwa, and M. Okado, Lett. Math. Phys. 14 (1987) 123. [7) J. Fuchs and P. van Driel, J. Math. Phys. 31 (1990) 1770; D. Altschiiler, M. Bauer, and C. Itzykson, Commun. Math. Phys. 132 (1990) 349. [8) A. Kuniba and T. Nakanishi, "Level-rank duality in fusion RSOS models," preprint, to appear in proceedings of Int. ColI. on Modem Quantum Field Theory, TIFR, Bombay, India; A. Kuniba, T. Nakanishi, and J. Suzuki, "Ferro- and Antiferro-magnetizations in RSOS models," preprint.

232 (9) H. Saleur and D. AltschUler, "Level-rank duality in Quantwn Groups," Saclay preprint, SPhT-90-041. (10) F. Goodman and H. Wenzl, Adv. Math. 82 (1990) 244. [11) T. Nakanishi and A. Tsuchiya, Nagoya preprint NU-MATH-002, July 1990. (12) E. Verlinde, Nud. Phys. B300 [FS22] (1988) 360. [13] E. Witten, Comm. Math. Phys. 121 (1989) 351; Nud. Phys. B322 (1989) 351; Nud. Phys. B330 (1990) 285; M. Atiyah, The Geometry and Physics of Knots, Cambridge University Press, Cambridge (1990); L. Kaufmann, Knots and Physics, World-Scientific, Singapore (1991). [14) V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B24'T (1984) 83. [15) S. Martin, Nud. Phys. B338 (1990) 244. [16) I. Bars, Physica D 15 (1985) 42; Lee. Appl. Math. 21 (1985) 17. [17] V. Kac and J. van de Leuer, Ann. de L'Institute Fourier3'T (1987) 99. [18) L. Alvarez-Gawne, C. Gomez, and G. Sierra, Nud. Phys. B330 (1990) 347.

233

NONCRITICAL STRINGS BEYOND c=1 ° Joseph Polchinski Theory Group, Department of Physics University of Texas, Austin, Texas 78712 joe@utaphy INTRODUCTION My talk was concerned with an attempt to understand the recent exact results in low dimensional string theories, looking at the Liouville theory from a spacetime point of view. I have not made much progress on this subject beyond what is already published,!l] so here I would like to expand upon a different aspect of the Liouville theory: how to make sense of the theory in dimensions greater than one. The first part of these notes is concerned with the standard Liouville theory. It is based on recent work with Djordje Minic and Zhu Yang,!2] and the reader should consult that paper for more details and a more extended list of references. The second is based on work in progress with Andy Strominger, on effective string theories for long NielsenOlesen and QCD strings. The third part presents some simple-minded remarks about the possible relation between QCD and string theory at shorter distances. LIOUVILLE THEORY BEYOND c=l Consider a sum over surfaces in 1 < D < 25 dimensions, defined by the Polyakov path integral. In conformal gauge, with units 0/ = 2, the action becomes!3]

S

= 4~

f dzdz {8X 8X + 84>84> + V(4))} + Sghosh P

p

(1)

and the energy-momentum tensor

(2) Conformal invariance determines Q = (25 - D)1/2/3 1/ 2 and also

(3) °Research supported in part by the Robert A. Welch Foundation and NSF Grant PRY 9009850.

234 The notation ,~, means that the indicated form holds only to linear order in the the Liouville potential-that is, when V is small, so the theory'is approximately free. Now, there are a number of puzzles here. Some are connected with the attempt to interpret this theory as a physical sum over surfaces: 1. The potential V(4)) is supposed to be the properly renormalized world sheet density but the latter is positive definite while the former oscillates in sign.

vg,

2. The lightest state in the string theory (1) is tachyonic, with mass m 2 so the theory is unstable.

= (1-D)/12,

3. The world-sheet field 4> has all the physical properties of a semi-infinite embedding dimension. It cannot be interpreted as a spacetime dimension, because Lorentz transformations involving this dimension are broken. It cannot be interpreted as an internal degree of freedom because there are too many states (a continuous spectrum from the 4> zero mode). The other puzzles arise even if one wishes to consider this as an abstract string theory with no special physical interpretation: 4. What is the nature of the physics as 4> -> oo? In D :5 1 the Liouville potential rises exponentially and prevents propagation to 00. Here, the naive form has oscillations on top of the exponential. 5. Related to the previous question, is there a weak coupling expansion for the string theory? The log of the string coupling grows linearly with 4>, from the curvature coupling associated with the improvement term in the energy momentum tensor (2). The theory becomes strongly coupled as 4> -> 00, unless (as in D :5 1) the potential effectively prevents propagation to that region. 6. As 4> increases, the naive potential (3) has ever deeper oscillations. This suggests that in addition to the continuous spectrum from 4> -> -00, there is a family of discrete levels in each well, below the continuum. The last three puzzles at least go beyond the validity of the linear form (3) and require that we understand higher order corrections to the conformal invariance. From the D + 1 dimensional spacetime point of view, the Liouville potential is a tachyon background, and we are in a region where the tachyon-tachyon interaction is important. The behavior of large tachyon fields is an old problem in string theory. Fortunately, we now have some exact information on this subject, as a result of the progress in random matrix models, These models are restricted thus far to D :5 1, but we can transform the present problem in the following way. Leave the Liouville theory unchanged, but replace the D embedding dimensions with a single spacelike dimension X, with the energy momentum tensor T

= --18 X 8 X + -QX82 X 2 2

- -184> 8 if> + -Q 8 2 4> 2 2

(4)

and Q2 + Qk = 8. We have simply replaced the matter theory with a different one having the same central charge. The condition for conformal invariance of the Liouville theory is unchanged, as long as we are only considering string tree level. Now, define the new Liouville field q to be the linear combination which appears in the improvement term and the new embedding dimension t to be the orthogonal

235 combination, q

(Qt/> + QxX)/4 (Qxt/> - QX)/4.

(5)

This is then the familiar continuum theory of the D = 1 noncritical string, except that the tachyon is not a function only of the Liouville field, but of a linear combination of the Liouville field q and the (Euclidean) time t-it is a 'tilted' solution. The exact classical field theory of the tachyon is known from the collective field method,[4] and the general solution described in refs. [5]. The tilted solutions, as well as others relevant to other ranges of central charge, were studied in ref. [2]. The details of the collective field theory and its solutions are too lengthy to present here, but the physical conclusions are simply stated.1 Using the same numbering as above:

4. The tachyon does not oscillate indefinitely as t/> ..... 00. Rather, past a certain point the eigenvalue density ends, so the range of string propagation is limited. Therefore also 5. There is a weak coupling expansion for the string theory. 6. From a study of fluctuations around the full nonlinear solution, one finds that the oscillations stop before they are deep enough to bind: there is only the continuous spectrum. This improved understanding of the Liouville theory does not explain the discrepancy with physical expectations, however. For all solutions there is a semi-infinite range where the linearized form (3) holds, and therefore: 1. The potential still oscillates in sign.

2. The lightest state is still iachyonic. 3. The zero mode and spectrum are still continuous. The solution to these problems will be given in the next section. EFFECTIVE STRING THEORY Consider the effective world-sheet field theory describing the fluctuations of a very long QCD or Nielsen-Olesen flux tube. These tubes have energy proportional to their length, so the leading behavior is a string theory with Nambu-Goto action. However, for the purpose of describing such physical strings none of the ltandard Itring quantizationl il correct. To see why this is so consider the following properties of these theories. First, they are Lorentz invariant with positive Hilbert space, since these properties are inherited from the underlying field theory. Second, the strings have only transverse oscillations, D - 2 in all. This is because the oscillations, from the world-sheet point of view, are the Goldstone bosons (collective coordinates) of the transverse translational symmetries, which are broken by a long straight tube. The tube is lIt is not clear to me how this approach relates to the more algebraic approach Gervais and collaborators-see ref. [6) and references therein.

236 invariant under longitudinal and time translations, so there are no Goldstone bosons for these, and it would be unnatural for additional massless degrees of freedom to appear on the world-sheet. In the case of QCD one might argue that it is a strongly coupled theory and one cannot be sure, but in the Nielsen-Olesen case one can take the couplings to be arbitrarily small and then one is certain about the spectrum of collective coordinates. 2 Consider the standard string quantizations. The light-cone quantization spoils Lorentz invariance outside the critical dimension. The Virasoro quantization leads to longitudinal oscillators when the central charge is less than 26, as does the Polyakov quantization. 3

If none of these is the string theory we seek, what is? Consider the following operator: 68

=

I

dzdz (oXl'oXV

- illJWoX. oX) (axl'ax. . - ~llJWax. aX)e-Y,p,

with "Y

= -h/Q2 + 8 + ~Q.

(6)

(7)

The matter part is a (2,2) tensor and the Liouville part a (-1, -1) tensor, so the whole is conformally invariant and can be consistently added to the action (1). Why has this not been included previously? First, it has four derivatives and 50 is nonrenormalizeable. Now, though, we are considering an effective theory, and should not exclude it. Second, it couples the matter and Liouville theories together. In deriving the Liouville theory from the Polyakov, one might assume that the measures for the metric and embedding are independent, so that the field theory factorizes. In fact this is not a good assumption, as we will explain further below. So this term should appear. What happens when 68 is added to the original action (1)? We are oversimplifying, because there is infinity of additional terms to add, and because the action will not be a simple sum due to interactions. But we can get a good idea by looking at the qualitative form of 8 + 68. In the long string, the gradient oGXI' has an expectation value w:;. By the Virasoro constraint w~ and w: are null, but w • . w~ will have an expectation value of order the string length L squared. The term 68 then produces a pot ential (8) for the Liouville field. Noting that "y is negative, this prevents the string from propagating in the direction of negative ¢. Combined with the discovery in the previous section that the usual Liouville potential prevents propagation to large positive 4>, we see that the effective Liouville potential now has a minimum! The problem with the semi-infinite range is gone, as are the unwanted oscillations since these are now massive. The value of ¢ at the minimum is solved for in terms of the induced metric 'In their original work!7] Nielsen and Olesen considered the rather speculative limit of .trong conpling, hoping to take the string thickness to zero. Here we are less ambitious, merely taking the string length to be great. SIn fact the Polyakov and Virasoro quantizations are equivalent, except for the Liouville zero mode,!8} The Polyakov quantization adds an extra field, but haz the correct· central charge and 10 has oft'seUing null states.

237 consistent with the interpretation of t/> as a metric. (The tachyon problem is not relevant, since our analysis applies only to long strings.)

11,

Of course, the Liouville field, being massive, should not even be considered part of the effective theory. It can be integrated out, leaving an effective theory containing only the X". The surviving theory must be a conformal field theory of central charge 26, yet is a Poincare-invariant theory in D < 26 dimensions. What is it? The new feature is that integrating out t/> leaves a theory which need not be analytic in OX .OX; this is acceptable because we are expanding around a state where this operator has an expectation value. When one enlarges the space of allowed theories in this way, one does indeed find a CFT with the right properties. More details can be found in ref. [9]. This low energy theory can in principle be found in an entirely different way: derive the Nambu-Goto theory with correct path integral measure by carrying out the collective coordinate quantization of the Nielsen-Olesen flux tube, and then gaugefix with care. The subtlety which has been ignored in the past is that the measure obtained from the collective coordinate method depends on the physical motion of the string in spacetime and so is built out of the induced world-sheet metric OX· OX. The Polyakov determinants will then also involve this metric. But this leads to inverse powers of the induced metric in the conformal field theory (or something more complicated after renormalization), just as we have concluded above by different means. This also indi~ates why one should not expect the measures for the intrinsic metric and the embedding to separate, as remarked above. Incidentally, nontrivial couplings between the Liouville and embedding degrees of free40m have recently been considered in various contexts: the vanishing null states of ref. [1,10], and the extra physical states refs. [11]. The former, like the operator 58, grow as t/> --+ _00. 4 The whole picture appears to hang together quite well: Nambu-Goto string with correct measure

= Conformal field theory of XI' with nonanalytic dependence on OX . OX

= Low energy limit of Liouville + X,. field theory with general nonrenormalizable interactions. COULOMB FORCES FROM FLUX TUBES? One of the fascinating features of string theory is its many applications: as a fundamental theory, as a theory of physical extended objects such as flux tubes, and as a model for quantum gravity. There has been much cross-fertilization between these subjects. The Landau (now New Jersey) school seems to have been largely motivated by the challenge of QCD strings, but their results have been invaluable to the iundamental theory. As another example, the supermembrane action was discovered by contemplation of extended objects in supersymmetric gauge theories. For this and other reasons, it may be a good time to reconsider QCD, particularly 4Incidentally there appears to be a duality between the vanishing null states and extra physical states. Physical states must be orthogonal to all spurious states. When one of the latter w.nishes, there is one fewer condition and so one more solution is expected.

238 at large N, and ask if recent years' progess in string theory gives us any new ideas. The results of the previous section may be useful, but are not necessarily encouraging. If QCD is a string theory, we now know the low energy limit of the world-sheet theory. But this low energy limit could result from one of many possible world-sheet field theories (=spacetime string theories) at short distance, or from no field theory at all. It is interesting to contemplate this from the point of view of a two-dimensional physicist, who has found the analog of pion theory (a nonrenormalizable theory with the right sy=etries and degrees of freedom) and who is trying to find the short-distance theory. The group-theoretic part oflarge-N QCD is tantalizingly planar,(12) but it is not yet clear whether this translates into a world-sheet structure in spacetime. Attempts to address this always collide with the fact that QCD does not look very stringy at short distance. For example, long QCD flux tubes look like strings, but they have a thickness. If QCD is a string theory, then we should think of this thickness as coming about from averaging over the wiggles of an infinitely thin string. At distances short compared to the string thickness, QCD becomes free and the field looks Coulombic. Consider a quark-anti quark pair at very short distances, with the color electric field forming a dipole pattern between them. In a string theory the flux would travel from quark to antiquark along an infinitely thin tube. Can we think of the dipole field as being due to the averaging over all possible flux tube paths? We can try to make this question more quantitative. Let us study first a different question. Consider a free charged Klein-Gordon field , and consider the product i(z)(y), which creates a charge at z and destroys it at y. We want to ask whether the charge flows from z to y along an infinitely thin path. To study the spacetime structure introduce a background gauge field AI' and consider the propagator in this background:

(9) with D being the covariant derivative. Make a functional Fourier transform in terms of a variable jl': p(j) = J[dAle-iJddmA~j~ < '(z)(y) >A (10)

< '(z)¢(y) >A= J[djleiJdda:A~j~p(j).

(11)

The function p(j) measures the weight of a given current distribution, and the question is whether the support is restricted to infinitely thin paths. In this case we already know the answer: the propagator can be written as a path sum

The Fourier transform (10) can now be carried out, with the result

(13)

239 where

j"[z]

=[

dr'i:I'(r').5(z - z(r'».

(14)

The delta-function in eq. (14) means that the support of p is restricted to infinitely thin paths: p(j) vanishes on smooth functions. The background gauge field is being used as a sort of theoretical cloud chamber to make the path visible. Of course we cheated, using the known path-sum representation for the propagator. But if we had not known this, by carrying out the functional Fourier transform (10) we could have discovered that the propagator is secretly a sum over paths. Let us therefore try the same thing for the gauge theory. We are interested in very short distances where QCD is free, so we will study a free Abelian theory. The analog of tPt(z)tP(y) is the Wilson loop W[C], and we are interested in the path ofthe flux so we will couple an external field BIU' to the flux FlU" Thus:

< W[C] >B= ![dA]eifcd".Ae-fd''''(~F•• F'.-iB •• F•• ).

(15)

To discover the weight for a given pattern of flux make a Fourier transform analogous to (10),

(16) But in a free theory < W[C] >B will simply be Gaussian in B, so its Fourier transform will be Gaussian in f, with support on all smooth functions. Thus the Coulomb field is not secretly an average over flux tubes. Of course this is far from conclusive. Even at weak coupling it is probably incorrect to approximate the non-Abelian theory by an Abelian one, since the nonAbelian nature is essential to the large-N picture. Also, the flux to which we couple is not gauge invariant in a non-Abelian theory. We need perhaps to consider the flux measured by the 't Hooft 100p)13j Also, even in the Abelian case the issue is not as well-posed as it may have seemed. In a pointwise sense a smooth function is very different from one with support only on a curve, but as a distribution the smooth function can be approximated arbitrarily well by a curve which traces back and forth many times. In conclusion, the ideas in this section are very preliminary, but I believe that it is a good time to reexamine this subject.

240 REFERENCES 1. J. Polchinski, "Ward Identities in Two Dimensional Gravity," Texas preprint UTTG-39-90. 2. D. Minie, J. Polchinski, and Z. Yang, "Translation-Invariant Backgrounds in 1+1 Dimensional String Theory," Texas preprint UTTG-16-91. 3. A. M. Polyakov, Phys. Lett. B163 (1981) 207; T. L. Curtright and C. B. Thorn, Phys. Rev. Lett. 48 (1982) 1309; J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509; F. David, Mod. Phys. Lett. A3 (1988) 1651. 4. S. Das and A. Jevicki, Mod. Phys. Lett. A5 (1990) 1639. 5. J. Polchinski, "Classical Limit of 1+ 1 Dimensional String Theory," Texas preprint UTTG-06-91; J. Avan and A. Jevicki, "Classical Integrability and Higher Symmetries of Collective String Field Theory," Brown preprint IAS-PUB-BROWN-HET-801 (1991). 6. J.-L. Gervais, Phys. Lett. B255 (1991) 22. 7. H. B. Nielsen and P. Olesen, Nucl. Phys. B61 (1973) 45. 8. R. Marnelius, Phys. Lett. B172 (1986) 337. 9. J. Polchinski and A. Strominger, Texas/Santa Barbara preprint UTTG-17-91 (1991). 10. K. Hamada, "On Structures of W and Virasoro Constraints in Liouville Gravity," Tokyo preprint UT-Komaba 91-13; R. Leigh and J. Polchinski, unpublished. 11. A. M. Polyakov, "Self-tuning Fields and Resonant Correlations in 2d-Gravity," Landau/Princeton preprint (1991); G. Mandal, A. M Sengupta, and S. R. Wadia, "Classical Solutions of 2-Dimensional String Theory," lAS preprint IASSNS-HEP/91-10 (1991); E. Witten, "On String Theory and Black Holes," lAS preprint IASSNS-HEP91/12 (1991). 12. G. 't Hooft, Nucl. Phys. B72 (1974) 461; A. A. Migdal, Phys. Rep. 102 (1983) 199. 13. G. 't Hooft, Nucl. Phys. B138 (1978) 1.

241

SUBCruTICAL STRINGS IN 1 < D :S 25 * Christian R. Preitschopf and Charles B. Thorn Department of Physics University of Florida Gainesville, FL 32611

ABSTRACT We consider the problem of constructing multiloop amplitudes for open string dual resonance models in space-time dimension D less than 26, such that open string unitarity is automatic. For general subcritical dimension, we show how to remove redundant states by including an extra scalar world sheet field (the free field incarnation of the Liouville degree of freedom) to make the BRST charge nilpotent, and then constructing a vertex function invariant under this modified charge. Our results are summarized by giving a Witten-style action for open sub critical string fields. We consider, with partial success, the implications of this work for the interactions of sub critical closed strings.

INTRODUCTION Interest in subcritical string theory[l,2]has recently been rekindled by connecting string theory in less than one space-time dimension to large N matrix models!3-7] Although this approach has been successfully extended to the case of one spacetime dimension [8,9] it does not seem to work in higher dimensions. In Ref. [10] it was suggested that some insight into the higher dimensional case might be gained by studying the sub critical (open string) dual resonance models. These models are consistent at tree level. The one-loop dual amplitude then implies a continuous mass spectrum for the closed strings coupled to open strings!lO) a result also implied [11) by the Polyakov approach to subcritical string theory. Here we outline the study of subcritical dual models at the multi-loop level. A full account is given in Ref. [12).

* Work supported in part by the Department of Energy, contract DE-FG05-86ER--40272.

242 The ultimate aim is to understand sub critical closed strings by analyzing the multiIoop diagrams of the open string theory. For example, the three closed string coupling enters first at two loops. Actually, this program was attempted in another guise nearly two decades ago. In 1972 Neveu and Scherk[13] proposed to construct currents (off-shell amplitudes) for dual models by factorizing the "Pomeron" (closed string) cut in multiloop dual models. In effect, we are interpreting their "currents" as on-shell closed strings with a continuous mass spectrum. The program was later pursued by Cremmer and Scherk!14] with only partial results. The stumbling block was the lack of a systematic method of ensuring that redundant states don't propagate in amplitudes with two or more loops. Thus while they were able to obtain expressions for the coordinate part of the loop integrand, there was an unknown "measure" factor to be determined by the removal of spurious states. We apply the modem BRST approach to this problem. In Ref.lO it was shown that the subcritical one loop dual amplitude could be interpreted as a world sheet path integral over coordinates, ghosts, and an additional free pseudoscalar field 1{1 obeying certain Dirichlet boundary conditions. This additional field was interpreted as the Backlund transform of the Liouville field!15,16] The 1{1 field on a cylinder differs from an ordinary coordinate in that the zero mode sector is non degenerate: the 1{1 state space is restricted to the Verma module built on the odd parity zero mode vacuum state. In Ref.10 this halving of the closed string spectrum was linked to the presence of an extra factor of (1 - w) in the open string partition function for 1{1:

Tr[wLo]

= (1 -

n--. 00

w)

=1

1

1-wR

One aspect of this factor of (1- w) is that the "photon" of the open subcritical string remains transverse. For us the state space of the Liouville sector of the open string[17-19] will be taken simply as a free world sheet scalar tP (dual to 1{1), introduced in such a way that the multiloop amplitudes satisfy unitarity on the physical open string states. The Virasoro generators for 4> contain an "improvement" term which is chosen to give the correct central charge 26 - D for the Liouville field. To achieve our goal, we find it necessary to restrict the zero mode "momentum" of tP to have a fixed value. Further, for D < 25 the construction of BRST invariant vertices requires the state space for 4> to be restricted to the Verma module built on the SL(2, R) invariant state. With this restriction the cohomology of Q is isomorphic to the physical state space of subcritical dual models. Note that this treatment of the open string Liouville theory is not quite the same as that of Marnelius[18] or of Gervais and Neveu!19] We use the Liouville field as a tool for removing redundant states from multiIoop amplitudes, and it does that job by construction. The neatest way to present our results is in terms of an action for the open string field, from which the multiloop dual amplitudes can be derived. In fact, all of the ingredients of Witten's version of open string field theory!20] including an associative * product, can be constructed for the sub critical case. The associativity of the * product ensures duality. The trees and one loopdiagrarns following from this action coincide with those of the subcritical dual model. This of course means that the closed string implied by the string field theory has a continuous mass spectrum.

243 We then begin the study of two loop diagrams and the implied three closed string coupling. Instead of trying to evaluate the full two loop diagram, we consider the diagram factored on two closed string singularities. Following earlier work[21) we construct the sub critical open string-closed string transition amplitude. Then we glue two of them together with the open string propagator to obtain an expression for the amplitude for two closed strings coupling to any number of open strings. Although we obtain a definite formula for the Liouville factor, we are unable to evaluate it in closed form.

THE STRING FIELD ACTION AND MULTILOOP AMPLITUDES For clarity we shall employ the formalism of string field theory [20.22) to summarize our results for multi-loop dual amplitudes. For subcritical dual models we must introduce the Liouville degree of freedom in such a way as to make the Witten vertex BRST invariant, and the * product associative. These features will then guarantee that the unitarity sums obtained by cutting open string lines of the multiloop diagrams include only the physical states of the dual resonance model. We. take the coordinate and ghost parts of the Witten vertex to be identical to those of the critical stringi23,24) BRST invariance· can be arranged if we choose the Liouville part of the vertex to satisfy particular conformal Ward identities of the type

(V123I(L~m + em +

L: M;'~L~') = 0 n~O

with r = 1,2, 3 labelling the three strings and r + 3 == r. These identities are identical to those satisfied by the coordinate part!26) except for the c-number term, which cancels that of the coordinates and ghosts. For the Liouville Ward identities, the c-number should have D replaced by 26 - D, so that the identities for coordinate plus Liouville will be identical to those of the critical string. The Virasoro generators for the Liouville field include an improvement term!l.11)corresponding to a coupling (-r/21r) J dAt/JR(2) of t/J to a background world sheet curvature R(2):

Ln

.

12

1

= 'Int/Jn + "2 6n,0 + 2L: :t/J-"t/Jn+lo :

(1).

"

The background charge I' describing the coupling to the world-sheet curvature, is given by

_J25-D 12'

1 -

In addition, the requirement thai the mass spectrum of the free string be that of the subcritical dual model * implies that t/Jo

*

= ±il •

For D < 25, other choices require, to avoid a mass squared spectrum Cor the open string unbounded below, operator insertions (analogous to P, P below) Cor which duamy (associativity) is not manifest.

244 A peculiar feature of the fact that the value of q,o is imaginary is that the L,., with . the zero modes replaced by their values, do not satisfy the usual hermiticity relations

(2) To handle this situation we make use of the two vacua 10) and Ic) with 4>0 eigenvalues i-yand -i-y respectively. Then (el and (01 have eigenvalues i-y and -i-y respectively and satisfy (eIO) = (Ole) = 1 With these definitions,

(010)

= (clc) = o.

q,o can be treated as hermitian, and the L,. satisfy Eq.(2).

In constructing a string field formalism, we have a choice of identifying the string field A with the Fock space built on 10) or that built on Ie). We find it more convenient to choose the former, its principal virlue being that its vacuum is the 8L(2, R) invariant state. To build a string field action, we must find a two string verlex (Val and a three string verlex (V123 I that are BRST invariant and accept kets from the Fock space built on 10). Then the string field action will be 1 AI. 8 = 2(V12IA)QIA) + a(V123IA)lA)lA).

(3)

The three string verlex can be used to define the * product of string fields. For this purpose we need a third object, the double ket, li12 ) which accepts bras built on (cl. then

IT Q is a derivation with respect to the under the gauge transformation [20] ~A

* algebra and * is associative, 8 is invariant

= QIA) + IA *~) -I~ * A).

We first tum to the construction of tile Liouville part of lin)' It should satisfy the Ward identities

The standard choice for this verlex 00

q,l 4>2

1112) = exp{- ~ (-)"~}(10,I)1c,2) + Ie, 1)l0, 2» =1 n satisfies, in addition (q,~ + q,~)II12)

= 0,

so that it couples (e, 11 to (0,21 or vice verla. In order to get a nonvanishing coupling

245 between (c,11 and (c,21 we need an operator P with the properties

[Ln,P) = 0 (1)0 - i-y)P = P(1)o + i-y) = O. One can show that the construction

(4) with the matrix M defined by M{~}.M = (elL;' L;'", L~ftL':.';•... L~~L~'210)

= (OIL;' L;' ... L~ft L~';.··· L~33L~'2Ie), satisfies these properties. In addition,

(5) is unique up to a multiplicative constant. It is significant that li12 } couples only to a subspace of the full 1> Fock space. This accounts for the fact that the massless spin 1 particle present in sub critical dual models is transverse. Next we discuss the two string vertex ()l121 with which one constructs the kinetic term of the string field action. Its Liouville part should satisfy

As with lIn) the standard choice for this vertex 00

(V121 = «(0,11(c,21

+ (c,11(0,21)exp{- L(-t

1>11>2

:

n}

n=l

couples Ic, I} to 10,2} or vice versa. Thus we seek an operator

P with the properties

[LmP) = 0 (1)0

+ i-y)p = p(1)o - i-y) = O.

Then (V12 IA would be a vertex coupling 10,1) to 10,2} with the correct Ward identities. The operator 1 P- = Ll'a - n ' " LI"I -2 C}M{I'},{~} (IL~' C 2

•••

L~· n

has the required properties in the weaker sense: (OIL~' .. ~ L~· [Lm P)L~';. . " L~~ 10) =

o.

Thus we can construct a suitable vertex if we demand that the Liouville part of the string field belong to the Verma module V built on 10). With this restriction ()l121 is

246 uniquely (up to a multiplicative constant) given by

(6) It satisfies the Ward identities weakly: (Vd(L~ - (-)" L:,,) IV, I} IV, 2}

= O.

Next we turn to the construction of the Liouville part of the three string vertex. It is well known how to construct the Witten vertex (V123 1 for world sheet fields with a background charge!23,24,26! One must apply a curvature insertion e"l(1C/2) to the state (V~2sl describing the Witten overlap for vanishing background charge, just as for bosonizedghosts. Such a vertex satisfies the desired Ward identities, but implies an anomalous conservation law for the CPo:

=

Thus this vertex does not connect three states with CPo il' One of the states must have 4>0 = -il' In order to get a vertex with nonvanishing coupling for three states built on 10) we need to apply the operator P to one leg of the vertex. Since P only has the required properties restricted to matrix elements between states in V, it is important to check that (V123 IV,1)1V,2) belongs to the Verma module built on (01. Fortunately, this is true: a string field vertex with, for instance, the third leg carrying CPo = - i l satisfies

(V12310, I) 10, 2}L_dc, 3}

= 0,

(7)

for the general class of string field vertices considered in Ref. [26). They include the lightcone-like vertices and SL(2)-vertices, and of course Witten's vertex. Hence the modified Witten vertex

couples three states with CPo

= +il

(V12S I(L: m + em +

and satisfies the Ward identities in the weak form

:E M:;:"L~)IV, 1) IV, 2) IV, 3) = O. ,,~o

Using only the Ward identities one can completely determine all couplings

(V 123 IV, l}IV, 2)IV, 3) up to an overall multiplicative constant. In particular, restricted to these states, the vertex is cyclically symmetric in the three strings, so it doesn't matter which leg we choose to carry CPo = -il' By arranging the Liouville parts of the two-string and three-string functions to satisfy the proper (weak) Ward identities, it follows that the total vertices satisfy (weak) Ward identities isomorphic to those of the critical string, and hence, with the Liouville parts of the string fields restricted to V the vertices are BRST invariant.

247 Fortunately, the * product is consistent with this restriction because li12 ) belongs to V1 x V2 and is BRST invariant without restictions. IA * B) belongs to V if A and B do. Moreover, the * product satisfies the derivation property with respect to Q: for A,B E V. The associativity of the * product is essentially four string function duality: the s channel four string diagram should equal the t channel diagram when the intermediate propagators have zero length. This is just a feature of the geometrical nature of the Witten vertex. However, for our subcritical vertices, there are various operators P and/or P on the intermediate lines, which could spoil associativity. To check associativity (duality), we must show that these operators are harmless. In terms of explicit formulas, the duality or associativity condition is

This relation is true provided the states attached to legs 1, 2, 3, and 4 are restricted to the Verma module built on 10). To see this, write out the vertices choosing the legs 5 and 3 for the attachment of the operator P. Then the duality condition reads

where

is a projection operator satisfying 'P2 has the decomposition

= 'P.

One can show that the identity operator

1= 'P + L: L:-;. ... L:\lc)(OIX{~}. ~'oFo

Thus by Eq.(7), if the states attached to legs 1, 2, and 4 are in the Verma module built on 10), 'Ps can be replaced by the identity, and the duality/associativity condition is proved, since it reduces to that for the ordinary Witten vertex. The above line of argument is useful in the evaluation of tree and one loop amplitudes with physical states on the external legs. One constructs a general multiloop open string diagram in Siegel gauge by tying three string vertices together with propagators

(L~bo).1112 ) = J"" dT (boe-L'T)· • 11

12 ).

1

(8)

o

With L~ = L. +Lk +Lth we denote the total Virasoro generators including Liouville, coordinates, and ghosts. For trees, one can write out the vertices and propagators in a way for which all

'P operators can be replaced by the identity. Then by conformal invariance, one can transform to the disk with external lines represented by vertex operators independent

248 of f/J attached to the boundary. The factor associated with the Liouville degree of freedom then reduces to a constant, and the ghost and coordinate factors reduce to the Koba-Nielsen integrand for the subcritical dual model. For one loop diagrams all but one 'P can be removed. Conformally transforming to the Hat cylinder metric with vertex operators chosen as for trees, the Liouville factor reduces to

(9) with f(w)

= II(l -

wn), and the standard result [27.28] is reproduced.

CLOSED STRING FACTORIZATION We shall now proceed to study multiloop amplitudes in our subcritical string field theory in more detail. In particular, we shall show how to extract information about the structure of closed strings in D ::; 25. As a first step, we consider at one loop the turret diagram~29] It is given by the expression

(10)

We now factorize it on closed string poles, i.e. we take the limit T2,T3 ~ O. As discussed in Ref. [21], the factors are identity string fields[30]with physical primary closed string vertex operator insertions at the midpoint Vi31 ] We denote them by {TIS}, where IS} is the closed string state associated with the inserted operator. We have the Ward identities

(11) If we choose the closed string state to be a simple product of coordinate-ghost and Liouville factors, the vertex also satisfies

(12) Here c is the conformal anomaly of the Liouville sector of the theory, and (h, Ii.) are the conformal weights of the Liouville part of the closed string state. By construction the total vertex is a product of the matter, ghost and Liouville parts. We therefore have two ways of deriving the Liouville part of {TIS}. We either take the limit of (10), or we solve the Ward identities (12) directly, for c = 26-D and h = Ii. = 1-p2 /8-R. Here p2/8 + R is the conformal weight of the left-moving as well as the right-moving matter part of the vertex operator. The solution has a unique action on the Verma module built on IO}. We have to make choices if we wish to write an open string closed string vertex on the complete open string Fock space and for off-shell closed

249 string insertions. We find it convenient to simply extend the vertex of Ref. [21] to the Liouville sector. In the language of Ref. [26], it is defined on open string states IA} and closed string states IB, fJ} as follows:

(TIB,fJ}IA}

= (g[B]g[B]

I[A]}

(13)

with the maps

2z l(z)=1_z 2 '

.1 + w 1-w

.1+w

9 () w =1-, 1--w

() =-1--. gw

(14)

The closed string vertex operator that we have to insert in order to obtain (12) is then, for example,

S( z, z) = c( z )1:( z)e ipX (z •• )+(ip /2+-r)+(z)+( -ill/2+-rji(z)

(15)

with

so that S(z, z) transforms as a scalar. For different matter primaries we would have to replace p2/4 in J.! and Ji with their conformal weight h = It respectively (the equality is enforced by factorization). The Liouville part of S(z, z) is not a local operator. We may write it as

+ iW,p(z, z)) . and 'I/1(z,z) = 1/2(\p(z) -

exp (2/X(z, z)

(16)

The fields X(z,z) = 1/2(\P(z) + ~(z)) ~(z)) are related by a duality transformation. We encountered '1/1 already in Ref. [10]. (16) is the vertex operator corresponding to the momentum eigenstate IJ.!} of the field '1/1 on the cylinder. We can employ the technique we used to analyze the one-loop diagram also in the two-loop case. Thus we now contemplate deriving the coupling between three on-shell closed strings from the open string two-loop amplitudes. In order to simplify the geometry of the diagrams we consider, we will replace the external open strings by local vertex operators on the world-sheet boundary and obtain

(17) and 00

J

.. (

dT1 dT2 dT3 (V123 !(V456 ! bo e-

L'T) (bo e- L'T) 2 (bo e- L'T)· . . 3!I14}II2s}!I36} 0

1

1

02

0 3

o

(18) respectively. In order to insert external open string states, we simply replace each propagator by the appropriate combination of local vertex operators and multiple propagators. We are interested in the limit T1 , T2 , T3 -> o. In principle the calculation

250 is straightforward, but in practice it seems to require numerical methods. We can make some headway if we use the results of our one-loop calculation above. This means that we take the limit T1 , Ts -+ first. We then arrive at the formula

°

(19)

We now simplify the calculation by mapping to a different conformal frame. For that purpose, let us first rewrite (TIS) as

where v(z) = vnz- n- 1 is the generating vector field for the map f(z) in (14), and S( i, -i) is the closed string vertex. In Ref. [26] the gluing procedure for such objects was discussed at length. We obtain for (19) the expression

Here h is the smoothing function, which is essentially the map from Fig. 1 to the upper half plane. We now use the conformal invariance of P to write hlP] = P, the property f(±i) = 10 f(±i) = ±i, and the fact that the ghosts combine with the propagator into a density on moduli space, to write (20) as

J T.

dT (OISl(i, -i)P 60

e-L~T Sa(i, -i)IO)

,

(21)

o

where To is determined by h( z). We are interested in the limit T -+ 0 of the integrand. If we insert N open string vertices at the boundary, we obtain (after an SL(2,R) transformation)

nJ

dy, (YN-2 - YN-l)(YN-2 - YN )(YN-l - YN)

• =1

n

l$:i 0, ITI > 1, -t

and F

: : ; Tl < t.

The

summation over spin structures is included in dT. The contribution of the matter sector are given in terms of the characters of the superconformal algebra. We list results for the discrete SCFT here. The character of a representation of superconformal algebra defined by (3.2) will give· us the counting of states on the torus. It is well-known for representations given in (2.1)(2.2) that the characters are given by[13]

x:.:. = jNS(q)r!~p"

jNS(q)

= q .~ 71- 1 (q)

:fi

(1

+ qn-!)

(3.3)

n=l

for the NS sector and 00

R -_ jR( q)r21 Xp,p' p,p"

jR(q)

= qt.71-1(q) II (1 + qn)

(3.4)

n=l

for the R sector where 1J(q) is the Dedekind function, 1 ::; P ::; m - 1,1 ::; p' ::; m

+ 1 and

p - p' is even in the NS sector and odd in the R sector. The functions r;~p' are defined by

r;~p,(q)

=L

(q,,;:.,(n) _

q6;:.,(n»

(3.5)

nEZ

where 21

( ) _

ap,p' n b21 ( ) p,p' n

[41(1 + l)n + p(1 + 1) - p'W 81(1 +1)

= [41(1 + l)n + P(I + 1) + p'lj2 8/(1 + 1)

(3.6)

261 We also have the projected characters in the NS sector as

X:':' = tr( -ll qLo-1o = jNs(q)f;~p"

jNs(q)

= q -;i

IT

(1 - qn-t),.,-l(q)

(3.7)

n=l

where F is the fermion number operator and the functions

f;~p,(q)

=L

f

are given by

(qa;:.,(n) - (-1)PP' l;:.' (n»)

(3.8)

n€Z

The holomorphic contributions of the ghost fields to the partition function for different spin structures are well-known except for a phase factor which is absent in the final theories. The holomorphic contributions from the oscillators part of the super-Liouville theory to the partition function for different spin structures are the same as those of one free scalar superfield. They are given in the following table

ghost

Liouville

:

(fNS(q»-2

fNS(q)

(-,+) :

(jNS(q»-2

jNS(q)

(+,-) :

(fR(q»-2

(-,-)

(3.9)

fR(q)

The contribution of the zero mode of the super-Liouville field is the same for all the spin structures and denoted as T2

Z%).

Collecting contributions from matter, Liouville and ghost sectors with the same spin structure and summing over spin structures, the torus partition function is given by Zl =N

r d T ~LMi,jf'j(T)ri(T)

1F

(3.10)

2

yT2

where the i, j refer to both spin structures and (p,p'). The exact form of

Zf

is not

important for our purpose and has been computed by Bershadsky and Klebanov[14) to be independent of T. All string theories arising from

d < 1 SCFT coupled to 2-d supergravity are modular

invariant solutions Mi,j of (3.10). Here we will give a series of generic solutions. Let's define functions e and () as

e21~ = p,p

{

t [r!~;. + f!~;.] - rr:p"+l_.p,

1[r21- + r~21-] 2 p,p' p,p' - r21l-p,l+l-p'

(}21-, = { t [r!~;. - f!:;'] + rr:p,I+l+p' p,p

12

[r

2l - _

p,p'

2l -] f p,pl

1 ::; p ::; 1 - 1,1 ::; p' ::; 1,

p,p' odd

,

p,p even p,p' odd p,p' even

(3.11)

262 and

1)2'+ 1',1" where

= !2

21 + - f'21+] [rP,P' p,,'

± r 21 p,p'

= r21p,p' ± r 21p,2(1+1)-p'

p,p' even

r± are defined by (3.12)

The torus partition function of the non-critical superstrings can be written in terms of those functions as

Z1

f"o.J

1

cP T Zf cr {

:F

+

V T2

"L...J

1S,~;t-1.1S.'SI 1',,' odd

E 1$.$21-1.1$.'$.,+1 p,,' ."en

(9.. 21 -e21 - + (P1-(}21-) p,p' .. p,p'

p,p' p,p'

jp l+(}21+ } p,p' p,p'

(3.13)

2

The modular invariance is easily verified. Furthermore we have identities e;~;; = 0 and = 0, therefore the torus partition function vanishes identically. The theories are therefore "space-time" supersymmetric, i.e. superstrings. This concludes our discussion of (};~;.

the consistency of the theories. Several comments are in order. The fermions(2.17) enter partition function with a minus sign and are consistent with statistics. Only a subset of the physical states of (2.16)(2.17) is contained in the theory, namely Vp~p" V,~P,1+1-P" V:'p: and V,~p,'+1-p' with p, pi odd. Since the vanishing of partition function is a consequence of holomorphic identities one could construct non-critical heterotic superstrings[15] from the level matching consideration[6]. There are other solutions to the condition of modular invariance o~ (3.10). The modular invariant spin models are solutions. They have only bosonic states but nevertheless are interesting string theories. Generically, the R fields in those theories have smaller dressing coefficients and therefore set the scale of the theories. The critical exponents(2.20) for the mth theory are therefore modified to be "Y

= { - 2';+1' 0,

m odd,

m even

!p(m+2)-p'm!-1 ,_ { 2m+1 Cl.p,p !p(m+2)-f'm! 2(m+1 '

,m

0

dd (3.14)

m even

Notice that there are two choice of the sign for the odd spin structure, i.e. the sign for the R sector with (_l)F inserted. For m even, it results in two different theories with different

263 scaling. For the minus sign, the particular R operator is projected out and therefore the scale is set by the NS operator (2.11)(2.15). It is in agreement with the super-Kdv resu1t[16]. The theory with plus sign will have scaling behavior (3.14). The matching condition is more general than tlle requirement of "space-time" supersymmetry. It can also be used to construct non-critical fermionic strings analogy to the

80(16) ® 80(16) heterotic string[17]. References [1] V. Knizhnik, A.M. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819; F. David, Mod. Phys. Lett. A3 (1988) 1213; J. Distler and H. Kawai, Nucl. Phys. B321 (1989) 509. [2] E. Brezin and V. Kazakov, Phys. Lett. B236 (1990) 14; M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 123; D. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [3] M. B. Green, J. H. Schwarz and E. Witten "Superstring Theory", Cambridge University Press (1987). [4] A. M. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 1213. [5] J. Distler, Z. mousek and H. Kawai, Int. J. Mod. Phys. A5 (1990) 391. [6] Z. Qiu, "Non-Critical Superstrings from Two Dimensional N=l Supergravity", University of Florida preprint IFT-HEP-90-28." [7] D. Friedan, Z. Qiu and S. H. Shenker, Phys. Rev. Lett. 52 (1984) 1575; Phys. Lett. 151B (1985)37. [8] A. M. Polyakov, Phys. Lett. 103B (1981) 311. [9] T. L. Curtright and C. B. Thorn, Phys. Rev. Lett. 48 (1982); E. Braaten, T. Curtright and C. B. Thorn Phys. Lett. U8B (1982) 115; Ann. Phys. 147 (1983) 365; E. Braaten, T. Curtright, G. Chandour and C. B. Thorn Phys. Rev. Lett. 51 (1983) 19. [10] J. Polchinski "Remarks on the Liouville Field theory", UTTG-15-90; N. Seiberg, "Notes on Quantum Liouville Theory and Quantum Gravity", Rutgers preprint RU-90-19. [11] D. Friedan, E. Martinec and Shenker, Nucl. Phys. B271 (1986) 93. [12] Z. Qiu, unpublished. [131 P. Goddard, A. Kent and D. Olive, Comm. Math. Phys. 103 (1986) 105. [14] M. Bershadsky and I. R. Klebanov, Phys. Rev. Lett. 65 (1990) 3088. [15) D. J. Gross, J. A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B256 (1985) 253; Nucl. Phys. B267 (1986) 75. [16] P. Di Francesco, J. Distler, D. Kutasov Mod. Phys. Lett. AS (1990) 2135 [17) L. Dixon and J. Harvey, Nucl. Phys. B274 (1986) 678;Alvarez-Gaume, P. Ginsparg, G. Moore and C. Vafa, Phys. Lett. 171B (1986) 155.

264

QUANTUM SUPERGROUPS N. Reshetikhin Department of Mathematics University of California Berkeley, CA 94270, USA

ABSTRACT A construction of the double of Z2-graded Hopf algebras is given and described explicitly for quantized universal enveloping algebras of rank=l Lie superalgebras. A completion of these algebras by Weyl elements is given. The relation to the Alexander polynomial of links is discussed.

Introduction Quantization, or, defonnation of universal enveloping algebras of Lie su~ peralgebras (for details about Lie superalgebras see [K,Sc]) is the next and natural step after the theory of quantum groups [D,J,FRTJ. It is natural to define this generalization as a theory of quantum supergroups. Some results in this direction are already obtained [K,KR, CK,FSV, DegJ. In this note a detailed explanation of the double construction for rank=l Lie superalgebras is given. It is shown that quantizations of Ugl(lll) and Uosp(112) (universal enveloping algebras of Lie superalgebras gl(111) and osp(112» can be considered naturally as doubles of its Borel subalgebras. It is shown also, how to complete these algebras by the Weyl element. In §1 a definition is given of Hopf superalgebras, and of a double construction for Hopf superalgebras. §2 is devoted to the analysis of Lie superalgebra of rank one. In §3 we discuss the relation between Uhgl(111) and the Alexander polynomial. A similar analysis of the quantized simple Lie superalgebras will be given in a separate publication.

265

1

Hope superalgebras

Let Ie be a commutative ring and A a Z2-graded Ie-algebra. As a linear space A = Ao + AI. Define p(a) = 0 if a E Ao and p(a) = 1 if E AI.

Definition 1.1. A tensor product of two Z2-graded algebras A and B is a Z2 graded algebra which is isomorphic to A ® B as a linear space and

(a ® b)(c ® d) = (-l)p. transformations are a similar combination of '>.- Einstein' and '>.- Weyl' symmetries[4,5), whose geometrical meaning is unclear at this moment. Without h and B fields, (8) is invariant under chiral symmetries with parameters f+(X-) and >.++(x-). To see this note that the variation of the first term in (8) is equal to f o_tpio+ [t+O_tpit5 ii +>.++diiko_tpio_tpk]. For chiral parameters one can rewrite this, using the total symmetry of t5ii and d'ilc, as

-!

(10) For local 10+ and >.++ there are extra variations proportional to 0+10+ and 0+>'++ which are taken care of by the t5h++ = O+f+ variation in the second equation of (9), and the t5B+++ = 0+>'++ variation in the last equation of (9). All other 10+ or >.++ variations

288

are then proportional to h++ and B+++, and cancel by themselves, provided the ,file symbols satisfy the following quadratic relation

(11) In this paper we shall always assume that this relation holds, and, furthermore, assume that the ,file symbols are traceless (d symbols with these properties were systematically studied in [16]). With traceless ,file fewer Feynman diagrams are possible; one important property is that in this case all diagrams contributing to the induced action r[h, B) are one-particle irreducible. In what follows we shall often omit + and - subscripts where no confusion should arise. The local gauge algebra of f and .\ symmetries reads

[{fl),c5(f2») = 15(£'),

t' = -f1lLf2 + f28_fl'

(12)

[c5(f),c5(.\») = 15(.\'),

'\' = -f8_.\ + 2.\8_f,

(13)

(14) The first commutator shows that f transformations have the same group structure as (one-dimensional) general relativity, while (13) and (14) show that f and .\ symmetries form on-shell a so-called symmetric algebra. The last commutator shows that the algebra does not close off-shell, and produces an equation of motion symmetry c5'P "" -c51/c5h and c5h "" +c5I/c5'P. BRST QUANTIZATION According to the method of BRST quantization, one adds gauge fixing and· ghost terms to the classical action such that the final action is invariant under BRST rules. The fields in the classical action ('Pi, h++ and B+++ in our case) transform under BRST transformations as under local gauge transformations, but with the local gauge parameters (,,+, .\++) replaced by ghosts

(15) Here A is the constant anticommuting real BRST parameter. All other transformation laws, as well as the terms in the quantum action beyond the classical action, follow from the requirement that the action be BRST invariant.

289 Since the local gauge algebra only closes modulo field equations, we must follow the BRST quantization for gauge theories with open gauge algebras. The quantization of gauge theories with open algebras was first performed in ref. [17], where hamiltonian methods, diagrammatic methods and lagrangian BRST methods were employed, respectively. We shall use the BRST method, for a review see ref. [18]. Since those early days, a new framework for BRST quantization has been erected(10), which contains all previous approaches, and deals also with theories which contain so-called ghosts-forghosts. Although we do not have ghosts-for-ghosts in our case, we shall still follow this so-ca.lled BY formalism for BRST quantization, because it allows a uniform treatment of all aspects. It should, however, be noted that although the BY formalism is a convenient book-keeping device, it contains no new quantization rules. For example, the so-ca.lled 'antifields' in the BY formalism are the anticommuting sources which in the standard treatment are added to quantum Yang-Mills theory in order to prove its renormalizability(19). The BY formalism facilitates the analysis of anomalies and Ward identities in cases where the local ga.uge algebra. does not close[ll). According to the BRST-BY quantization scheme, one first constructs a minimal action 8(min) depending on all minimal fields til = { 1 it seems unlikely that such additional charges could exist because the equations that determine the relative motion of the various parts of the object cannot be solved exactly for arbitrary initial conditions. For p = 1, however, the motion can be solved exactly (at least for T :f. 0 for which the motion is periodic) and so the prognosis seems more favourable. In an early version of this contribution I claimed to have found an additional infinite set of conserved charges of the form (33), but this was the result of a computational error. Despite the current lack of positive evidence in favour of this possibility it remains an attractive one. For example, it was argued in Ref. (6), on the basis of a connection with a possible phase transition at the Hagedorn temperature, that the quanti sed null string has a unit S-matrix. This would certainly suggest the existence of an infinite number of symmetries (other than those associated with the centre-of-mass motion). Such an infinite symmetry group might allow the coupling of the null string to a conformal-higher-spin theory with the same (infinite) multiplicity as found in the string spectrum. In this way the null string and the conformal-higher-spin approaches to the unbroken phase of string theory might turn out to be equivalent.

Acknowledgements: I thank the Direccion General de Investigacion Cientifica y Tecnica of the Spanish Ministry for Education and Science for financial support.

References [1]

U. Lindstrom, B. Sundborg and G. Theodoridis, Phys. Lett. B258 (1991) 331.

[2]

A. Schild, Phys. Rev. D16 (1977) 1722.

323

[3]

A. Karlhede and U. Lindstrom, Class. and Quantum Grav. 3 (1986) L73.

[4]

A.A. Zheltukhin, Sov. J. Nuel. Phys. 48(2) (1988) 375.

[5]

F. Lizzi, B. Rai, G. Sparano and A Srivastava, Phys. Lett. B182 (1986) 326.

[6]

J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Nuel. Phys. B338 (1990) 143.

[7]

E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin, X. Shen and K.S. Stelle, Phys. Lett. B243 (1990) 350.

[8]

E.S. Fradkin and V.Ya. Linetsky, Mod. Phys. Lett. A4 (1989) 731.

[9]

C.N. Pope and P.K. Townsend, Phys. Lett. B225 (1989) 245.

[10] E.S. Fradkin and V.Ya. Linetsky, Mod. Phys. Lett. A4 (189) 2363. [11] E.S. Fradkin and V.Ya. Linetsky, Phys. Lett. B231 (1989) 97. [12] M.A. Vasiliev, Phys. Lett. B243 (1990) 378. [13] P.K. Townsend, Class. Quantum. Grav. 8 (1991) 1231. [14] W. Rindler and R. Penrose, Spinors and Space-time, Vo1.2 (C.U.P. 1986); T. Shirafuji, Prog. Theor. Phys. 70 (1983) 18. [15] I. Bakas, Phys. Lett. B228 (1989) 57; A. Bilal, Phys. Lett. B227 (1989) 406; C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B236 (1990) 173. [16] Q.-Han Park, Phys. Lett. B238 (1990) 287. [17] J. Hoppe, Ph.D. Thesis, MIT (1982); E. Bergshoeff, M.P. Bleneowe and K.S. Stelle, Commun. Math. Phys. 128 (190) 213. [18] A. Ferber, Nuel. Phys. B132 (1978) 55. [19] M. Kaku, P.K. Townsend and P. Van Nieuwenhuizen, Phys. Lett. B69 (1977) 304; Phys. Rev. D17 (1978) 3179.

324

INTEGRABLE FIELD THEORY OF SELF-AVOIDING POLYMERS IN 20 A. B. Zamolodchikov Department of Physics and Astronomy Rutgers University, P. O. Box 849 Piscataway, NJ 08855-0849, USA ABSTRACT We propose the exact factorizable S-matrix of the Field Theory corresponding to the scaling limit of the n-vector model with -2 < n < 2 away from the critical point. The limiting case n = 0 describes the scaling properties of the long selfavoiding polymer chains.

The self-avoiding polymer model is one of the basic models in polymer physics (see e.g. [1]). The polymer is considered as a flexible chain of L equal monomers which interact through the excluded volume effect. The partition function is written as Z(fL) = L:N(L)e-I'L, (1) L

where fL is the chemical potential and N(L) is the "microcanonical" partition sum, i.e. the volume of the configuration space of the self- avoiding polymer of the length L (the polymer chain may be closed or open depending on the problem of interest). At large L the function N(L) of, say, closed polymer behaves as

(2) with some constant fLc which depend on the details of interaction of the monomers and universal exponent v, and the partition function (1) is singular at fL -+ fLc

(3) The universal behavior of the system in the scaling domain fL - fLc -+ 0 is controlled by the euclidean (non-unitary) Quantum Field Theory (QFT). The collection of fields of this QFT include the local quantities like u(:c) or f(:C), where u(:c) is associated wi th the termination point of the open polymer and f(:C) correspond to "pinning" the polymer chain at the point :c (see Fig.I). The field u has the properties close to

325 that of the "order parameter" in the magnetic systems while f couples to T in (3) and so it is called the "energy density". The correlation functions of these fields, for example

(u(R)u(O» (€(R)€(O)}

= R- 4 tl.

6

G,,( ~)j

= R-4tl.'G.(~),

(4a)

(4b)

(which are illustrated by self-explanatory pictures in Fig.2) are expressed in terms of the universal exponents 11",11., ... (the anomalous dimensions ofthe fields u, f, ... ) and the universal scaling functions G", G., ... j here R. ~ T- V is the "correlation length" which defines the mass scale of this QFT. I want to show in this note that in the case of 2D space this QFT is integrable. I'll conjecture also the corresponding factorizable S-matrix and give some predictions about the scaling functions. In study of the self-avoiding polymers the following generalization of the model is known [1,2] to be very useful. Instead of a single polymer loop (as in (1» one considers the ensemble of non-intersecting and self-avoiding loops (Fig.3) with the partition function 00

Z(J.L,n) = LLnr e-I'L

(5)

r=l Gf'

where r is the number of connected polymer loops in the configuration Gr and L is the totallengthj n is the parameter ("loop fugacity"). The sum (5) is convergent at J.L > J.Lc, where J.Lc is some (non-universal) number, and in this domain it defines the so called "dilute polymer" phase of the system. In the limit n -+ 0 one recovers (1),

d

Z(J.L) = dn Z(J.L,n) In=o .

(6)

For n =po6itive integer the partition function (5) is related to the high-temperature phase of the n-vector model [1,2], i.e. the statistical syste describing the ensemble of unit vectors S., E sn-l ,the vector S., being associated with the site z of, say, honeycomb lattice. The probability distribution is

P[S]

= Z(K,ntl II (1 + KS"S1I)'

(7)

(.,.11)

where the product is taken over all nearest neighbors on this lattice. This system evidently exhibits the O( n) symmetry. The high-temperature expansion of the partition function Z(K,n) can be written in the form (5) with K ~ e-I'. Eq.5 can be considered as the definition of the analytic If -2 < n transition at [2]

< 2, the model (5) is known to exhibit the second order phase (8)

and the long-range fluctuations at the critical point K = Kc are described by the Conformal Field Theory [3]. Based on the coulomb-gas realization of the universality class of (5) proposed by Nienhuis [2], Dotsenko and Fateev [4] (see also [5]) have found the corresponding Virasoro central charge 6 c=I---pep + 1)

(9)

326 where the parameter p is related to n as n

7r

= 2cos(-)j p

1

< p < 00,

(10)

and identified some of the local operators. In particular, the "energy density" the microscopic quantity f"

= LS"S~,

(11)

y

(the sum is taken over the nearest neighbors of z), and admits the above interpretation in terms of "pinning" the polymer chain in the point z, has the conformal dimension 2 ~

=1---

(12)

• p+l It coincides with the degenerate primary field 41(1,3)[3] i.e. it satisfies the null-vector equation 2 1 3 ] [L_ 3 - ~ + 1 L-IL_2 + ~(~ + l)L_l f(Z) = 0 (13) Therefoce, the scaling behaviour of the model (5) away from the critical point can be described by the Quantum Field Theory 1

A = Ac + T

J

41(1,3)( Z )d 2 z

(14)

where Ae is the action of eFT with the central charge (4) and T

=

(Ke - K) ( ) Ke '" I' - I'e

(15)

In what follows we analyze the QFT (14) in order to describe the off-critical scaling behavior of model (5). There is every reason to believe that at T > 0 the Field Theory (14) develops the finite correlation length Re = T-I', l/ = ~ tt, and so it describes the interaction of the massive particles with the mass m = R;1 2. Hence, the QFT (14) with T > 0 is completely characterized by the S-matrix. On the other hand the QFT (14) is known to possess a number of nontrivial local integrals of motion [8], i.e. it is integrable 3. Therefore the corresponding S-matrix is "purely elastic" and factorizable in terms of two-particle scattering amplitudes [9].

HI -

In order to obtain this S-matrix one first needs to know the particle content of this scattering theory. The form (5) of the partition function seems to be rather suggestive. It is natural to think of loops entering (5) as the trajectories of certain 'Of course, this QFT is not unitary unless p is an integer greater then 2 [6]. 2The QFT (14) in the domain T < 0 (where it corresponds to the so called "dense phase" of the polymer system) is expected to be massless as it is related to the Renormali.ation Group trajectory going down to the "dense polymer" fixed point c = 1 - P(P~,)" Although the QFT (14) is integrable at T < 0 as well as at T > 0, the following analysis does not directly apply to this (perhaps the most interesting) domain. I believe this "crossover domain" could be treated by the "massless S-matrix" technique proposed recently in [7]. 3The integrals of motion of [8] does not give a proof, but rather give a strong evidence of the integrability of (14). Anyway, the integrals of motion constructed in [8] are enough to prove the factorisability of the S-matrix.

327 particles. The following two properties of the particles are evident from this picture. First, factor n r in (5) suggests that there are n sorts of particles A. = A 1 ,A2 , ••• ,A.. of the same mass m which form the vector multiplet of O(n). Evidently, at this d). The two-particle S-matrix associated with the process

where the 6's indicate rapidities Pa

= msh(6a)

(16)

of the particles, have the following general O( n) covariant from

sl:t(6)

= So(6)61:61: + SI(6)61:6!; + S2(6)6i ,i. 6M•

(17)

(6 = 61 - ( 2 ) with some scalar functions So, SI, S2. Second, as the loops entering the ensemble in (5) are assumed to be "self-avoiding", the particles A. are "impenetrable" each for another. The property would manifest in the S-matrix (17) if one assumes

So(6) = O.

(18)

The two particle S-matrix (17) have to satisfy the Yang-Baxter (or "factorization") equations (see e.g.[9]). Under the assumption (18) these equations reduce to the single functional equation

= SI(6)S2(6

+ 6')Sl(6').

(19)

The general solution of (19) satisfying the crossing symmetry relation [9)

(20) is

i7l" - 6 SI(6) = ish(--)R(6); P

S2(6)

= ish(~)R(6),

(21)

P

where p is given exactly by the same equation (10) and R( 6) is an arbitrary function satisfying (22) R(6) = R(i7l" - 6). The S-matrix satisfies also the unitarity condition

4

4The non-unitarity of the field theory does not generally mean that the equation sl s = I fails; rather sl in this equation has to be defined by specific conjugation which generally is not Hermitian. See the discussion of the point in Ref. [10].

328 (23) The second of these equations is automatically satisfied by (21) whereas the first one leads to the further limitation on R( 9) (24) The "minimal" solution to (22) and (24) is R(9)

1

= sinll'( 1 P

00

r(~ - i!,)r(1 +~

-

r(1 -.,!...) '''P .,!... ) r(1 + .,!... ) 11rp t1l'p

i!,)reA.;1

+ r,;,)r(1 + ll!f + i!,)

II r(~P + "!"')r(1 + ~P + "!"')r(aI!::..l"!"')r(1 + aI!:1_ .,!...) "=1 '''I' '''P P '''I' P'''I'

(25)

We propose (17,18,21,25) as the exact S-matrix ofthe Field Theory associated with the scaling limit of (5). Let us stress that this conjecture concerns only the domain -2 < n < 2. For n > 2 the S-matrix of [11] seems to be more appropriate to describe the continious limit of (7). Note that at n = 2 the above S-matrix coincides with that of [11]. Another nontrivial check can be made for n = 1; at this value of n the model (7) reduces to the Ising Model (on the hexagonal lattice). One can verify thatatn=l (26)

In fact, writing (25) we have used this particular case to fix the overall sign of the S-matrix. As was mentioned above, the n -+ 0 limit of (5) is related to the self avoiding polymer problem. In this case the S-matrix (20) takes the form

81 (9)

= -ch 29 R(9);

82(9) =

-ish~R(9)

(26)

with (27) As in this case the "particle trajectories" are interpreted as the polymers themselves the amplitudes 81 and 8 2 are associated with the two possible ways of the interactions of the polymers shown in Fig.4. The S-matrix has no direct interpretation in terms of standard observables in statistical mechanics. However, the S-matrix concentrated all the physical information about the field theory and all other characteristics can be in principle deduced

329 from it. In particular, the "formfactor bootstrap" program [12] gives a way to reconstruct the correlation functions (see[13)). Here we present only the main large distance asymptotics of the correlation function

= (0(R)0(0))

G(R) for the "self-avoiding polymer" case n

= O.

(28)

Here

(29) is the trace component of the stress-energy tensor, which is proportional to the energy density (11). In writing (28) we have assumed the following normalization ofthe field

E(:C ) (£(:c )£(0))

-+1 :c I-t

as 1:c

1-+ 0

(30)

The correlation function (23) is related to the distribution of monomers constituting a long closed polymer ring as it is seen from any given point on this ring. Our result is

where m

= const.r1

(32)

with yet unknown constant factor, and

(33)

(010(0) 1A.(B1 )A.i(B2 ))'n = o'.iF(B 1 specified to the case n = O.

( 2)

(34)

330 REFERENCES [1] P. G. De Gennes, Phys. Lett. 38A (1972) 399. [2] B. Nienhuis, J. Stat. Phys. 34 (1984) 731; see Ref. 3. [3] "Conformal Invariance and Applications to Statistical Mechanics", C.Itzykson, H. Saleur, J.B. Zuber eds., World Scientific 1988. [4] V. Dotsenko, V. Fateev, Nucl. Phys. B240 [FS 12] (1984) 312. [5] P. di Francesco, H. Saleur, J. B. Zuber, J. Stat. Phys. 49 (1987) 57. [6] D. Friedan, Z. Qiu, S. Shenker, Phys. Rev. Letters 43 (1984) 1556. [7] Ai. B. Zamolodchikov, "From Tricritical Ising to Critical Ising by Thermodynamic Bethe Ansatz", Preprint ENS-LPS-327, 1991. [8] A. B. Zamolodchikov, Advanced Studies in Pure Mathematics 19 (1989) 1. [9] A. B. Zamolodchikov and Ai. B. Zamolodchikov, Ann. Phys. (N.Y.) 120 (1979) 253. [10] J. Cardy, G. Mussardo, Phys. Lett. B225 (1989) 243. [11] A. B. Zamolodchikov and Ai. B. Zamolodchikov, Nuci. Phys. B133 (1978) 525. [12] F. A. Smirnov, J. Phys. A17 (1984) L873; TMF, 67 (1986) 40; TMF, 71 (1987) 341; A. N. Kirillov, F. A. Smirnov, Phys. Lett B198 (1987) 506; Int. J. Mod. Phys. A3 (1988) 731. [13] Ai. B. Zamolodchikov, "Two Point Correlation Function in Scaling Lee-Yang Model", SIS SA Preprint, Trieste 1990.

331

o

=

?I'.,clauical

= fJ"t/J -

e'.sinh(gt/J),

fJ..t/J =

?I'.p,clmical

= fJ"t/>

e'. cosh(gt/J ).

(16)

Integrability conditions for these equations follow from alternately differentiating with respect to u and T. Consistency requires that the classical t/> and t/J fields must obey the Liouville and free wave equations. 4m 2

(fJ.. 2- fJ,,2) t/> + -- e2g• = 0, 9

(17)

For the classical Liouville field theory, the Ba.cklund equations may be integrated to obtain all solutions for the interacting field t/> in terms of the free field t/J. Hence, all interacting field functionals g[t/>, ?I'.) may be expressed in terms of free field functionals, G[t/J, ?I'.p), and the classical theory is completely solved. However, there remains a long way to go to arrive at a quantum Liouville theory, even with a complete set of classical solutions. There are at least three major routes one can take to arrive at a quantized Liouville theory. These involve either operator methods [7), Schrodinger functional techniques [8), or path integrals [9). Among these three major rolites, there are also several minor variations, and while all these approaches may indeed be equivalent in principle, they are not necessarily equivalent in practice. For example, one operator approach to quantize the theory is to convert· the classical relations between t/> and t/J into well-defined operator expressions in such a way that the locality and conformal transformation properties expected of the expressions

337 do indeed hold. This is a laborious procedure, but it has been carried out for many of the classical relations between Liouville and free fields [10]. Unfortunately, even when valid operator relations have been obtained, there still remains the difficult task of using those operator results to evaluate correlation functions. Here we shall follow the canonical functional methods pedagogically discussed in the previous section for the linear potential. We will use the generating functional F[4>, t/J) within the Schrodinger equal-time functional formalism to construct energy eigenfunctionals if1E[4>] for the quantum Liouville theory from well-known free field energy eigenfunctionals I) E[t/J]. We then approach the task of evaluating correlation functions in a fashion completely analogous to the evaluation of the propagator for the linear potential. As in that simple case, the end result for the Liouville theory, at least in principle, is to reduce the problem to the evaluation of free field functional integrals. Indeed, in the weak-coupling limit, we immediately obtain a series of Gaussian functional integrals which evaluates to reproduce results obtained using operator methods [ll). Again, as in the case of the propagator for the linear potential, this functional approach to correlation functions should be compared to that using path integrals (9). Unfortunately, here we will be forced to leave a detailed comparison with path integral results as an exercise for the interested student. Suffice it to say that the functional methods seem to be easier to implement than operator methods, and they may provide a useful bridge between operator methods and path integral techniques. Let us return to the Backlund transformations, Eqn.(16). These first-order functional derivative relations are most conveniently written for the quantum Liouville theory as (18) D:l:(u) eiF = 0, where

D:l: () U

=

.( 8 ± 8) (8 .1.( ) 84>(u) 8t/J(u) - tT'I' U

-I

:r: T

8~.J.(~)) + 2m eg.(tT):I:g~(tT) • .'1' 9 v

(19)

Considering two such functional derivatives, in the form D:l: (uI)D:l: (u2)e iF as UI -+ U2, leads to the conclusion that the Liouville and free field Hamiltonians have equivalent effects on eiF , just as in the case of the linear potential, Eqn.(6). Consequently, the Liouville energy eigenfunctionals are functional transforms of the free field eigenfunctionals.3 (20) As in the previous quantum mechanics example, the classical form for the generating functional serves to provide an exact transformation between interacting and free theories. However, there is one important difference between the simple quantum mechanics example and the Liouville theory. The parameters in Fare renorrnalized, as explained below. In any case, correlation functions are now given in terms of functional integrals. For example, the expectation of an exponentiated Liouville field is

3N.B. The field integrations in this result, and in the expectation of exp(ag4», are over all field configurations at fixed time. That is, f d4> and f d.p here, and elsewhere in this paper, are not path integrals, but rather Schriidinger functional integrals.

338

=N(EdON(E2)

Jdl/l Jdl/l2 W'E.[I/Il] O[l/IhI/l2] 1

WE. [1/12]

(21)

where the effective operator on the space of 1/1 functionals is

(22) In principle, this should be equivalent to the operator results of Braaten et a1. [10], but in practice, we believe that this functional form for the expectation value can sometimes lead to more immediate results. For example, perturbation theory in the non-zero mode effects for expectations between low energy states is immediately developed from the functional expression. The traditional separation of the fields into zero (q and Q) and non-zero (~ and ~) modes at fixed T is given by [10]

= q + ~(u),

(23)

= Q + ~(u),

(24)

~(U) I/I(u)

However, one should keep in mind that the expansion coefficients in the functional formalism used here are not operator valued. For the free field, we may explicitly display the time dependence of the modes to obtain right-movers . (T,U ) = ~ i" 1 B n e -in(-T-") , ( 1/1* L..J25) y4?r n;o!O n and left-movers

. .1. 'I-'~ ( T,U )

i

=~

~ 1 Ane-m . (T+" ) . -

(26)

y4?r n#O n

For ~, the time-dependence is not so simple. Rather, it is convenient to combine the left- and right-moving modes for fixed time using projection operators. We first define these projections for the free field.

~(U)

= ~ I/In ein" = ~+(u) + ~_(u),

~±(U)

= P± ~(u) = ~ I/I±n e±incr,

(27)

n>O

n#O

(28)

We then note that this separation of the modes is a well-defined procedure even for the interacting field.

~(U)

= ~ ~n ein" = ~+(U) + ~_(U),

~±(U)

= P± ~(U) = ~ ~±.. e±i..".

..#0

(29)

.. >0

Using this separation into zero and non-zero modes, we may now parallel the operator approach in Braaten et a1. [11] and perform a perturbative analysis of the effects of the non-zero modes for low energy expectation values of exponentiated Liouville fields. The non-zero mode free field vacuum functional is (30)

339 where 18,,1 t/J(q) == 18,,1 tb(q) == En>o Inl t/J±n e±in". From this we construct low energy free field wave functionals, IJ!k(t/J) == IJ!k(Q)lJ!vacuum(~)' with E g2P /47r and IJ!k(Q) == Nk exp(i gkQ) , where it is understood that 9 is small and k is of order unity. The functional transform (20) then yields low energy Liouville wave functionals.

=

~k[4»

== Nk

. J'dt/J JdQ exp(lgkQ)

• e''F[o/>",] • IJ!vacuum[t/J).

(31)

We now split-off the non-zero mode contributions by writing

(32) where the interaction between zero and non-zero modes is contained in

Fint == - ; e9 (q+Q) We shift ~ to obtain ~k[4»

-+

~

l"

dq (e9 (4>+,j,) -

1) - ;

l"

dq (e9(~-,j,) -

1) .

(33)

+ 4>+ - 4>-, complete the square, and eliminate the I dq 4> 8,,~ term

1 f21r. .) == Nk exp ( -210 dq 4>(q) 18,,1 4>(q)

x

e9(q-Q)

JdQ (i gkQ -""92 47rmi

exp

e99 sinh(gQ)

Jd~ exp (iFint[4>, t/J + 4>+ - 4>-)- ~ l" dq ~(q) 18,,1 tb(q)) .

)

(34)

After the shift, the interaction becomes

- ; e9 (q+Q)

l"

dq (e9(2~++,j,) -

1) - ;

e9 (q-Q)

l"

du (eg(2~--,j,) -

1) .

(35)

Now expand exp(iFint) in powers of Fint to obtain a series of Gaussian integrals

Jd~ ==

exp (iFint[4>, t/J + 4>+

- 4>+]- ~

ESJ

l"

dq

~(u) 18,,1 ~(q))

dtb e-H:- d, t/J + ~+ -

~-lr

(36)

Thus ~k[4» reduces to a series of ~ Gaussian integrals with coefficients which depend on the zero modes, q and Q, Subsequent integration over Q yields a q and 4> dependent series.

(37) n=O

In this series, ~(n) is obtained by keeping only the term involving (Find n in the previous expansion. The lowest order non-trivial results for the expectation of e",go/> are obtained by keeping terms up to and including (Fint}3 in this series. The evaluation of the resulting Gaussian integrals is straightforward. To 0(g6) we obtain

x

340

=N q + (a(2 -

NI:, Zk,l:,(a,m,g) ea2g2c, (N)/21r

(1 +8 (a- (k~ + k~»)

(::r

(2

a)(~ + ~a + a 2)+ 2a(2 - a)(k~ + kn - (k~ _ kn2) (::) 3 (3 + 0(g8») , (38)

where the zero-mode matrix element is defined by

= 2eg3r(a) (I:,- kd

(L)a Ir (a +

i(kl 2

+ k2 ») 121r

(a +

») 12

i(kl - k 2 2' . (39) and where Nk is a normalization factor required by the zero-mode wave function. The exact form of Nk is not important for the present discussion, but the possibility of such normalization factors is important, as we shall see. The usual ultraviolet divergence in the vacuum expectation value of an "un-normal-ordered" exponential is present here and is given by Z

10,10.

(

a,m,g

)

1r

211"m

(40) where we have imposed a mode cutoff, N , to write (l(N)

= L:~=l ~ .

Actually, to obtain (38), it is necessary to remove a similar divergence due to the exponentials in the generating functional. To this end we have replaced egJ, appearing in (33) by e- g• C,(N)/l1r egJ,. We may think of this as a renormalization of the "mass" appearing in Fint . This is not an unexpected renormalization. However, it is not the whole story for mass renormalization, as is evident from the 0(g4) terms in (38). The a(2 term on the RHS is a problem (the quantum version of the equations of motion would fail) and must be eliminated. This may be achieved by making an additional finite renormalization of the mass in F , which acquires an a dependence and cancels the a(2 term through its appearance in the zero-mode expression (39). The structure of certain terms in the perturbation series suggests that this finite mass renormalization is of the form :. sin(g2/2), although we have only checked this fully to 0(g4). Nonetheless, this form is supported to all orders by comparison with the operator results in Braaten et al.[10]. In summary, the mass appearing in F is renormalized according to

(41) Based on exact results within the operator formalism and for the effective potential of the Liouville theory, we also anticipate another quantum correction to the exponentials in F, in the form of a finite renormalization of g. 9

-+-g-

1+~'

(42)

2..

However, we have not checked this correction within the functional formalism. To do so to lowest order would require perturbative results beyond 0(g6). Finally, other anomalous 0(g4) non-zero mode contributions in (38) can be eliminated through changes in normalizations. The terms (kl + kn (~) 2 (2 may be removed by changing the normalizations of the states 4i k, [~l and 4i1:, [~l . At this time, we have no firm conjectures about the form for such normalization changes in higher orders.

341

After these renormalizations, the results in (38) agree to 0(g6) with those of Braaten et al. [11) obtained through the use of operator techniques. Perhaps it will be possible to extend these perturbative results to obtain correct closed-form expressions to all orders in 9 within the functional framework. Or, perhaps it will be possible to numerically evaluate the functional expressions to obtain non-perturbative results. We leave these as open problems for the interested reader. LIOUVILLE THEORY ON CURVED SURFACES Perhaps functional methods are also useful when the (r,u) = (ZO,Zl) manifold is not intrinsically flat. The classical relations between