Topology and physics 9789813278493, 9789813278509

Table of contents :
Preface Early examples of topological concepts in physics......Page 6
Contents......Page 8
1. Introduction......Page 10
2. Topology and Physics of Algebraic Surfaces......Page 13
3. Valley of Stability......Page 17
4. Intersection Form......Page 19
5. Other Surfaces......Page 21
6. Conclusions......Page 23
References......Page 24
Chapter 2 Developments in topological gravity......Page 26
1. Introduction......Page 27
2. Weil{Petersson Volumes and Two-Dimensional Topological Gravity......Page 28
3. Open Topological Gravity......Page 39
4. Interpretation via Matrix Models......Page 73
References......Page 86
1. Introduction......Page 90
2. Braids......Page 94
3. Braiding Operators and Universal Quantum Gates......Page 96
4. Fermions, Majorana Fermions and Braiding Fermion Algebra.......Page 98
6. Majorana Fermions Generate Universal Braiding Gates......Page 106
7. The Bell-Basis Matrix and Majorana Fermions......Page 107
References......Page 115
1. Fermat's Principle......Page 118
2. Diophantine Geometry and Gauge Theory......Page 119
3. Principal Bundles and Number Theory: Weil's Constructions......Page 121
4. Arithmetic Gauge Fields......Page 122
Lemma 4.1.......Page 123
5. Homotopy and Gauge Fields......Page 126
6. The Local to Global Problem, Reciprocity Laws, and Euler Lagrange Equations......Page 129
7. The Tate Shafarevich Group and Abelian Gauge Fields......Page 132
8. Non-Abelian Gauge Fields and Diophantine Geometry......Page 134
Conjecture 8.1 (Ref.6).......Page 136
and Chern Simons Actions......Page 138
References......Page 142
5.1. General Background......Page 144
5.2. The Big Bang and Gravitational Collapse......Page 147
5.3. Collapse to a Black Hole......Page 149
5.4. The Problem of Generic Gravitational Collapse......Page 153
5.5. Causal Structure of Space-Times......Page 156
5.6. Geodetic Maximality......Page 163
5.7. Collapse to Singularity......Page 168
Bibliography......Page 178
1. Introduction......Page 182
2. 2D Non-Abelian Objects......Page 183
3. Three-Dimensional Topological Physics Beyond Anyons......Page 186
4. Four-Dimensional Topological Physics......Page 187
References......Page 189
1. Four Revolutions in Physics......Page 190
2. It From Qubit, Not Bit A Second Quantum Revolution......Page 198
matter = information......Page 199
Eight wonders:......Page 200
3. A String-Net Liquid of Qubits and A Uni cation of Gauge Interactions and Fermi Statistics......Page 201
References......Page 211
Appendix | SO4 symmetry in a Hubbard model......Page 224

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11217_9789813278493_tp.indd 1

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Yang, Chen Ning, 1922– editor. | Ge, M. L. (Mo-Lin), editor. | He, Yang-Hui, 1975– Title: Topology and physics / editors, Chen Ning Yang (Institute for Advanced Study (IAS), Tsinghua University, China), Mo-Lin Ge (Chern Institute of Mathematics, China), Yang-Hui He (City University of London, UK). Description: New Jersey : World Scientific, 2018. | Includes bibliographical references. Identifiers: LCCN 2018045467| ISBN 9789813278493 (hardcover : alk. paper) | ISBN 9789813278509 (pbk : alk. paper) Subjects: LCSH: Topology. | Physics. Classification: LCC QA611 .T65725 2018 | DDC 514--dc23 LC record available at https://lccn.loc.gov/2018045467

editor.

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

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Preface Early examples of topological concepts in physics∗

C. N. Yang Institute for Advanced Study, Tsinghua University, P. R. China

In the mid-1940s, S. S. Chern published an “intrinsic proof” of a generalization of the Gauss–Bonnet Theorem to 4-dimensions. The paper led to the Chern Class and Chern Numbers, to the new exciting field of global differential geometry, and to new important topological concepts in other areas of mathematics. Andrei Weil was one mathematician who was greatly impressed. He wrote an enthusiastic review of the paper which became very influential. A few years later, in 1946–1949, several totally unexpected new elementary particles were discovered by experimental physicists. They were of different kinds, with very different quantum numbers, and quickly became physicist’s center of attention. One day in 1948, I was present at a lunch in which Weil told Fermi his speculation that these new particles might be related to some topological classification ideas in geometry. Neither Fermi, nor I, nor others at that lunch, understood what Weil had meant that day by his speculation across the boundary of math–physics. Many years later, in the mid-1970s, after I learned from Jim Simon elements of fiber bundle geometry and related concepts, I realized Weil maybe speculating that day about possible relationships between the new particles plus their new quantum number with topological concepts such as the Chern Numbers. For details please see Ref. 1. **************************** In an article published in 2012,2 I discussed in some detail the following early entry of topology into physics: • The Aharonov–Bohm experiment proposed theoretically in 1959, and verified experimentally by Tonomura in 1983–1986. • In the early 1950s physicists used the new computers to calculate the vibrational frequency distribution of crystals, and were surprised to find unexplained ups ∗ This

chapter also appeared in Modern Physics Letters A, Vol. 33, No. 22 (2018) 1830009. DOI: 10.1142/S0217732318300094.

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and downs in the spectra. Were they real? Or just quirks of the computation? The puzzle was resolved in a 1953 paper of Van Hove which introduced topology, viz. Morse Theory, into physics. **************************** That topological concepts are important in physics is now well known, especially in phenomena/problems involving Abelian or non-Abelian phases. Here is an example which shows that in one problem in classical Maxwell theory, topology already plays an essential role: Consider An EM field interacting with both an electric charge e and a magnetic charge g, a problem which had been considered by Dirac in 1931.3 The electromagnetic potential (i.e. the connection), when analytically continued, forms a complicated nontrivial manifold. The action integral a is then definable only modulo 4πeg.4 If one tries to quantize this theory, `a la Feynman’s path integral, one would be dealing with the quantity exp(ia /h) , which is meaningful only if 2eg/h = an integer . This condition, first given by Dirac, is thus a consequence of the topology of classical Maxwell theory. References 1. 2. 3. 4.

C. N. Yang, Phys. Today 65, 33 (2012). C. N. Yang, Int. J. Mod. Phys. A 27, 1230035 (2012). P. A. M. Dirac, Proc. R. Soc. London A 133, 60 (1931). T. T. Wu and C. N. Yang, Phys. Rev. D 14, 437 (1976).

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Contents

Preface — Early examples of topological concepts in physics C. N. Yang

v

1.

Complex geometry of nuclei and atoms M. F. Atiyah and N. S. Manton

1

2.

Developments in topological gravity Robbert Dijkgraaf and Edward Witten

17

3.

Majorana Fermions and representations of the braid group Louis H. Kauffman

81

4.

Arithmetic gauge theory: A brief introduction Minhyong Kim

109

5.

Singularity theorems Roger Penrose

135

6.

Beyond anyons Zhenghan Wang

173

7.

Four revolutions in physics and the second quantum revolution — A unification of force and matter by quantum information Xiao-Gang Wen

181

Topological insulators from the perspective of first-principles calculations Haijun Zhang and Shou-Cheng Zhang

205

8.

Appendix — SO4 symmetry in a Hubbard model C. N. Yang and Shou-Cheng Zhang

215

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Chapter 1 Complex geometry of nuclei and atoms∗

M. F. Atiyah School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK [email protected] N. S. Manton Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK [email protected]

We propose a new geometrical model of matter, in which neutral atoms are modelled by compact, complex algebraic surfaces. Proton and neutron numbers are determined by a surface’s Chern numbers. Equivalently, they are determined by combinations of the Hodge numbers, or the Betti numbers. Geometrical constraints on algebraic surfaces allow just a finite range of neutron numbers for a given proton number. This range encompasses the known isotopes. Keywords: Atoms; nuclei; algebraic surfaces; 4-manifolds. PACS numbers: 02.40.Tt, 02.40.Re, 21.60.−n

1. Introduction It is an attractive idea to interpret matter geometrically, and to identify conserved attributes of matter with topological properties of the geometry. Kelvin made the pioneering suggestion to model atoms as knotted vortices in an ideal fluid.1 Each atom type would correspond to a distinct knot, and the conservation of atoms in physical and chemical processes (as understood in the 19th century) would follow from the inability of knots to change their topology. Kelvin’s model has not survived because atoms are now known to be structured and divisible, with a nucleus formed

∗ This

chapter also appeared in International Journal of Modern Physics A, Vol. 33, No. 24 (2018) 1830022. DOI: 10.1142/S0217751X18300223.

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of protons and neutrons bound together, surrounded by electrons. At high energies, these constituents can be separated. It requires of order 1 eV to remove an electron from an atom, but a few MeV to remove a proton or neutron from a nucleus. Atomic and nuclear physics has progressed, mainly by treating protons, neutrons and electrons as point particles, interacting through electromagnetic and strong nuclear forces.2 Quantum mechanics is an essential ingredient, and leads to a discrete spectrum of energy levels, both for the electrons and nuclear particles. The nucleons (protons and neutrons) are themselves built from three pointlike quarks, but little understanding of nuclear structure and binding has so far emerged from quantum chromodynamics (QCD), the theory of quarks. These point particle models are conceptually not very satisfactory, because a point is clearly an unphysical idealisation, a singularity of matter and charge density. An infinite charge density causes difficulties both in classical electrodynamics3 and in quantum field theories of the electron. Smoother structures carrying the discrete information of proton, neutron and electron number would be preferable. In this paper, we propose a geometrical model of neutral atoms where both the proton number P and neutron number N are topological and none of the constituent particles are pointlike. In a neutral atom the electron number is also P , because the electron’s electric charge is exactly the opposite of the proton’s charge. For given P , atoms (or their nuclei) with different N are known as different isotopes. A more recent idea than Kelvin’s is that of Skyrme, who proposed a nonlinear field theory of bosonic pion fields in 3 + 1 dimensions with a single topological invariant, which Skyrme identified with baryon number.4,5 Baryon number (also called atomic mass number) is the sum of the proton and neutron numbers, B = P + N . Skyrme’s baryons are solitons in the field theory, so they are smooth, topologically stable field configurations. Skyrme’s model was designed to model atomic nuclei, but electrons can be added to produce a model of a complete atom. Protons and neutrons can be distinguished in the Skyrme model, but only after the internal rotational degrees of freedom are quantised.6
This leads to a quantised 1 “isospin,” with the proton
having isospin up I3 = 2 and the neutron having 1 isospin down I3 = − 2 , where I3 the third component of isospin. The model is consistent with the well-known Gell-Mann–Nishijima relation7 Q=

1 B + I3 , 2

(1.1)

where Q is the electric charge of a nucleus (in units of the proton charge) and B is the baryon number. Q is integral, because I3 is integer-valued (half-integervalued) when B is even (odd). Q equals the proton number P of the nucleus and also the electron number of a neutral atom. The neutron number is N = 21 B − I3 . The Skyrme model has had considerable success providing models for nuclei.8–11 Despite the pion fields being bosonic, the quantised Skyrmions have half-integer spin if B is odd.12 But a feature of the model is that proton number and neutron number are not separately topological, and electrons have to be added on.

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The Skyrme model has a relation to 4-dimensional fields that provides some motivation for the ideas discussed in this paper. A Skyrmion can be well approximated by a projection of a 4-dimensional Yang–Mills field. More precisely, one can take an SU(2) Yang–Mills instanton and calculate its holonomy along all lines in the (Euclidean) time direction.13 The result is a Skyrme field in 3-dimensional space, whose baryon number B equals the instanton number. So a quasi-geometrical structure in 4-dimensional space (a Yang–Mills instanton in flat R4 ) can be closely related to nuclear structure, but still there is just one topological charge. A next step, first described in Ref. 14, was to propose an identification of smooth, curved 4-manifolds with the fundamental particles in atoms — the proton, neutron and electron. Suitable examples of manifolds were suggested. These manifolds were not all compact, and the particles they modelled were not all electrically neutral. One of the more compelling examples was Taub-NUT space as a model for the electron. By studying the Dirac operator on the Taub-NUT background, it was shown how the spin of the electron can arise in this context.15 There has also been an investigation of multi-electron systems modelled by multi-TaubNUT space.16,17 However, there are some technical difficulties with the models of the proton and neutron, and no way has yet been found to geometrically combine protons and neutrons into more complicated nuclei surrounded by electrons. Nor is it clear in this context what exactly should be the topological invariants representing proton and neutron number. A variant of these ideas is a model for the simplest atom, the neutral hydrogen atom, with one proton and one electron. This appears to be well modelled by CP2 , the complex projective plane.a The fundamental topological property of CP2 is that it has a generating 2-cycle with self-intersection 1. The second Betti number is 2 − b2 = 1, which splits into b+ 2 = 1 and b2 = 0. A complex line in CP represents this cycle, and in the projective plane, two lines always intersect in one point. A copy of this cycle together with its normal neighbourhood can be interpreted as the proton part of the atom, whereas the neighbourhood of a point dual to this is interpreted as the electron. The neighbourhood of a point is just a 4-ball, with a 3-sphere boundary, but this is the same as in the Taub-NUT model of the electron, which is topologically just R4 . The 3-sphere is a twisted circle bundle over a 2-sphere (the Hopf fibration) and this is sufficient to account for the electron charge. In this paper, we have a novel proposal for the proton and neutron numbers. The 4-manifolds we consider are compact, to model neutral atoms. Our previous models always required charged particles to be noncompact so that the electric flux could escape to infinity, and this is an idea we will retain. We also restrict our manifolds to be complex algebraic surfaces, and their Chern numbers will be related to the proton and neutron numbers. There are more than enough examples to model all currently known isotopes of atoms. We will retain CP2 as the model for the hydrogen atom. a CP2

had a different interpretation in Ref. 14.

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Fig. 1. The Hodge diamond for a general complex surface (left) and its entries in terms of Betti numbers for an algebraic surface (right).

2. Topology and Physics of Algebraic Surfaces Complex surfaces18 provide a rich supply of compact 4-manifolds. They are principally classified by two integer topological invariants, denoted c21 and c2 . For a surface X, c1 and c2 are the Chern classes of the complex tangent bundle. c2 is an integer because X has real dimension 4, whereas the (dual of the) canonical class c1 is a particular 2-cycle in the second homology group, H2 (X). c21 is the intersection number of c1 with itself, and hence another integer. There are several other topological invariants of a surface X, but many are related to c21 and c2 . Among the most fundamental are the Hodge numbers. These are the dimensions of the Dolbeault cohomology groups of holomorphic forms. In two complex dimensions the Hodge numbers are denoted hi,j with 0 ≤ i, j ≤ 2. They are arranged in a Hodge diamond, as illustrated in Fig. 1. Serre duality, a generalisation of Poincar´e duality, requires this diamond to be unchanged under a 180◦ rotation. For a connected surface, h0,0 = h2,2 = 1. Complex algebraic surfaces are a fundamental subclass of complex surfaces.19,20 A complex algebraic surface can always be embedded in a complex projective space CPn , and thereby acquires a K¨ ahler metric from the ambient Fubini–Study metric on CPn . For any K¨ ahler manifold, the Hodge numbers have an additional symmetry, hi,j = hj,i . For a surface, this gives just one new relation, h0,1 = h1,0 . Not all complex surfaces are algebraic: some are still K¨ahler and satisfy this additional relation, but some are not K¨ ahler and do not satisfy it. Particularly interesting for us are the holomorphic Euler number χ, which is an alternating sum of the entries on the top right (or equivalently, bottom left) diagonal of the Hodge diamond, and the analogous quantity for the middle diagonal, which we denote θ. More precisely, χ = h0,0 − h0,1 + h0,2 , 1,0

θ = −h

1,1

+h

1,2

−h

(2.1) .

(2.2)

(Note the sign choice for θ.) The Euler number e and signature τ can be expressed in terms of these as e = 2χ + θ ,

(2.3)

τ = 2χ − θ .

(2.4)

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Fig. 2.

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Hodge diamonds for the projective plane CP2 (left) and for a K3 surface (right).

The first of these formulae reduces to the more familiar alternating sum of Betti numbers e = b0 − b1 + b2 − b3 + b4 , because each Betti number is the sum of the entries in the corresponding row of the Hodge diamond. The second formula is the less trivial Hodge index theorem. τ is more fundamentally defined by the splitting − of the second Betti number into positive and negative parts, b2 = b+ 2 + b2 . Over the reals the intersection form on the second homology group H2 (X) is nondegenerate − and can be diagonalised. b+ 2 is then the dimension of the positive subspace, and b2 − the dimension of the negative subspace. The signature is τ = b+ 2 − b2 . The Chern numbers are related to χ and θ through the formulae c21 = 2e + 3τ = 10χ − θ ,

c2 = e = 2χ + θ .

(2.5)

1 2 12 (c1

+ c2 ), which is always integral. Their sum gives the Noether formula χ = For an algebraic surface, there are just three independent Hodge numbers and − they are uniquely determined by the Betti numbers b1 , b+ 2 and b2 . The Hodge diamond must take the form shown on the right in Fig. 1, which gives the correct values for b1 , e and τ . Note that b1 must be even and b+ 2 must be odd. χ and θ are now given by χ=

1 (1 − b1 + b+ 2 ), 2

(2.6)

θ = 1 − b1 + b− 2 .

(2.7)

If X is simply connected, which accounts for many examples, then b1 = 0. Hodge diamonds for the projective plane CP2 and for a K3 surface, both of which are simply connected, are shown in Fig. 2. For the projective plane χ = 1 and θ = 1, so e = 3 and τ = 1, and for a K3 surface χ = 2 and θ = 20, so e = 24 and τ = −16. Our proposal is to model neutral atoms by complex algebraic surfaces and to interpret χ as proton number P , and θ as baryon number B. So neutron number is N = θ − χ. This proposal fits with CP2 having P = 1 and N = 0. We will see later that for each positive value of P there is an interesting, finite range of allowed N values. In terms of e and τ , P =

1 (e + τ ) , 4

B=

1 (e − τ ) , 2

N=

1 (e − 3τ ) . 4

(2.8)

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Note that for a general, real 4-manifold, these formulae for P and N might be fractional, and would need modification. It is also easy to verify that in terms of P and N , c21 = 9P − N ,

(2.9)

c2 = e = 3P + N ,

(2.10)

τ = P −N.

(2.11)

The simple relation of signature τ to the difference between proton and neutron numbers is striking. If we write N = P + Nexc , where Nexc denotes the excess of neutrons over protons (which is usually zero or positive, but can be negative), then τ = −Nexc . If an algebraic surface X is simply connected then b1 = 0, and in terms of P and N , b+ 2 = 2P − 1 ,

b− 2 = P + N − 1 = 2P − 1 + Nexc .

(2.12)

These formulae will be helpful when we consider intersection forms in more detail. The class of surfaces that we will use, as models of atoms, are those with c21 and c2 non-negative. Many of these are minimal surfaces of general type. Perhaps the most important results on the geometry of algebraic surfaces are certain inequalities that the Chern numbers of minimal surfaces of general type have to satisfy. The basic inequalities are that c21 and c2 are positive. Also, there is the Bogomolov– Miyaoka–Yau (BMY) inequality which requires c21 ≤ 3c2 , and finally there is the Noether inequality 5c21 − c2 + 36 ≥ 0. These inequalities can be converted into the following inequalities on P and N : P > 0,

0 ≤ N < 9P ,

N ≤ 7P + 6 .

(2.13)

All integer values of P and N satisfying these are allowed. The allowed region is shown in Fig. 3, and corresponds to the allowed region shown on page 229 of Ref. 18, or in the article, Ref. 21. There are also the elliptic surfaces (including the Enriques surface and K3 surface) where c21 = 0 and c2 is non-negative, and we shall include these among our models. Here, P ≥ 0 and N = 9P , so c2 = 12P and τ = −8P . CP2 is also allowed, even though it is rational and not of general type, because c21 and c2 are positive. In addition to CP2 , there are further surfaces on the BMY line c21 = 3c2 ,22 which have P > 1 and N = 0. Physicists usually denote an isotope by proton number and baryon number, where proton number P is determined by the chemical name, and baryon number is P + N . For example, the notation 56 Fe means the isotope of iron with P = 26 and N = 30. The currently recognised isotopes are shown in Fig. 4. The shape of the allowed region of algebraic surfaces qualitatively matches the region of recognised isotopes, and this is the main justification for our proposal.

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Fig. 3. Proton numbers P , and neutron numbers N , for atoms modelled as algebraic surfaces. The allowed region is limited by inequalities on the Chern numbers, as discussed in the text. Note the change of slope from 9 to 7 at the point P = 3, N = 27 on the boundary. The line N = P corresponds to surfaces with zero signature, i.e. τ = 0.

Fig. 4. Nuclear isotopes. The horizontal axis is proton number P (Z in physics notation) and the vertical axis is neutron number N . The shading (colouring online) indicates the lifetime of each isotope, with black denoting stability (infinite lifetime).

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For example, for P = 1, the geometric inequalities allow N to take values from 0 up to 9. This corresponds to a possible range of hydrogen isotopes from 1 H to 10 H. Physically, the well-known hydrogen isotopes are the proton, deuterium and tritium, that is, 1 H, 2 H and 3 H respectively, but nuclear physicists recognise isotopes of a quasi-stable nature (resonances) up to 7 H, with N = 6. The minimal models for the common isotopes, the proton alone, and deuterium, each bound to one electron, are CP2 and the complex quadric surface Q. The quadric is the product Q = CP1 × CP1 , with e = 4 and τ = 0. We shall say more about its intersection form below. For P = 2, N is geometrically allowed in the range 0 to 18. The corresponding algebraic surfaces should model helium isotopes from 2 He to 20 He. Isotopes from 3 He up to 10 He are physically recognised. All of these potentially form neutral atoms with two electrons. The helium isotope 2 He with no neutrons is not listed in some nuclear tables, but there does exist an unbound diproton resonance, and diprotons are sometimes emitted when heavier nuclei decay. The most common, stable helium isotope is 4 He, with two protons and two neutrons, but 3 He is also stable. 4 He nuclei are also called alpha-particles, and play a key role in nuclear processes and nuclear structure. It is important to have a good geometrical model of an alpha-particle, which ideally should match the cubically symmetric B = 4 Skyrmion that is a building block for many larger Skyrmions.9,11,23,24 3. Valley of Stability Running through the nuclear isotopes is the valley of stability.2 In Fig. 4, this is the irregular curved line of stable nuclei marked in black. On either side, the nuclei are unstable, with lifetimes of many years near the centre of the valley, reducing to microseconds further away. Sufficiently far from the centre are the nuclear drip lines, where a single additional proton or neutron has no binding at all, and falls off in a time of order 10−23 seconds. For small nuclei, for P up to about 20, the valley is centred on the line N = P . In the geometrical model, this line corresponds to surfaces with signature τ = 0. For larger P , nuclei in the valley have a neutron excess, Nexc , which increases slowly from just a few when P is near 20 to over 50 for the quasi-stable uranium isotopes with P = 92, and slightly more for the heaviest artificially produced nuclei with P approaching 120. In standard nuclear models, the main effect explaining the valley is the Pauli principle. Protons and neutrons have a sequence of rather similar 1-particle states of increasing energy, and just one particle can be in each state. For given baryon number, the lowest-energy state has equal proton and neutron numbers, filling the lowest available states. If one proton is replaced by one neutron, the proton state that is emptied has lower energy than the neutron state that is filled, so the total energy goes up. An important additional effect is a pairing energy that favours protons to pair up and neutrons to pair up. Most nuclei with P and N both odd are unstable as a result.

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For larger values of P , the single-particle proton energies tend to be higher than the single-particle neutron energies, because in addition to the attractive, strong nuclear forces which are roughly the same for protons and neutrons, there is the electrostatic Coulomb repulsion that acts between protons alone. This effect becomes important for nuclei with large P , and favours neutron-rich nuclei. It also explains the instability of all nuclei with P larger than 83. These nuclei simply split up into smaller nuclei, either by emitting an alpha-particle, or by fissioning into larger fragments. However, the lifetimes can be billions of years in some cases, which is why uranium, with P = 92, is found in nature in relatively large quantities. Note that if N = P , then the electric charge is half the baryon number, and according to formula (1.1), the third component of isospin is zero. By studying nuclear ground states and excited states, one can determine the complete isospin, and it is found to be minimal for stable nuclei. So nuclei with N = P have zero isospin. When the baryon number is odd, the most stable nuclei have N just one greater than P (if P is not too large), and the isospin is 21 . Within the Skyrme model, isospin arises from the quantisation of internal degrees of freedom, associated with an SO(3) symmetry acting on the pion fields. There is an energy contribution proportional to the squared isospin operator I2 , analogous to the spin energy proportional to J2 . In the absence of Coulomb effects, the energy is minimised by fixing the isospin to be zero or 12 . The Coulomb energy competes with isospin, and shifts the total energy minimum towards neutron-rich nuclei. These are the general trends of nuclear energies and lifetimes. However there is a lot more in the detail. Each isotope has its own character, depending on its proton and neutron numbers. This is most clear in the energy spectra of excited states, and the spins of the ground and excited states. Particularly interesting is the added stability of nuclei where either the proton or neutron number is magic. The smaller magic numbers are 2, 8, 20, 28, 50. It is rather surprising that protons and neutrons can be treated independently with regard to the magic properties. This appears to contradict the importance of isospin, in which protons and neutrons are treated as strongly influencing each other. Particularly stable nuclei are those that are doubly magic, like 4 He, 16 O, 40 Ca and 48 Ca. 40 Ca is the largest stable nucleus with N = P . 48 Ca is also stable, and occurs in small quantities in nature, but is exceptionally neutron-rich for a relatively small nucleus. The important issue for us here is to what extent our proposed geometrical model based on algebraic surfaces is compatible with these nuclear phenomena, not forgetting the electron structure in a neutral atom. There are some broad similarities. First there is the “geography” of surfaces we have discussed above, implying that the geometrical inequalities restrict the range of neutron numbers. Algebraic geometers also refer to “botany,” the careful construction and study of surfaces with particular topological invariants. The patterns are very complicated. Some surfaces are simple to construct, others less so, and their internal structure is very variable. This is analogous to the complications of the nuclear landscape, and

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the similar complications (better understood) of the electron orbitals and atomic shell structure. Rather remarkable is that the line of nuclear stability where N = P corresponds to the simple geometrical condition that the signature τ is zero. We have not yet tried to pinpoint an energy function on the space of surfaces, but clearly it would be easy to include a dominant contribution proportional to τ 2 , whose minimum would be in the desired place. Mathematicians have discovered that it is much easier to construct surfaces on this line, and on the neutron-rich side of it, where τ is negative, than on the proton-rich side. There are always minimal surfaces on the neutron-rich side which are simply connected, but not everywhere on the proton-rich side. The geometry of surfaces therefore distinguishes protons from neutrons rather clearly. This is attractive for the physical interpretation, as it can be regarded as a prediction of an asymmetry between the proton and neutron. In standard nuclear physics it is believed that in an ideal world with no electromagnetic effects, there would be an exact symmetry between the proton and neutron, but in reality they are not the same, partly because of Coulomb energy, but more fundamentally, because their constituent up (u) and down (d) quarks are not identical in their masses, making the proton (uud) less massive than the neutron (udd), despite its electric charge. The geometrical model would need an energy contribution that favours neutrons over protons for the larger nuclei and atoms. One possibility has been explored by LeBrun.25,26 This is the infimum, over complex surfaces with given topology, of the L2 norm of the scalar curvature. For surfaces with b1 even, including all surfaces that are simply connected, this infimum is simply a constant multiple of c21 . The scalar curvature can be zero for surfaces on the line c21 = 0, for example the K3 surface, which is the extreme of neutron-richness, with P = 2 and N = 18. It would be interesting to consider more carefully the energy landscape for an energy that combines τ 2 and a positive multiple of the L2 norm of scalar curvature. 4. Intersection Form A complex surface X is automatically oriented, so any pair of 2-cycles has an unambiguous intersection number.27 Given a basis αi of 2-cycles for the second homology group H2 (X), the matrix Ωij of intersection numbers is called the intersection form of X. Ωij ≡ Ω(αi , αj ) is the intersection number of basis cycles αi and αj , and the self-intersection number Ωii is the intersection number of αi with a generic smooth deformation of itself. Ω is a symmetric matrix of integers, and by Poincar´e duality it is unimodular (of determinant ±1). Over the reals, such a symmetric matrix is diagonalisable, and the diagonal entries are either +1 or −1. The numbers of each − of these are b+ 2 and b2 , respectively, and we have already given an interpretation of them for simply-connected algebraic surfaces X in terms of P and N in Eq. (2.12) above. However, diagonalisation over the reals does not make sense for cycles, because one can end up with fractional cycles in the new basis. One may only change the

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basis of cycles using an invertible matrix of integers, whose effect is to conjugate Ω by such a matrix. The classification of intersection forms is finer over the integers than the reals. − For almost all algebraic surfaces, Ω is indefinite. b+ 2 is always positive, and b2 is positive too, except for surfaces with b1 = 0 and B = θ = 1. So the only surfaces for which the intersection form Ω is definite are CP2 , and perhaps additionally the fake projective planes, for which we have not found a physical interpretation. For CP2 , with P = 1 and N = 0, the intersection form is the 1 × 1 matrix Ω = (1). Nondegenerate, indefinite forms over the integers have a rather simple classification. The basic dichotomy is between those that are odd and those that are even. An odd form is one for which at least one entry Ωii is odd, or more invariantly, Ω(α, α) is odd for some 2-cycle α. An odd form can always be diagonalised, with entries +1 and −1 on the diagonal. Even forms are more interesting. Here Ω(α, α) is even for any cycle α. The simplest example is   0 1 Ω= . (4.1) 1 0 This is the intersection form of the quadric Q, with the two CP1 factors as basis cycles, α1 and α2 . If α = xα1 + yα2 then Ω(α, α) = 2xy, so is always even. Over the reals this form can be diagonalised and has entries +1 and −1 (the eigenvalues). So it has zero signature. But the diagonalisation involves fractional matrices, and is not possible over the integers. The intersection form (4.1) is called the “hyperbolic plane.” A second ingredient in even intersection forms is the matrix −E8 . This is the negative of the Cartan matrix of the Lie algebra E8 (with diagonal entries −2). It is even and unimodular. By itself this form is negative definite, but when combined with hyperbolic plane components, the result is indefinite, as needed. The most general (indefinite) even intersection form for an algebraic surface can be brought to the block diagonal form   0 1 Ω=l ⊕ m(−E8 ) , (4.2) 1 0 with l > 0 and m ≥ 0. l must be odd, and the Betti numbers are b+ 2 = l and b− = l + 8m. The signature is τ = −8m. 2 For most surfaces, the signature is not a multiple of 8, so the intersection form is odd. If the signature is a multiple of 8, it may be even. For given Betti numbers, there could be two distinct minimal surfaces (or families of these), one with an odd intersection form, and the other with an even intersection form. We do not know if surfaces with both types of intersection form always occur. We can reexpress these conditions in terms of the physical numbers P and N . If Nexc = N − P is neither zero nor a positive multiple of 8, then the intersection form must be odd. If N = P , then the intersection form can be of the hyperbolic   plane type l 01 10 , with l = 2P − 1, or it might still be odd. Notice that l is odd,

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as it must be. The isotopes for which even intersection forms are possible therefore include all those with N = P . These are numerous. In addition to the stable isotopes with N = P that occur up to 40 Ca, with P = 20, there are many that are quasistable, like 52 Fe, with P = 26. The heaviest recognised isotopes with N = P are 100 Sn and perhaps 108 Xe, with P = 50 and P = 54. Our geometrical model suggests that the additional stability of these isotopes is the result of the nontrivial structure of an even intersection form. If Nexc = 8m then the intersection form can be of type (4.2), again with l = 2P −1, but it might also be odd. Examples are the Enriques surface, for which l = 1 and m = 1, and the K3 surface, for which l = 3 and m = 2. The potential isotopes corresponding to these surfaces are 10 H and 20 He. These are both so neutronrich that they have not been observed, but there are many heavier nuclei (and corresponding atoms) for which the neutron excess Nexc is a multiple of 8. There is some evidence that nuclei whose neutron excess is a multiple of 8 have additional stability. The most obvious example is 48 Ca, but this is conventionally attributed to the shell model, as P = 20 and N = 28, both magic numbers. A more interesting and less understood example is the heaviest known isotope of oxygen, 24 O, with 8 protons and 16 neutrons. This example and others do not obviously fit with the shell model. The most stable isotope of iron is 56 Fe, whose neutron excess is 4, but it is striking that 60 Fe, whose neutron excess is 8, has a lifetime of over a million years. Here P = 26 and N = 34. 64 Ni, also with a neutron excess of 8, is one of the stable isotopes of nickel. There are also striking examples of stable or relatively stable isotopes with neutron excesses of 16 or 24. Some of these are outliers compared to the general trends in the valley of stability. An example is 124 Sn, the heaviest stable isotope of tin, with Nexc = 24. A more careful study would be needed to confirm if the additional stability of isotopes whose neutron excess is a multiple of 8 is statistically significant. There is no evidence that a neutron deficit of 8 has a stabilising effect. In fact, almost no nuclei with such a large neutron deficit are recognised. The only candidate is 48 Ni, with the magic numbers P = 28 and N = 20. 5. Other Surfaces In addition to the minimal surfaces of general type there are various other classes of algebraic surface. Do these have a physical interpretation? On a surface X it is usually possible to “blow up” one or more points. The result is not minimal, because a minimal surface, by definition, is one that cannot be constructed by blowing up points on another surface. Blowing up one point increases c2 by 1 and decreases c21 by 1. This is equivalent, in our model, to increasing N by 1, leaving P unchanged. In other words, one neutron has been added. Topologically, blowing up is a local process, equivalent to attaching (by connected sum) a copy of CP2 . This adds a 2-cycle that has self-intersection −1, but no intersection with any other 2-cycle. The rank (size) of the intersection form Ω increases by 1, with an

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extra −1 on the diagonal, and the remaining entries of the extra row and column all zero. This automatically makes the intersection form odd, so any previously even form now becomes diagonalisable. The physical interpretation seems to be that a neutron has been added, well separated from any other neutron or proton. This adds a relatively high energy, more than if the additional neutron were bound into an existing nucleus. Minimal algebraic surfaces, and especially those with even intersection forms, should correspond to tightly bound nuclei and atoms, having lower energy. The simplest example is the blow up of one point on CP2 . The result is the Hirzebruch surface H1 , which is a nontrivial CP1 bundle over CP1 . Its intersection   form is 10 −10 . The Hirzebruch surface and quadric are both simply connected and − have the same Betti numbers, b+ 2 = b2 = 1, corresponding to P = 1 and N = 1, but the intersection form is odd for the Hirzebruch surface and even for the quadric. The proposed interpretation is that the Hirzebruch surface represents a separated proton, neutron and electron, whereas the quadric represents the deuterium atom, with a bound proton and neutron as its nucleus, orbited by the electron. There is an inequality of LeBrun for the L2 norm of the Ricci curvature supporting this interpretation.25,26 The norm increases if points on a minimal surface are blown up, the increase being a constant multiple of the number of blown-up points. This indicates that both the norm of the Ricci curvature and the norm of the scalar curvature, possibly with different coefficients, should be ingredients in the physical energy. So far, we have not considered any surfaces X that could represent a single neutron, or a cluster of neutrons. Candidates are the surfaces of Type VII. These have c21 = −c2 , with c2 positive, equivalent to P = 0 and arbitrary positive N . These surfaces are complex, but are not algebraic and do not admit a K¨ahler metric. They are also not simply connected. It is important to have a model of a single neutron. The discussion of blow-ups suggests that CP2 is another possible model. In this case a single neutron would be associated with a 2-cycle with self-intersection −1, mirroring the proton inside CP2 being represented by a 2-cycle with self-intersection +1. A free neutron is almost stable, having a lifetime of approximately 10 minutes. There is considerable physical interest in clusters of neutrons. There is a dineutron resonance similar to the diproton resonance. Recently there has been some experimental evidence for a tetraneutron resonance, indicating some tendency for four neutrons to bind.28 Octaneutron resonances have also been discussed, but no conclusive evidence for their existence has yet emerged. Neutron stars consist of multitudes of neutrons, accompanied perhaps by a small number of other particles (protons and electrons), but their stability is only possible because of the gravitational attraction supplementing the nuclear forces. Standard Newtonian gravity is of course negligible for atomic nuclei. Products of two Riemann surfaces (algebraic curves) of genus 2 or more are examples of minimal surfaces of general type, but they are certainly not simply

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connected. Their interpretation as atoms should be investigated. Other surfaces, for example ruled surfaces, may have some physical interpretation, but our formulae would give them negative proton and neutron numbers. They do not model antimatter, that is, combinations of antiprotons, antineutrons and positrons, because antimatter is probably best modelled using the complex conjugates of surfaces modelling matter. Also bound states of protons and antineutrons, with positive P and negative N , do not seem to exist. 6. Conclusions We have proposed a new geometrical model of matter. It goes beyond our earlier proposal14 in that it can accommodate far more than just a limited set of basic particles. In principle, the model can account for all types of neutral atom. Each atom is modelled by a compact, complex algebraic surface, which as a real manifold is 4-dimensional. The physical quantum numbers of proton number P (equal to electron number for a neutral atom) and neutron number N are expressed in terms of the Chern numbers c21 and c2 of the surface, but they can also be expressed in terms of combinations of the Hodge numbers, or of the Betti numbers − b1 , b+ 2 and b2 . Our formulae for P and N were arrived at by considering the interpretation of just a few examples of algebraic surfaces — the complex projective plane CP2 , the quadric surface Q, and the Hirzebruch surface H1 . Some consequences, which follow from the known constraints on algebraic surfaces, can therefore be regarded as predictions of the model. Among these are that P is any positive integer, and that N is bounded below by 0 and bounded above by the lesser of 9P and 7P + 6. This encompasses all known isotopes. A most interesting prediction is that the line N = P , which is the centre of the valley of nuclear stability for small and − medium-sized nuclei, corresponds to the line τ = 0, where τ = b+ 2 − b2 is the signature. Surfaces with τ positive and τ negative are known to be qualitatively different, which implies that in our model there is a qualitative difference between proton-rich and neutron-rich nuclei. For simply connected surfaces with b1 = 0 (or more generally, if b1 is held fixed) then an increase of P by 1 corresponds to an increase of b+ 2 by 2. The interpretation is that there are two extra 2-cycles with positive self-intersection, corresponding to the extra proton and the extra electron. This matches our earlier models, where a proton was associated with such a 2-cycle,14 and where multi-Taub-NUT space with n NUTs modelled n electrons.16,17 On the other hand, an increase of N by 1 corresponds to an increase of b− 2 by 1. This means that a neutron is associated with a 2-cycle of negative self-intersection, which differs from our earlier ideas, where a neutron was modelled by a 2-cycle with zero self-intersection. It appears now that the intersection numbers are related to isospin (whose third component is 21 for a proton and − 12 for a neutron) rather than to electric charge (1 for a proton and 0 for a neutron).

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Clearly, much further work is needed to develop these ideas into a physical model of nuclei and atoms. We have earlier made a few remarks about possible energy functions for algebraic surfaces. Combinations of the topological invariants and nontopological curvature integrals should be explored, and compared with the detailed information on the energies of nuclei and atoms in their ground states. It will be important to account for the quantum mechanical nature of the ground and excited states, their energies and spins. Discrete energy gaps could arise from discrete changes in geometry, for example, by replacing a blown-up surface with a minimal surface, or by considering the effect of changing b1 while keeping P and N fixed, or by comparing different embeddings of an algebraic surface in (higherdimensional) projective space. In some cases there should be a discrete choice for the intersection form. There are also possibilities for finding an analogue of a Schr¨ odinger equation using linear operators, like the Laplacian or Dirac operator, acting on forms or spinors on a surface. Alternatively, the right approach may be to consider the continuous moduli of surfaces as dynamical variables, and then quantise these. The moduli should somehow correspond to the relative positions of the protons, neutrons and electrons. Some of the ideas just mentioned have already been investigated in the context of single particles, modelled by the TaubNUT space or another noncompact 4-manifold.15,29 Further physical processes, for example, the fission of larger nuclei, and the binding of atoms into molecules, also need to be addressed. Before these investigations can proceed, it will be necessary to decide what metric structure the surfaces need. Previously, we generally required manifolds to have a self-dual metric, i.e. to be gravitational instantons, but this now seems too rigid, as there are very few compact examples. Requiring a K¨ahler–Einstein metric may be more reasonable, although these do not exist for all algebraic surfaces.30,31 For further developments of these ideas, see Refs. 32 and 33. Acknowledgments We are grateful to Chris Halcrow for producing Figs. 1–3, and Nick Mee for supplying Fig. 4. References 1. 2. 3. 4. 5.

W. Thomson, On vortex atoms, Trans. R. Soc. Edin. 6, 94 (1867). J. Lilley, Nuclear Physics: Principles and Applications (Wiley, Chichester, 2001). F. Rohrlich, Classical Charged Particles, 3rd edn. (World Scientific, Singapore, 2007). T. H. R. Skyrme, A nonlinear field theory, Proc. R. Soc. London A 260, 127 (1961). T. H. R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31, 556 (1962). 6. G. S. Adkins, C. R. Nappi and E. Witten, Static properties of nucleons in the Skyrme model, Nucl. Phys. B 228, 552 (1983). 7. D. H. Perkins, Introduction to High Energy Physics, 4th edn. (Cambridge University Press, Cambridge, 2000).

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8. R. A. Battye et al., Light nuclei of even mass number in the Skyrme model, Phys. Rev. C 80, 034323 (2009). 9. P. H. C. Lau and N. S. Manton, States of carbon-12 in the Skyrme model, Phys. Rev. Lett. 113, 232503 (2014). 10. C. J. Halcrow, Vibrational quantisation of the B = 7 Skyrmion, Nucl. Phys. B 904, 106 (2016). 11. C. J. Halcrow, C. King and N. S. Manton, A dynamical α-cluster model of 16 O, Phys. Rev. C 95, 031303(R) (2017). 12. D. Finkelstein and J. Rubinstein, Connection between spin, statistics and kinks, J. Math. Phys. 9, 1762 (1968). 13. M. Atiyah and N. S. Manton, Skyrmions from instantons, Phys. Lett. B 222, 438 (1989). 14. M. Atiyah, N. S. Manton and B. J. Schroers, Geometric models of matter, Proc. R. Soc. London A 468, 1252 (2012). 15. R. Jante and B. J. Schroers, Dirac operators on the Taub-NUT space, monopoles and SU(2) representations, J. High Energy Phys. 01, 114 (2014). 16. G. Franchetti and N. S. Manton, Gravitational instantons as models for charged particle systems, J. High Energy Phys. 03, 072 (2013). 17. G. Franchetti, Harmonic forms on ALF gravitational instantons, J. High Energy Phys. 12, 075 (2014). 18. W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces (Springer, Berlin, Heidelberg, 1984). 19. P. Griffiths and J. Harris, Principles of Algebraic Geometry (Wiley Classics, New York, Chichester, 1994). 20. C. Voisin, Hodge Theory and Complex Algebraic Geometry, Vol. I (Cambridge University Press, Cambridge, 2002). 21. Enriques–Kodaira classification, Wikipedia, 2016. 22. D. I. Cartwright and T. Steger, Enumeration of the 50 fake projective planes, C.R. Acad. Sci. Paris, Ser. I 348, 11 (2010). 23. E. Braaten, S. Townsend and L. Carson, Novel structure of static multisoliton solutions in the Skyrme model, Phys. Lett. B 235, 147 (1990). 24. R. A. Battye, N. S. Manton and P. M. Sutcliffe, Skyrmions and the α-particle model of nuclei, Proc. R. Soc. London A 463, 261 (2007). 25. C. LeBrun, Four-manifolds without Einstein metrics, Math. Res. Lett. 3, 133 (1996). 26. C. LeBrun, Ricci curvature, minimal volumes, and Seiberg–Witten theory, Invent. Math. 145, 279 (2001). 27. S. K. Donaldson and P. B. Kronheimer, Geometry of Four-Manifolds (Oxford University Press, Oxford, 1990). 28. K. Kisamori et al., Candidate resonant tetraneutron state populated by the 4 He(8 He, 8 Be) reaction, Phys. Rev. Lett. 116, 052501 (2016). 29. R. Jante and B. J. Schroers, Spectral properties of Schwarzschild instantons, Class. Quantum Grav. 33, 205008 (2016). 30. T. Ochiai (ed.), K¨ ahler Metric and Moduli Spaces, Advanced Studies in Pure Mathematics, Vol. 18-II (Kinokuniya, Tokyo, 1990). 31. G. Tian, K¨ ahler–Einstein metrics with positive scalar curvature, Invent. Math. 137, 1 (1997). 32. M. Atiyah and M. Marcolli, Anyons in geometric models of matter, J. High Energy Phys. 07, 076 (2017). 33. M. F. Atiyah, Geometric models of helium, Mod. Phys. Lett. A 32, 1750079 (2017).

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Chapter 2 Developments in topological gravity

Robbert Dijkgraaf and Edward Witten Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 USA

This note aims to provide an entr´ ee to two developments in two-dimensional topological gravity — that is, intersection theory on the moduli space of Riemann surfaces — that have not yet become well-known among physicists. A little over a decade ago, Mirzakhani discovered [1, 2] an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler [3] (with further developments in [4–6]) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint — it corresponds to adding vector degrees of freedom to the matrix model — constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved. Contents 1 2

3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Weil–Petersson Volumes and Two-Dimensional Topological Gravity . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 Background and initial steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 A simpler problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 How Maryam Mirzakhani cured modular invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25 2.4 Volumes and intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Open Topological Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Relation to condensed matter physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4 Two realizations of theory T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Boundary conditions in theory T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 Boundary anomaly of theory T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6.1 The trivial case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.2 Boundary condition in condensed matter physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.6.3 Boundary condition in two-dimensional gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Anomaly cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.8 Branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8.1 The ζ-instanton equation and compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8.2 Boundary condition in the ζ-instanton equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.8.3 Orientations and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.8.4 Quantizing the string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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3.9 Boundary degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.10 Computations of disc amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Interpretation via Matrix Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.1 The loop equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2 Double-scaling limits and topological gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 Branes and open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

1. Introduction There are at least two candidates for the simplest model of quantum gravity in two space–time dimensions. Matrix models are certainly one candidate, extensively studied since the 1980s. These models were proposed in [7–11] and solved in [12–14]; for a comprehensive review with extensive references, see [15]. A second candidate is provided by topological gravity, that is, intersection theory on the moduli space of Riemann surfaces. It was conjectured some time ago that actually two-dimensional topological gravity is equivalent to the matrix model [16, 17]. This equivalence led to formulas expressing the intersection numbers of certain natural cohomology classes on moduli space in terms of the partition function of the matrix model, which is governed by KdV equations [18] or equivalently by Virasoro constraints [19]. These formulas were first proved by Kontsevich [20] by a direct calculation that expressed intersection numbers on moduli space in terms of a new type of matrix model (which was again shown to be governed by the KdV and Virasoro constraints). A little over a decade ago, Maryam Mirzakhani found a new proof of this relationship as part of her Ph.D. thesis work [1, 2]. (Several other proofs are known [21,22].) She put the accent on understanding the Weil–Petersson volumes of moduli spaces of hyperbolic Riemann surfaces with boundary, showing that these volumes contain all the information in the intersection numbers. A hyperbolic structure on a surface Σ is determined by a flat SL(2, R) connection, so the moduli space M of hyperbolic structures on Σ can be understood as a moduli space of flat SL(2, R) connections. Actually, the Weil–Petersson symplectic form on M can be defined by the same formula that is used to define the symplectic form on the moduli space of flat connections on Σ with structure group a compact Lie group such as SU (2). For a compact Lie group, the volume of the moduli space can be computed by a direct cut and paste method [23] that involves building Σ out of simple building blocks (three-holed spheres). Naively, one might hope to do something similar for SL(2, R) and thus for the Weil–Petersson volumes. But there is a crucial difference: in the case of SL(2, R), in order to define the moduli space whose volume one will calculate, one wants to divide by the action of the mapping class group on Σ. (Otherwise the volume is trivially infinite.) But dividing by the mapping class group is not compatible with any simple cut and paste method. Maryam Mirzakhani overcame this difficulty in a surprising and elegant way, of which we will give a glimpse in Sec. 2. Matrix models of two-dimensional gravity have a natural generalization in which vector degrees of freedom are added [24–29]. This generalization is related, from a

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physical point of view, to two-dimensional gravity formulated on two-manifolds Σ that carry a complex structure but may have a boundary. We will refer to such two-manifolds as open Riemann surfaces (if the boundary of Σ is empty, we will call it a closed Riemann surface). It is natural to hope that, by analogy with what happens for closed Riemann surfaces, there would be an intersection theory on the moduli space of open Riemann surfaces that would be related to matrix models with vector degrees of freedom. In trying to construct such a theory, one runs into immediate difficulties: the moduli space of open Riemann surfaces does not have a natural orientation and has a boundary; for both reasons, it is not obvious how to define intersection theory on this space. These difficulties were overcome by Pandharipande, Solomon, and Tessler in a rather unexpected way [3] whose full elucidation involves introducing spin structures in a problem in which at first sight they do not seem relevant [4–6]. In Sec. 3, we will explain some highlights of this story. In Sec. 4, we review matrix models with vector degrees of freedom, and show how they lead — modulo a slightly surprising twist — to precisely the same Virasoro constraints that have been found in intersection theory on the moduli space of open Riemann surfaces. The matrix models we consider are the direct extension of those studied in [12–14]. The same problem has been treated in a rather different approach via Gaussian matrix models with an external source in [30] and in chapter 8 of [31]. See also [32] for another approach. For an expository article on the relation of matrix models and intersection theory, see [33]. 2. Weil–Petersson Volumes and Two-Dimensional Topological Gravity 2.1. Background and initial steps Let Σ be a closed Riemann surface of genus g with marked points1 p1 , . . . , pn , and let Li be the cotangent space to pi in Σ. As Σ and the pi vary, Li varies as the fiber of a complex line bundle — which we also denote as Li — over Mg,n , the moduli space of genus g curves with n punctures. In fact, these line bundles extend naturally over Mg,n , the Deligne–Mumford compactification of Mg,n . We write ψi for the first Chern class of Li ; thus ψi = c1 (Li ) is a two-dimensional cohomology class. For a non-negative integer d, we set τi,d = ψid , a cohomology class of dimension 2d. The usual correlation functions of 2d topological gravity are the intersection numbers Z Z hτd1 τd2 . . . τdn i = τ1,d1 τ2,d2 · · · τn,dn = ψ1d1 ψ2d2 · · · ψndn , (2.1) Mg,n

Mg,n

where d1 , . . . , dn is any n-plet of nonnegative integers. The right-hand side of Eq. Pn (2.1) vanishes unless i=1 di = 3g−3+n. To be more exact, what we have defined in Eq. (2.1) is the genus g contribution to the correlation function; the full correlation 1 The

marked points are labeled and are required to be always distinct.

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function is obtained by summing over g ≥ 0. (For a given set of di , there is at most Pn one integer solution g of the condition i=1 di = 3g − 3 + n, and this is the only value that contributes to hτd1 τd2 . . . τdn i.) Let us now explain how these correlation functions are related to the Weil– Petersson volume of Mg . In the special case n = 1, we have just a single marked point p and a single line bundle L and cohomology class ψ. We also have the forgetful map π : Mg,1 → Mg that forgets the marked point. We can construct a two-dimensional cohomology class κ on Mg by integrating the four-dimensional class τ2 = ψ 2 over the fibers of this forgetful map: κ = π∗ (τ2 ).

(2.2)

More generally, the Miller–Morita–Mumford (MMM) classes are defined by κd = π∗ (τd+1 ), so κ is the same as the first MMM class κ1 . κ is cohomologous to a multiple of the Weil–Petersson symplectic form ω of the moduli space [34, 35]: ω = κ. (2.3) 2π 2 Because of (2.2), it will be convenient to use κ, rather than ω, to define a volume form. With this choice, the volume of Mg is Z Z κ3g−3 = exp(κ). (2.4) Vg = Mg (3g − 3)! Mg The relation between κ and τ2 might make one hope that the volume Vg would be one of the correlation functions of topological gravity:

3g−3  1 ? Vg = τ . (2.5) (3g − 3)! 2 Such a simple formula is, however, not true, for the following reason. To compute the right-hand side of Eq. (2.5), we would have to introduce 3g − 3 marked points on Σ, and insert τ2 (that is, ψi2 ) at each of them. It is true that for a single marked point, κ can be obtained as the integral of τ2 over the fiber of the forgetful map, as in Eq. (2.2). However, when there is more than one marked point, we have to take into account that the Deligne–Mumford compactification of Mg,n is defined in such a way that the marked points are never allowed to collide. Taking this into account leads to corrections in which, for instance, two copies of τ2 are replaced by a single copy of τ3 . The upshot is that Vg can be expressed in terms of the correlation functions of topological gravity, and thus can be computed using the KdV equations or the Virasoro constraints, but the necessary formula is more complicated. See Subsec. 2.4 below. For now, we just remark that this approach has been used [36] to determine the large g asymptotics of Vg , but apparently does not easily lead to explicit formulas for Vg in general. Weil–Petersson volumes were originally studied and their asymptotics estimated by quite different methods [37]. Mg,n likewise has a Weil–Petersson volume Vg,n of its own, which likewise can be computed, in principle, using a knowledge of the intersection numbers on Mg,n0

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a)

b)

Fig. 1. (a) A marked point in a hyperbolic Riemann surface is treated as a cusp: it lies at infinity in the hyperbolic metric. (b) Instead of a cusp, a hyperbolic Riemann surface might have a geodesic boundary, with circumference any positive number b.

for n0 > n. Again this gives useful information but it is difficult to get explicit general formulas. Mirzakhani’s procedure was different. First of all, she worked in the hyperbolic world, so in the following discussion Σ is not just a complex Riemann surface; it carries a hyperbolic metric, by which we mean a Riemannian metric of constant scalar curvature R = −1. We recall that a complex Riemann surface admits a unique Kahler metric with R = −1. We recall also that in studying hyperbolic Riemann surfaces, it is natural2 to think of a marked point as a cusp, which lies at infinity in the hyperbolic metric (Fig. 1). Instead of a marked point, we can consider a Riemann surface with a boundary component. In the hyperbolic world, one requires the boundary to be a geodesic in the hyperbolic metric. Its circumference may be any positive number b. Let us consider, rather than a closed Riemann surface Σ of genus g with n labeled marked points, an open Riemann surface Σ also of genus g, but now with n labeled boundaries. In the hyperbolic world, it is natural to specify n positive numbers b1 , . . . , bn and to require that Σ carry a hyperbolic metric such that the boundaries are geodesics of lengths b1 , . . . , bn . We denote the moduli space of such hyperbolic metrics as Mg;b1 ,b2 ,...,bn or more briefly as Mg,~b , where ~b is the n-plet (b1 , b2 , . . . , bn ). As a topological space, M ~ is independent of ~b. In fact, M ~ is an orbifold, g,b

g,b

and the topological type of an orbifold cannot depend on continuously variable data such as ~b. In the limit that b1 , . . . , bn all go to zero, the boundaries turn into cusps and Mg,~b turns into Mg,n . Thus topologically, Mg,~b is equivalent to Mg,n for any ~b. Very concretely, we can always convert a Riemann surface with a boundary

component to a Riemann surface with a marked point by gluing a disc, with a marked point at its center, to the given boundary component. Thus we can turn a Riemann surface with boundaries into one with marked points without changing the parameters the Riemann surface can depend on, and this leads to the topological equivalence of Mg,~b with Mg,n . If we allow the hyperbolic metric of Σ to develop 2 This

is natural because the degenerations of the hyperbolic metric of Σ that correspond to Deligne–Mumford compactification of Mg,n generate cusps. Since the extra marked points that occur when Σ degenerates (for example to two components that are glued together at new marked points) appear as cusps in the hyperbolic metric, it is natural to treat all marked points that way.

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cusp singularities, we get a compactification Mg,~b of Mg,b which coincides with the Deligne–Mumford compactification Mg,n of Mg,n . Mg,n and Mg,~b have natural Weil–Petersson symplectic forms that we will call ω and ω~b (see [38]). Since Mg,n and Mg,~b are equivalent topologically, it makes sense to ask if the symplectic form ω~b of Mg,~b has the same cohomology class as the symplectic form ω of Mg,n . The answer is that it does not. Rather, one has (see [2], Theorem 4.4) n

ω~b = ω +

1X 2 b ψi . 2 i=1 i

(2.6)

(This is a relationship in cohomology, not an equation for differential forms.) From this it follows that the Weil–Petersson volume of Mg,~b is3 ! Z n 1X 2 1 exp ω + b ψi . (2.7) Vg,~b = (2π 2 )3g−3+n Mg,~b 2 i=1 i Equivalently, since compactification by allowing cusps does not affect the volume integral, and the compactification of Mg,~b is the same as Mg,n , one can write this as an integral over the compactification: ! Z n 1 1X 2 Vg,~b = exp ω + b ψi . (2.8) (2π 2 )3g−3+n Mg,n 2 i=1 i This last result tells us that at ~b = 0, Vg,~b reduces to the volume Vg,n = R 2 3g−3+n eω of Mg,n . Moreover, Eq. (2.8) implies that Vg,~b is a polyno(1/2π ) Mg,n mial in ~b2 = (b2 , . . . , b2 ) of total degree 3g − 3 + n. In evaluating the term of top 1

n

degree in Vg,~b , we can drop ω from the exponent in Eq. (2.8). Then the expansion in powers of the bi tells us that this term of top degree is n i X Y 1 b2d i hτd1 τd2 · · · τdn i . (2.9) (2π 2 )3g−3+n 2di di ! d1 ,...,dn i=1 P (Only terms with i di = 3g − 3 + n make nonzero contributions in this sum.) In other words, the correlation functions of two-dimensional topological gravity on a closed Riemann surface appear as coefficients in the expansion of Vg,~b . Of course, Vg,~b contains more information,4 since we can also consider the terms in Vg,~b that are subleading in ~b. Thus Mirzakhani’s approach to topological gravity involved deducing the correlation functions of topological gravity from the volume polynomials Vg,~b . We will 3 The reason for the factor of 1/(2π 2 )3g−3+n is just that we defined the volumes in Eq. (2.4) using κ rather than ω. 4 This additional information in principle is not really new. Using facts that generalize the relationship between Vg,n and the correlation functions of topological gravity that we discussed at the outset, one can deduce also the subleading terms in Vg,~b in terms of the correlation functions of topological gravity. However, it appears difficult to get useful formulas in this way.

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give a few indications of how she computed these volume polynomials in Subsec. 2.3, after first recalling a much simpler problem. 2.2. A simpler problem Before explaining how to compute the volume of Mg,~b , we will describe how volumes can be computed in a simpler case. In fact, the analogy was noted in [2]. Let G be a compact Lie group, such as SU (2), with Lie algebra g, and let Σ be a closed Riemann surface of genus g. Let Mg be the moduli space of homomorphisms from the fundamental group of Σ to G. Equivalently, Mg is the moduli space of flat g-valued flat connections on Σ. Then [38,39] Mg has a natural symplectic form that in many ways is analogous to the Weil–Petersson form on Mg . Writing A for a flat connection on Σ and δA for its variation, the symplectic form of Mg can be defined by the gauge theory formula Z 1 Tr δA ∧ δA, (2.10) ω= 4π 2 Σ where (for G = SU (2)) we can take Tr to be the trace in the two-dimensional representation. Actually, the Weil–Petersson form of Mg can be defined by much the same formula. The moduli space of hyperbolic metrics on Σ is a component5 of the moduli space of flat SL(2, R) connections over Σ, divided by the mapping class group of Σ. Denoting the flat connection again as A and taking Tr to be the trace in the two-dimensional representation of SL(2, R), the right hand side of Eq. (2.10) becomes in this case a multiple of the Weil–Petersson symplectic form ω on Mg . There is also an analog for compact G of the moduli spaces Mg,~b of hyperbolic Riemann surfaces with geodesic boundary. For ~b = (b1 , . . . , bn ), M ~ can be interg,b

preted as follows in the gauge theory language. A point in Mg,~b corresponds, in the gauge theory language, to a flat SL(2, R) connection on Σ with the property that the holonomy around the ith boundary is conjugate in SL(2, R) to the group element diag(ebi , e−bi ). In this language, it is clear how to imitate the definition of Mg,~b for a compact Lie group such as SU (2). For k = 1, . . . , n, we choose a conjugacy class in SU (2), say the class that contains Uk = diag(eiαk , e−iαk ), for some αk . We write α ~ for the nplet (α1 , α2 , . . . , αn ), and we define Mg,~α to be the moduli space of flat connections on a genus g surface Σ with n holes (or equivalently n boundary components) with 5 The moduli space of flat SL(2, R) connections on Σ has various components labeled by the Euler class of a flat real vector bundle of rank 2 (transforming in the two-dimensional representation of SL(2, R)). One of these components parametrizes hyperbolic metrics on Σ together with a choice of spin structure. If we replace SL(2, R) by P SL(2, R) = SL(2, R)/Z2 (the symmetry group of the hyperbolic plane), we forget the spin structure, so to be precise, Mg is a component of the moduli space of flat P SL(2, R) connections. This refinement will not be important in what follows and we loosely speak of SL(2, R). In terms of P SL(2, R), one can define Tr as 1/4 of the trace in the three-dimensional representation.

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a)

b)

Fig. 2. (a) A three-holed sphere or “pair of pants.” (b) A Riemann surface Σ, possibly with boundaries, that is built by gluing three-holed spheres along their boundaries. Each boundary of one of the three-holed spheres is either an external boundary — a boundary of Σ — or an internal boundary, glued to a boundary of one of the three-holed spheres (generically a different one). The example shown has one external boundary and four internal ones.

the property that the holonomy around the k th hole is conjugate to Uk . With a little care,6 the right hand side of the formula (2.10) can be used in this situation to define the Weil–Petersson form κ~b of Mg,~b , and the analogous symplectic form ωα~ of Mg,~α . Thus in particular, Mg,~α has a symplectic volume Vg,~α . Moreover, Vg,~α is a polynomial in α ~ , and the coefficients of this polynomial are the correlation functions of a certain version of two-dimensional topological gauge theory — they are the intersection numbers of certain natural cohomology classes on Mg,~α . These statements, which are analogs of what we described in the case of gravity in Subsec. 2.1, were explained for gauge theory with a compact gauge group in [40]. Moreover, for a compact gauge group, various relatively simple ways to compute the symplectic volume Vg,~α were described in [23]. None of these methods carry over naturally to the gravitational case. However, to appreciate Maryam Mirzakhani’s work on the gravitational case, it helps to have some idea how the analogous problem can be solved in the case of gauge theory with a compact gauge group. So we will make a few remarks. First we consider the special case of a three-holed sphere (sometimes called a pair of pants; see Fig. 2(a)). In the case of a three-holed sphere, for G = SU (2), M0,~α is either a point, with volume 1, or an empty set, with volume 0, depending on α ~ . The volumes of the three-holed sphere moduli spaces can also be computed (with a little more difficulty) for other compact G, but we will not explain the details as the case of SU (2) will suffice for illustration.

the gravity side, Mirzakhani’s proof that the cohomology class of κ~b is linear in ~b2 did not use Eq. (2.10) at all, but a different approach based on Fenchel–Nielsen coordinates. On the gauge theory side, in using Eq. (2.10), it can be convenient to consider a Riemann surface with punctures (i.e., marked points that have been deleted) rather than boundaries. This does not affect the moduli space of flat connections, because if Σ is a Riemann surface with boundary, one can glue in to each boundary component a once-punctured disc, thus replacing all boundaries by punctures, without changing the moduli space of flat connections. For brevity we will stick here with the language of Riemann surfaces with boundary.

6 On

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Now to generalize beyond the case of a three-holed sphere, we observe that any closed surface Σ can be constructed by gluing together three-holed spheres along some of their boundary components (Fig. 2(b)). If Σ is built in this way, then the corresponding volume Mg,~α can be obtained by multiplying together the volume functions of the individual three-holed spheres and integrating over the α parameters of the internal boundaries, where gluing occurs. (One also has to integrate over some twist angles that enter in the gluing, but these give a trivial overall factor.) Thus for a compact group it is relatively straightforward to get formulas for the volumes Vg,~α . Moreover, these formulas turn out to be rather manageable. If we try to imitate this with SU (2) replaced by SL(2, R), some of the steps work. In particular, if Σ is a three-holed sphere, then for any ~b, the moduli space M0,~b is a point and V0,~b = 1. What really goes wrong for SL(2, R) is that, if Σ is such that M0,~b is not just a point, then the volume of the moduli space of flat SL(2, R) connections on Σ is infinite. For SU (2), the procedure mentioned in the last paragraph leads to an integral over the parameters α ~ . Those parameters are angular variables, valued in a compact set, and the integral over these parameters converges. For SL(2, R) (in the particular case of the component of the moduli space of flat connections that is related to hyperbolic metrics), we would want to replace the angular variables α ~ with the positive parameters ~b. The set of positive numbers is not compact and the integral over ~b is divergent. This should not come as a surprise as it just reflects the fact that the group SL(2, R) is not compact. The relation between flat SL(2, R) connections and complex structures tells us what we have to do to get a sensible problem. To go from (a component of) the moduli space of flat SL(2, R) connections to the moduli space of Riemann surfaces, we have to divide by the mapping class group of Σ (the group of components of the group of diffeomorphisms of Σ). It is the moduli space of Riemann surfaces that has a finite volume, not the moduli space of flat SL(2, R) connections. But here is precisely where we run into difficulty with the cut and paste method to compute volumes. Topologically, Σ can be built by gluing three-holed spheres in many ways that are permuted by the action of the mapping class group. Any one gluing procedure is not invariant under the mapping class group and in a calculation based on any one gluing procedure, it is difficult to see how to divide by the mapping class group. Dealing with this problem, in a matter that we explain next, was the essence of Maryam Mirzakhani’s approach to topological gravity. 2.3. How Maryam Mirzakhani cured modular invariance Let Σ be a hyperbolic Riemann surface with geodesic boundary. Ideally, to compute the volume of the corresponding moduli space, we would “cut” Σ on a simple closed geodesic `. This cutting gives a way to build Σ from hyperbolic Riemann

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b)

a)

Fig. 3. A “cut” of a Riemann surface with boundary along an embedded circle may be separating as in (a) or non-separating as in (b).

surfaces that are in some sense simpler than Σ. If cutting along ` divides Σ into two disconnected components (Fig. 3(a)), then Σ can be built by gluing along ` two hyperbolic Riemann surfaces Σ1 and Σ2 of geodesic boundary. If cutting along ` leaves Σ connected (Fig. 3(b)), then Σ is built by gluing together two boundary components of a surface Σ0 . We call these the separating and nonseparating cases. In the separating case, we might naively hope to compute the volume function Vg,~b for Σ by multiplying together the corresponding functions for Σ1 and Σ2 and integrating over the circumference b of `. Schematically, Z ∞ ? VΣ = db VΣ1 ,b VΣ2 ,b , (2.11) 0

where we indicate that Σ1 and Σ2 each has one boundary component, of circumference b, that does not appear in Σ. In the nonseparating case, a similarly naive formula would be Z ∞ ? VΣ = db VΣ0 ,b,b , (2.12) 0 0

where we indicate that Σ , relative to Σ, has two extra boundary components each of circumference b. The surfaces Σ1 , Σ2 , and Σ0 are in a precise sense “simpler” than Σ: their genus is less, or their Euler characteristic is less negative. So if we had something like (2.11) or (2.12), a simple induction would lead to a general formula for the volume functions. The trouble with these formulas is that a hyperbolic Riemann surface actually has infinitely many simple closed geodesics `α , and there is no natural (modularinvariant) way to pick one. Suppose, however, that there were a function F (b) of a positive real number b with the property that X F (bα ) = 1, (2.13) α

where the sum runs over all simple closed geodesics `α on a hyperbolic surface Σ, and bα is the length of `α . In this case, by summing over all choices of embedded simple closed geodesic, and weighting each with a factor of F (b), we would get a corrected version of the above formulas. In writing the formula, we have to remember that cutting along a given `α either leaves Σ connected or separates a genus g surface Σ

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into surfaces Σ1 , Σ2 of genera g1 , g2 such that g1 + g2 = g. In the separating case, the boundaries of Σ are partitioned in some arbitrary way between Σ1 and Σ2 and each of Σ1 , Σ2 has in addition one more boundary component whose circumference we will call b0 . So denoting as ~b the boundary lengths of Σ, the boundary lengths of Σ1 and Σ2 are respectively ~b1 , b0 and ~b2 , b0 , where ~b = ~b1 t ~b2 (here ~b1 t ~b2 denotes the disjoint union of two sets ~b1 and ~b2 ) and Σ is built by gluing together Σ1 and Σ2 along their boundaries of length b0 . This is drawn in Fig. 3(a), but in the example shown, the set ~b consists of only one element. In the nonseparating case of Fig. 3(b), Σ is made from gluing a surface Σ0 of boundary lengths ~b, b0 , b0 along its two boundaries of length b0 . The genus g 0 of Σ0 is g 0 = g − 1. Assuming the hypothetical sum rule (2.13) involves a sum over all simple closed geodesics `α , regardless of topological properties, the resulting recursion relation for the volumes will also involve such a sum. This recursion relation would be Z ∞ X X ? 1 db0 F (b0 )Vg1 ,~b1 ,b0 Vg2 ,~b2 ,b0 Vg,~b = 2 0 g1 ,g2 |g=g1 +g2 ~b1 ,~b2 |~b1 t~b2 =~b Z ∞ + db0 F (b0 )Vg−1,~b,b0 ,b0 . (2.14) 0

In the first term, the sum runs over all topological choices in the gluing; the factor of 1/2 reflects the possibility of exchanging Σ1 and Σ2 . The factors of F (b0 ) in the formula compensate for the fact that in deriving such a result, one has to sum over cuts on all simple closed geodesics. By induction (in the genus and the absolute value of the Euler characteristic of a surface), such a recursion relation would lead to explicit expressions for all Vg,~b . There is an important special case in which there actually is a sum rule [41] precisely along the lines of Eq. (2.13) and therefore there is an identity precisely along the lines of Eq. (2.14). This is the case that Σ is a surface of genus 1 with one boundary component. The general case is more complicated. In general, there is an identity that involves pairs of simple closed geodesics in Σ that have the property that — together with a specified boundary component of Σ — they bound a pair of pants (Fig. 4). This identity was proved for hyperbolic Riemann surfaces with punctures by McShane in [41] and generalized to surfaces with boundary by Mirzakhani in [1], Theorem 4.2. This generalized McShane identity leads to a recursive formula for Weil– Petersson volumes that is similar in spirit to Eq. (2.14). See Theorem 8.1 of Mirzakhani’s paper [1] for the precise statement. The main difference between the naive formula (2.14) and the formula that actually works is the following. In Eq. (2.14), we imagine building Σ from simpler building blocks by a more or less arbitrary gluing. In the correct formula — Mirzakhani’s Theorem 8.1 — we more specifically build Σ by gluing a pair of pants onto something simpler, as in Fig. 4. There is a function F (b, b0 , b00 ), analogous to F (b0 ) in the above schematic discussion, that enters in the generalized McShane identity and therefore in the recursion relation.

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Σ′

Fig. 4. Building a hyperbolic surface Σ by gluing a hyperbolic pair of pants with geodesic boundary onto a simpler hyperbolic surface Σ0 . Σ and Σ0 both have geodesic boundary. (Shown here is the case that Σ0 is connected.)

It compensates for the fact that in deriving the recursion relation, one has to sum over infinitely many ways to cut off a hyperbolic pair of pants from Σ. In this manner, Mirzakhani arrived at a recursive formula for Weil–Petersson volumes that is similar to although somewhat more complicated than Eq. (2.14). Part of the beauty of the subject is that this formula turned out to be surprisingly tractable. In [1], section 6, she used the recursive formula to give a new proof — independent of the relation to topological gravity that we reviewed in Subsec. 2.1 — that the volume functions Vg,~b are polynomials in b21 , . . . , b2n . In [2], she showed that these polynomials satisfy the Virasoro constraints of two-dimensional gravity, as formulated for the matrix model in [19]. Thereby — using the relation between volumes and intersection numbers that we reviewed in Subsec. 2.1 and to which we will return in a moment — she gave a new proof of the known formulas [17, 20] for intersection numbers on the moduli space of Riemann surfaces, or equivalently for correlation functions of two-dimensional topological gravity.

2.4. Volumes and intersection numbers We conclude this section by briefly describing the formula that relates Weil– Petersson volumes to correlation functions of topological gravity. Given a surface Σ with n + 1 marked points, there is a forgetful map π : Mg,n+1 → Mg,n that forgets one of the marked points p. If we insert the class τd+1 at p and integrate over the fiber of π, we get the Miller–Morita–Mumford class κd = π∗ (τd+1 ), which is a class of degree 2d in the cohomology of Mg,n . Qk As a first step in evaluating a correlation function hτd+1 j=1 τnj i, one might to try to integrate over the choice of the point at which τd+1 is inserted. Integrating over the fiber of π : Mg,n+1 → Mg , one might hope to get a formula * τd+1

k Y j=1

+ τnj

* ?

=

κd

k Y

+ τnj

.

(2.15)

j=1

This is not true, however. The right version of the formula has corrections that involve contact terms in which τd+1 collides with τnj for some j. Such a collision generates a correction that is an insertion of τd+nj . For a fuller explanation, see [17].

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Taking account of the contact terms, one can express correlation functions of the τ ’s in terms of those of the κ’s, and vice versa. A special case is the computationR of volumes. As before, we write just κ for κ1 , and we define the volume of Mg as Mg κ3g−3 /(3g − 3)!. This can be expressed in terms of correlation functions of the τ ’s, but one has to take the contact terms into account. As an example, we consider the case of a closed surface of genus 2. The volume of the compactified moduli space M2 is V2 =

1 hκκκi, 3!

(2.16)

and we want to compare this to topological gravity correlation functions such as 1 hτ2 τ2 τ2 i. 3!

(2.17)

By integrating over the position of one puncture, we can replace one copy of τ2 with κ, while also generating contact terms. In such a contact term, τ2 collides with some τs , s ≥ 0, to generate a contact term τs+1 . Thus for example hτ2 τ2 τ2 i = hκτ2 τ2 i + 2hτ3 τ2 i,

(2.18)

where the factor of 2 reflects the fact that the first τ2 may collide with either of the two other τ2 insertions to generate a τ3 . The same process applies if factors of κ are already present; they do not generate additional contact terms. For example, hκτ2 τ2 i = hκκτ2 i + hκτ3 i = hκκκi + hκτ3 i.

(2.19)

hτ2 τ3 i = hκτ3 i + hτ4 i.

(2.20)

Similarly

Taking linear combinations of these formulas, we learn finally that hκκκi = hτ2 τ2 τ2 i − 3hτ2 τ3 i + hτ4 i.

(2.21)

This is equivalent to saying that V2 , which is the term of order ξ 3 in hexp(ξκ)i , is equally well the term of order ξ 3 in    ξ2 ξ3 exp ξτ2 − τ3 + τ4 . 2! 3! The generalization of this for higher genus is that !+   * ∞ X (−1)k ξ k−1 exp (ξκ) = exp τk . (k − 1)! k=2

(2.22)

(2.23)

(2.24)

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The volume of Mg is the coefficient of ξ 3g−3 in the expansion of either of these formulas. To prove Eq. (2.24), we write W (ξ) for the right-hand side, and we compute that * ! !+ ∞ ∞ r−2 k k−1 X X d ξ (−1) ξ W (ξ) = τ2 + τr exp τk (−1)r . (2.25) dξ (r − 2)! (k − 1)! r=3 k=2

Next, one replaces the explicit τ2 term in the parentheses on the right-hand side with κ plus a sum of contact terms between τ2 and the τk ’s that appear in the exponential. These contact terms cancel the τr ’s inside the parentheses, and one finds * !+ ∞ X (−1)k ξ k−1 d W (ξ) = κ exp τk . (2.26) dξ (k − 1)! k=2

Repeating this process gives for all s ≥ 0 * !+ ∞ X ds (−1)k ξ k−1 s W (ξ) = κ exp τk . dξ s (k − 1)!

(2.27)

k=2

Setting ξ = 0, we get ds = hκs i , W (ξ) dξ s ξ=0

(2.28)

and the fact that this is true for all s ≥ 0 is equivalent to Eq. (2.24). Eq. (2.24) has been deduced by comparing matrix model formulas to Mirzakhani’s formulas for the volumes [42]. We will return to this when we discuss the spectral curve in Subsec. 4.2. For algebro-geometric approaches and generalizations see [43, 44]. It is also possible to obtain similar formulas for the volume of Mg,~b . 3. Open Topological Gravity 3.1. Preliminaries In this section, we provide an introduction to recent work [3–6] on topological gravity on open Riemann surfaces, that is, on oriented two-manifolds with boundary. From the point of view of matrix models of two-dimensional gravity, one should expect an interesting theory of this sort to exist because adding vector degrees of freedom to a matrix model of two-dimensional gravity gives a potential model of two-manifolds with boundary.7 We will discuss matrix models with vector degrees of freedom in Sec. 4. Here, however, we discuss the topological field theory side of the story. 7 Similarly,

by replacing the usual symmetry group U (N ) of the matrix model with O(N ) or Sp(N ), one can make a model associated to gravity on unoriented (and possibly unorientable) two-manifolds. It is not yet understood if this is related to some sort of topological field theory.

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Let Σ be a Riemann surface with boundary, and in general with marked points or punctures both in bulk and on the boundary. Its complex structure gives Σ a natural orientation, and this induces orientations on all boundary components. If p is a bulk puncture, then the cotangent space to p in Σ is a complex line bundle L, and as we reviewed in Subsec. 2.1, one defines for every integer d ≥ 0 the cohomology class τd = ψ d of degree 2d. The operator τ0 = 1 is associated to a bulk puncture, and the τd with d > 0 are called gravitational descendants. A boundary puncture has no analogous gravitational descendants, because if p is a boundary point in Σ, the tangent bundle to p in Σ is naturally trivial. It has a natural real subbundle given by the tangent space to p in ∂Σ, and this subbundle is actually trivialized (up to homotopy) by the orientation of ∂Σ. So c1 (L) = 0 if p is a boundary puncture. Thus the list of observables in 2d topological gravity on a Riemann surface with boundary consists of the usual bulk observables τd , d ≥ 0, and one more boundary observable σ, corresponding to a boundary puncture. Formally, the sort of thing one hopes to calculate for a Riemann surface Σ with n bulk punctures and m boundary punctures is Z ψ1d1 ψ2d2 · · · ψndn , (3.1) hτd1 τd2 · · · τdn σ m iΣ = M

where M is the (compactified) moduli space of conformal structures on Σ with n bulk punctures and m boundary punctures. The di are arbitrary nonnegative Qn integers, and we note that the cohomology class i=1 ψidi that is integrated over M is generated only from data at the bulk punctures (and in fact only from those bulk punctures with di > 0). The boundary punctures (and those bulk punctures with di = 0) participate in the construction only because they enter the definition of M, the space on which the cohomology class in question is supposed to be integrated. Similarly to the case of a Riemann surface without boundary, to make the integral (3.1) nonzero, Σ must be chosen topologically so that the dimension of M is the same as the degree of the cohomology class that we want to integrate: n X dim M = 2di . (3.2) i=1

Assuming that we can make sense of the definition in Eq. (3.1), the (unnormalized) correlation function hτd1 τd2 · · · τdn σ m i of 2d gravity on Riemann surfaces with boundary is then obtained by summing hτd1 τd2 · · · τdn σ m iΣ over all topological choices of Σ. (If Σ has more than one boundary component, the sum over Σ includes a sum over how the boundary punctures are distributed among those boundary components.) It is also possible to slightly generalize the definition by weighting a surface Σ in a way that depends on the number of its boundary components. For this, we introduce a parameter w, and weight a surface with h boundary components with a factor of 8 wh . 8 Still

more generally, we could introduce a finite set S of “labels” for the boundaries, so that each

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Introducing also the usual coupling parameters ti associated with the bulk observables τi , and one more parameter v associated with σ, the partition function of 2d topological gravity on a Riemann surface with boundary is then formally * !+ ∞ X ∞ X X h Z(ti ; v, w) = w . (3.3) exp ti τi + vσ h=0 Σ

i=0

Σ

The sum over Σ runs over all topological types of Riemann surface with h boundary components, and specified bulk and boundary punctures. The exponential on the right hand side is expanded in a power series, and the monomials of an appropriate degree are then evaluated via Eq. (3.1). There are two immediate difficulties with this formal definition: Q (1) To integrate a cohomology class such as i ψidi over a manifold M , that manifold must be oriented. But the moduli space of Riemann surfaces with boundary is actually unorientable. (2) For the integral of a cohomology class over an oriented manifold M to be well-defined topologically, M should have no boundary, or the cohomology class in question should be trivialized along the boundary of M . However, the compactified moduli space M of conformal structures on a Riemann surface with boundary is itself in general a manifold with boundary. Dealing with these issues requires some refinements of the above formal definition [3–6]. The rest of this section is devoted to an introduction. We begin with the unorientability of M. Implications of the fact that M is a manifold with boundary will be discussed in Subsec. 3.9. 3.2. The anomaly The problem in orienting the moduli space of Riemann surfaces with boundary can be seen most directly in the absence of boundary punctures. Thus we let Σ be a Riemann surface of genus g with h holes or boundary components and no boundary punctures, but possibly containing bulk punctures. First of all, if h = 0, then Σ is an ordinary closed Riemann surface, possibly with punctures. The compactified moduli space M of conformal structures on Σ is then a complex manifold (or more precisely an orbifold) and by consequence has a natural orientation. This orientation is used in defining the usual intersection numbers on M, that is, the correlation functions of 2d topological gravity on a Riemann surface without boundary. boundary is labeled by some s ∈ S. Then for each s ∈ S, one would have a boundary observable σs corresponding to a puncture inserted on a boundary with label s, and a corresponding parameter vs to count such punctures. Eq. (3.3) below corresponds to the case that w is the cardinality of the set S, and vs = v for all s ∈ S. This generalization to include labels would correspond in Q Eq. (4.49) below to modifying the matrix integral with a factor s∈S det(zs − Φ). Similarly, one Q could replace wh by s∈S wshs , where hs is the number of boundary components labeled by s and there is aQseparate parameter ws for each s. This corresponds to including in the matrix integral a factor s∈S (det(zs − Φ))ws . Such generalizations have been treated in [45].

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This remains true if Σ has punctures (which automatically are bulk punctures since so far Σ has no boundary). Now let us replace some of the punctures of Σ by holes. Each time we replace a bulk puncture by a hole, we add one real modulus to the moduli space. If we view Σ as a two-manifold with hyperbolic metric and geodesic boundaries, then the extra modulus is the circumference b around the hole. By adding h > 1 holes, we add h real moduli b1 , b2 , . . . , bh , which we write collectively as ~b. We denote the corresponding compactified moduli space as Mg,n,~b (here n is the number of punctures that have not been converted to holes). A very important detail is the following. In defining Weil–Petersson volumes in Sec. 2, we treated the bi as arbitrary constants; the “volume” was defined as the volume for fixed ~b, without trying to integrate over ~b. (Such an integral would have been divergent since the volume function Vg,n,~b is polynomial in ~b. Moreover, what naturally enters Mirzakhani’s recursion relation is the volume function defined for fixed ~b.) In defining two-dimensional gravity on a Riemann surface with boundary, the bi are treated as full-fledged moduli — they are some of the moduli that one integrates over in defining the intersection numbers. Hopefully this change in viewpoint relative to Sec. 2 will not cause serious confusion. If we suppress the bi by setting them all to 0, the holes turn back into punctures and Mg,n,~b is replaced by Mg,n+h . This is a complex manifold (or rather an orbifold) and in particular has a natural orientation. Restoring the bi , Mg,n,~b is a fiber bundle9 over Mg,n+h with fiber a copy of Rh+ parametrized by b1 , . . . , bh . (Here R+ is the space of positive real numbers and Rh+ is the Cartesian product of h copies of R+ .) Orienting Mg,n,~b is equivalent to orienting this copy of Rh+ . If we were given an ordering of the holes in Σ up to an even permutation, we would orient Rh+ by the differential form Ω = db1 db2 · · · dbh .

(3.4)

However, for h > 1, in the absence of any information about how the holes should be ordered, Rh+ has no natural orientation. Thus Mg,n,~b has no natural orientation for h > 1. In fact it is unorientable. This follows from the fact that a Riemann surface Σ with more than one hole has a diffeomorphism that exchanges two of the holes, leaving the others fixed. (Moreover, this diffeomorphism can be chosen to preserve the orientation of Σ.) Dividing by this diffeomorphism in constructing the moduli space Mg,n,~b ensures that this moduli space is actually unorientable. We can view this as a global anomaly in two-dimensional topological gravity on an oriented two-manifold with boundary. The moduli space is not oriented, or even 9 This

assertion actually follows from a fact that was exploited in Sec. 2: for fixed ~b, Mg,n,~b is

isomorphic to Mg,n as an orbifold (their symplectic structures are inequivalent, as we discussed in Sec. 2). Given this equivalence for fixed ~b, it follows that upon letting ~b vary, Mg,n,~b is a fiber bundle over Mg,n+h with fiber paramerized by ~b.

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orientable, so there is no way to make sense of the correlation functions that one wishes to define. As usual, to cancel the anomaly, one can try to couple two-dimensional topological gravity to some matter system that carries a compensating anomaly. In the context of two-dimensional topological gravity, the matter system in question should be a topological field theory. To define a theory that reduces to the usual topological gravity when the boundary of Σ is empty, we need a topological field theory that on a Riemann surface without boundary is essentially trivial, in a sense that we will see, and in particular is anomaly-free. But the theory should become anomalous in the presence of boundaries. These conditions may sound too strong, but there actually is a topological field theory with the right properties. First of all, we endow Σ with a spin structure. (We will ultimately sum over spin structures to get a true topological field theory that does not depend on the choice of an a priori spin structure on Σ.) When ∂Σ = ∅, we can define a chiral Dirac operator on Σ (a Dirac operator acting on positive chirality spin 1/2 fields on Σ). There is then a Z2 -valued invariant that we call ζ, namely the mod 2 index of the chiral Dirac operator, in the sense of Atiyah and Singer [46,47]. ζ is defined as the number of zero-modes of the chiral Dirac operator, reduced mod 2. ζ is a topological invariant in that it does not depend on the choice of a conformal structure (or metric) on Σ. A spin structure is said to be even or odd if the number of chiral zero-modes is even or odd (in other words if ζ = 0 or ζ = 1). For an introduction to these matters, see [48], especially Subsec. 3.2. We define a topological field theory by summing over spin structures on Σ with each spin structure weighted by a factor of 12 (−1)ζ . The reason for the factor of 1 2 is that a spin structure has a symmetry group that acts on fermions as ±1, with 2 elements. As in Faddeev–Popov gauge-fixing in gauge theory, to define a topological field theory, one needs to divide by the order of the unbroken symmetry group, which in this case is the group Z2 . This accounts for the factor of 21 . The more interesting factor, which will lead to a boundary anomaly, is (−1)ζ . It may not be immediately apparent that we can define a topological field theory with this factor included. We will describe two realizations of the theory in question in Subsec. 3.4, making it clear that there is such a topological field theory. We will call it T . On a Riemann surface of genus g, there are 12 (22g + 2g ) even spin structures and 1 2g − 2g ) odd ones. The partition function of T in genus g is thus 2 (2     1 1 2g 1 2g g g Zg = (2 + 2 ) − (2 − 2 ) = 2g−1 . (3.5) 2 2 2 This is not equal to 1, and thus the topological field theory T is nontrivial. However, when we couple to topological gravity, the genus g amplitude has a factor 2g−2 gst , where gst is the string coupling constant.10 The product of this with Zg is 10 In

mathematical treatments, gst is often set to 1. There is no essential loss of information, as

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2 g−1 (2gst ) . Thus, as long as we are on a Riemann surface without boundary,√coupling topological gravity to T can be compensated11 by absorbing a factor of 2 in the definition of gst . In that sense, coupling of topological gravity to T has no effect, as long as we consider only closed Riemann surfaces. Matters are different if Σ has a boundary. On a Riemann surface with boundary, it is not possible to define a local boundary condition for the chiral Dirac operator that is complex linear and sensible (elliptic), and there is no topological invariant corresponding to ζ. Thus theory T cannot be defined as a topological field theory on a manifold with boundary. It is possible to define theory T on a manifold with boundary as a sort of anomalous topological field theory, with an anomaly that will help compensate for the problem that we found above with the orientation of the moduli space. To explain this, we will first describe some more physical constructions of theory T . First we discuss how theory T is related to contemporary topics in condensed matter physics.

3.3. Relation to condensed matter physics Theory T has a close cousin that is familiar in condensed matter physics. One considers a chain of fermions in 1 + 1 dimensions with the property that in the bulk of the chain there is an energy gap to the first excited state above the ground state, and the further requirement that the chain is in an “invertible” phase, meaning that the tensor product of a suitable number of identical chains would be completely trivial.12 There are two such phases, just one of which is nontrivial. The nontrivial phase is called the Kitaev spin chain [50]. It is characterized by the fact that at the end of an extremely long chain, there is an unpaired Majorana fermion mode, usually called a zero-mode because in the limit of a long chain, it commutes with the Hamiltonian.13 The Kitaev spin chain is naturally studied in condensed matter physics from a Hamiltonian point of view, which in fact we adopted in the last paragraph. From a relativistic point of view, the Kitaev spin chain corresponds to a topological field theory that is defined on an oriented two-dimensional spin manifold Σ, and the dependence on gst carries the same information as the dependence on the parameter t1 in the generating function. This follows from the dilaton equation, that is, the L0 Virasoro constraint. 11 This point was actually made in [49], as a special case of a more general discussion involving rth roots of the canonical bundle of Σ for arbitrary r ≥ 2. 12 Triviality here means that by deforming the Hamiltonian without losing the gap in the bulk spectrum, one can reach a Hamiltonian whose ground state is the tensor product of local wavefunctions, one on each lattice site. 13 A long but finite chain has a pair of such Majorana modes, one at each end. Upon quantization, they generate a rank two Clifford algebra, whose irreducible representation is two-dimensional. As a result, a long chain is exponentially close to having a two-fold degenerate ground state. In condensed matter physics, this degeneracy is broken by tunnelling effects in which a fermion propagates between the two ends of the chain. In the idealized model considered below, the degeneracy is exact.

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whose partition function if Σ has no boundary is (−1)ζ . We will see below how this statement relates to more standard characterizations of the Kitaev spin chain. Our theory T differs from the Kitaev spin chain simply in that we sum over spin structures in defining it, while the Kitaev model is a theory of fermions and is defined on a two-manifold with a particular spin structure. Moreover, as we discuss in detail below, when Σ has a boundary, the appropriate boundary conditions in the context of condensed matter physics are different from what they are in our application to two-dimensional gravity. Despite these differences, the comparison between the two problems will be illuminating. Because we are here studying two-dimensional gravity on an oriented twomanifold Σ, time-reversal symmetry, which corresponds to a diffeomorphism that reverses the orientation of Σ, will not play any role. The Kitaev spin chain has an interesting time-reversal symmetric refinement, but this will not be relevant. Theory T has another interesting relation to condensed matter physics: it is associated to the high temperature phase of the two-dimensional Ising model. In this interpretation [51], the triviality of theory T corresponds to the fact that the Ising model in its high temperature phase has only one equilibrium state, which moreover is gapped. 3.4. Two realizations of theory T We will describe two realizations of theory T , one in the spirit of condensed matter physics, where we get a topological field theory as the low energy limit of a physical gapped system, and one in the spirit of topological sigma models [52], where a supersymmetric theory is twisted to get a topological field theory. First we consider a massive Majorana fermion in two spacetime dimensions. It is convenient to work in Euclidean signature. One can choose the Dirac operator to be Dm = γ 1 D1 + γ 2 D2 + mγ,

(3.6)

where one can choose the gamma matrices to be real and symmetric, for instance     01 1 0 1 2 γ = , γ = . (3.7) 10 0 −1 This ensures that γ = γ 1 γ 2 is real and antisymmetric:   0 −1 γ= . 1 0

(3.8)

These choices ensure that the Dirac operator Dm is real and antisymmetric. We call m the mass parameter; the mass of the fermion is actually |m|. Formally, the path integral for a single Majorana fermion is Pf(Dm ), the Pfaffian of the real antisymmetric operator Dm . The Pfaffian of a real antisymmetric operator is real, and its square is the determinant; in the present context, the determinant

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det Dm = (Pf(Dm ))2 can be defined satisfactorily by, for example, zeta-function regularization. However, the sign of the Pfaffian is subtle. For a finite-dimensional real antisymmetric matrix M , the sign of the Pfaffian depends on an orientation of the space that M acts on. In the case of the infinite-dimensional matrix Dm , no such natural orientation presents itself and therefore, for a single Majorana fermion, there is no natural choice of the sign of Pf(Dm ). Suppose, however, that we consider a pair of Majorana fermions with the same (nonzero) mass parameter m. Then the path integral is Pf(Dm )2 and (since Pf(Dm ) is naturally real) this is real and positive. This actually ensures that the topological field theory obtained from the low energy limit of a pair of massive Majorana fermions of the same mass parameter is completely trivial. Without losing the mass gap, we can take |m| → ∞, and in that limit, Pf(Dm )2 produces no physical effects at all except for a renormalization of some of the parameters in the effective action.14 To get theory T , we consider instead a pair of Majorana fermions, one of positive mass parameter and one of negative mass parameter. Just varying mass parameters, to interpolate between this theory and the equal mass parameter case, we would have to let a mass parameter pass through 0, losing the mass gap. This suggests that a theory with opposite sign mass parameters might be in an essentially different phase from the trivial case of equal mass parameters. To establish this and show the relation to theory T , we will analyze what happens to the partition function of the theory when the mass parameter of a single Majorana fermion is varied between positive and negative values. The absolute value of Pf(Dm ) does not depend on the sign of m. This follows 2 is invariant under m → −m. (The determinant from the fact that the operator Dm 2 of −Dm is a power of Pf(Dm ), and the fact that it is invariant under m → −m implies that Pf(Dm ) is independent of sign(m) up to sign.) Therefore the partition functions of the two theories with opposite masses or with equal masses have the same absolute value. They can differ only in sign, and this sign is what we want to understand. To determine the sign, we ask what happens to Pf(Dm ) when m is varied from large positive values to large negative ones. To change sign, Pf(Dm ) has to vanish, and it vanishes only when Dm has a zero-mode. This can only happen at m = 0. So the question is just to determine what happens to the sign of Pf(Dm ) when m passes through 0. Zero-modes of Dm at m = 0 are simply zero-modes of the massless Dirac operator D = γ 1 D1 + γ 2 D2 . Such modes appear in pairs of equal and opposite chirality. To be more precise, let γ b = iγ be the chirality operator, with eigenvalues 1 and −1 for fermions of positive or negative chirality. What we called the chiral Dirac 14 In

two dimensions, when we integrate out a massive neutral field, the only parameters that have to be renormalized are the vacuum energy, which corresponds to a term in the R R effective √ √ action proportional to the volume Σ d2 x g of Σ, and the coefficient of another term Σ d2 x gR proportional to the Euler characteristic of Σ (here R is the Ricci scalar of Σ).

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operator when we defined the mod 2 index in section 3.2 is the operator D restricted to act on states of γ b = +1. Since γ b is imaginary and D is real, complex conjugation of an eigenfunction reverses its γ b eigenvalue while commuting with D; thus zero-modes of D occur in pairs with equal and opposite chirality. Restricted to such a pair of zero-modes of D, the antisymmetric operator Dm looks like   0 −m , (3.9) m 0 and its Pfaffian is m, up to an m-independent sign that depends on a choice of orientation in the two-dimensional space. The important point is that this Pfaffian changes sign when m changes sign. If there are s pairs of zero-modes, the Pfaffian in the zero-mode space is ms , and so changes in sign by (−1)s when m changes sign. But s is precisely the number of zero-modes of positive chirality, and (−1)s is the same as (−1)ζ , where ζ is the mod 2 index of the Dirac operator. Now we can answer the original question. Since the partition function for the theory with two equal mass parameters is trivial (up to a renormalization of some of the low energy parameters), the partition function of the theory with one mass of each sign is (−1)ζ (up to such a renormalization). Thus we have found a physical realization of theory T . The result we have found can be interpreted in terms of a discrete chiral anomaly. At the classical level, for m = 0, the Majorana fermion has a Z2 chiral symmetry15 ψ→γ bψ. The mass parameter is odd under this symmetry, so classically the theories with positive or negative m are equivalent. Quantum mechanically, one has to ask whether the fermion measure is invariant under the discrete chiral symmetry. As usual, the nonzero modes of the Dirac operator are paired up in such a way that the measure for those modes is invariant; thus one only has to test the zero-modes. Since ψ → γ bψ leaves invariant the positive chirality zero-modes and multiplies each negative chirality zero-mode by −1, this operation transforms the measure by a factor (−1)s = (−1)ζ , where s is the number of negative (or positive) chirality zero-modes, and ζ is the mod 2 index. Finally, we will describe another though closely related way to construct the same topological field theory. The extra machinery required will be useful later. We consider in two dimensions a theory with (2, 2) supersymmetry and a single complex chiral superfield Φ. We work in flat spacetime to begin with and assume a familiarity with the usual superspace formalism of (2, 2) supersymmetry and its 15 With

our conventions, the operator γ b is imaginary in Euclidean signature and one might wonder if this symmetry makes sense for a Majorana fermion. However, after Wick rotation to Lorentz signature (in which γ 0 acquires a factor of i), γ b becomes real, and it is always in Lorentz signature that reality conditions should be imposed on fermion fields and their symmetries. Thus actually ψ → γ bψ is a physically meaningful symmetry and ψ → γψ (which may look more natural in Euclidean signature) is not. Under the latter transformation, the massless Dirac action actually changes sign, so it is indeed ψ → γ bψ and not ψ → γψ that is a symmetry at the classical level.

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twisting to make a topological field theory. Φ can be expanded Φ = φ + θ− ψ+ + θ+ ψ− + θ+ θ− F.

(3.10)

Here φ is a complex scalar field; ψ+ and ψ− are the chiral components of a Dirac fermion field; and F is a complex auxiliary field. We consider the action Z Z Z i i 2 S = d2 xd4 θΦΦ + d2 xd2 θ mΦ2 − d2 xd2 θ mΦ . (3.11) 2 2 Thus the superpotential is W (Φ) = imΦ2 /2. In general, here m is a complex mass parameter, but for our purposes we can assume that m > 0. After integrating over the θ’s and integrating out the auxiliary field F , the action becomes16 Z
S = d2 x ∂µ φ∂ µ φ + m2 |φ|2 + ψγ µ ∂µ ψ + imαβ (ψα ψβ − ψ α ψ β ) . (3.12) If we expand the √Dirac fermion ψ in terms of a pair of Majorana fermions χ1 , χ2 by ψ = (χ1 + iχ2 )/ 2, we find that χ1 and χ2 are massive Majorana fermions with a mass matrix that has one positive and one negative eigenvalue, as in our previous construction of theory T . The massive field φ does not play an important role at low energies: its path integral is positive definite, and in the large m or low energy limit, just contributes renormalization effects. So at low energies the supersymmetric theory considered here gives another realization of theory T . However, the supersymmetric machinery gives a way to obtain theory T without taking a low energy limit, and this will be useful later. Because the superpotential W = imΦ2 /2 is homogeneous in Φ, the theory has a U (1) R-symmetry that acts on the superspace coordinates as θ± → eiα θ± . Because W is quadratic in Φ, one has to define this symmetry to leave ψ invariant and to transform φ by φ → eiα φ. When one “twists” to make a topological field theory, the spin of a field is shifted by one-half of its R-charge. In the present case, as ψ is invariant under the R-symmetry, it remains a Dirac fermion after twisting, but φ acquires spin +1/2 (it transforms under rotations like the positive chirality part of a Dirac fermion). The twisted theory can be formulated as a topological field theory on any Riemann surface Σ with any metric tensor. We use the phrase “topological field theory” loosely since the twisted theory, as it has fields of spin 1/2, requires a choice of spin structure. To get a true topological field theory, one has to sum over the choice of spin structure. The supersymmetry of the twisted theory ensures that the path integral over φ cancels the absolute value of the path integral over ψ, leaving only the sign (−1)ζ . Thus the twisted theory is precisely equivalent to theory T , without taking any low energy limit. In [49], this last statement is deduced in another way as a special case of an analysis of a theory with Φr superpotential for any r ≥ 2. 16 Here

αβ is the Levi-Civita antisymmetric tensor in the two-dimensional space spanned by ψ+ , ψ− .

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For our later application, it will be useful to know that the condition for a configuration of the φ field to preserve the supersymmetry of the twisted theory is ∂φ + imφ = 0.

(3.13)

The generalization of this equation for arbitrary superpotential is ∂φ +

∂W = 0, ∂φ

(3.14)

which has been called the ζ-instanton equation [53]. A small calculation shows that   φ 1 if we set φ = φ1 + iφ2 with real φ1 , φ2 , and set φb = , then Eq. (3.13) is −φ2 equivalent to Dm φb = 0,

(3.15)

with Dm the massive Dirac operator (3.6). 3.5. Boundary conditions in theory T Our next task is to consider theory T on a manifold with boundary. Here of course we must begin by discussing possible boundary conditions. In this section, we will use the realization of theory T in terms of a pair of Majorana fermions with opposite masses. The main requirement for a boundary condition is that it should preserve the antisymmetry of the operator Dm . If tr denotes the transpose, then the antisymmetry means concretely that Z
χtr Dm ψ + (Dm χ)tr ψ = 0. (3.16) In verifying this, one has to integrate by parts, and one encounters a surface term, which is the boundary integral of χtr γ⊥ ψ, where γ⊥ is the gamma matrix normal to the boundary. This will vanish if we impose the boundary condition γk ψ = ηψ, (3.17) where η = +1 or −1, γk is the gamma matrix tangent to the boundary, and | represents restriction to the boundary. Just to ensure the antisymmetry of the operator Dm , either choice of sign will do. With either choice of sign, Dm is a real operator, so its Pfaffian Pf(Dm ) remains real. The boundary conditions γk ψ = ±ψ have a simple interpretation. Tangent to the boundary, there is a single gamma matrix γk . It generates a rank 1 Clifford algebra, satisfying γk2 = 1. In an irreducible representation, it satisfies γk = 1 or γk = −1. Thus the spin bundle of Σ, which is a real vector bundle of rank 2, decomposes along ∂Σ as the direct sum of two spin bundles of ∂Σ, namely the subbundles defined respectively by γk ψ = ψ and by γk ψ = −ψ. These two spin bundles of ∂Σ are isomorphic, since they are exchanged by multiplication by γ⊥ ,

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which is globally-defined along ∂Σ. Thus the spin bundle of Σ decomposes along ∂Σ in a natural way as the direct sum of two copies of the spin bundle of ∂Σ, and the boundary condition says that along the boundary, ψ takes values in one of these bundles. We will write S for the spin bundle of Σ and E for the spin bundle of ∂Σ. Now let us discuss the behavior near the boundary of a Majorana fermion that satisfies one of these boundary conditions. We work on a half-space in R2 , say the half-space x1 ≥ 0 in the x1 x2 plane. For a mode that is independent of x2 , the Dirac equation Dm ψ = 0 becomes   d + mγ2 ψ = 0, (3.18) dx1 with solution ψ = exp(−mx1 γ2 )ψ0 ,

(3.19)

for some ψ0 . In this geometry, γ2 is the same as γk . We see that if ψ satisfies the boundary condition γk ψ = ηψ, then this mode is normalizable if and only if mη > 0.

(3.20)

If mη < 0, the theory remains gapped, with a gap of order m, even along the boundary. But if mη > 0, the mode that we have just found propagates along the boundary as a (0 + 1)-dimensional massless Majorana fermion. We will now use these results to study the boundary anomaly of theory T , with several possible boundary conditions. 3.6. Boundary anomaly of theory T Let us first recall that for a real fermion field with a real antisymmetric Dirac operator such as Dm , in general there is an anomaly in the sign of the path integral Pf(Dm ). The anomaly is naturally described mathematically by saying that there is a real Pfaffian line PF associated to the Dirac operator, and the fermion Pfaffian Pf(Dm ) is well-defined as a section of PF . In our problem, there are two Majorana fermions, say ψ1 and ψ2 , with possibly different masses and possibly different boundary conditions. Correspondingly there are two Pfaffian lines, say PF 1 and PF 2 , and the overall Pfaffian line is the tensor product17 PF = PF 1 ⊗ PF 2 . 17 There

is a potential subtlety here. If a fermion field has an odd number of zero-modes, its Pfaffian line should be considered odd or fermionic. Accordingly, if ψ1 and ψ2 each have an odd number of b zero-modes, then PF 1 and PF 2 are both odd and the correct statement is that PF = PF 1 ⊗PF 2, b is a Z2 -graded tensor product (this notion is described in Subsec. 3.6.2). We will not where ⊗ encounter this subtlety, because always at least one of ψ1 and ψ2 will satisfy one of the boundary conditions (3.17). A fermion field obeying one of those boundary conditions has an even number of zero-modes, since there are none at all if mη < 0 and the number is independent of m mod 2. Note that on a Riemann surface with boundary, there is no notion of the chirality of a zero-mode and we simply count all zero-modes. By contrast, the mod 2 index that is used in defining theory T on a surface without boundary is defined by counting positive chirality zero-modes only.

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In general, the Pfaffian line of a Dirac operator does not depend on a fermion mass, but it may depend on the boundary conditions. Indeed, as we will see, there is such a dependence in our problem and it will play an essential role. We will now consider the boundary path integral and boundary anomaly in our problem for several choices of boundary condition. 3.6.1. The trivial case The most trivial case is that the two masses and also the two boundary conditions are the same. Moreover, we choose the masses and the signs so that mη < 0. Since the two boundary conditions are the same, PF 1 is canonically isomorphic to PF 2 , and therefore PF = PF 1 ⊗ PF 2 is canonically trivial. Since the two Majorana fermions have the same mass and boundary condition, the combined Dirac operator D of the two modes is just the direct sum of two copies of the same Dirac operator Dm . Thus the fermion path integral Pf(D) satisfies Pf(D) = Pf(Dm )2 , and in particular Pf(D) is naturally positive (relative to the trivialization of PF that reflects the isomorphism PF 1 ∼ = PF 2 ). Since mη < 0, there are no low-lying modes near the boundary and the theory has a uniform mass gap of order m along the boundary as well as in the bulk. Therefore, after renormalizing a few constants in the low energy effective action, the path integral Pf(D) is just 1. In other words, with equal boundary conditions for the two modes, the trivial theory with equal masses remains trivial along the boundary. Assuming we allow ourselves to make a generic relevant deformation of the theory (as we would certainly do in condensed matter physics, for example), this is still true if we pick the boundary conditions for the two Majorana fermions to be equal but such that mη > 0. Then we generate two (0 + 1)-dimensional massless Majorana fermions, say χ1 , χ2 . But given any such pair of Majorana modes in (0 + 1) dimensions, one can add a mass term iµχ1 χ2 to the Hamiltonian (or the Lagrangian), with some constant µ, removing them from the low energy theory. The theory becomes gapped and the renormalized partition function is again 1. Fermi statistics do not allow the addition of a mass term for a single massless 1d Majorana fermion. Hence the number of 1d Majorana modes along the boundary is a topological invariant mod 2. We will discuss next the case that this invariant is nonzero. 3.6.2. Boundary condition in condensed matter physics For theory T , or for the Kitaev spin chain, we consider two Majorana fermions, with opposite signs of m. In the context of condensed matter physics, to study the theory on a manifold with boundary, we want a boundary condition that makes the theory fully anomaly-free. In other words, we want to ensure that the Pfaffian line bundle PF = PF 1 ⊗ PF 2 remains canonically trivial. This is straightforward: since PF is

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in general independent of the masses, we simply use the same boundary condition as in Subsec. 3.6.1 — namely the same sign of η for both Majorana fermions — and then PF remains canonically trivial, regardless of the masses. However, since the two Majorana fermions have opposite signs of m, we see now that regardless of the common choice of η, precisely one of them has a normalizable zero-mode (3.19) along the boundary. This means that the mass gap of the theory breaks down along the boundary. Although it is gapped in bulk, there is a single (0 + 1)-dimensional massless Majorana fermion propagating along the boundary. As we noted in Subsec. 3.3, this is regarded in condensed matter physics as the defining property of the Kitaev spin chain. Now let us discuss a consequence of this construction that has been important in mathematical work [3–6] on 2d gravity on a manifold with boundary. We will see later the reason for its importance. In general, suppose that Σ has h boundary components ∂1 Σ, ∂2 Σ, . . . , ∂h Σ. On each boundary component, one makes a choice of sign in the boundary condition, and this determines a real spin bundle Ei of ∂i Σ. Along each ∂i Σ, there propagates a massless 1d Majorana fermion χi . In propagating around ∂i Σ, χi may obey either periodic or antiperiodic boundary conditions. Indeed, on the circle ∂i Σ, there are two possible spin structures, which in string theory are usually called the Neveu-Schwarz or NS (antiperiodic) spin structure and the Ramond (periodic) spin structure. The NS spin structure is bounding and the R spin structure is unbounding. The underlying spin bundle S of Σ determines whether Ei is of NS or Ramond type. For general S, the only general constraint on the Ei is that the number R of boundary components with Ramond spin structure is even. The field χi , in propagating around the circle ∂i Σ, has a zero-mode if and only if Ei is of Ramond type. This is not an exact zero-mode, but it is exponentially close to being one if m is large (compared to the inverse of the characteristic length scale of Σ). Let us write νi , i = 1, . . . , R, for these modes. The νi have much smaller eigenvalues of Dm than any other modes of ψ1 and ψ2 , so there is a consistent procedure in which we integrate out all other modes and leave an effective theory of the νi only. Since the underlying theory was chosen to be anomaly-free, it must determine a well-defined measure for the νi . This condition is not as innocent as it may sound. A measure on the space parametrized by the νi is something like dν1 dν2 · · · dνR .

(3.21)

However, a priori, this expression does not have a well-defined sign. First of all, its sign is obviously changed if we make an odd permutation of the νi , that is of the Ramond boundary components. But in addition, we should worry about the signs of the individual νi . Since the νi are real, we can fix their normalization up to sign by asking them to have, for example, unit L2 norm. But there is no natural way to choose the signs of the νi , and obviously, flipping an odd number of the signs will reverse the sign of the measure (3.21).

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There is no natural way to pick the signs of the νi up to an even number of sign flips, and likewise, there is no natural way to pick an ordering of the νi up to even permutations. However, the fact that there is actually a well-defined measure on the space spanned by ν1 , . . . , νR means that one of these choices determines the other. This fact (originally proved in a very different way) is an important lemma in [3–6]. The existence of a natural measure on the space spanned by the νi can be expressed in the following mathematical language. For i = 1, . . . , R, let εi be the onedimensional real vector space generated by νi . The Z2 -graded tensor product18 of b i εi , is equivalent to the ordinary tensor product once an ordering of the εi , denoted ⊗ the εi is picked, by an isomorphism that reverses sign if two of the εi are exchanged. The lemma that we have been describing is equivalent to the statement that the Z2 -graded tensor product of the εi is canonically trivial: b Ri=1 εi ∼ ⊗ = R.

(3.22)

3.6.3. Boundary condition in two-dimensional gravity For the application of theory T to two-dimensional gravity — or at least to the theory studied in [3–6] — we need a different boundary condition. In this application, we want theory T to remain gapped along the boundary as well as in bulk. But it will have an anomaly that will help in canceling the gravitational anomaly. Thus, the two Majorana fermions must remain gapped along the boundary, even though they have opposite masses. To achieve this, we must give the two Majorana fermions opposite boundary conditions, so that mη < 0 for each of the two modes. Given that the theory has a uniform mass gap of order m even near the boundary, its path integral, after renormalizing a few parameters in the effective action, is of modulus 1. Moreover, this path integral is naturally real. Thus it is fairly natural to write the path integral as (−1)ζ , just as we did in the absence of a boundary.19 However, (−1)ζ is no longer a number ±1; it now takes values in the real line bundle PF . In fact, since it is everywhere nonzero, (−1)ζ is a trivialization of PF . This theory actually challenges some of the standard terminology about anomalies. The line bundle PF is clearly trivial, because the renormalized partition function (−1)ζ provides a trivialization. However, because this trivialization is provided by the path integral itself, rather than by more local or more elementary considerations, it is not natural to call the theory anomaly-free. When we say that a theory 18 Since

this notion may be unfamiliar, we give Q an example, following P. Deligne. Let Si , i = 1, . . . , t be a family of circles, and let T be the torus ti=1 Si . Then εi = H 1 (Si , R) is a one-dimensional vector space, as is α = H n (T, R). There can be no natural isomorphism between α and the ordinary tensor product ⊗i εi , since the exchange of two of the circles acts trivially on ⊗i εi , while b i εi . acting on α as −1. But there is a canonical isomorphism α ∼ =⊗ 19 This is a path integral for a particular spin structure. As usual, to make the partition function of theory T , we sum over spin structures and divide by 2.

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is anomaly-free, we usually mean that its path integral can be defined as a number, rather than as a section of a line bundle; that is not the case here. In our problem, PF cannot be trivialized by local considerations. Rather, local considerations will give an isomorphism b Ri=1 εi , PF ∼ =⊗

(3.23)

where the product is over all boundary components with Ramond spin structure. This claim is consistent with the claim that PF is trivial, because we have shown b Ri=1 εi is trivial. in Eq. (3.22) that ⊗ To explain what we mean in saying that (3.23) can be established by local considerations, first set V = ⊕Ri=1 εi .

(3.24)

Then the statement (3.23) is equivalent to PF ∼ = det V,

(3.25)

where for a vector space V , det V is its top exterior power. (Note that exchanging two summands εi and εj in V acts as −1 on det V , and likewise acts as −1 on the Z2 -graded tensor product in (3.23).) We will use the following fact about Pfaffian line bundles. Consider a family of real Dirac operators parametrized by some space W (in our case, W represents the choice of metric on Σ). As long as the space of zero-modes of the Dirac operator has a fixed dimension, it furnishes the fiber of a vector bundle V → W . The Pfaffian line bundle PF → W is then det V , the top exterior power of V . More generally, instead of considering zero-modes, we can consider any positive number a that (in a given portion of W ) is not an eigenvalue of iDm , and let V be the space spanned by eigenvectors of the Dirac operator with eigenvalue less than a in absolute value. One still has an isomorphism PF ∼ = det V . Furthermore, the Pfaffian line bundle PF is independent of fermion masses. This means that to compute PF in our problem, instead of considering the case that the masses are opposite and the signs in the boundary conditions are also opposite, we can take the masses to be the same while the boundary conditions remain opposite. In this situation, one of the fields ψ1 , ψ2 has positive mη and one has negative mη. So although the interpretation is different, we are back in the situation considered in Subsec. 3.6.2: one fermion has a mass gap m that persists even along the boundary, and the other has a single low-lying made along each Ramond boundary component. The space of low-lying fermion modes is thus V = ⊕Ri=1 εi , and this leads to Eq. (3.23). Eq. (3.23) will suffice for our purposes, but it is perhaps worth pointing out that it has the following generalization, which is analogous to Theorem B in [54]. Instead of flipping the boundary condition simultaneously along all boundary components of Σ, it makes sense to flip the boundary condition along one boundary component at a time. Let S be a particular boundary component of Σ and let PF and PF 0 be

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the Pfaffian line bundles before and after flipping the boundary condition of one fermion along S. If the spin structure along S is of NS type, then PF 0 ∼ = PF ,

(3.26)

that is, changing the boundary condition has no effect. But if it is of Ramond type, then20 b PF 0 ∼ , = ε⊗PF

(3.27)

where ε is the space of fermion zero-modes along S. Repeated application of these rules, starting with the fact that PF is trivial if ψ1 and ψ2 have the same sign of η, leads to Eq. (3.23) for the case that they have opposite signs of η. To justify the statements (3.26) and (3.27), we use the fact that by the excision property of index theory, the change in the Pfaffian line when we flip the boundary condition along S depends only on the geometry along S and not on the rest of Σ. Thus we can embed S in any convenient Σ of our choice. It is convenient to take Σ to be the annulus S × I, where I = [0, 1] is a unit interval, and we consider S to be embedded in S × I as the left boundary S × {0}. We want to compute the effect of flipping the boundary condition at S × {0}, keeping it fixed at S × {1}. We can take the fermion mass to be 0, so the Dirac operator becomes conformally invariant and we can take the metric on the annulus to be flat. A fermion zero-mode is then simply a constant mode that satisfies the boundary conditions. For the case of an NS spin structure, the fermions are antiperiodic in the S direction and so have no zero-modes. Thus the space of zero-modes is V = 0, so that det V = R. This justifies (3.26) in the NS case. In the R case, flipping the boundary condition at one end adds or removes a zero-mode (depending on the boundary condition at the other end). The relevant space of zero-modes is V ∼ = ε, so that det V ∼ = ε, leading to (3.27). 3.7. Anomaly cancellation We are finally ready to explain how the anomaly that we described in Subsec. 3.2 has been canceled in [3–6]. We consider first the case that all boundaries of Σ are of Ramond type, and to start with, we omit boundary punctures. We denote the circumference of the ith boundary as bi . We recall that the reason for the anomaly is that there is no natural sign of the differential form Ω = db1 db2 · · · dbR (Eq. (3.4)). However, after coupling to theory T , what needs to have a natural sign is the product of this with (−1)ζ , the path integral of theory T : b = db1 db2 · · · dbR (−1)ζ . Ω 20 This

(3.28)

formula shows that if we flip the boundary condition for one of the Majorana fermions along just one of the Ramond boundaries or more generally along some but not all of them, then the Pfaffian line becomes nontrivial and the theory becomes genuinely anomalous.

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b i εi , where εi is a one-dimensional We recall, in addition, that (−1)ζ takes values in ⊗ th vector space of zero-modes along the i Ramond boundary. b i is a Z2 -graded tensor product, meaning that ⊗ b i εi changes sign if any Here ⊗ two of the boundary components are exchanged. But the original anomaly was that db1 db2 · · · dbR likewise changes sign if any two boundary components are exchanged. b does not change sign under permutations of The upshot then is that the product Ω boundary components. It naturally takes values in the ordinary tensor product of the εi : b ∈ ⊗Ri=1 εi . Ω

(3.29)

What have we gained? The anomaly has not disappeared, but it has become local: it has turned into an ordinary tensor product of factors associated with individual boundary components; because it is an ordinary tensor product, it can be canceled by a local choice made independently on each boundary component. The last step in canceling the anomaly is to say that a boundary of Σ is not just a “bare” boundary: it comes with additional structure. Let Si be the ith boundary component of Σ, and let Ei be its spin structure. In the theory developed in [3–6] (but for the moment still ignoring boundary punctures) each Si is endowed with a trivialization of Ei , up to homotopy. For the moment we consider Ramond boundaries only. Since Ei is a real line bundle, and is trivial on a Ramond boundary, it has two homotopy classes of trivialization over each Ramond boundary. In addition to summing over spin structures on Σ and integrating over its moduli, one is supposed to sum over (homotopy classes of) trivializations of Ei for each Ramond boundary Si . A fermion zero-mode on Si is a “constant” mode that is everywhere nonvanishing, so the choice of such a zero-mode gives a trivialization of Ei . This means that, still in the absence of boundary punctures, trivializations of Ei correspond to choices of the sign of the zero-mode on Si . Hence once we trivialize all the Ei , the b acquires a well-defined sign. right-hand side of (3.29) is trivialized and Ω Thus once theory T is included and the boundaries are equipped with trivializations of their spin bundles, the problem with the orientation of the moduli space is solved. However, without some further ingredients, all correlation functions would vanish. Indeed, summing over the signs of the trivializations of the εi will imply b summing over the sign of Ω. Moreover, what we have said does not make sense for boundaries with NS spin structure, since their spin bundles cannot be globally trivialized. The additional ingredient that has to be considered is a boundary puncture. One postulates that locally, away from punctures, Ei is trivialized, but that this trivialization changes sign in crossing a boundary puncture. With this rule, it is possible to incorporate NS as well as Ramond boundaries. A simple example of a boundary component with NS spin structure is the boundary of a disc (Fig. 5). Its spin structure is of NS or antiperiodic type, and cannot be trivialized globally. It can be trivialized on the complement of one point, but then

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the trivialization changes sign in crossing that point. In the theory of [3–6], that point would be interpreted as a boundary puncture. So an NS boundary with one boundary puncture is possible in the theory, but an NS boundary with no boundary punctures is not. More generally, the number of punctures on a given NS boundary can be any positive odd number, since the spin structure of an NS boundary can have a trivialization that jumps in sign any odd number of times in going around the boundary circle. As an example, a disc with n boundary punctures and m bulk ones has a moduli space of dimension 2m + n − 3. The fact that n is odd means that this number is even. is actually a necessary condition for some of the correlation functions R Q This di (Eq. (3.1)) to be nonzero, since the cohomology classes ψi are all of even ψ i i M degree. The spin structure of a Ramond puncture is globally trivial, so it is possible to have a Ramond boundary with no boundary punctures. Of course, this is the case we started with. More generally, a Ramond boundary can have any even number of punctures. On any given boundary component of either NS or Ramond type, there are two allowed classes of piecewise trivialization of the spin structure. One can pick an arbitrary trivialization at a given starting point (not one of the punctures), and then the extension of this over the rest of the circle is uniquely determined by the condition that the trivialization jumps in sign whenever a boundary puncture is crossed.

Fig. 5. A disc with five boundary punctures. The spin bundle of the boundary circle S is a real line bundle that is inevitably of NS type. This real line bundle is not trivial globally over S, but — since the number of boundary punctures is odd — it can be trivialized on the complement of the boundary punctures in such a way that the trivialization changes sign whenever one crosses a boundary puncture.

This description of boundaries and their punctures may seem bizarre at first, but we will see in Subsec. 3.8 that it is not too difficult to give it a plausible physical interpretation. But first, let us ask whether incorporating boundary punctures has reintroduced any problem with the orientation of moduli space. We will deal with this question by describing a consistent recipe [3–6] for dealing with the sign ques-

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tions. We expect that this recipe could be deduced from the framework of Subsec. 3.8, but we will not show this. First let us consider the case of a boundary component S with NS spin structure. It has a circumference b and it has an odd number n of boundary punctures that have a natural cyclic order. Let us pick an arbitrary starting point p ∈ S and relative to this label the punctures in ascending order by angles α1 < α2 < · · · < αn . So b and α1 , . . . , αn are the moduli that are associated to S. To orient this parameter space, we can use the differential form Υ = db dα1 dα2 · · · dαn .

(3.30)

We note that Υ has a natural sign: because the number of α’s is odd, moving a dα from the end of the chain to the beginning does not affect the sign of Υ. Also, since Υ is of even degree, it commutes with similar factors associated to other boundary components. Therefore, an NS boundary component raises no problem in orienting the moduli space. Now let S have Ramond spin structure. In this case, n is even. This has two consequences. First, we get a sign change if we move a dα from the end of the chain to the beginning. However, just as in the case n = 0 that we started with, the sign of (−1)ζ depends on how one trivializes the spin structure of a Ramond boundary. A consistent recipe is to define the sign of (−1)ζ using the trivialization that is in effect just to the right of the starting point p ∈ S relative to which we measured the α’s. Then moving one of the boundary punctures from the end of the chain to the beginning will reverse the sign of Υ while also reversing the sign of (−1)ζ . Also, because n is even, Υ is of odd degree in the case of a Ramond boundary. Therefore the Υ factors associated to different Ramond boundaries anticommute with each other. Just as we discussed for the case n = 0, this compensates for the fact that (−1)ζ is odd under exchanging any two Ramond boundaries. 3.8. Branes 3.8.1. The ζ-instanton equation and compactness In the present section, we will attempt to interpret the possibly strange-sounding picture just described in terms of the physics of branes. For this, it will be helpful to use the second realization of theory T that was presented in Subsec. 3.4. This was based on topologically twisting a two-dimensional theory with (2, 2) supersymmetry and a complex chiral superfield Φ. The bottom component of Φ is a complex field φ. The theory also has a holomorphic superpotential, which in our application is W (Φ) = 2i m2 Φ2 , but we will write some formulas for a more general W (Φ). The condition for a configuration of the φ field to be supersymmetric is the ζ-instanton equation ∂φ ∂W + = 0. ∂z ∂φ

(3.31)

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This equation can be written ∂φ + dz∂φ W , so it can be defined on a Riemann surface Σ with a distinguished, everywhere nonzero (0, 1)-form dz. (For example, such a form exists globally if Σ is a Riemann surface of genus 1 or a domain in the complex plane.) If W is quasihomogeneous, as in our case, the equation is conformally-invariant if φ is suitably interpreted. The conformally-invariant version of the equation can be formulated on any Riemann surface. This conformallyinvariant form of the equation is what one gets when one topologically twists the theory using the R-symmetry that exists for quasi-homogeneous W . For example, in the case of a quadratic W , after topological twisting, φ has to be interpreted as a section of a chiral spin bundle L → Σ, a square root of the canonical bundle K → Σ. In [49], a more general case W ∼ Φr was considered, and then in the topologically twisted version of the theory, φ is a section of an rth root of K (this rth root may have singularities at specified points in Σ where “twist fields” are inserted). Certain important properties hold whenever the ζ-instanton equation can be defined, whether in a topologically-twisted version or simply in a naive version in which φ is a complex field. In particular, if Σ has no boundary, then the ζ-instanton equation has only “trivial” solutions. This is proved in a standard way: take the absolute value squared of the equation, integrate over Σ, and then integrate by parts, to show that any solution satisfies ! 2 Z 2 Z
∂W ∂W ∂φ = + ∂z W + ∂z W |d2 z| |dφ|2 + . (3.32) + 0= |d2 z| ∂z ∂φ ∂φ Σ Σ If Σ has no boundary, we can drop the total derivatives ∂z W and ∂z W , and we learn that on a closed surface Σ, any solution has dφ = 0 and ∂W/∂φ = 0; in other words, φ must be constant and this constant must be a critical point of W . For a large class of W ’s, this implies that, on a surface Σ without boundary, the space of solutions of the ζ-instanton equation is compact (and in fact “trivial”). This compactness is an important ingredient in the well-definedness of the twisted topological field theory constructions related to the ζ-instanton equation. 3.8.2. Boundary condition in the ζ-instanton equation If Σ has a boundary, then we have to pick a boundary condition on the ζ-instanton equation. Let us first ignore the twisting and treat φ as an ordinary complex scalar field. If we also set W to 0, the equation for φ becomes the Cauchy–Riemann equation saying that φ is holomorphic. The topological σ-model associated to counting solutions of this equations is then an ordinary A-model. Though the topological field theory associated to theory T is not an ordinary A-model — because of the superpotential and because φ is twisted to have spin 1/2 — it will be useful to first discuss this more familiar case. A boundary condition for the Cauchy–Riemann equations that is sensible (elliptic) at least locally can be obtained by picking an arbitrary curve ` ∈ C

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and asking that the boundary values of φ should lie in `. Adding a superpotential to get the ζ-instanton equation does not affect this statement, which only depends on the “leading part” of the equation (the terms with the maximum number of derivatives). Here we may loosely call ` a brane, although to be more precise, it is the support of a brane. As we will discuss later, there can be more than one brane with support `. More generally, as is usual in brane physics, we may impose such a boundary condition in a piecewise way. For this, we pick several branes `α , we decompose the boundary ∂Σ as a union of intervals Iα that meet only at their endpoints, and for each α, we require that Iα should map to `α . (A common endpoint of Iα and Iβ must then map to an intersection point of `α and `β .) What sort of ` should we use? At first sight, it may seem that the A-model is most obviously well-defined if ` is compact. Actually, a compact closed curve in C is a boundary, and with such a choice of `, the A-model with target C is actually anomalous, as explained from a physical point of view in [53], Subsec. 13.5. This anomaly is an ultraviolet effect that is related to a boundary contribution to the fermion number anomaly on a Riemann surface. More intuitively, if ` is a closed curve in the plane, that it can be shrunk to a point and is not interesting topologically. Thus we should consider noncompact `, for example a straight line in C. With such a choice, we avoid the ultraviolet issues mentioned in the last paragraph, but the noncompactness of ` raises potential infrared problems. The space of solutions of the Cauchy–Riemann equation ∂φ = 0, with boundary values in the noncompact space `, is in general not compact, and this poses difficulties in defining the A-model with target C. There are a number of approaches to resolving these difficulties, depending on what one wants. One approach leads mathematically to the “wrapped Fukaya category.” For our purposes, we want to use the superpotential W to prevent φ from becoming large. This corresponds mathematically to the Fukaya-Seidel category [55]; for a physical interpretation, see [53], especially Subsecs. 11.2.6 and 11.3. To see the idea, let us return to the identity (3.32), but now allow for the possibility that Σ has a boundary. For instance, we can take Σ to be the upper half z-plane. Setting z = x1 + ix2 , the identity becomes Z 0= Σ

2 ! Z ∂W 2 2 +2 |d z| |dφ| + ∂φ

dx1 Im W.

(3.33)

∂Σ

Now it becomes clear what sort of brane we should consider. We should choose ` so that Im W → ∞ at ∞ along `. Then the boundary term in the identity will ensure that φ cannot become large along ∂Σ, and given this, the bulk terms in the identity ensure that φ cannot become large anywhere. That is an essential technical step toward being able to define the A-model. Let us implement this in our case that W (φ) = 2i mφ2 , with m > 0 and φ = 2 2 φ1 + iφ2 . We have Im W (φ) = m 2 (φ1 − φ2 ). Thus near infinity in the complex φ

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plane, there are two regions with Im W → +∞: this happens near the positive real φ axis and also near the negative axis. A noncompact 1-manifold ` is topologically a copy of the real line, with two ends. To ensure that Im W → ∞ at ∞ along `, we should pick ` so that each of its ends is in one of the good regions near the positive or negative φ axis. Beyond this, the precise choice of ` does not matter, because of the fact that the A-model is invariant under Hamiltonian symplectomorphisms of C. All that really matters is whether φ tends toward +∞ or −∞ at each of the two ends of `. Moreover, if φ tends to infinity in the same direction at each end of `, it is “topologically trivial” in the sense that it can be pulled off to infinity in the φ-plane while preserving the fact that Im W → ∞ at ∞ along `. So the only interesting case is that φ tends to −∞ at one end of ` and to +∞ at the other. Further details do not matter. Therefore, we may as well simply take21 ` to be the real φ axis. In other words, the boundary condition on φ is that it is real along ∂Σ, or in other words if φ = φ1 + iφ2 , then φ2 = 0 at x2 = 0. This has an interesting interpretation in the topologically twisted model that we are really interested in. We recall that in this model, φ is a section of the chiral spin bundle L of Σ. The fiber of L at a point in Σ is a complex vector space of dimension 1. This is actually the same as a real vector space of rank 2. Thus, we can alternatively view the complex line bundle L → Σ as a rank 2 real vector bundle S → Σ. The resulting S is simply the real, nonchiral spin bundle of Σ. Thus, it is possible to view the real and imaginary parts of φ as a two-component real spinor field over Σ. In fact, we have already made much the same statement in Eq. (3.15), where we asserted that  the ζ-instanton equation for φ is equivalent to the massive  φ 1 Dirac equation for φb = . −φ2 Now recall that in Subsec. 3.5, we defined a rank 1 real spin bundle E → ∂Σ by saying that a section of E is a section φb of the rank 2 spin bundle S of Σ (restricted b (The opposite sign in this relation, γk φb = −φ, to ∂Σ) that satisfies γk φb = φ. defines another equivalent real spin bundle of ∂Σ.) For Σ the upper half plane, the tangential gamma matrix is γk = γ1 , and the representation that we have used of the gamma matrices (Eq. (3.7)) is such that γ1 φb = φb is equivalent to φ2 = 0. Thus, we can state the boundary condition that we have found in a way that makes sense in general for the twisted topological field theory under study. In bulk, that is away from ∂Σ, φ is a section of the chiral spin bundle L → Σ. The boundary condition satisfied by φ is that along ∂Σ, it is a section of the real spin bundle E → ∂Σ. The merit of this boundary condition is the same as it is in the ordinary A-model, which we used as motivation: it ensures that the surface terms in Eq. (3.32) vanish, and therefore that the only solution of the ζ-instanton equation on a Riemann surface Σ with boundary is φ = 0. 21 This

` can be described as a Lefschetz thimble for the superpotential W associated to its unique critical point at φ = 0. In general, in the Fukaya-Seidel category, the most basic objects are such Lefschetz thimbles.

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We can gain some more insight by comparison to the ordinary A-model. To construct a brane with support `, we need to pick an orientation of `. There are two possible orientations, so there are two possible branes, which we will call B 0 and B 00 . Neither one is distinguished relative to the other. In the ordinary A-model, we could at our discretion introduce B 0 or B 00 or both. The twisted model that is related to theory T , in which φ is a chiral spinor rather than a complex-valued field, is different in this respect. The reason it is different is that B 0 and B 00 represent choices of orientation of the real spin bundle E → ∂Σ, but in general this real spin bundle is unorientable. Thus, if one goes all the way around a component of ∂Σ with NS spin structure, then B 0 and B 00 are exchanged. Accordingly, in the model relevant to theory T , if we introduce one of these branes, we have to also introduce the other. Once we introduce branes B 0 and B 00 , we are very close to the picture developed in the mathematical literature [3–6]. The boundary of Σ is decomposed as a union of intervals Iα that have only endpoints in common, and each interval is labeled by B 0 or B 00 . This labeling here means simply a chosen orientation of E → ∂Σ. Since E is a real vector bundle of rank 1, a choice of orientation of E is (up to homotopy) the same as a trivialization of E, the language used in Subsec. 3.7. There is really just one more puzzle. In the theory developed in [3–6], whenever one crosses a boundary puncture, the orientation of E jumps. Why is this true? A quick answer is the following. In general, for any brane B, (B, B) strings in the A-model correspond to local operators that can be inserted on the boundary of the string in a region of the boundary that is labeled by brane B. Our model is only locally equivalent to an A-model, but this is good enough to discuss local operators. In the case of the branes B 0 and B 00 , as ` is contractible, the only interesting local (B 0 , B 0 ) or (B 00 , B 00 ) operator is the identity operator. However, in topological string theory, what we add to the action along the boundary of the string worldsheet is really a descendant of a given local operator. In the case of a boundary local operator O, what we want is the 1-form operator V that can be deduced from O via the descent procedure. If O is the identity operator, then V = 0. (Recall that V is characterized by {Q, V} = dO, where Q is the BRST operator of the theory; if O is the identity operator, then dO = 0 so V = 0.) Therefore we cannot get anything interesting from (B 0 , B 0 ) or (B 00 , B 00 ) strings. The analogy with the standard A-model indicates that the space of (B 0 , B 00 ) or 00 (B , B 0 ) strings is also one-dimensional (see Subsecs. 3.8.3 and 3.8.4), but now a (B 0 , B 00 ) or (B 00 , B 0 ) string corresponds to a local operator that causes a jumping in the brane that labels the boundary, and this is certainly not the identity operator. Thus the gravitational descendant will not vanish. Another crucial detail concerns the statistics of the operators. The identity operator is bosonic, so its 1-form descendant, if not zero, would be fermionic. A fermionic boundary puncture operator is not what we need for the theory of [3–6], in which the coupling parameters and correlation functions are all bosonic. The

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analogy with the standard A-model indicates (Subsec. 3.8.3) that the (B 0 , B 00 ) and (B 00 , B 0 ) local operators are fermionic, so that their 1-form descendants are bosonic. There is also an important detail on which the analogy to the standard Amodel is a little misleading, because it is only valid locally. In an A-model with branes B 0 and B 00 , the (B 0 , B 00 ) and (B 00 , B 0 ) local operators would be independent operators, and we would potentially include them (or their 1-form descendants) with independent coupling parameters. In the present context, there is not really any way to say which is which of B 0 and B 00 ; one can only say that they differ by the orientation of the real spin bundle.22 So there is really only one type of boundary puncture, which one can think of as (B 0 , B 00 ) or (B 00 , B 0 ), and correspondingly there is only one boundary coupling. It follows, incidentally, that even if the identity (B 0 , B 0 ) or (B 00 , B 00 ) operator had a nontrivial 1-form gravitational descendant, it could not play a role. We would have to identify these two operators, so we would have a single such operator with a fermionic coupling constant υ. As the correlation functions of topological gravity are bosonic, they could not depend on a single fermionic variable υ.

3.8.3. Orientations and statistics Consider a brane B 0 in an arbitrary A-model with some target space X. The support of B 0 is a Lagrangian submanifold L ⊂ X. Take B 0 to have trivial Chan-Paton bundle.23 If we consider N copies of brane B 0 , we get an effective U (N ) gauge theory along L. Another M copies of brane B 0 would similarly support by themselves a U (M ) gauge theory. If we combine N copies of B 0 with M more copies, we get a U (N +M ) gauge theory. Now consider another brane B 00 that differs from B 0 only by reversing the orientation of L. M copies of B 00 would support a U (M ) gauge theory. However, if one combines M copies of B 00 with N copies of B 0 , one does not get a gauge group U (N + M ). Instead, one gets the supergroup U (N |M ) [56]. We will give a simple example to explain why this must be the case. For a familiar setting, take X to be a Calabi–Yau three-fold. The effective gauge theory for N copies of a brane is actually a U (N ) gauge theory. Let us denote the gauge field as A. The theory also has a 1-form field φ in the adjoint representation, which describes fluctuations in the position of the brane. The effective action is a multiple

example, B0 and B00 are exchanged in going all the way around a circle with NS spin structure. Perhaps more fundamentally, orienting the real spin bundle of one boundary of Σ does not in general tell us how to choose such an orientation for other boundaries. So we can say locally how B0 and B00 differ but there is no global notion of which is which. 23 For example, L might be topologically trivial (as it is in our application, with L = `). We will ignore various subtleties related to the K-theory interpretation of branes; these are not relevant for our purposes.

22 For

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of the Chern–Simons three-form for the complex connection A = A + iφ: Z 1 I= CS(A). (3.34) gst L
Here CS(A) = Tr A ∧ dA + 32 A ∧ A ∧ A is the Chern–Simons three-form and gst is the string coupling constant. There is no problem, given L purely as a bare three-manifold, so define the three-form CS(A). But to integrate a three-form over L requires an orientation of L. There is no natural choice, but a choice is part of the definition of a brane with support L. That is one way to understand the fact that in order to define a brane B 0 or B 00 with support L, one needs to endow L with an orientation; and there are in fact two A-branes B 0 and B 00 with the same support L that differ only by which orientation is chosen. The sign of the effective action I is opposite for B 0 relative to B 00 . Now if we bring together N branes supporting a U (N ) Chern–Simons theory to M more branes supporting a U (M ) Chern–Simons theory with the same sign of the action, the two Chern–Simons theories can merge into a U (N + M ) Chern– Simons theory. (The expectation value of the field φ can describe the breaking of U (N +M ) down to U (N )×U (M ).) However, if the U (M ) and U (N ) Chern–Simons actions have opposite signs, they cannot possibly combine to a U (N + M ) Chern– Simons theory. Instead, they can combine to a U (N |M ) supergroup Chern–Simons theory. We recall that the supertrace of an N |M -dimensional matrix is defined, in an obvious notation, as   U V Str = Tr U − Tr X, (3.35) W X where the relative minus sign is just what we need so that the supertrace of a Chern–Simons three-form of U (N |M ) leads to opposite signs for the U (N ) and U (M ) parts of the action. A consequence of going from U (N + M ) to U (N |M ) is that the statistics of the off-diagonal blocks V and W is reversed. At the end of Subsec. 3.8.2, that is what we needed so that the (B 0 , B 00 ) strings are fermionic, and have bosonic 1-form descendants. The situation just described does not usually arise in physical string theory, because there one usually is interested in branes that satisfy a stability condition involving the phase of the holomorphic volume form of the Calabi–Yau manifold, restricted to the brane. For a given Lagrangian submanifold, this condition is satisfied at most for one orientation. 3.8.4. Quantizing the string In the standard A-model, the space of local operators of type (B1 , B2 ), for any branes B1 and B2 that may or may not be the same, is the same as the space of physical states found by quantization on an infinite strip with boundary conditions set by B1 at one end and by B2 at the other end. Here we will explain the analog

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of this for the model under consideration here, which is only locally equivalent to a standard A-model. We will work on the strip 0 ≤ x2 ≤ a in the x1 x2 plane, for some a, and will treat x1 as Euclidean “time.” In Eq. (3.33), there is now a boundary contribution at x2 = a, as well as the one at x2 = 0 that was discussed previously. The two contributions have opposite signs, and to achieve compactness the boundary condition at x2 = a should ensure that Im W → −∞ at infinity. Thus we take the boundary condition at x2 = a to be φ1 = 0, while at x2 = 0 it is φ2 = 0, as before.24 To find the space of physical states with these boundary conditions, the first step is to find the space of classical ground states. With x1 viewed as “time,” these are the x1 -independent solutions of the ζ-instanton equation that satisfy the boundary conditions at the two ends. For solutions that depend only on x2 , the dφ + mφ = 0. The only solution of this linear ζ-instanton equation reduces to dx 2 first-order equation with φ2 = 0 at x2 = 0 and φ1 = 0 at x2 = a is φ = 0. Moreover, this solution is nondegenerate, meaning that when we linearize around it, the linearized equation has trivial kernel. (In the present case, this statement is trivial since the ζ-instanton equation is already linear.) A nondegenerate classical solution corresponds upon quantization to a single state. If there were multiple classical vacua, we would have to consider possible tunnelling effects to identity the quantum states that really are supersymmetric ground states. With only one classical vacuum, this step is trivial. So in our problem, there is just one supersymmetric ground state. One might be slightly puzzled that we seem to have used different boundary conditions and thus different branes at x2 = a relative to x2 = 0. However, if we conformally map the strip to the upper half plane x2 ≥ 0, mapping x2 = −∞ in the strip to the origin x1 = x2 = 0 in the boundary of the upper half plane, then this difference disappears. What we have done, on both boundaries, is to require that φ should restrict on ∂Σ to a section of the real spin bundle E → ∂Σ. The space of supersymmetric ground states that we just obtained corresponds to the space of local operators of type (B 0 , B 0 ), (B 00 , B 00 ), or (B 0 , B 00 ) that can be inserted at x1 = x2 . Since we did not have to orient the spin bundles of the boundaries of the strip in order to determine that there is a 1-dimensional space of physical states on the strip, the spaces of local operators of type (B 0 , B 0 ), (B 00 , B 00 ), or (B 0 , B 00 ) are the same if understood just as vector spaces. But these operators have different statistics, as explained in Subsec. 3.8.3. 3.9. Boundary degenerations So far we have concentrated on questions concerning the orientation of the moduli space. However, as explained in Subsec. 3.1, in trying to define topological gravity 24 This

difference to something that will be explained in Subsec. √ in boundary condition is related √ 3.9: at x2 = 0, dz is real, while at x2 = a, −dz is real.

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on Riemann surfaces with boundary, there is a second serious problem, which is that the moduli space of Riemann surfaces with boundary, with its Deligne–Mumford compactification, itself has aR boundary. Because of this, intersection numbers such Q as the correlation functions M i ψidi of topological gravity (Eq. (3.1)) are a priori not well-defined from a topological point of view. We will explain schematically how this difficulty has been overcome, going just far enough to describe the simplest concrete computations. For full explanations, see [3–6]. First let us give a simple example to illustrate the problem. A disc Σ with n boundary punctures (and no bulk punctures) has a moduli space M of real dimension n−3. The disc can degenerate in real codimension 1 by forming a narrow neck (Fig. 6(a)), which then pinches off (Fig. 6(b)) to make a singular Riemann surface Σ that can be obtained by gluing together two discs Σ1 and Σ2 (Fig. 6(c)). This occurs in real codimension 1, and thus Fig. 6(b) describes a component of ∂M, the boundary of M. As a check, let us confirm that the configuration in Fig. 6(b) has precisely n − 4 real moduli, so that it is of real codimension 1 in M. Σ1 and Σ2 inherit the boundary punctures of Σ, say n1 for Σ1 and n2 for Σ2 with n1 + n2 = n. In addition, Σ1 and Σ2 have one more boundary puncture p1 or p2 where the gluing occurs. So in all, Σ1 and Σ2 have respectively n1 +1 and n2 +1 boundary punctures, and moduli spaces of dimension n1 − 2 and n2 − 2. The singular configuration in Fig. 6(b) thus has a total of (n1 − 2) + (n2 − 2) = n − 4 real moduli, as claimed. Thus, we have confirmed the assertion that moduli spaces of Riemann surfaces with boundary are themselves manifolds (or orbifolds) with boundary. This presents a problem for defining intersection numbers. Now let us reexamine this assuming that Σ is endowed with a spin bundle S and that the induced real spin bundle E of ∂Σ is piecewise trivialized along ∂Σ, as described in Subsec. 3.7. We immediately run into something interesting. If Σ is a disc, the spin bundle E → ∂Σ is always of NS type, and the number n of boundary punctures on a disc will have to be odd. But when Σ degenerates to the union of two branches Σ1 and Σ2 , with n1 + 1 punctures on one side and n2 + 1 on the other side, inevitably either n1 + 1 or n2 + 1 is even. But in the theory that we are describing here, a disc is always supposed to have an odd number of boundary punctures.

a)

b)

c) p1

Σ

Σ1

Σ2

Σ1

p2 Σ2

Fig. 6. (a) A disc Σ with n boundary punctures that develops a narrow neck. (b) The neck collapses and Σ degenerates to the union of two discs Σ1 and Σ2 glued at a point. (c) The picture of part (b) can be recovered by gluing p1 ∈ Σ1 to p2 ∈ Σ2 . The original boundary punctures of Σ are divided in some way between Σ1 and Σ2 .

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What this means in practice is that either p1 or p2 does not really behave as a boundary puncture in the sense of this theory: the piecewise trivializations of the real spin bundles E1 → Σ1 and E2 → Σ2 jump in crossing either p1 or in crossing p2 , but not both. This is explained more explicitly shortly. As a result, the cohomology classes ψi whose products we want to integrate to get the correlation functions have the property that when restricted to ∂M, they are pullbacks from a quotient space in which either p1 or p2 is forgotten. Effectively, then, ∂M behaves as if it is of real codimension 2 and the intersection numbers are well-defined. Now let us explain these assertions in more detail. First we introduce a useful language. In the following, Σ will be a Riemann surface, possibly with boundary. We write K for the complex canonical bundle of Σ and S for its chiral spin bundle. So K is a complex line bundle over Σ, and S is a complex line bundle over Σ with a linear map w : S ⊗ S → K that establishes an isomorphism between S ⊗ S and K. Along ∂Σ, it is meaningful to say that a one-form is real, and thus K, restricted to ∂Σ, has a real subbundle. Moreover, the Riemann surface Σ is oriented and this induces an orientation of ∂Σ. As a result, it is meaningful to say that a section of K, when restricted to ∂Σ, is real and positive. For example, if Σ is the upper half of the complex z-plane, so that ∂Σ is the real z axis, then the complex 1-form dz is real and positive when restricted to ∂Σ. But if Σ is the lower half of the z-plane, then its boundary is the real z axis now with the opposite orientation, and so in this case, −dz is real and positive along ∂Σ. This gives a convenient framework in which to describe the real spin bundle E of ∂Σ. We say that a local section ψ of S → Σ is real along ∂Σ if the 1-form w(ψ ⊗ ψ) is real and positive when restricted to ∂Σ. In this case, we say that the restriction of ψ to ∂Σ is a section of E. This serves to define E. For example, if Σ is the upper half of the complex z-plane, then a section ψ of S with the property that w(ψ ⊗ ψ) = dz is real along ∂Σ, and its restriction √ to ∂Σ provides a section of E. We describe this more informally by writing ψ = dz. Note that since (−ψ) ⊗ (−ψ) = ψ ⊗ ψ, in this situation we also have w((−ψ) ⊗ (−ψ)) = dz. So just like the square root of a number, a square root of dz is only uniquely determined up to sign. If Σ is the lower half of the complex z plane, then a section ψ of S that satisfies w(ψ ⊗ ψ) √ = −dz is −dz or real and is a section of E. We describe this informally by writing ψ = ± √ ψ = ±i dz. A trivialization of the real spin bundle E → ∂Σ is given by any nonzero section of E. For example, if Σ is the upper half z plane, then E → ∂Σ can be trivialized √ by ψ = ± √dz, and if Σ is the lower half z plane, then E → ∂Σ can be trivialized by ψ = ±i dz. With this in place, we can return to our problem. In Fig. 7, we show the same open-string degeneration as in Fig. 6, but now we zoom in on the important region where the degeneration occurs and do not specify what the Riemann surface Σ looks like outside this region. The open-string degeneration is drawn in the figure

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a)

b) !

c) !1

z

!1 p 1

z

z

!2 p 2

!2

Fig. 7. (a) The complement of the shaded region of the complex z-plane is a Riemann surface Σ with boundary. It consists of an upper and lower half plane connected through a narrow neck. (b) In real codimension 1, the neck collapses and Σ degenerates to a pair of branches Σ1 and Σ2 glued together along a double point. (c) In this picture, the two branches have been separated. Now Σ1 and Σ2 are upper and lower half-planes, respectively, with distinguished boundary punctures p1 and p2 . Gluing p1 to p2 will return us to the singular configuration in (b).

±

√

dz

√ ∓ i dz

√

dz

z

√ i dz

Fig. 8. Here we repeat Fig. 7(a), but now providing information on the trivialization of the spin bundle of ∂Σ. On the upper and lower left and right of the figure, ∂Σ is parallel to the √ real z √ axis and so the spin structure is trivialized by a choice of ± z (the upper regions) or ±i dz (the lower regions).

ignoring spin structures and their trivializations. In figure 8, we repeat Fig. 7(a), but now providing information about the trivializations of spin structures. First of all, as there are no boundary punctures in this picture,25 the real spin bundle of ∂Σ is supposed to be trivialized everywhere in the picture. The trivializations are easy to describe in the regions — the upper and lower left and right in the figure — in which ∂Σ is parallel to the real z axis. We will use the fact that as Σ is a region in the complex z plane, the complex 1-form dz is defined√everywhere on Σ; similarly it is possible to make a global choice of sign of ψ = dz, though such √ a ψ will not be everywhere real on ∂Σ. The overall sign of what we mean by dz will not be important in what follows. √ We begin on the upper right of the picture with E trivialized by ψ = dz. (It √ would add √ nothing essentially new to use − dz in the starting point, as the overall sign of dz is anyway arbitrary.) Now on the upper left of the picture, we pick a √ trivialization ± dz. This sign is meaningful, given that we used the trivialization √ + dz on the upper right. Now we continue through the narrow neck into the lower part of the picture. As we do this, the boundary of ∂Σ bends counterclockwise by 25 The

Deligne–Mumford compactification is defined in such a way that a degeneration never occurs at the location of an already existing puncture. Hence in Fig. 7(a), which shows the part of Σ in which an open-string degeneration occurs, we can assume that there are no boundary punctures.

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an angle π on the right of the figure and by an angle −π on the left. As a result, a section of S → ∂Σ has√to acquire a phase in order to remain√ real. The trivialization of E that is defined as dz on the upper right will evolve to i dz on the lower right, √ and the trivialization of E that is defined as ± dz on the upper left will evolve to √ ∓i dz on the lower left. We see that with one choice of sign on the left part of the picture, the trivializations agree on the upper left and upper right of the figure but not on the lower left and lower right; with the other choice of sign, matters are reversed. So when Σ degenerates to the union of two branches Σ1 and Σ2 that are to be joined by gluing a point p1 ∈ ∂Σ1 to a point p2 ∈ ∂Σ2 , as in Fig. 7(c), the trivialization of the spin structure of the boundary jumps in crossing p1 but not in crossing p2 or in crossing p2 but not in crossing p1 . In the construction studied in [3–6], precisely one of p1 and p2 plays no role and can be forgotten. This is the basic reason that the boundary of M behaves as if it is of real codimension two and the correlation functions are well-defined. We provide more detail momentarily. 3.10. Computations of disc amplitudes Several concrete methods to compute in this framework have been deduced [3–6]. Here we will just describe the simplest computations of disc amplitudes. First let us discuss the proper normalization of a disc amplitude. We write gst for the string coupling constant in topological gravity of closed Riemann surfaces with its usual normalization, and e gst for the string coupling constant in the present theory. 2g−2 In the standard approach, genus g amplitudes are weighted by a factor of gst . 2g−2 g−1 g−1 With theory T included, this is replaced by e gst 2 , where 2 is the partition function of theory T (Eq. (3.5)). The relation between the two is thus gst (3.36) e gst = √ . 2 gst = √ A disc has Euler characteristic 1, so a disc amplitude is weighted by 1/e 2/gst . The partition function of theory T on a disc is 1/2 (as a disc has only one spin structure). However, for any given set of boundary punctures, there are two possible piecewise trivializations of the spin structure of the boundary, with the requisite jumps across boundary punctures. These two choices will contribute equally in the simple computations we will discuss, so we can take them into account by including a factor of 2. √ √ The factors discussed so far combine to 2 · 12 2/g √ st = 2/gst . In addition, in [3] it was found convenient to include a factor of 1/ 2 for every boundary puncture. Thus, let Σ be a disc with m boundary punctures and n bulk punctures labeled by integers d1 , . . . , dn ; let M be the compactified moduli space of conformal structures on Σ. Then refining Eq. (3.1), the general disc amplitude is Z 2(1−m)/2 hτd1 τd2 . . . τdn σ m iD = ψ1d1 ψ2d2 · · · ψndn . (3.37) gst M

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This formula agrees with Eq. (18) in [3]. We have included factors of gst in this explanation, because that helps determine the factors of 2 that are needed to ensure that the theory is consistent with the standard normalization in the case that a surface Σ has no boundary. However, in mathematical treatments, gst is often set to 1, and we will do so in the rest of this section. (No topological information is lost, since a given correlation function receives contributions only from surfaces with a given Euler characteristic, and this determines the power of gst .) In interpreting Eq. (3.37), we consider the boundary punctures to be inequivalent and labeled, and we sum over all possible cyclic orderings. For example, let us compute hσσσi, which receives a contribution only from a disc with three boundary punctures labeled 1,2,3. There are two cyclic orderings (namely 123 and 132), and R for each cyclic ordering, M is just a point, with M 1 = 1. So after setting gst = 1, Eq. (3.37) with n = 0, m = 3, and including a factor of 2 from the sum over cyclic orderings, gives hσ 3 i = 1.

(3.38) √ Getting this formula was the motivation to include a factor 1/ 2 for each boundary puncture. Another simple formula is hτ0 σi = 1.

(3.39)

This is again easy because the moduli space is a point. With boundary punctures only, Eq. (3.38) is the only nonzero amplitude, for dimensional reasons, and similarly (3.39) is the only additional nonzero disc amplitude with insertions of σ and τ0 only. The simplest method to compute arbitrary disc amplitudes is given by the recursion relations in Theorem 1.5 of [3], and indeed the first of these relations is sufficient. To explain it, first we recall the genus 0 recursion relations of [17]. It is convenient to define * !+ ∞ X hhτd1 τd2 · · · τds ii = τd1 τd2 · · · τds exp tn τn . (3.40) n=0

Thus hhτd1 τd2 · · · τds ii is an amplitude with specified insertions as shown, with all possible additional insertions weighted by powers of the tn . We also write hhτd1 τd2 · · · τds ii0 for the genus 0 contribution to hhτd1 τd2 · · · τds ii. Then one has the genus 0 recursion relation hhτd1 τd2 τd3 ii0 = hhτd1 −1 τ0 ii0 hhτ0 τd2 τd3 ii0 .

(3.41)

The proof goes roughly as follows. For a smooth genus 0 surface Σ, we take the complex z-plane plus a point at infinity. We denote the specified punctures as z1 , z2 , z3 . We will construct a convenient section λ of the line bundle L1 → M whose fiber is the cotangent bundle to Σ at z1 . Let ρ be the 1-form ρ = (z2 − z3 )

dz . (z − z2 )(z − z3 )

(3.42)

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a)

b) z2

z3

z1

z2 z3

Fig. 9. (a) If the two-sphere Σ degenerates to two branches with punctures z2 and z3 on opposite sides, then the 1-form ρ = dz/(z − z2 )(z − z3 ) has poles on each branch, so in particular it is nonzero on each branch. (When Σ degenerates, ρ also acquires poles at the double point that the two branches have in common, with equal and opposite residues on the two sides.) λ also remains nonzero. (b) If instead z2 and z3 are on the same branch, then all poles of ρ are on that branch and in fact ρ = 0 on the other branch. Since λ is defined by setting z = z1 in ρ, λ vanishes if, as sketched here, z1 is on the branch on which ρ is identically zero.

It has poles at z = z2 , z3 , with residues 1 and −1, and elsewhere is regular and nonzero. These properties characterize ρ uniquely, so ρ does not depend on the coordinates used in writing the formula. Upon setting z = z1 in ρ, we get a holomorphic section λ of L1 → M; the divisor D of the zeroes of this section represents c1 (L1 ). But λ never vanishes when Σ is smooth, because ρ has no zeroes on the finite z-plane or at z = ∞. If Σ degenerates to two components with z2 and z3 on opposite sides (Fig. 9(a)), λ is still everywhere nonzero. But if z2 and z3 are contained in the same component (Fig. 9(b)), then λ vanishes on the other component. Finally, then, ρ vanishes precisely if, as in the figure, z1 is contained in the opposite component from the one containing z2 and z3 . Moreover, this is a simple zero (because ρ has a simple zero at z2 = z3 ). So in τd1 = c1 (L1 )d1 , we can replace one factor of c1 (L1 ) with a restriction to the divisor D that is depicted in Fig. 9(b). After making this substitution, we are left with an insertion of τd1 −1 on one branch and insertions of τd2 and τd3 on the other; in addition, a new puncture corresponding to an insertion of τ0 appears on each branch, where the two branches meet. All this leads to the right hand side of Eq. (3.41). It is not difficult to see that this recursion relation uniquely determines all genus zero amplitudes, modulo the statement that the only nonzero amplitude with insertions of τ0 only is hτ03 i0 = 1. The disc recursion relation that we aim to describe can be formulated and proved in almost the same way. Similarly to the previous case, we define * !+ ∞ X m m hhτd1 τd2 · · · τds σ ii = τd1 τd2 · · · τds σ exp tn τn + vσ , (3.43) n=0

m

and write hhτd1 τd2 · · · τds σ iiD for the disc contribution. The desired recursion relation is hhτn σiiD = hhτn−1 τ0 ii0 hhτ0 σiiD + hhτn−1 iiD hhσ 2 iiD .

(3.44)

Given a knowledge of eqns. (3.38) and (3.39) and vanishing of hτ0n σ m iD for other

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values of n, m, it is not difficult to see that Eq. (3.44) determines all disc amplitudes in terms of the genus 0 amplitudes. These in turn can be determined, for example, from (3.41). The proof of Eq. (3.44) is rather similar to the proof of the genus zero recursion relation (3.41). However, we will have to explain more fully what is meant in saying that one of the punctures in an open-string degeneration should be forgotten. Roughly speaking, we are going to again compute c1 (L1 ), for one of the bulk punctures, from the zeroes of a convenient section λ of L1 . However, here because M has a boundary, we have to discuss how to relate c1 (L1 ) to the zeroes of a section. As discussed in section 3.9, the boundary ∂M of M has a forgetful map in which precisely one of the extra boundary punctures that appears at an openstring degeneration is forgotten. Let us write N for the remaining moduli space when this puncture is forgotten, so that the forgetful map is π : ∂M → N . Simplifying a little,26 the recipe [3] is that c1 (L1 ) can be represented by the zeros of any section s of L1 that is nonvanishing everywhere along ∂M, and whose restriction to ∂M is a pullback from N . Alternatively, one can still calculate c1 (L1 ) using any section s of L1 that is everywhere nonzero along the boundary, even if its restriction to the boundary is not a pullback. But in this case, c1 (L1 ) is represented by a sum of two contributions, one involving in the usual way the zeroes of s, and the second measuring the failure of the restriction of s to be a pullback. Setting z = x + iy, we take a smooth disc D to be the closed upper halfplane y ≥ 0 plus a point at infinity. On the left-hand side of Eq. (3.44), we see a distinguished bulk puncture that we place at z1 = x1 + iy1 , y1 > 0, and a distinguished boundary puncture that we place at x0 . In the present case, there is a convenient section λ of L1 that is everywhere nonzero along the boundary, but whose restriction to the boundary is not a pullback. To construct it, rather as before, we set ρ = (z 1 − x0 )

dz . (z − z 1 )(z − x0 )

(3.45)

This 1-form is regular and nonzero throughout D, except at the boundary point x0 . Evaluating ρ at z = z1 , we get a section λ of L1 that is regular and nonzero as long as D is smooth. At a closed-string degeneration, where D splits up into the union of a two-sphere and a disc (Fig. 10(a)), λ has a simple zero if and only if z1 is on the two-sphere component. This is responsible for the first term on the right-hand side of the 26 The

general recipe has two further complications. First, in general one is allowed to compute using a multisection rather than a section. This is important because the conditions on a section that we Qare about dto state are difficult to satisfy. Second, the general procedure allows one to i define n i=1 c1 (Li ) , without defining the individual c1 (Li ), by picking a multisection s of E = ⊕di n ⊕i=1 Li . This multisection should obey conditions analogous to the ones that we will state momentarily.

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b)

a)

z1

z1

p1

p2

x0

z¯1

x0

z¯1 Fig. 10. (a) A disc D splits up into the union of a disc and a sphere (upper half of the drawing). If the bulk puncture z1 is contained in the sphere, then the section λ vanishes. To see this, take the closed oriented double cover, obtained here by adding additional components (lower half of the drawing, sketched with dotted lines). It is a union of three spheres connected at double points. The differential ρ has poles only on the bottom two components and vanishes identically on the top component. So, setting z = z1 to define λ, we learn that, with z1 being in the top component, λ vanishes. (b) The same disc D splits into a union of two discs, again comprising the upper half of the drawing. The interesting case is that z1 and x0 are on opposite sides, as shown. The oriented double cover (the full drawing including the bottom half) is a union of two spheres. ρ has poles at x0 and z 1 and so is nonzero on both branches; hence λ 6= 0 along this divisor. On the branch containing z1 , ρ has an additional pole at the point labeled p1 where the two branches meet. Therefore λ depends on p1 , and, if p1 is the boundary puncture that is forgotten by the forgetful map π : ∂M → N , then along this component of the boundary, λ is not a pullback.

recursion relation (3.44). At an open-string degeneration, where D splits up into the union of two discs (Fig. 10(b)), λ remains everywhere nonzero. However, in case the boundary puncture that is supposed to be forgotten is in the same component as z1 , λ restricted to ∂M is not a pullback from N . The second term on the right hand side of Eq. (3.44) corrects for this failure. See Fig. 10 for an explanation of the statements about the behavior of λ at degenerations. 4. Interpretation via Matrix Models 4.1. The loop equations Let us now briefly recapitulate the representation of topological gravity in terms of random matrix models. The simplest models are single matrix models of the form   Z 1 1 dΦ · exp − Tr W (Φ) (4.1) Z= vol(U (N )) gst

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65

(b)

Fig. 11. (a) Feynman diagrams of the matrix model are naturally ribbon graphs. The two sides of a ribbon represent the flow of the two “indices” of an N × N matrix M i j , i, j = 1, . . . , N . The edges of the ribbons form closed loops. Gluing a disc to each such loop, the union of the ribbons and the discs is a two-manifold Σ without boundary on which the given Feynman diagram can be drawn. (The edges of the ribbon are oriented — not shown here — because the two indices transform according to inequivalent, dual representations of U (N ). As a result, Σ has a natural orientation. (A similar model with symmetry group O(N ) or Sp(N ) leads to unoriented twomanifolds.) (b) New variables Ψ, Ψ transforming in the N -dimensional representation of U (N ) and its dual are added to the matrix model. Because Ψi and Ψj , i, j = 1, . . . , N carry only a single “index” — rather than the two indices of the matrix M i j — their propagator is naturally represented by a single line rather than the double line of the matrix propagator. These single lines provide boundaries of the surface Σ, so now we get a ribbon graph on Σ with Ψ propagating on the boundary of Σ, as shown. For the model described in the text, the Ψ propagtor is 1/z and this gives a factor 1/z L where L is the length of the boundary.

Here Φ is a Hermitian N × N matrix integrated with the Euclidean measure for each matrix element, W (x) is a complex polynomial, say of degree d + 1, and gst is the string coupling constant. Since we divide by the volume of the “gauge group” U (N ), this integral should be considered the zero-dimensional analogue of a gauge theory — we integrate over matrices Φ modulo gauge transformations Φ → U · Φ · U −1 .

(4.2)

In general, if Re W is not bounded below, one needs to complexify the matrix Φ and pick a suitable integration contour in the space of complex matrices to make the integral well-defined. For a formal expansion in powers of gst and even for the formal expansion in powers of 1/N that we will make shortly, this is not necessary and we can consider (4.1) as a formal expression. In a perturbative expansion near a critical point of W (Φ), the Feynman diagrams become so-called “fat” or ribbon graphs that can be conveniently represented (see Fig. 11(a)) by a double line [57]. These are graphs, in general with ` loops, that can be naturally drawn on some oriented two-manifold of genus g. The contribution of

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such a graph to the expansion of the matrix integral is weighted by a factor 2g−2 (gst N )` gst .

(4.3)

The large N or ’t Hooft limit is obtained by taking the rank N of the matrix to infinity and simultaneously the coupling gst to zero, keeping fixed the combination µ = gst N.

(4.4)

In the limit, all graphs with a fixed genus and an arbitrary number of holes contribute in the same order, so the matrix integral has an asymptotic expansion of the form   X Z ∼ exp  gst 2g−2 Fg  , (4.5) g≥0

where Fg is the contribution of ribbon graphs of genus g. In general, the matrix integral depends on the coefficients of the potential W and the particular critical point around which the expansion is made. We describe the critical points at the end of this section. Matrix integrals are governed by Virasoro constraints that are associated to the ∂ . Though these constraints can be deduced directly vector fields Ln ∼ −Tr Φn+1 ∂Φ from that representation of Ln , a fuller understanding with details that we will need below can be obtained by diagonalizing the matrix as Φ = U ΛU −1 , with U unitary and Λ = diag(λ1 , λ2 , . . . , λN ). The integral over U cancels the factor of 1/vol(U (N )) in the definition of the matrix integral, and the integral becomes ! Z Y X 1 2 Z = dNλ W (λI ) . (4.6) (λI − λJ ) exp − gst I 0, finding a straight line graph coming down to the value 0 at the affine parameter 1 1 value τ = ρ3 . Comparing this with the actual graph for ∆ 3 with D2 ∆ 3 ≤ 0, we 1

find that ∆ 3 necessarily becomes zero at some point q of γ, given by positive value of τ ≤ ρ3 . Now, we can only obtain a zero value for ∆ at a point q on γ where the vectors (Jacobi fields) v1 , v2 , v3 , defining ∆ become linearly dependent, so that this linear combination of them would be a Jacobi field vanishing at q, so that q is indeed a conjugate point to R, as required. The argument for the null case is 1 1 identical to this, but with ∆ 2 replacing ∆ 3 . We are now in a position to establish the first singularity theorem, concerning the collapse of an over-massive star, as envisaged at the end of §5.4, or alternatively, the collapsing together of a large collection of stars. The picture of a collapsing dust cloud, as provided by Oppenheimer and Snyder and described in §5.3 (see Fig. 5.1), provides a plausible overall description of the kind of initial situation in a gravitational collapse. Such an initial picture might well be generally plausible, even if minor deviations from spherical symmetry may be present, and also if the O–S assumption of “dust” (describing a gravitationally collapsing “perfect fluid” of noninteracting particles) may be regarded as a rather crude approximation. Moreover, if one considers a large, roughly spherical, collection of stars to be involved in an overall gravitational in-fall, the “Schwarzschild radius” of the whole system could well encompass a great number of the stars, without the densities of the material approaching anything like the somewhat exotic densities unevolved in neutron stars or even in white dwarfs. This follows merely from the way that physical quantities scale in general relativity. For a large enough total mass, the Schwarzschild radius can arise, and be crossed, by material of density as small as we please.

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However, as the collapse progresses, it could well be expected that the O–R picture gets less and less reliable. Even small deviations from spherical symmetry would be expected to become more and more enhanced as the material gets compressed into irregular shapes, with matter swirling around in complicated ways, and, moreover, the “dust” approximation is hardly likely to remain plausible for long. What our singularity theorem is able to achieve is to circumvent all this complication, by concentrating on a certain key feature of the collapse that signifies the passing of a “point of no return”. Fig. 5.6 depicts an O–S type collapse, as in Fig. 5.1, but within the region inside the horizon, and just outside the collapsing material, a trapped surface is delineated, as the pair of points at the bottom of the shaded region — where we must bear in mind that the 4-space is obtained by rotating the picture through a sphere S 2 about the central vertical axis, so this pair of points itself represents a spherical spacelike 2-surface T . The key feature of this 2-surface is that the two families of null normals (the ingoing and outgoing ones — where we notice that the “outgoing” ones are also falling inwards, but at a lesser rate than the “ingoing” ones) are both converging into the future, which we see by the fact that they both enter regions of decreasing r values, so that for both families the convergence ρ is positive at T , as is illustrated in Fig. 5.7. This property of the spacelike 2-surface T , together with

Fig. 5.6. Spherically symmetrical collapse (one space dimension surpressed). The diagram essentially also serves for the discussion of the asymmetrical case. The coordinate u is that of the E–F metric of (5.4.2).

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Fig. 5.7.

For a trapped surface, the rays normal to it on both sides start converging.

Fig. 5.8. The past light cones of two spacelike-separated points p and q in Minkowski space intersect in a spacelike 2-surface that is locally trapped, but not compact.

the fact that T is closed (compact without boundary) is what characterizes T as a trapped surface. It might be felt that this property of having the rays orthogonal to a spacelike surface converging on both sides is a strange local behaviour for a surface, but this is not so at all. This situation occurs wherever two past light cones intersect, in Minkowski space M (see Fig. 5.8). What is odd about a trapped surface is that this property of converging null normal on both sides holds globally over a closed surface and it is this feature that leads to singular behaviour of the space-time, as we see in the following theorem. Theorem 5.1. Let M be a globally hyperbolic space-time with a non-compact Cauchy hypersurface S. Assume that M is null Ricci positive. Suppose there is a trapped surface T to the future of S. Then M contains a null-singular TIP. Recall that a null-singular TIP is a TIP generated by a ray segment of finite affine length; see (5.5.1) and (5.7.5). Proof. Let B = ∂I + [T ]. By Lemma 5.1, the topological 3-surface B is generated by ray segments each of which either has a past end-point on T or else is pastendless, while remaining on B. The latter possibility cannot occur here because, by

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(5.5.2), any past-endless ray in int D+ (S) (which here is the whole of I + [S], by M’s global hyperbolicity) must meet S, impossible because B ⊂ I + [S]. Now, since T is trapped, we have ρ > 0 at the initial point of every ray generating the 3-surface B, so by Lemma 5.5 there must be a conjugate point to T on each of these generators, by an affine distance no greater than ρ2 , to the future of its initial point on T , so long as that generator continues, within M to such a parameter value. If it does not, this would be a situation not normally envisaged for a non-singular space-time, and, indeed, such an “incomplete” ray segment would generate a null-singular TIP. Let us suppose, therefore, that such null-singular TIPs are absent in M. Then, we infer from Lemma 5.3 — or more explicitly from the comments following Lemma 5.3, in the final paragraph of §5.5 — that such a ray continuing into the future would have to leave B, and enter the interior of I + [T ]. Thus, if M is free of null-singular TIPs, then B must be entirely generated by ray segments of finite affine length, where these ray segments join together at their end-points at various places, namely at T , at their past end-points, and at caustics and crossing points at their future endpoints. Overall, being a topological 3-manifold, and being composed entirely of an S 2 ’s worth of finite line segments, coming together at various places, T must be a compact topological 3-manifold (i.e. closed without boundary). We now invoke a general property of any Lorentzian time-oriented manifold, namely that it admits a smooth global timelike vector field ha , the integral curves of which can be used 4 to map the compact achronal 3-surface B down to the initial 3-surface S, injectively, to obtain a homoeomorphic image B 0 of B. This is not possible, because B 0 would have to be compact without boundary, whereas S is a non-compact topological manifold, of the same dimension. We deduce that M must therefore contain a null-singular TIP, and the theorem is proved. It may be remarked that this theorem (basically [Penrose, R. (1965)]) tells us somewhat more than the mere existence of singularities, under the assumptions stated. Lemma 5.5, as used here, already tells us how far to the future of S (namely 2 ρ ), in terms of affine distance, we must expect singularities to have arisen. Moreover, we can consider that the singularities arising in the collapse provide a family of singular TIFs, of which the null-singular TIPs in the considerations above form a subset. The projection down to S, by the vector field ha gives some kind of image of the space of these singular TIFs. The singularities would provide “holes” in B that would render B’s homoeomorphic image B 0 to be non-compact, and whose boundary points, when mapped back upwards by the ha vector field would correspond to generators of singular TIPs. As we consider a family of trapped surfaces like T but moved continuously outwards until, in the limit, they reach the horizon, we can envisage that the TIPs arising give us some kind of picture of the entire singular boundary of the space-time. 4 This

simplification to my original argument of [Penrose, R. (1965)], was suggested to me by Charles W. Misner in the later 1960s

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Following the publication of what was essentially the above theorem in early 1965, Stephen Hawking produced a series of papers aimed primarily at the cosmological big-bang singularity. His first paper [Hawking, S.W. (1965)], was based simply on the observation that the time-reverse of a trapped surface could occur at the very distant regions of an expanding universe. Subsequently, by developing new techniques, many of which have been incorporated into the above discussions (in §5.5–§5.7), he was able to provide results that eliminated the need for a Cauchy hypersurface, non-compact or otherwise, the study of Cauchy horizons providing an important input into the arguments [Hawking, S.W. (1966a)], [Hawking, S.W. (1966b)], [Hawking, S.W. (1967)]. Finally, we got together to supply a very general version [Hawking, S.W. and Penrose, R. (1970)], encompassing most of what had gone before. Yet, what might be regarded as a possibly significant disadvantage of this later work, was its dependence on Ricci positivity, rather than the weaker assumption of null Ricci positivity, which can be argued to have a firmer foundational status. A natural question to ask, at this point, is how do these results square with the work of Lifshitz and Khalatnikov [Lifshitz, E.M. and Khalatnikov, I.M. (1963)], referrer to in §5.3? The answer is that after becoming acquainted with these singularity theorems, with the aid of Vladimir Alekseevich Belinski, they were able to locate an error in their earlier work and were able to identify a much larger class of singularity types that could well be of a completely general character, (see [Belinskii, V.A., Khalatnikov, I.M. and Lifshitz, E.M. (1970)], [Belinskii, V.A., Khalatnikov, I.M. and Lifshitz, E.M. (1972)] and also the related work of Misner [Misner, C.W. (1969)]). The conflict with the singularity theorems was thereby resolved. Of course, the usual response to the classical singularity theorems described in this article and elsewhere, was that when curvatures become extraordinarily large, quantum-gravitational considerations must take over, and perhaps some kind of non-singular quantized theory could provide us with finite answers. However, as quantized theory currently stands, we are not in a position to make trustworthy statements about this (but see [Ashtekar, A. (2005)], [Bojowald, M. (2005)]). The point should be made, nevertheless, that even the very slight modifications to classical gravitational theory that are involved in the well-founded phenomenon of the Hawking evaporation of large, very classically behaving black holes, an extremely tiny violation of even the null energy condition is needed in order that a black holes horizon can slowly shrink owing to its loss of energy due to Hawking evaporation [Hawking, S.W. (1975)]. For astrophysical black holes, this requires a very tiny violation of null Ricci positivity. This can be attributed to an allowed small negative energy flux from quantum fields. Yet this does not at all involve large space-time curvatures, and it is hard to see how such considerations can provide a qualitative change to the purely classical arguments presented in this article.

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Penrose, R. (1969) Gravitational collapse: the role of general relativity, Rivista del Nuovo Cimento Serie I, Vol. 1; Numero speciale, 252–276. Reprinted: Gen. Rel. Grav. 34(7), July 2002, 1141–1165. Geroch, R and Horowitz, G.T. (1987) Global structure of spacetimes, in S.W. Hawking and W. Israel (eds.) General Relativity an Einstein Centenary Survey, Cambrdge University Press, Cambridge. Geroch, R. (1970) Domain of dependence, J. Math. Phys. 11(2), 437–449. Choquet-Bruhat, Y. and Geroch, R. (1969) Global aspects of the Cauchy problem in general relativity, Commun. Math. Phys. 14(4), 329–335. Hawking, S.W. (1967) The occurrence of singularities in cosmology III. Causality and singularities, Proc. Roy. Soc. (London) A300, 187–201. The Road to Reality: A Complete Guide to the Laws of the Universe, Vintage. ISBN: 9780-679-77631-4. Raychaudhuri, A.K. (1955) Relativistic cosmology. I, Phys. Rev. 98(4), 1123–1126. Komar, A. (1956) Necessity of singularities in the solution of the field equations of general relativity, Phys. Rev. 104(2), 544–546. Myers, S.B. (1941) Riemannian manifolds with positive mean curvature, Duke Math. J. 8(2), 401–404. Penrose, R. (1965) Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14(3), 57–59. Hawking, S.W. (1965) Occurrence of singularities in open universes, Phys. Rev. Lett. 15(17), 689–690. Hawking, S.W. (1966a) The occurrence of singularities in cosmology, Proc. Roy. Soc. (London) A294(1439), 511–521. Hawking, S.W. (1966b) The occurrence of singularities in cosmology II, Proc. Roy. Soc. (London) A295(1443), 490–493. [Adams Prize Essay: Singularities in the geometry of space-time.] Hawking, S.W. and Penrose, R. (1970) The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. (London) A314(1519), 529–548. Belinskii, V.A., Khalatnikov, I.M. and Lifshitz, E.M. (1970) Oscillatory approach to a singular point in the relativistic cosmology, Usp. Fiz. Nauk 102, 463–500. Engl. transl. in Adv. in Phys. 19(80), 525–573. Belinskii, V.A., Khalatnikov, I.M. and Lifshitz, E.M. (1972) Construction of a general cosmological solution of the Einstein equation with a time singularity, Zh. Eksp. Teor. Fiz. 62, 1606–1613. English transl. in Soviet Phys. JETP 35(5), 838–841. Misner, C.W. (1969) Mixmaster universe, Phys. Rev. Lett. 22(20), 1071–1074. Ashtekar, A. (2005) Quantum geometry and its ramifications, in Abhay Ashtekar (ed.) 100 Years of Relativity. Space-Time Structure: Einstein and Beyond, World Scientific, Singapore. Bojowald, M. (2005) Loop quantum cosmology, in Abhay Ashtekar (ed.) 100 Years of Relativity. Space-Time Structure: Einstein and Beyond, World Scientific, Singapore. Hawking, S.W. (1975) Particle creation by black holes, Commun. Math. Phys. 43, 199–220.

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Chapter 6 Beyond anyons∗

Zhenghan Wang Microsoft Research Station Q and Department of Mathematics, University of California, Santa Barbara, CA, USA [email protected]

The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature. To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in the toric code. A more speculative and interesting direction is the study of Banados–Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this short survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then, I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity. Keywords: Anyon; topological phase; symmetry defect; gapped boundary. PACS Nos.: 71.10.-w, 73.20.-r, 75.60.ch

1. Introduction Systematic application of quantum topology in condensed matter physics accelerated significantly after the 2003 Workshop Topological Phases in Condensed Matter Physics.8 Anyons, elementary excitations in 2D topological phases of matter, play a central role in this new period of interactions between topology and physics in 3d spacetimea (see Refs. 13, 16 and references therein). Conceptually, anyons provide an interpolation between bosons and fermions through the topological spins of Abelian anyons: eiθ for some θ’s with θ = 0 bosons and θ = π fermions. I see a striking parallel of the application of topology in physics to the application of topology in differential geometry — global differential geometry. Progress in science and

∗ This

chapter also appeared in Modern Physics Letters A, Vol. 33, No. 28 (2018) 1830011. DOI: 10.1142/S0217732318300112. a I will use the convention that nD means the D-dimensional space and nd the d-dimensional spacetime, so (n + 1)d = nD + 1.

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technology has made topological physics inevitable: the inherently discrete nature of quantum mechanics, the continuing miniaturization of quantum devices, and the maturity of local classical physics. At the forefront of topological physics is the application to quantum computing: topological quantum computing with the promise of an inherently fault-tolerant universal quantum computer.9 However, to build a real topological quantum computer, topological physics has to be coupled to conventional physics such as during the initialization and read-out. Therefore, it is natural to consider how to extend topological physics beyond anyons. One obvious direction is from 2D to 3D. But the difference between 2D and 3D could be enormous as the topology of 3d and 4d manifolds is dramatically different. For example, we have a rather complete classification of 3d spacetime manifolds due to the geometrization theorem of Pereleman– Thurston, while there is no reasonable conjectured picture of smooth 4d spacetime manifolds. The theory of anyon systems, as modular functors topologically and unitary modular tensor categories algebraically, is mature.13 To go beyond anyons, our first step is the interplay of anyons with conventional group symmetry due to the paramount importance of group symmetry in physics. This led to the theory of symmetry-enriched topological order. Another direction is the boundary physics of topological phases, both gapless as in the fractional quantum Hall physics and gapped as in toric code. A more speculative and interesting direction is the study of Banados– Teitelboim–Zanelli (BTZ) black holes and quantum gravity in 3d. The clearly defined physical and mathematical issues require a far-reaching generalization of anyons and seem to be within reach. In this survey, I will first cover the extensions of anyon theory to symmetry defects and gapped boundaries. Then I will discuss a desired generalization of anyons to anyon-like objects — the BTZ black holes — in 3d quantum gravity. 2. 2D Non-Abelian Objects Non-Abelian means the order in a sequence of things is important such as the order of letters in words: NO is not the same as ON. If many non-Abelian objects X1 , . . . , Xn are lined up in a line, then their states can be changed by exchanging any two of them. If two different exchanges are performed sequentially, the order of the two exchanges becomes important. The fundamental prerequisite for such a phenomenon is that the states of such n non-Abelian objects are not unique, i.e. the ground states have degeneracy, which is more fundamental than statistics.14 The best understood non-Abelian objects are non-Abelian anyons. The last two decades of research in condensed matter physics has yielded remarkable progress in the understanding of non-Abelian anyons in topological phases of matter. Recently, other non-Abelian objects are discovered such as the Majorana zero modes, which lead essentially to the same non-Abelian physics as non-Abelian anyons. These new

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non-Abelian objects — symmetry defects and gapped boundaries — open doors to new approaches to topological quantum computation. 2.1. Symmetry defects In the absence of any symmetry, gapped quantum systems at zero temperature can still form distinct phases of matter — topological phases of matter (TPMs), which are characterized by their topological order. A TPM H = {H} is an equivalence class of gapped Hamiltonians H which realizes a TQFT at low energy. Elementary excitations in a TPM H are point-like anyons. Anyons can be modeled algebraically as simple objects in a unitary modular tensor category (UMC) B, which will be referred to as the topological order of the TPM H. The interplay of symmetry with topological order has generated intense research. In the presence of symmetries, TPMs acquire a finer classification and fractional quasi-particles of a topologically ordered state can acquire fractional quantum numbers of the global symmetry. When a Hamiltonian for a TPM possesses a global symmetry, it is natural to consider the topological order that is obtained when this global symmetry is promoted to a local, gauge symmetry. This gauging procedure is useful in many ways. 2.2. Algebraic model of symmetry defects In the real world, TPMs are always coupled to conventional degrees of freedom. TPMs with conventional group symmetries are called symmetry enriched topological phases of matter (SETs). When the intrinsic topological order is trivial, SETs become symmetry protected topological phases (SPTs). Important examples of SPTs are topological insulators and topological superconductors. Let G be a finite group and C a UMC, also called an anyon model. 2.2.1. Topological symmetry Promoting G to a categorical-group G, we denote by Autbr ⊗ (C) the categorical-group of braided tensor auto-equivalences of the UMC C, which is the full topological symmetry of C. Definition 2.1. A finite group G is a topological symmetry of the UMC C if there is a monoidal functor ρ: G → Autbr ⊗ (C). The topological symmetry is denoted as (ρ, G) or simply ρ.

2.2.2. Symmetry defects The invertible module categories over C form the Picard categorical-group Pic(C) of C. The Picard categorical-group Pic(C) of the UMC C is monoidally equivalent

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to the categorical-group Autbr ⊗ (C). This one–one correspondence between braided tensor auto-equivalences and invertible module categories underlies the relation between symmetry and defect. Hence, given a topological symmetry (ρ, G) of the UMC br C and an isomorphism of Pic(C) with Autbr ⊗ (C), each ρ(g) ∈ Aut⊗ (C) corresponds to an invertible bi-module category Cg ∈ Pic(C). Definition 2.2. An extrinsic topological defect of flux g ∈ G is a simple object in the invertible module category Cg ∈ Pic(C) corresponding to the braided tensor auto-equivalence ρ(g) ∈ Autbr ⊗ (C). 2.2.3. Gauging topological symmetry Let G be the categorical 2-group, and Autbr ⊗ (C) be the categorical 2-group of braided tensor auto-equivalences. Definition 2.3. A topological symmetry ρ: G → Autbr ⊗ (C) can be gauged if ρ can be lifted to a categorical 2-group functor ρ: G → Autbr ⊗ (C). The physical and mathematical theory of symmetry defects can be found in Refs. 1 and 2. Symmetry defects can be used to enhance the computing power of anyons. Study of topological quantum computation with symmetry defects can be found in Ref. 7. 2.3. Gapped boundaries A second direction for non-Abelian objects beyond anyons is gapped boundaries in TPMs which are Drinfeld centers Z(C) of unitary fusion categories C, called doubled theory in physics. The Levin–Wen (LW) model in 2D is a lattice Hamiltonian realization of Turaev–Viro type TQFTs based on unitary fusion categories. The conceptual underpinning of the 2D LW model is two mathematical theorems: The Drinfeld center Z(C) of a unitary fusion category C is always modular, and the Turaev–Viro TQFT based on C is equivalent to the Reshetikhin–Turaev TQFT based on Z(C). Therefore, LW model is a lattice Hamiltonian implementation of both theorems simultaneously. Those rigorously solvable models provide the best playground for the theoretical study of TPMs. Realistically, samples of TPMs have boundaries. The interplay of the boundary with the interior or bulk contains rich physics as exemplified by the famous holographic principle. In the categorical formalism, the bulk of a doubled TQFT is given by a UMC B = Z(C) for some unitary fusion category C, and a (gapped) hole is a Lagrangian algebra A = ⊕a na a in B. In the case of Dijkgraaf–Witten theories, we have C = VecG . For most purposes, A can be regarded as a (composite) non-Abelian anyon of quantum dimension dA . Gapped boundaries are conjectured to be in one-to-one correspondence to indecomposable module categories Mi over C. Then, elementary

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excitations on Mi are the simple objects in the functor fusion category Cii = FunC (Mi , Mi ), and simple boundary defects between two gapped boundaries Mi and Mj are the simple objects in the bimodule category Cij = FunC (Mi , Mj ). The collections of fusion categories Cii and their bimodule categories Cij form a multifusion category C. From this multi-fusion category, we can find quantum dimensions of both boundary excitations and the defects between gapped boundaries. Topological quantum computation with gapped boundaries is investigated in Ref. 3 and a striking example is the universal gate set from a purely Abelian TPM.4 Topological quantum computation with boundary defects are also very interesting, but a systematical study has not been initiated. 3. Three-Dimensional Topological Physics Beyond Anyons Classically the universe has three spacial dimensions, but nano-technology makes the study of low-dimensional physics exciting such as anyons in 2D. Therefore both as a toy model for 3D physics and potentially realistic low-dimensional physics, it is interesting to consider all possible 3d physics including the Yang–Mills theory. One salient feature of 3d is the Chern–Simons (CS) action which could be coupled to Yang–Mills. A fascinating direction is 3d quantum gravity. Classical 3d gravity is the same as a doubled CS theory with gauge group SL(2, R), but how to quantize doubled SL(2, R)-CS theory, which has a non-compact gauge group, is very challenging. This is an excellent example for the generalization of anyon systems to topological systems with infinitely many elementary excitation types, closely related to BTZ black holes. The geometry, topology, and physics in and around 3d pure quantum gravity with negative cosmological constant center on the relation between quantum 3D gravity and quantum doubled CS gauge theory (complicated by the invertibility of viebeins), the existence of BTZ black holes, and the asymptotic Virasoro algebra discovered by Brown and Henneaux, which is a precursor of the Ads/CFT correspondence. 3.1. 3d gravity Let X 3 be a closed oriented 3d spacetime manifold and g a gravitational field. The Einstein–Hilbert action is Z √ I(g) = d3 x g(R − 2Λ) , X3

where R is the scalar curvature and Λ the cosmological constant. The equation of motion gives rise to the Einstein equation: 1 Rµν − Rgµν + Λgµν = 0 , 2 where Rµν is the Ricci curvature. The 3d anti-de Sitter space is the subspace of R4 defined as {−x21 − x22 + x23 + x24 = −l2 } ,

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for some constant l > 0 with the metric ds2 = −dx21 − dx22 + dx23 + dx24 . Direct computation gives Rµν = −

2 gµν , l2

R=−

6 , l2

therefore the gravitational field ds2 = −dx21 − dx22 + dx23 + dx24 is a solution of the Einstein equation if we choose the negative cosmological constant Λ = − l12 . Topologically the anti-de Sitter space is simply S 1 × R2 . The tangent bundle of X 3 is trivial so T X ∼ = X 3 × R3 . A framing e: T X ∼ = 3 3 X × R is a choice of such an identification, which is called a vierbein (really dreibein) in physics. Let ω be a spin connection, then (ω, e) can be made into an SO(2, 2) gauge field. Let A± = ω ± el , then the Einstein–Hilbert action becomes a doubled CS action I(X, g) = k4πL CS(A+ ) − k4πR CS(A− ). Therefore, classical 3d gravity with Λ < 0 is the same as doubled CS theory with levels kL = kR = 3G 2l , 18 where G is the Newton constant. The quantum theory of 3d gravity is more subtle as the correspondence with CS theory is not exact due to the difference between gauge transformations and non-invertible vierbeins. But if 3d quantum gravity can be defined mathematically, it should be some irrational TQFT. Then solving 3d quantum gravity would be to find the corresponding conformal field theory in a sense, presumably irrational too. Speculatively, then BTZ back holes would be anyon-like objects. 3.2. Volume conjecture One profound implication of 3d quantum gravity is the possibility of a volume conjecture.18 The volume conjecture is made precise for hyperbolic knots.11 Our interest is on closed hyperbolic 3-manifolds. It can be easily seen that the naive generalization from knots to closed 3-manifolds using rational unitary TQFTs cannot be correct as the 3-manifold invariants grow only polynomially as the level goes to infinity. The most promising version is to use non-unitary rational TQFTs as in Ref. 5. It is puzzling how the non-unitarity arises from the unitary 3d quantum gravity. 4. Four-Dimensional Topological Physics Two interesting families of (3 + 1)-TQFTs are discrete gauge theories and BF theories. Both families are related to the Crane–Frenkel–Yetter (CFY) (3 + 1)TQFTs based on unitary pre-modular categories, which can be realized by lattice models. A more general framework for the CFY TQFTs are the Mackaay’s TQFTs based on spherical 2-categories.12 Cui generalized the CFY construction to G-crossed braided fusion categories.6 The resulting new (3 + 1)-TQFTs do not fit into Mackaay’s notion of spherical 2-categories. Therefore, one problem is to formulate a

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higher category theory that underlies all these theories, and study their application in 3D TPMs. Cui’s TQFTs can also be realized with exactly solvable lattice models.17 4.1. Crane Frenkel Yetter TQFTs and tetra-categories The lattice models for (3 + 1)-TQFTs are generalizations of the LW models in 2D. We expect a tetra-category, which is some double of the input tri-category, to describe all elementary excitations. Algebraic formulation of general tri- and tetra-categories are extremely complicated. The physical intuition we gain from the lattice models would help us to understand these special tetra-categories. 4.2. Representations of motion groups and statistics of extended objects One lesson that we learned from 2D is that deep information about TQFTs is encoded in their associated representations of the mapping class groups, in particular the braid groups. The braid groups are motion groups of points in the 2D disk. In 3D, given a 3-manifold M and a (not necessarily connected) sub-manifold N , we can define the motion group of N in M . The first interesting cases will be for links in S 3 . While the partition functions of the CFY (3 + 1)-TQFTs are not necessarily interesting topological invariants, their induced representations of the motion groups could be more interesting. The Mueger center describes the pointed excitations in lattice models. More interesting in 3D are the loop excitations or in general excitations of the shape of any closed surface. Consider the simplest case for n unlinked unknotted oriented closed loops in R3 , we obtain the n-component loop braid group LB n . The elementary exchange of two loops leads to a subgroup isomorphic to the permutation group Sn , and the passing-through operation to a subgroup isomorphic to the braid group Bn . It follows that the loop braid group for n unlinked unknotted oriented closed loops in R3 is some product of the permutation group Sn with the braid group Bn . By general TQFT properties, each Crane–Frenkel–Yetter TQFT will lead to a representation of the loop braid group. Physically, we are computing the generalized statistics of loop excitations in the lattice model. There is no reason to consider only unknotted loops. Similarly we can consider statistics of knotted loop excitations. 4.3. Fracton physics Another question in 4d physics is how topological is the fracton physical systems.10,15 The low energy effective theories of fracton systems are not TQFTs as the ground state degeneracy of a fixed space manifold grows as the lattice size grows. Therefore, a new framework beyond anyons is necessary to capture these new systems.

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5. Applications An immediate application of topological phases of matter is the construction of a scalable fault-tolerant quantum computer. In quantum computation, qubit is an abstraction of all 2-level quantum systems, though it is not the same as any particular one just like the number 1 is not the same as an apple. Quantumness should be a new source of energy, and a quantum computer is a machine that converts quantum resources such as superposition and entanglement into useful energy. We need new constants similar to the Planck constant or Boltzmann constant to quantify the energy in superposition and entanglement. Acknowledgment This work was partially supported by NSF DMS under Grant No. 1411212. References 1. M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, arXiv:1410.4540. 2. S.-X. Cui, C. Galindo, J. Plavnik and Z. Wang, Commun. Math. Phys. 348, 1043 (2016). 3. I. Cong, M. Cheng and Z. Wang, arXiv:1609.02037. 4. I. Cong, M. Cheng and Z. Wang, Phys. Rev. Lett. 119, 170504 (2017), arXiv:1707. 05490. 5. Q. Chen and T. Yang, arXiv:1503.02547. 6. S.-X. Cui, arXiv:1610.07628. 7. C. Delaney and Z. Wang, Symmetry defects and their application to topological quantum computing, in preparation. 8. https://aimath.org/pastworkshops/topquantum.html. 9. M. Freedman, A. Kitaev, M. Larsen and Z. Wang, B. Am. Math. Soc. 40, 31 (2003). 10. J. Haah, Phys. Rev. A 83, 042330 (2011). 11. R. Kashaev, Lett. Math. Phys. 39, 269 (1997). 12. M. Mackaay, Adv. Math. 143, 288 (1999). 13. E. C. Rowell and Z. Wang, arXiv:1705.06206. 14. E. C. Rowell and Z. Wang, Phys. Rev. A 93, 030102 (2016). 15. S. Vijay, J. Haah and L. Fu, Phys. Rev. B 94, 235157 (2016). 16. Z. Wang, Topological Quantum Computation, Issue 112 of Regional Conference Series in Mathematics (American Mathematical Soc., 2010). 17. D. Williamson and Z. Wang, Ann. Phys. 377, 311 (2017). 18. E. Witten, Nucl. Phys. B 311, 46 (1988).

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Chapter 7 Four revolutions in physics and the A unification second quantum revolution of force and matter by quantum information∗

Xiao-Gang Wen Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Newton’s mechanical revolution unifies the motion of planets in the sky and the falling of apples on Earth. Maxwell’s electromagnetic revolution unifies electricity, magnetism, and light. Einstein’s relativistic revolution unifies space with time, and gravity with space–time distortion. The quantum revolution unifies particle with waves, and energy with frequency. Each of those revolution changes our world view. In this article, we will describe a revolution that is happening now: the second quantum revolution which unifies matter/space with information. In other words, the new world view suggests that elementary particles (the bosonic force particles and fermionic matter particles) all originated from quantum information (qubits): they are collective excitations of an entangled qubit ocean that corresponds to our space. The beautiful geometric Yang–Mills gauge theory and the strange Fermi statistics of matter particles now have a common algebraic quantum informational origin.

Symmetry is beautiful and rich. Quantum entanglement is even more beautiful and richer.

1. Four Revolutions in Physics We have a strong desire to understand everything from a single or very few origins. Driven by such a desire, physics theories were developed through the cycle of discoveries, unification, more discoveries, and bigger unification. Here, we would like to review the development of physics and its four revolutions.a We will see that the

∗ This

chapter also appeared in International Journal of Modern Physics A, Vol. 32, No. ?? (2018) 1830010. DOI: 10.1142/S0217979218300104. a Here we do not discuss the revolution for thermodynamical and statistical physics.

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history of physics can be summarized into three stages: (1) all matter is formed by particles; (2) the discovery of wave-like matter; (3) particle-like matter = wave-like matter. It appears that we are now entering the fourth stage: matter and space = information (qubits), where qubits emerge as the origin of everything.1–8 In other words, all elementary bosonic force particles and fermionic matter particles can be unified by quantum information (qubits). 1.1. Mechanical revolution Although the downpull by the earth was realized even before human civilization, such a phenomenon did not arouse any curiosity. On the other hand, the planet motion in the sky has aroused a lot of curiosity and led to many imaginary fantasies. However, it was only after Kepler found that planets move in a certain way described by a mathematical formula (see Fig. 1) that people started to wonder: Why are planets so rational? Why do they move in such a peculiar and precise way? This motivated Newton to develop his theory of gravity and his laws of mechanical motion (see Fig. 2). Newton’s theory not only explains the planets’ motion; it also explains the downpull that we feel on earth. The planets’ motion in the sky and the apple falling on earth look very different (see Fig. 3); however, Newton’s theory unifies the two seemingly unrelated phenomena. This is the first revolution in physics — the mechanical revolution.

Fig. 1. Kepler’s Laws of Planetary Motion: (1) The orbit of a planet is an ellipse with the Sun at one of the two foci. (2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. (3) The square of the orbital period of a planet is proportional to the cube of the semimajor axis of its orbit.

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Fig. 2. Newton laws: (a) The more force, the more acceleration, no force leads to no acceleration. (b) Action force = reaction force. (c) Newton’s universal gravitation: F = G m1r2m2 , where G = 3

m . 6.674 × 10−11 kg s2

Fig. 3. The perceived trajectories of planets (Mars and Saturn) in the sky. The falling of an apple on earth and the motion of a planet in the sky are unified by Newton’s theory.

Mechanical revolution All matter is formed by particles, which obey Newton’s laws. Interactions are instantaneous over distance. After Newton, we view all matter as formed by particles, and use Newton’s laws for particles to understand the motion of all matter. The success and the completeness of Newton’s theory gave us a sense that we understood everything. 1.2. Electromagnetic revolution But, later we discovered that two other seemingly unrelated phenomena, electricity and magnetism, can generate each other (see Fig. 4). Our curiosity about electricity and magnetism led to another giant leap in science, which is summarized by Maxwell’s equations. Maxwell’s theory unifies electricity and magnetism and reveals that light is merely an electromagnetic wave (see Fig. 5). We gain a much deeper understanding of light, which is so familiar and yet so unexpectedly rich and

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(a)

(b)

(c)

Fig. 4. (a) A changing magnetic field can generate an electric field around it, that drives electric current in a coil. (b) An electric current I in a wire can generate a magnetic field B around it. (c) A changing electric field E (just like electric current) can generate a magnetic field B around it.

Fig. 5. Three very different phenomena, electricity, magnetism, and light, are unified by Maxwell’s theory.

complex in its internal structure. This can be viewed as the second revolution — electromagnetic revolution. Electromagnetic revolution The discovery of a new form of matter — wave-like matter: electromagnetic waves, which obey Maxwell equation. Wave-like matter causes interaction. However, the true essence of Maxwell theory is the discovery of a new form of matter — wave-like (or field-like) matter (see Fig. 6), the electromagnetic wave. The motion of this wave-like matter is governed by Maxwell’s equation, which is very different from the particle-like matter governed by Newton’s equation F = ma. Thus, the sense that Newtonian theory describes everything is incorrect. Newtonian theory does not apply to wave-like matter, which requires a new theory — Maxwell’s theory. Unlike particle-like matter, the new wave-like matter is closely related to a kind of interaction — electromagnetic interaction. In fact, electromagnetic interaction can be viewed as an effect of the newly discovered wave-like matter.

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(a)

(b)

185

(c)

Fig. 6. (a) Magnetic field revealed by iron powder. (b) Electric field revealed by glowing plasma. (c) They form a new kind of matter: light — a wave-like matter.

1.3. Relativity revolution After realizing the connection between the interaction and wave-like matter, one naturally asks: does gravitational interaction also correspond to a wave-like matter? The answer is yes. First, people realized that Newton’s equation and Maxwell’s equation have different symmetries under the transformations between two frames moving against each other. In other words, Newton’s equation F = ma is invariant under Galilean transformation, while Maxwell’s equation is invariant under Lorentz transformation (see Fig. 7). Certainly, only one of the above two transformations is correct. If one believes that physical laws should be the same in different frames, then the above observation implies that Newton’s equation and Maxwell’s equation are incompatible, and one of them must be wrong. If Galilean transformation is correct, then the Maxwell theory is wrong and needs to be modified. If Lorentz transformation is correct, then the Newton theory is wrong and needs to be modified. The Michelson–Morley experiment showed that the speed of light is the same in all the frames, which implied the Galilean transformation to be wrong. So, Einstein chose

(a)

(b)

(c)

Fig. 7. (a) A rest frame and a moving frame with velocity v. An event is recorded with coordinates (x, y, z, t) in the rest frame and with (x0 , y 0 , z 0 , t0 ) in the moving frame. There are two opinions on how (x, y, z, t) and (x0 , y 0 , z 0 , t0 ) are related: (b) Galilean transformation or (c) Lorentz transformation where c is the speed of light. In our world, the Lorentz transformation is correct.

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Bob

Alice

earth Fig. 8. The equivalence of the gravitational force of the earth and the force experienced in an accelerating elevator, leads to a geometric way of understanding gravity: gravity = distortion in space. In other words, the “gravitational force” in an accelerating elevator is related to a geometric feature: the transformation between the coordinates in a still elevator and in an accelerating elevator.

to believe in the Maxwell equation. He modified Newton’s equation and developed the theory of special relativity. Thus, the Newton theory is not only incomplete, it is also incorrect. Einstein has gone further. Motivated by the equivalence of gravitational force and the force experienced in an accelerating frame (see Fig. 8), Einstein also developed the theory of general relativity.9 Einstein’s theory unifies several seeming unrelated concepts, such as space and time, as well as interaction and geometry. Since gravity is viewed as a distortion of space and since the distortion can propagate, Einstein discovered the second wave-like matter — gravitational wave (see Fig. 9). This is another revolution in physics — relativity revolution. Relativity revolution A unification of space and time. A unification of gravity and space–time distortion. Motivated by the connection between interaction and geometry in gravity, people went back to reexamine the electromagnetic interaction, and found that the electromagnetic interaction is also connected to geometry. Einstein’s general relativity views gravity as a distortion of space, which can be viewed as a distortion

Fig. 9. Gravitational wave is a propagating distortion of space: a circle is distorted by a gravitational wave.

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Fig. 10. A curved space can be viewed as a distortion of local directions of the space: parallel moving a local direction (represented by an arrow) around a loop in a curved space, the direction of the arrow does not come back. Such a twist in local direction corresponds to a curvature in space.

of local directions of space (see Fig. 10). Motivated by such a picture, in 1918, Weyl proposed that the unit that we used to measure physical quantities is relative and is defined only locally. A distortion of the unit system can be described by a vector field which is called gauge field. Weyl proposed that such a vector field (the gauge field) is the vector potential that describes the electromagnetism. Although the above proposal turns out to be incorrect, Weyl’s idea is correct. In 1925, the complex quantum amplitude was discovered. If we assume the complex phase is relative, then a distortion of the unit system that measures local complex phase can also be described by a vector field. Such a vector field is indeed the vector potential that describes electromagnetism. This leads to a unified way to understand gravity and electromagnetism: gravity arises from the relativity of spatial directions at different spatial points, while electromagnetism arises from the relativity of complex quantum phases at different spatial points. Furthermore, Nordstr¨om, M¨oglichkeit, Kaluza, and Klein showed that both gravity and electromagnetism can be understood as a distortion of space–time provided that we think of the space–time as five-dimensional with one dimension compactified into a small circle.10–12 This can be viewed as an unification of gravity and electromagnetism. Those theories are so beautiful. Since that time, the geometric way to view our world has dominated theoretical physics. 1.4. Quantum revolution However, such a geometric view of the world was immediately challenged by new discoveries from the microscopic world.b Experiments in the microscopic world tell us that not only is Newtonian theory incorrect, even its relativistic modification b Many

people have ignored such challenges and the geometric world view becomes mainstream.

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(a)

(b)

Fig. 11. (a) An electron beam passing through a double-slit can generate an interference pattern, indicating that electrons are also waves. (b) Using light to eject electrons from a metal (the photoelectric effect) shows that the higher the light wave frequency (the shorter the wave length), the higher the energy of the ejected electron. This reveals that a light wave of frequency f can be 2

viewed a beam of particles of energy E = hf , where h = 6.62607004 × 10−34 m skg .

is incorrect. This is because Newtonian theory and its relativistic modification are theories for particle-like matter. But through experiments on very tiny things, such as electrons, people found that particles are not really particles. They also behave like waves at the same time. Similarly, experiments also reveal that light waves behave like a beam of particles (photons) at the same time (see Fig. 11). So, the matter in our world is not what we thought it was. Matter is neither particle nor wave, and both particle and wave. So, the Newton theory (and its relativistic modification) for particle-like matter and the Maxwell/Einstein theories for wavelike matter cannot be the correct theories for matter. We need a new theory for the new form of existence: particle–wave-like matter. The new theory is the quantum theory that explains the microscopic world. The quantum theory unifies particle-like matter and wave-like matter. Quantum revolution There is no particle-like matter nor wave-like matter. All the matter in our world is particle–wave-like matter. From the above, we see that quantum theory reveals the true existence in our world to be quite different from the classical notion of existence in our mind. What exist in our world are not particles or waves but both particle and wave. Such a picture is beyond our wildest imagination, but reflects the truth about our world and is the essence of quantum theory. To understand the new notion of existence more clearly, let us consider another example. This time it is about a bit (represented by spin-1/2). A bit has two possible states of classical existence: |1i = | ↑i and |0i = | ↓i. However, quantum theory also allows a new kind of existence | ↑i + | ↓i. One may say that | ↑i + | ↓i is also a classical existence since | ↑i + | ↓i = |→i that describes a spin in x-direction. So, let us consider a third example of two bits. Then there will be four possible states of classical existence: | ↑↑i, | ↑↓i, | ↓↑i, and

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Fig. 12. To observe two points of distance l apart, we need to send in light of wave length λ < l. The corresponding photon has an energy E = hc/λ. If l is less than the Planck length l < lP , then the photon will make a back hole of size larger then l. The black hole will swallow the two points, and we can never measure the separation of two points of distance less than lP . What cannot be measured cannot exists. So the notion of “two points less than lP apart” has no physical meaning and does not exist.

| ↓↓i. Quantum theory allows a new kind of existence | ↑↑i + | ↓↓i. Such a quantum existence is entangled and has no classical analogues. Although the geometric way to understand our world is mainstream in physics, here we will take a position that the geometric understanding is not good enough and will try to advocate a very different non-geometric understanding of our world. Why is the geometric understanding not good enough? First the geometric understanding is not self-consistent. It contradicts with quantum theory. The consideration based on quantum mechanics and Einstein’s gravity indicates that two points separated by a distance less than the Planck length r ~G lP = = 1.616199 × 1035 m (1) c3 cannot exist as a physical reality (see Fig. 12). Thus, the foundation of the geometric approach — manifold — simply does not exist in our universe, since manifold contains points with arbitrarily small separation. This suggests that geometry is an emergent phenomenon that appears only at long distances. So, we cannot use geometry and manifold as a foundation to understand fundamental physical problems. Second, Maxwell’s theory of light and Einstein’s theory of gravity predict light waves and gravitational waves. But the theories fail to tell us what is waving. Maxwell’s theory and Einstein’s theory are built on top of geometry. They fail to answer the question of what the origin of the apparent geometry that we see is. In other words, the Maxwell theory and Einstein theory are incomplete, and they should be regarded as effective theories at long distances. Since geometry does not exist in our world, we say the geometric view of world is challenged by quantum theory. The quantum theory tell us such a point of view is wrong at length scales of order Planck length. So, the quantum theory represents the most dramatic revolution in physics. 2. It From Qubit, Not Bit

A Second Quantum Revolution

After realizing that even the notion of existence is changed by quantum theory, it is no longer surprising to see that quantum theory also blurs the distinction between

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information and matter. In fact, it implies that information is matter, and matter is information. This is because the frequency is an attribute of information. Quantum theory tells us that frequency is energy E = hf , and relativity tells us that energy is mass m = E/c2 . Both energy and mass are attributes of matter. So matter = information. This represents a new way to view our world.

The essence of quantum theory The energy–frequency relation E = hf implies that matter = information.

The above point of view of “matter = information” is similar to Wheeler’s “it from bit,” which represents a deep desire to unify matter and information. In fact, such a unification has happened before at a small scale. We introduced electric and magnetic field to informationally (or pictorially) describe electric and magnetic interaction. But later, electric/magnetic field became real matter with energy and momentum, and even has a particle associated with it. However, in our world, “it” are very complicated. (1) Most “it” are fermions, while “bit” are bosonic. Can fermionic “it” come from bosonic “bit”? (2) Most “it” also carry spin-1/2. Can spin-1/2 arise from “bit”? (3) All “it” interact via a special kind of interaction — gauge interaction. Can “bit” produce gauge interaction? Can “bit” produce waves that satisfy the Maxwell equation? Can “bit” produce a photon? In other words, to understand the concrete meaning of “matter from information” or “it from bit,” we note that matter is described by Maxwell’s equation (photons), Yang–Mills equation (gluons and W/Z bosons), as well as Dirac and Weyl equations (electrons, quarks, neutrinos). The statement “matter = information” means that those wave equations can all come from qubits. In other words, we know that elementary particles (i.e. matter) are described by gauge fields and anti-commuting fields in a quantum field theory. Here we try to say that all those very different quantum fields can arise from qubits. Is this possible? All the waves and fields mentioned above are waves and fields in space. The discovery of the gravitational wave strongly suggested that space is a deformable dynamical medium. In fact, the discovery of the electromagnetic wave and the Casimir effect already strongly suggested that space is a deformable dynamical medium. As a dynamical medium, it is not surprising that the deformation of space gives rise to various waves. But the dynamical medium that describes our space must be very special, since it should give rise to waves satisfying the Einstein equation (gravitational wave), Maxwell equation (electromagnetic wave), Dirac equation (electron wave), etc. But what is the microscopic structure of space? What kind of microscopic structure can, at the same time, give rise to waves that satisfy the Maxwell equation, Dirac/Weyl equation, and Einstein equation?

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Let us view the above questions from another angle. Modern science has made many discoveries and has also unified many seemingly unrelated discoveries into a few simple structures. Those simple structures are so beautiful and we regard them as wonders of our universe. They are also very mysterious since we do not understand where they come from and why they have to be the way they are. At the moment, the most fundamental mysteries and/or wonders in our universe can be summarized by the following short list: Eight wonders: (1) (2) (3) (4) (5) (6) (7) (8)

Locality. Identical particles. Gauge interactions.13–15 Fermi statistics.16,17 Tiny masses of fermions (∼ 10−20 of the Planck mass).2,18,19 Chiral fermions.6,7,20,21 Lorentz invariance.22 as Gravity.9

In the current physical theory of nature (such as the standard model), we take the above properties for granted and do not ask where they come from. We put those wonderful properties into our theory by hand, for example, by introducing one field for each kind of interaction or elementary particle. However, here we would like to question where those wonderful and mysterious properties come from. Following the trend of science history, we wish to have a single unified understanding of all the above mysteries. Or more precisely, we wish that we can start from a single structure to obtain all the above wonderful properties. The simplest element in quantum theory is qubit |0i and |1i (or | ↓i and | ↑i). Qubit is also the simplest element in quantum information. Since our space is a dynamical medium, the simplest choice is to assume space to be an ocean of qubits. We will give such an ocean a formal name, “qubit ether.” Then the matter, i.e. the elementary particles, are simply the waves, “bubbles” and other defects in the qubit ocean (or qubit ether). This is how we get “it from qubit” or “matter = information.” Qubit, having only two states | ↓i and | ↑i, is very simple. We may view the many-qubit state with all qubits in | ↓i as the quantum state that corresponds to the empty space (the vacuum). Then the many-qubit state with a few qubits in | ↑i corresponds to a space with a few spin-0 particles described by a scalar field. Thus, it is easy to see that a scalar field can emerge from qubit ether as a density wave of up-qubits. Such a wave satisfies the Euler equation, but not the Maxwell equation or Yang–Mills equation. So the above particular qubit ether is not the one that corresponds to our space. It has the wrong microscopic structure and cannot carry waves satisfying the Maxwell equation and Yang–Mills equation. But this

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line of thinking may be correct. We just need to find a qubit ether with a different microscopic structure. However, for a long time, we do not know how waves satisfying the Maxwell equation or Yang–Mills equation can emerge from any qubit ether. The anticommuting wave that satisfies the Dirac/Weyl equation seems even more impossible. So, even though quantum theory strongly suggests “matter = information,” trying to obtain all elementary particles from an ocean of simple qubits is regarded as impossible by many and has never become an active research effort. So, the key to understand “matter = information” is to identify the microscopic structure of the qubit ether (which can be viewed as space). The microscopic structure of our space must be very rich, since our space can not only carry gravitational wave and electromagnetic wave, it can also carry electron wave, quark wave, gluon wave, and the waves that correspond to all elementary particles. Is such a qubit ether possible? In condensed matter physics, the discovery of fractional quantum Hall states23 brings us into a new world of highly entangled many-body systems. When the strong entanglement becomes long range entanglement,24 the systems will possess a new kind of order–topological order,25,26 and represent new states of matter. We find that the waves (the excitations) in topologically ordered qubit states can be very strange: they can be waves that satisfy the Maxwell equation, Yang–Mills equation, or Dirac/Weyl equation. So the impossible become possible: all elementary particles (the bosonic force particles and fermionic matter particles) can emerge from long range entangled qubit ether and be unified by quantum information.2–8,27,28 We would like to stress that the above picture is “it from qubit,” which is very different from Wheeler’s “it from bit.” As we have explained, our observed elementary particles can only emerge from long range entangled qubit ether. The requirement of quantum entanglement implies that “it cannot from bit.” In fact “it from entangled qubits.” 3. A String-Net Liquid of Qubits and A Unification of Gauge Interactions and Fermi Statistics In this section, we will consider a particular entangled qubit ocean — a string liquid of qubits. Such an entangled qubit ocean supports new kind of waves and their corresponding particles. We find that the new waves and the emergent statistics are so profound that they may change our view of the universe. Let us start by explaining a basic notion — the “principle of emergence.” 3.1. Principle of emergence Typically, one thinks the properties of a material should be determined by the components that form the material. However, this simple intuition is incorrect, since all materials are made of the same components: electrons, protons and neutrons. So, we cannot use the richness of the components to understand the richness of

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Liquids only have a compression wave — a wave of density fluctuations.

the materials. In fact, the various properties of different materials originate from various ways in which the particles are organized. Different orders (the organizations of particles) give rise to different physical properties of a material. It is the richness of the orders that gives rise to the richness of the material world. Let us use the origin of mechanical properties and the origin of waves to explain, in a more concrete way, how orders determine the physical properties of a material. We know that a deformation in a material can propagate just like a ripple on the surface of water. The propagating deformation corresponds to a wave traveling through the material. Since liquids can resist only compression deformation, liquids can only support a single kind of wave — the compression wave (see Fig. 13). (Compression wave is also called longitudinal wave.) Mathematically the motion of the compression wave is governed by the Euler equation ∂2ρ ∂2ρ − v 2 2 = 0, 2 ∂t ∂x

(2)

where ρ is the density of the liquid. A solid can resist both compression and shear deformations. As a result, solids can support both a compression wave and transverse wave. The transverse wave corresponds to the propagation of shear deformations. In fact, there are two transverse waves corresponding to two directions of shear deformations. The propagation of the compression wave and the two transverse waves in solids are described by the elasticity equation 2 j ∂ 2 ui ikl ∂ u − T =0 j ∂t2 ∂xk ∂xl

(3)

where the vector field ui (x, t) describes the local displacement of the solid. We would like to point out that the elasticity equation and the Euler equation not only describe the propagation of waves, they actually describe all small deformations in solids and liquids. Thus, the two equations represent a complete mathematical description of the mechanical properties of solids and liquids. But why do solids and liquids behave so differently? What makes a solid have a shape and a liquid have no shape? What are the origins of the elasticity equation and Euler equations? The answers to those questions must wait until the discovery

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Fig. 14. Drawing a grid on a solid helps us to see the deformation of the solid. The vector ui in (3) is the displacement of a vertex in the grid. In addition to the compression wave (i.e. the density wave), a solid also supports transverse waves (waves of shear deformation) as shown in the above figure.

(a)

(b)

Fig. 15. (a) Particles in liquids do not have fixed relative positions. They fluctuate freely and have a random but uniform distribution. (b) Particles in solids form a fixed regular lattice.

of atoms in the 19th century. Since then, we realized that both solids and liquids are formed by collections of atoms. The main difference between solids and liquids is that the atoms are organized very differently. In liquids, the positions of atoms fluctuate randomly [see Fig. 15(a)], while in solids, atoms organize into a regular fixed array [see Fig. 15(b)].c It is the different organizations of atoms that lead to the different mechanical properties of liquids and solids. In other words, it is the different organizations of atoms that make liquids able to flow freely and solids able to retain their shapes. How can different organizations of atoms affect mechanical properties of materials? In solids, both the compression deformation [see Fig. 16(a)] and the shear deformation [see Fig. 16(b)] lead to real physical changes of the atomic configurations. Such changes cost energies. As a result, solids can resist both kinds of deformations and can retain their shapes. This is why we have both the compression wave and the transverse wave in solids. In contrast, a shear deformation of atoms in liquids does not result in a new configuration since the atoms still have uniformly random positions. So, the shear deformation is a do-nothing operation for liquids. Only the compression deformation c The

solids here should be more accurately referred to as crystals.

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(a) Fig. 16.

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(b)

The atomic picture of (a) the compression wave and (b) the transverse wave in a crystal.

Fig. 17.

The atomic picture of the compression wave in liquids.

which changes the density of the atoms results in a new atomic configuration and costs energies. As a result, liquids can only resist compression and have only the compression wave. Since shear deformations do not cost any energy for liquids, liquids can flow freely. We see that the properties of the propagating wave are entirely determined by how the atoms are organized in materials. Different organizations lead to different kinds of waves and different kinds of mechanical laws. Such a point of view of having different kinds of waves/laws originate from different organizations of particles is a central theme in condensed matter physics. This point of view is called the principle of emergence. 3.2. String-net liquid of qubits unifies light and electrons The elasticity equation and the Euler equation are two very important equations. They lay the foundation of many branches of science such as mechanical engineering, aerodynamic engineering, etc. But, we have a more important equation, the Maxwell equation, that describes light waves in vacuum. When the Maxwell equation was first introduced, people firmly believed that any wave must correspond to the motion of something. So, people want to find out what is the origin of the Maxwell equation? What is the motion that gives rise to the electromagnetic wave? First, one may wonder: can the Maxwell equation come from a certain symmetry breaking order? Based on Landau symmetry-breaking theory, the different symmetry breaking orders can indeed lead to different waves satisfying different wave equations. So, maybe a certain symmetry breaking order can give rise to a wave that satisfies the Maxwell equation. But people have been searching for ether —

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a medium that supports the light waves — for over 100 years and have not been able to find any symmetry breaking states that can give rise to waves satisfying the Maxwell equation. This is one of the reasons why people have given up the idea of ether as the origin of light and Maxwell equation. However, the discovery of topological order25,26 suggests that Landau symmetrybreaking theory does not describe all possible organizations of bosons/spins. This gives us new hope: the Maxwell equation may arise from a new kind of organizations of bosons/spins that have non-trivial topological orders. In addition to the Maxwell equation, there is an even stranger equation, the Dirac equation, that describes the wave of electrons (and other fermions). Electrons have Fermi statistics. They are fundamentally different from the quanta of other familiar waves, such as photons and phonons, since those quanta all have Bose statistics. To describe the electron wave, the amplitude of the wave must be anticommuting Grassmann numbers, so that the wave quanta will have Fermi statistics. Since electrons are so strange, few people regard electrons and electron waves as collective motions of something. People accept without questioning that electrons are fundamental particles, one of the building blocks of all that exist. However, from a condensed matter physics point of view, all low energy excitations are collective motions of something. If we try to regard photons as collective modes, why can’t we regard electrons as collective modes as well? So maybe, the Dirac equation and associated fermions can also arise from a new kind of organization of bosons/spins that have non-trivial topological orders. A recent study provides a positive answer to the above questions.3,29,30 We find that if bosons/spins form large oriented strings and if those strings form a quantum liquid state, then the collective motion of the such organized bosons/spins will correspond to waves described by the Maxwell equation and Dirac equation. The strings in the string liquid are free to join and cross each other. As a result, the strings look more like a network (see Fig. 18). For this reason, the string liquid is actually a liquid of string-nets, which is called string-net condensed state. But why does the waving of strings produce waves described by the Maxwell equation? We know that the particles in a liquid have a random but uniform distri-

Fig. 18. A quantum ether: The fluctuation of oriented strings give rise to electromagnetic waves (or light). The ends of strings give rise to electrons. Note that oriented strings have directions which should be described by curves with arrows. For ease of drawing, the arrows on the curves are omitted in the above plot.

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The fluctuating strings in a string liquid.

Fig. 20. A “density” wave of oriented strings in a string liquid. The wave propagates in xdirection. The “density” vector E points in y-direction. For ease of drawing, the arrows on the oriented strings are omitted in the above plot.

bution. A deformation of such a distribution corresponds to a density fluctuation, which can be described by a scaler field ρ(x, t). Thus the waves in a liquid are described by the scalar field ρ(x, t) which satisfy the Euler equation (2). Similarly, the strings in a string-net liquid also have a random but uniform distribution (see Fig. 19). A deformation of string-net liquid corresponds to a change of the density of the strings (see Fig. 20). However, since strings have an orientation, the “density” fluctuations are described by a vector field E(x, t), which indicates there are more strings in the E direction on average. The oriented strings can be regarded as flux lines. The vector field E(x, t) describes the smeared average flux. Since strings are continuous (i.e. they cannot end), the flux is conserved: ∂ · E(x, t) = 0. The vector density E(x, t) of strings cannot change in the direction along the strings (i.e. along the E(x, t) direction). E(x, t) can change only in the direction perpendicular to E(x, t). Since the direction of the propagation is the same as the direction in which E(x, t) varies, thus the waves described by E(x, t) must be transverse waves: E(x, t) is always perpendicular to the direction of the propagation. Therefore, the waves in the string liquid have a very special property: the waves have only transverse modes and no longitudinal mode. This is exactly the property of the light waves described by the Maxwell equation. We see that “density” fluctuations of strings (which are described be a transverse vector field) naturally give rise to the light (or electromagnetic) waves and the Maxwell equation.2,3,30–33

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It is interesting to compare solid, liquid, and string-net liquid. We know that the particles in a solid are organized into a regular lattice pattern. The waving of such organized particles produces a compression wave and two transverse waves. The particles in a liquid have a more random organization. As a result, the waves in liquids lose two transverse modes and contain only a single compression mode. The particles in a string-net liquid also have a random organization, but in a different way. The particles first form string-nets and string-nets then form a random liquid state. Due to this different kind of randomness, the waves in string-net condensed state lose the compression mode and contain two transverse modes. Such a wave (having only two transverse modes) is just the electromagnetic wave. To understand how electrons appear from string-nets, we would like to point out that if we only want photons and no other particles, the strings must be closed strings with no ends. The fluctuations of closed strings produce only photons. If strings have open ends, those open ends can move around and just behave like independent particles. Those particles are not photons. In fact, the ends of strings are nothing but electrons. How do we know that ends of strings behave like electrons? First, since the waving of string-nets is an electromagnetic wave, a deformation of string-nets corresponds to an electromagnetic field. So, we can study how an end of a string interacts with a deformation of string-nets. We find that such an interaction is just like the interaction between a charged electron and an electromagnetic field. Also, electrons have a subtle but very important property — Fermi statistics, which is a property that exists only in quantum theory. Amazingly, the ends of strings can reproduce this subtle quantum property of Fermi statistics.29,34 Actually, string-net liquids explain why Fermi statistics should exist. We see that qubits that organize into string-net liquid naturally explain both light and electrons (gauge interactions and Fermi statistics). In other words, stringnet theory provides a way to unify light and electrons.3,30 So, the fact that our vacuum contains both light and electrons may not be a mere accident. It may actually suggest that the vacuum is indeed a string-net liquid. 3.3. More general string-net liquid and emergence of Yang Mills gauge theory Here, we would like to point out that there are many different kinds of string-net liquids. The strings in different liquids may have different numbers of types. The strings may also join in different ways. For a general string-net liquid, the waving of the strings may not correspond to light and the ends of strings may not be electrons. Only one kind of string-net liquids give rise to light and electrons. On the other hand, the fact that there are many kinds of string-net liquids allows us to explain more than just light and electrons. We can design a particular type of string-net liquids which not only gives rise to electrons and photons, but also gives rise to quarks and gluons.2,29 The waving of such type of string-nets corresponds

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to photons (light) and gluons. The ends of different types of strings correspond to electrons and quarks. It would be interesting to see if it is possible to design a string-net liquid that produces all elementary particles! If this is possible, the ether formed by such string-nets can provide an origin of all elementary particles.d We like to stress that the string-nets are formed by qubits. So, in the string-net picture, both the Maxwell equation and Dirac equation emerge from a local qubit model, as long as the qubits from a long-range entangled state (i.e. a string-net liquid). In other words, light and electrons are unified by the long-range entanglement of qubits! The electric field and the magnetic field in the Maxwell equation are called gauge fields. The field in the Dirac equation are Grassmann-number valued field.e For a long time, we thought that we had to use gauge fields to describe light waves that have only two transverse modes, and we thought that we had to use Grassmannnumber valued fields to describe electrons and quarks that have Fermi statistics. So gauge fields and Grassmann-number valued fields became the fundamental building blocks of quantum field theory that describe our world. The string-net liquids demonstrate we do not have to introduce gauge fields and Grassmann-number valued fields to describe photons, gluons, electrons, and quarks. It demonstrates how gauge fields and Grassmann fields emerge from local qubit models that contain only complex scalar fields at the cutoff scale. Our attempt to understand light has a long and evolving history. We first thought light to be a beam of particles. After Maxwell, we understand light as electromagnetic waves. After Einstein’s theory of general relativity, where gravity is viewed as curvature in space–time, Weyl and others tried to view the electromagnetic field as curvatures in the “unit system” that we used to measure complex phases. It leads to the notion of gauge theory. General relativity and the gauge theory are two cornerstones of modern physics. They provide a unified understanding of all four interactions in terms of a beautiful mathematical framework: all interactions can be understood geometrically as curvatures in space–time and in “unit systems” (or more precisely, as curvatures in the tangent bundle and other vector bundles in space–time). Later, people in high-energy physics and in condensed matter physics found another way in which the gauge field can emerge35–38 : one first cuts a particle (such as an electron) into two partons by writing the field of the particle as the product of the two fields of the two partons. Then one introduces a gauge field to glue the two partons back to the original particle. Such a “glue-picture” of gauge fields (instead of the fiber bundle picture of gauge fields) allows us to understand the emergence of gauge fields in models that originally contain no gauge field at the cutoff scale.

d So

far we can use string-net to produce almost all elementary particles, expect for the graviton that is responsible for gravity. In particular, we can even produce the chiral coupling between the SU (2) gauge boson and the fermions from the qubit ocean.6,7 e Grassmann numbers are anticommuting numbers.

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A string picture represents the third way understanding gauge theory. String operators appear in the Wilson-loop characterization39 of gauge theory. The Hamiltonian and the duality description of lattice gauge theory also reveal string structures.40–43 Lattice gauge theories are not local bosonic models and the strings are unbreakable in lattice gauge theories. String-net theory points out that even breakable strings can give rise to gauge fields.44 So we do not really need strings. Qubits themselves are capable of generating gauge fields and the associated Maxwell equation. This phenomenon was discovered in several qubit models1,27,33,37,45 before realizing their connection to the string-net liquids.31 Since gauge fields can emerge from local qubit models, the string picture evolves into the entanglement picture — the fourth way to understand gauge field: gauge fields are fluctuations of long-range entanglement. I feel that the entanglement picture captures the essence of gauge theory. Despite the beauty of the geometric picture, the essence of gauge theory was not the curved fiber bundles. In fact, we can view gauge theory as a theory for long-range entanglement, although the gauge theory was discovered long before the notion of long-range entanglement. The evolution of our understanding of light and gauge interaction is: particle beam → wave → electromagnetic wave → curvature in fiber bundle → glue of partons → wave in string-net liquid → wave in long-range entanglement; this represents the 200-year effort of the human race to unveil the mystery of universe (see Fig. 21).

(a)

(b)

(e)

(c)

(f)

(d)

(g)

Fig. 21. The evolution of our understanding of light (and gauge interaction): (a) particle beam, (b) wave, (c) electromagnetic wave, (d) curvature in fiber bundle, (e) glue of partons, (f) wave in string-net liquid and (g) wave in long-range entanglement of many qubits.

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Viewing gauge fields (and the associated gauge bosons) as fluctuations of longrange entanglement has an added bonus: we can understand the origin of Fermi statistics in the same way: fermions emerge as defects of long-range entanglement, even though the original model is purely bosonic. Previously, there are two ways to obtain emergent fermions from purely bosonic model: by binding gauge charge and gauge flux in (2 + 1)D,46,47 and by binding the charge and the monopole in a U (1) gauge theory in (3 + 1)D.48–52 But those approaches only work in (2 + 1)D or only for U (1) gauge field. Using long-range entanglement and their string-net realization, we can obtain the simultaneous emergence of both gauge bosons and fermions in any dimensions and for any gauge group.2,29,30,34 This result gives us hope that maybe every elementary particle is emergent and can be unified using local qubit models. Thus, long-range entanglement offers us a new option to view our world: maybe our vacuum is a long-range entangled state. It is the pattern of the long-range entanglement in the vacuum that determines the content and the structures of observed elementary particles. Such a picture has an experimental prediction that will be described in Subsec. 3.4. We like to point out that the string-net unification of gauge bosons and fermions is very different from the superstring theory for gauge bosons and fermions. In the string-net theory, gauge bosons and fermions come from the qubits that form the space, and “string-net” is simply the name that describe how qubits are organized in the ground state. So, string-net is not a thing, but a pattern of qubits. In the string-net theory, the gauge bosons are waves of collective fluctuations of the stringnets, and a fermion corresponds to one end of the string. In contrast, gauge bosons and fermions come from strings in the superstring theory. Both gauge bosons and fermions correspond to small pieces of strings. Different vibrations of the small pieces of strings give rise to different kind of particles. The fermions in superstring theory are put in by hand through the introduction of Grassmann fields. 3.4. A falsifiable prediction of string-net unification of gauge interactions and Fermi statistics In the string-net unification of light and electrons,3,30 we assume that space is formed by a collection of qubits and the qubits form a string-net condensed state. Light waves are collective motions of the string-nets, and an electron corresponds to one end of the string. Such a string-net unification of light and electrons has a falsifiable prediction: all fermionic excitations must carry some gauge charges.29,34 The U (1) × SU (2) × SU (3) standard model for elementary particles contains fermionic excitations (such as neutrons and neutrinos) that do not carry any U (1)× SU (2) × SU (3) gauge charge. So according to the string-net theory, the U (1) × SU (2) × SU (3) standard model is incomplete. According to the string-net theory, our universe not only has U (1) × SU (2) × SU (3) gauge theory, it must also contain other gauge theories. Those additional gauge theories may have a gauge group of Z2 or other discrete groups. Those extra discrete gauge theories will lead to new cosmic strings which appear in the very early universe.

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4. A New Chapter in Physics Our world is rich and complex. When we discover the inner workings of our world and try to describe it, we often find that we need to invent new mathematical language to describe our understanding and insight. For example, when Newton discovered his law of mechanics, the proper mathematical language was not invented yet. Newton (and Leibniz) had to develop calculus in order to formulate the law of mechanics. For a long time, we tried to use the theory of mechanics and calculus to understand everything in our world. As another example, when Einstein discovered the general equivalence principle to describe gravity, he needed a mathematical language to describe his theory. In this case, the needed mathematics, Riemannian geometry, had been developed, which led to the theory of general relativity. Following the idea of general relativity, we developed the gauge theory. Both general relativity and gauge theory can be described by the mathematics of fiber bundles. Those advances led to a beautiful geometric understanding of our world based on quantum field theory, and we tried to understand everything in our world in terms of quantum field theory. Now, I feel that we are at another turning point. In a study of quantum matter, we find that long-range entanglement can give rise to many new quantum phases. So long-range entanglement is a natural phenomenon that can happen in our world. They greatly expand our understanding of possible quantum phases and bring the research of quantum matter to a whole new level. To gain a systematic understanding of new quantum phases and long-range entanglement, we would like to know, what mathematical language should we use to describe long-range entanglement? The answer is not totally clear. But early studies suggest that tensor category29,53–59 and group cohomology60,61 should be a part of the mathematical framework that describes long-range entanglement. Further progresses in this direction will lead to a comprehensive understanding of long-range entanglement and topological quantum matter. However, what is really exciting in the study of quantum matter is that it might lead to a whole new point of view of our world. This is because long-range entanglement can give rise to both gauge interactions and Fermi statistics. In contrast, the geometric point of view can only lead to gauge interactions. So maybe we should not use geometric pictures, based on fields and fiber bundles, to understand our world. Maybe we should use entanglement pictures to understand our world. This way, we can get both gauge interactions and fermions from a single origin — qubits. We may live in a truly quantum world. So, quantum entanglement represents a new chapter in physics.

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Topological insulators from the perspective of first-principles calculations

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Haijun Zhang and Shou-Cheng Zhang* Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-404531, USA Received 28 September 2012, revised 22 November 2012, accepted 22 November 2012 Published online 30 November 2012 Keywords topological insulators, first-principles calculations, spin–orbit coupling, surface states *

Corresponding author: e-mail [email protected], Phone: +01-650-723-2894, Fax: +01-650-723-9389

Topological insulators are new quantum states with helical gapless edge or surface states inside the bulk band gap. These topological surface states are robust against weak timereversal invariant perturbations without closing the bulk band gap, such as lattice distortions and non-magnetic impurities. Recently a variety of topological insulators have been predicted by theories, and observed by experiments. First-principles calculations have been widely used to predict topological insulators with great success. In this review, we summarize the current progress in this field from the perspective of first-principles calculations. First of all, the basic concepts of topological insulators and the frequently-used techniques within first-principles calculations are briefly introduced. Secondly, we summarize general methodologies to search for new topological insulators. In the last part, based on the band inversion picture first introduced in the context of HgTe, we classify topological insulators into three types with s–p, p–p and d–f, and discuss some representative examples for each type.

Surface states of topological insulator Bi2Se3 consist of a single Dirac cone, as obtained from first-principles calculations.

© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction In two-dimensional electron systems at low temperature and strong magnetic field, the Hall conductance σ xy takes quantized values [1], called quantum Hall (QH), which proved to have a fundamental topological meaning [2]. σ xy can be expressed as an integral of the first Chern number over the magnetic Brillouin zone. Quantum anomalous Hall (QAH) is adiabatically equivalent to QH. Both QH and QAH systems are called Chern insulators due to the non-zero Chern number where the time-reversal symmetry (TRS) is broken. Recently quantum spin Hall (QSH) state was predicted in CdTe/HgTe quantum well [3] and soon observed experimentally [4]. Earlier theoretical models [5–7] provided important con-

ceptual framework. In this system TRS is present, and spin–orbit coupling (SOC) effect plays the role of Lorentz force in QH effect. The concept of QSH can be generalized to three-dimensional (3D) topological insulators with TRS [8, 9]. The electromagnetic response of a topological insulator is described by the topological θ term of Sθ = (θ /2π) (α /2π) Ú d 3 x dt E ◊ B with θ = π, where E and B are the external electromagnetic fields [9]. This indicates the physically measurable and topologically non-trivial response, which opens the door for experiments and potential applications of topological insulators. Both 2D (QSH) and 3D topological insulators have interesting physical properties [10–15]. In this review, we © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Reprinted with permission from Physica Status Solidi RRL, Vol. 7, No. 1–2, pp. 72–81 (2013), c 2013 by John Wiley and Sons. DOI: 10.1002/pssr.201206414

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focus on 3D topological insulators with TRS. In this field, an important task is to systematically search for all topological insulators. In this process, first-principles calculations played a crucial role. Up to now, most of topological insulators were predicted first by first-principles calculations, and observed subsequently by experiments. 2 Theories and methods 2.1 First-principles methods Density functional theory (DFT) is a formally exact theory based on the two Hohenberg–Kohn theorems (HK) [16], but the functional of the exchange and correlation interaction is unknown in Kohn–Sham (KS) equation [17]. In order to perform numerical calculations, the local-density approximation (LDA) [17] and Generalized Gradient Approximation (GGA) [18, 19] are usually used to approximate the exchange and correlation interaction in KS equation. Based on recent experiences, LDA and GGA work quite well for the study of topological insulators, because most topological insulators found to-date are weakly correlated electronic systems, such as, Bi2Se3 [20], TlBiSe2 [21–24] and etc. Haijun Zhang is a post-doctoral researcher in the Department of Physics at Stanford University, CA, USA. He received his BS degree from University of Science and Technology of China (USTC) in 2004, and his Ph.D. from Institute of Physics (IOP), Chinese Academy of Sciences (CAS), China in 2009. His interest is to discover and understand the novel phenomena in condensed matter physics with first-principles calculations. Recently his research focuses on new materials of topological insulators. He received Outstanding Science and Technology Team Achievement Award by Qiu Shi Science & Technologies Foundation in 2011 for the discovery of three-demensional topological insulators (Bi2Se3, Bi2Te3 and Sb2Te3). Shou-Cheng Zhang is the J. G. Jackson and C. J. Wood professor of physics at Stanford University. He received his BS degree from the Free University of Berlin in 1983, and his Ph.D. from the State University of New York at Stony Brook in 1987. He was a postdoc fellow at the Institute for Theoretical Physics in Santa Barbara from 1987 to 1989 and a research staff member at the IBM Almaden Research Center from 1989 to 1993. He joined the faculty at Stanford in 1993. He is a condensed matter theorist known for his work on topological insulators, spintronics and high-temperature superconductivity. He is a fellow of the American Physical Society and a fellow of the American Academy of Arts and Sciences. He received the Guggenheim fellowship in 2007, the Alexander von Humboldt research prize in 2009, Johannes Gutenberg research prize in 2010, the Europhysics prize in 2010, the Oliver Buckley prize in 2012 and the Dirac Medal and Prize in 2012 for his theoretical prediction of the quantum spin Hall effect and topological insulators.

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As we know, the conventional LDA and GGA firstprinciples calculations tend to underestimate the band gap [25, 26]. However the band gap is directly related to the possibility of the band inversion which is the key topological property [3]. For example, sometimes LDA and GGA predict a negative band gap, whereas the band gap is positive in reality [27]. This can cause the serious problem to predict topological insulators. So it is necessary to improve the calculations of the energy gap. The most effective method to calculate the band gap is GW approximation [28]. Simply saying, GW approximation considers the Hartree–Fock self-energy interaction with the screening effect. Though the GW method has been used to study topological insulators, for example, Hg chalcogenides, half-Heusler compounds, antiperovskite nitride, honeycomb-lattice chalcogenides, Bi2Se3 and Bi2Te3 [29–31], this method is very expensive. Besides the GW method, the modified Becke–Johnson exchange potential together with LDA (MBJLDA), proposed by Tran and Blaha in 2009 [32], costs as much as LDA and GGA, but it allows a band gap with similar accuracy to GW. MBJLDA potential can also recover LDA for the electronic system with a constant charge density, and mimic the behavior of orbitaldependent potentials as well. MBJLDA was successfully used to predict topological insulators with the chalcopyrite structure [33]. LDA + U [34], LDA + DMFT [35] and LDA + Gutzwiller [36] are employed to study strongly correlated electronic systems (d and f electrons), because LDA often fails for these systems. In strongly correlated electronic systems, the electrons are strongly localized, and have more features of atomic orbitals. This case requires proper treatment of atomic configurations and orbital dependence. Both LDA and GGA do not include the orbital dependence of the Coulomb and exchange interactions. This is why they fail to describe strongly correlated electronic systems. Based on this understanding, all of LDA + U, LDA + DMFT and LDA + Gutzwiller include the orbital-dependent feature in different ways. For example, the on-site interaction is treated in a static Hartree–Fock mean-field manner in LDA + U method which is the simplest and cheapest method. It is often used for strongly correlated systems, but it does not work well with intermediately correlated metallic systems. The self-energy of the LDA + DMFT method is obtained in a self-consistent way. Up to now LDA + DMFT is the most accurate and reliable method, but its computational costs are high. LDA + Gutzwiller based on Gutzwiller variational approach is recently developed. This method works well for intermediately correlated electronic systems, and it is cheaper than LDA + DMFT. Though it is still an open question how well these methods work on strongly correlated systems. It is true that these methods could reproduce some results of experiments,and that they can help to understand some novel results in strongly correlated electronic systems, for example, the LDA + DMFT study for topological insulator PuTe [37]. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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2.2 Spin–orbit coupling Generally SOC describes the interaction of a particle’s spin with its orbital motion. For example, in one atom, the interaction between one electron’s spin and the magnetic field produced by its orbit around the nucleus can cause shifts in the electron’s atomic energy levels, which is the typical SOC effect. SOC Hamiltonian is given as [38]

where

H soc = -

= σ ◊ p ¥ (—V0 ) , 4m02 c 2

(1)

where = is Planck’s constant, m0 is the mass of a free electron, c is the velocity of light and σ represents the Pauli spin matrices. H soc couples the potential V0 and the momentum operator p together. In the case of the single atomic system V0 is spherically symmetric, H soc can be simplified, H soc = λ L ◊ σ ,

(2)

where λ is the strength of SOC interaction. L represents the angular moment. But in solid systems, V0 is the periodic potential which can be quite complex in form. For convenience, it is sufficient for SOC effect to employ a second-variational procedure with a radial symmetric average around the atoms. SOC interaction is the key to the band topology, so all first-principles calculations to study topological insulators should be carried out with SOC. 2.3 The criterion of topological insulators There are four Z 2 invariants (ν 0 ; ν 1ν 2ν 3 ) for three-dimensional topological insulators, first proposed by Fu, Kane and Mele [8]. When ν 0 = 1, materials are strong topological insulators which have topologically protected gapless surface states consisting of odd number of Dirac cones. These surface state are robust against time-reversal-invariant (TRI) weak disorders. If ν 0 = 0 and at least one of ν 1,2,3 is non zero, the corresponding materials are weak topological insulators which have surface states with even number of Dirac cones on special surfaces. We can simply consider weak topological insulators to be stacked by layered twodimensional QSH materials. In the presence of disorder, the surface states of weak topological insulators can be destroyed. When all ν 0,1,2,3 are zero, materials are conventional insulators. 2.3.1 With inversion symmetry The calculation of Z 2 invariants is very simple for the compounds with inversion symmetry. The formula of Z 2 can be just expressed with the parity values at the eight time-reversal-invariant moments (TRIMs) [39], 8

( -1)ν 0 = ’ δ i

(3)

i =1

and ( -1)ν k =

’

nk =1; n j π k = 0 ,1

δ i = ( n ,n ,n ) , 1

2

3

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(4)

N

δ i = ’ ξ2m ( Ki ) ,

(5)

m =1

N is half of the number of occupied bands, and ξ 2 m ( K i ) is the parity eigenvalue of the 2m -th occupied energy band at TRIM Ki = ( n1n2 n3 ) = 12 (n1b1 + n2 b2 + n3 b3 ) where b1,2,3 represent primitive reciprocal lattice vectors.

2.3.2 Without inversion symmetry For the compounds without inversion symmetry, several methods are proposed to calculate Z 2 invariants [40–43]. Considering the simplicity for first-principles calculations, here we briefly introduce the proposal of Fukui et al. [40]. Firstly, Z 2 formula of QSH state can be expressed with the Berry connection and the Berry curvature, shown by Fu and Kane, Z2 =

1 È Ú A ( k ) - Ú F ( k ) ˘ mod 2 , 2 π ÍÎ ∂v ˙˚ BB-

(6)

with A(k ) = iΣ n ·un (k ) | — k un (k )Ò and

F (k ) = —k ¥ A(k )

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(7)

where B - and ∂B - indicate half of two-dimensional (2D) tori and its boundary, respectively. In order to do numerical calculations, Eq. (6) can directly be rewritten to its lattice version. Secondly, for 3D case, we can define six 2D tori as Z 0 (k x , k y , 0), Z1 ( k x , k y , π), Y0 (k x , 0, k z ), Y1 ( k x , π, k z ), X 0 (0, k y , k z ), and X 1 (π, k y , k z ). We can calculate the Z 2 based on Eq. (6) for each of these six tori, as z0 , z1 , y0 , y1 , x0 , and x1 . The four Z 2 invariants of topological insulators are obtained by ν 0 = x0 xπ , ν 1 = xπ , ν 2 = yπ and ν 3 = zπ . Xiao et al. first successfully using these formulas to evaluate the Z 2 invariants of half-Heusler compounds by first-principles calculations [44]. 2.3.3 Adiabatic argument Sometimes it is not necessary to directly calculate Z 2 for the compounds without inversion symmetry. One can start from a respective compound with inversion symmetry, and then adiabatically change this compound to that without inversion symmetry. If the energy gap does not close in an adiabatic process, the topological property will not change. For example, the space group of α-Sn is Fd3m (No. 227) and the inversion symmetry is held in this structure. We can easily know α-Sn is topologically non-trivial from the parity calculations [39]. The band gap of α-Sn defined by Eq. (9) is negative, which is the key for α-Sn to be topologically non-trivial. Then we assume to adiabatically change α-Sn to HgTe without closing this negative gap. Based on the adiabatic argument, one can conclude, HgTe is topologically non-trivial. Another example to understand this adiabatic argument is to take SOC strength as an adiabatic parameter [45]. The band gap of YBiTe3 stays open with adiabatically tuning SOC strength from 0 to 100%, which www.pss-rapid.com

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means YBiTe3 with SOC has the same topological property as the non-SOC case. So one can conclude that YBiTe3 is topologically trivial. 2.3.4 Surface states Gapless surface states of topological insulators must include the odd number of Dirac cones on one surface, and these surface states are robust against TRI weak disorders. So the calculation of surface states is another useful method to judge the band topology. The simplest way to calculate surface states is based on the free-standing structure. It is true that this is a very powerful method to calculate surface states, but only for the compounds with inversion symmetry and layered structure, such as, Bi, Sb, Bi2Se3 etc. For example, if the compounds do not have inversion symmetry, the polarization field might cause serious artificial effect, especially for the compounds with a small band gap. In addition, if the compounds are not layered structures, the dangling bonds on the surface might cause a number of complex topologically trivial chemical surface states which can mix with topologically non-trivial ones. The topological surface states originate from the topological property of the bulk electronic structure. Though the details of these surface states can be modified by the special dangling bonds and the reconstruction of the electronic structure on the surface, we address that the topological feature does not change, such as, the odd number of Dirac cones. The calculation of the freestanding model also costs a lot, because the vacuum layer and the material part both should be thick enough in order to avoid hybridization between the up and down surfaces. Besides the free-standing model, maximally localized Wannier function (MLWF) methods [46, 47] can be used to calculate the surface states [20, 48]. Essentially the MLWF method is a tight-binding method, but the difference from the conventional tight-binding method is that MLWF method can exactly reproduce the band structure of first-principles calculations. But it is not easy to obtain MLWFs, because the transformation from Bloch functions to Wannier functions is not unique due to the phase ambiguity of the Bloch functions used in first-principles calculations. Marzari and Vanderbilt reported an effective method to obtain MLWF by minimizing the spread function  (· r 2 Ò - · r Ò 2 ) [46]. In order to calculate surface n

states, first we carry out the first-principles calculations for 3D bulk structure and then transform Bloch functions to MLWFs. At the same time the hopping parameters H mn ( R) = · n0| Hˆ |mR Ò between Wannier functions are obtained. At the next step, we use these hopping parameters to construct the hopping parameters of the corresponding semi-infinite structure, and then iterative method can be used to solve the surface Green’s function, α ,α Gnn (k|| , ε + iη ) ,

(8)

where n denotes the unit cell along the surface normal, and

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method can predict surface states well for layered compounds. For example, the calculated surface states of Bi2Se3 with MLWFs method agree well with the ones of angle resolved photoelectron spectroscopy (ARPES) [20, 49]. Usually we do not expect to predict the exact dispersion of surface states, because this method does not include all complex situations on the surface. On the other hand, the surface states obtained from the MLWF method originate from the topological property of the bulk electronic structure, so this is an ideal method to judge whether one compound is topologically non-trivial or not. 3 Three-dimensional topological insulators After the initial discovery of the 2D topological insulator HgTe [3, 4], a number of 3D topological insulators are found with the great effort of theorists and experimentalists [10, 12, 13]. In the following, we classify the topological insulators by the type of the band inversion, because the band inversion has a clear and general physical picture for most topological insulators. Up to now, there are three basic types of band inversions (s–p, p–p, d–f) in topological insulators discovered so far. In the following discussions, we will take some representative compounds as examples for each type of topological insulators. 3.1 s–p type The most important s–p topological insulator is HgTe [3, 4] which has the zinc-blende structure with space group F43m (No. 216). Before HgTe was found to be a topologically non-trivial compound, it had been widely studied experimentally and theoretically [50– 52]. Unlike other zinc-blende compounds, HgTe is a semiconductor with symmetry-protected zero-energy band gap. The Hg has occupied shallow 5d levels which tends to be delocalized, so Hg has a large effective positive charge in its core. The Hg s level, which forms Γ 6 state in cubic symmetry, is pulled down below the Te p levels which split into Γ8 and Γ 7 , by this effective positive charge of Hg’s core. Finally the energy level sequence at Γ point shows the Γ8 -Γ6 -Γ 7 order, which we call the s–p-type band inversion. If we define the energy gap DE , DE = EΓ6 - EΓ8 ,

(9)

where the EΓ6 and EΓ8 are the energy levels for Γ 6 and Γ8 at the Γ point. HgTe has a negative DE because of the s–p-type band inversion, so it is well known as a negative gap semiconductor. The normal LDA and GGA can predict the band inversion between Γ 6 and Γ8 , but the exact band sequence of Γ8 -Γ 6 -Γ 7 cannot be obtained [52]. The LDA band structure with SOC shows the Γ8 -Γ7 -Γ6 sequence, shown in Fig. 1(a). As we addressed above, MBJLDA method can correct the error of LDA band structure. The band structure with the MBJLDA method is shown in Fig. 1(b), which perfectly shows the correct Γ8 -Γ 6 -Γ 7 sequence. Bernevig, Hughes and Zhang first identified the band inversion in HgTe to be the key ingredient of its topologi© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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Figure 1 (online colour at: www.pss-rapid.com) (a) and (b) Band structure of HgTe by LDA and MBJLDA methods, respectively. Γ 6, 7 ,8 represent the symmetry of energy levels at Γ point. The solid red circles indicate the projection of the s orbital of Hg. The LDA band structure shows the Γ8 -Γ 7 -Γ6 band sequence which is not correct, but MBJLDA can calculate the correct band sequence as Γ8 -Γ 6 -Γ7 .

cally non-trivial behavior [3]. Its topological invariant can also be obtained by an adiabatic argument [39]. As we know, if we replace Hg and Te by the same atom in the zinc-blende structure, the crystal structure will change to the diamond structure with the inversion symmetry. Luckily, in nature grey tin has the diamond structure with space group Fd3m, and it is also a semiconductor with a negative energy gap DE due to the s level below the p level. Be-

H. Zhang and S.-C. Zhang: A review of topological insulators

cause grey tin holds the inversion symmetry, its parity values at all TRIMs can be easily calculated. It is worth to note that though grey tin is a zero-band gap semiconductor, we still can define the topological property for all of its occupied bands. Based on the formulas proposed by Fu and Kane, its Z 2 invariants are calculated to be (1;000) which indicate topologically non-trivial. Here the key is that the s and p at Γ point have opposite parity values. The occupied s state forms Γ7- , whereas p states form Γ7+ and Γ8+ . Taking grey tin as the starting point, we assume that we make a thought experiment to adiabatically change grey tin to HgTe. In this process, the negative gap ( DE ) is never closed, which means grey tin and HgTe have the same topological property. So HgTe proves to be topologically non-trivial with Z 2 invariant (1;000). Besides this adiabatic argument, HgTe’s Z 2 invariants can also be directly calculated by the numerical method addressed above. Similar to HgTe, there are is a big family of compounds known as half-Heusler materials ( XYZ ) [53] which include more than 250 semiconductors and semimetals. Half-Heusler compounds consist of face-centered cubic (fcc) sublattices sharing the same space group with HgTe. Y and Z form zinc-blende structure which is stuffed by X. Usually X and Y are transition metal or rare earth elements, and Z is a main group element. Usually the 18-electron half-Heusler compounds are candidates for topological insulators due to the requirement of semiconducting. The band structure of these half-Heusler compounds at Γ point near the Fermi level is almost the same with that of HgTe case. s state forms Γ 6 , and p states split into Γ 7 and Γ8 . Some of half-Heusler compounds, such as ScPtSb with the band sequence Γ 6 -Γ8 -Γ 7 , are topologically trivial and some others, such as LaPtBi, with the inverted band sequence Γ8 -Γ 6 -Γ 7 , are topologically non-trivial. The interesting thing is that half-Heusler family were independ-

Figure 2 (online colour at: www.pss-rapid.com) (a) Crystal structure of chalcopyrite compounds (ABC2). (b) Energy gap DE for various chalcopyrite compounds as a function of the lattice constant. Open symbols mean the lattice constant has been reported. The lattice constants of the rest are obtained by first-principles total energy minimization. Squares represent topological insulators, and diamonds represent topological metals. From Ref. [33]. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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ently reported almost at the same time by three theorygroups [44, 54, 55]. Besides the topological property, halfHeusler compounds are a class of multifunctional materials [56, 57], such as, superconductivity and magnetism, due to transition metals and rare earth elements. So half-Heusler compounds might be the best platform to study the Majorana fermion in topological superconductors [58], dynamical axion field in topological anti-ferromagnetic phase [59], and quantum anomalous Hall effect (QAH) in topological ferromagnetic phase [60]. Recently, some ARPES and transport experiments already have been reported for half-Heusler compounds [61–63]. Generally due to the cubic symmetry, many topologically non-trivial compounds (HgTe and half-Heusler compounds) are zero-gap semiconductors with Fermi level through Γ8 level at Γ point, and a uniaxial strain is usually needed to break the cubic symmetry in order to open a finite energy gap [64]. Feng et al. reported that chalcopyrite structure can naturally break the cubic symmetry [33]. The chalcopyrite structure (ABC2) is the body-centered tetragonal structure with space group I42d (No. 122), which could be regarded as a superlattice of two cubic zinc-blende unit cells, AC and BC, seen in Fig. 2(a). In essence, the unit cell of chalcopyrite is the double unit cell of HgTe with naturally breaking the cubic symmetry, and we expect that these two class compounds might share the same topological property. Feng et al. found that it is true that some materials with chalcopyrite structure are topological insulators, shown in Fig. 2(b). Besides the compounds talked about above, there are a lot of other s–p-type topological insulators, such as, β-Ag2Te [65], KHgSb family [66, 67], Na3Bi [68], CsPbCl3 family [69] and so on. 3.2 p–p type Due to the simple surface states consisting of a single Dirac cone, Bi2Se3, Bi2Te3 and Sb2Te3 compounds [20, 49, 72–75] quickly became topological insula-

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tors extensively studied worldwide. Especially Bi2Se3 has a big energy gap of 0.3 eV which is much larger than the energy scale at room temperature. These compounds share the layered structure with a five-atom layer, called the quintuple layer (QL), as the unit cell with the space group R3m (No. 166). Two equivalent Se atoms, two equivalent Bi atoms and a third Se atom are in each QL. The coupling is the chemical bonding between neighboring atomic layers within one QL, but the van der Waals type, which is much weaker, between two QLs. It is worth to note that the inversion symmetry is held in the crystal structure. In the following, we briefly introduce the basic electronic structure of this family compounds by taking Bi2Se3 as an example. First of all, the band structure without SOC shows Bi2Se3 to be a narrow band gap insulator. Both the bottom of conduction band and the top of valence band are at Γ point, seen in Fig. 3(a). After SOC is turned on, the bottom of conduction band is pulled down below the top of valence band, and an interaction gap opens at the crossing of valence and conduction bands, seen in Fig. 3(b). Based on the parity calculations, Z 2 invariants of Bi2Se3 are calculated to be (1;000) which mean topologically non-trivial. The key for Bi2Se3 to be the topological insulator is the band inversion at Γ between the conduction and valence bands with opposite parity values. The schematic of the band sequence at Γ point clearly tells the band evolution starting from atomic levels with three stages, shown in Fig. 3(c). Because the s levels are much lower than p levels, we just start from the atomic p levels of Bi (6s26p3) and Se (4s24p4). At the stage (I), the bonding and anti-bonding effect between Bi and Se atoms are considered. All the atomic orbitals are recombined into P 0 -x , y , z , P1±x , y , z and P 2 ±x , y , z where ‘0’ represents the third Se, and ‘1’, ‘2’ represent Bi and the other two Se, respectively. ‘±’ represents the parity values. Because the third Se is exactly at the inversion center, it is different from the two other Se atoms which together can be classified by the parity. We use P 0

Figure 3 (online colour at: www.pss-rapid.com) Band structure of Bi2Se3 without (a) and with (b) SOC. The blue dashed line represents the Fermi level. (c) Evolution of the band sequence at Γ point starting from atomic levels. The three stages (I), (II) and (III) represent turning on chemical bonding, crystal field and SOC effects step by step. From Ref. [20]. www.pss-rapid.com

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Figure 4 (online colour at: www.pss-rapid.com) (a) Calculated surface states of Bi2Se3 with MLWFs tight-binding method for a semiinfinite structure with the surface normal (111). The red regions indicate bulk bands and the blue regions indicate the band gaps. The clear surface states with linear dispersion at T can be seen in the band gap. (b) ARPES result for Bi2Se3 along the Γ-M direction. From Ref. [20, 49].

to indicate the third Se. At the stage (II), after the crystal field is turned on, pxyz levels will split into pxy and pz . The levels of P1+z and P 2-z are nearest to the Fermi level. At the stage (III), SOC effect is further introduced. P1+z becomes two degeneracy levels ( P1+z ,≠Ø ), and P 2-z becomes two degeneracy levels ( P 2-z ,≠Ø ) due to the time-reversal symmetry. Though the · pz | H soc |pz Ò is zero, · p+ | H soc |pz Ò is not zero which acts like the level repulsion between px , y and pz orbitals, so SOC effect pulls P1+z ,≠Ø down and pushes P 2 -z ,≠Ø up. Finally, if SOC is strong enough, the p– p-type band inversion will happen between P1+z ,≠Ø and P 2 -z ,≠Ø. Due to the layered structure with inversion symmetry, both the free-standing model and the tight-binding model based on MLWFs can be used to calculate surface states. Figure 4(a) shows the clear surface states of Bi2Se3 with a single Dirac cone at Γ calculated by the MLWFs tight-binding model. Almost at the same time of Zhang et al.’s theory prediction [20], the Hasan group reported the topologically non-trivial surface states of Bi2Se3 by the ARPES experiment [49], shown in Fig. 4(b). Comparing the theory and experimental results, we have to agree that first-principles calculations can successfully predict topological insulators, including the details of surface states. Recently a lot of experimental studies of topological insulators are focusing on these compounds, because these compounds are easily to be grown by all kinds of experiments. The topological insulator Bi1–xSbx (0.07 < x < 0.22) alloy also belongs to the p–p type [39]. Bulk Bi and Sb share a rhombohedral R3m structure which holds the inversion symmetry, and they both are semimetals with some tiny Fermi pockets around the TRIM L and T points, but there is a direct gap at every k point through the whole Brillouin zone (BZ). So we can define an imaginary Fermi surface in the direct gap. Based on the parity calculations, we confirm that Bi is topologically trivial with Z 2 (0;000), and that Sb is topologically non-trivial with Z 2 (1;111). The key dif© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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ference of Bi and Sb is the band sequence of the conduction and valence bands at three L points. For example, the conduction band of Bi is L s , and the valence band is L a where ‘a/s’ indicates the –/+ parity. Differently, these two bands switched with each other in Sb. After carefully comparing the band structure between Bi and Sb, Fu and Kane predicted that the insulate phase of Bi1–xSbx (0.07 < x < 0.22) alloy must be a topological insulator. Subsequently, the Hasan group observed the topologically non-trivial property of Bi1–xSbx by the ARPES experiment [71]. But the details of surface states do not agree with the

Figure 5 (online colour at: www.pss-rapid.com) Schematic for the comparison of the surface states of (a) first-principles calculations [48], (b) tight-binding calculations [70], and (d) ARPES experiment [71]. (c) With the extra surface states with red dotted lines, the surface states from first-principles calculations may agree with those of ARPES experiment. From Ref. [48]. www.pss-rapid.com

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(No. 225), and the inversion symmetry is also held in this structure. All these compounds have been well studied by theories and experiments, known as mixed valence materials. Here we take AmN as an example to understand the band structure. The configuration of actinide Am is 5f77s26d0. are only found by tight-binding and first-principles calculaThe SOC interaction is stronger than Hund’s rule, so the f tions. Zhang et al. argued that the extra surface state Σ 3 orbitals split into high energy J = 7/2 and low energy might come from the imperfect surface, but this still is an J = 5/2 states. Approximately, in AmN, Am forms Am3+ open question up to now. with the configuration 5f67s06d0, the states of J = 5/2 Following Bi2Se3 family, a number of other p – p Bi- should be fully occupied, and J = 7/2 states are unoccupied. based topological insulators are predicted by theories and But due to the delocalization of 5f in Am, 5f states partly observed by experiments, such as, TlBiSe2 family [21–24], hybridize with 6d states with neighbor Am atoms. SnBi2Te4 and SnBi4Te7 family [76], and so on. In the fcc crystal field, d orbitals first split into t 2g and eg states, and t 2g level goes down to cross 5f below the 3.3 d–f type There is no clear evidence for the limit Fermi level along Γ-X direction, shown in Fig. 6. The (>0.3 eV) of the energy gap size for topological insulators. band inversion happens at three X points. If only LDA calHow could we find new topological insulators with bigger culations are used, the full energy gap cannot open through energy gap? One possible way to enhance the SOC energy the whole BZ. After the electron correlation is introduced gap is to consider the cooperation of the SOC interaction with LDA + U method, a band gap can open up with and other effects, such as, the electron–electron correlation. proper correlation parameter U. We have to address that In this idea, topological Kondo insulators were proposed, the electron correlation U is found to enhance the SOC in and SmB6 as an example was predicted to be a topological these compounds. Because there are three TRIM X points Kondo insulator [78]. Though due to 4f orbitals SmB6 is a in BZ, Z2 invariants of AmN must be topologically nonstrong correlated system. It only has a tiny energy gap. Re- trivial. Furthermore, our conclusion suggests that all the cently Zhang et al. predicted AmN and PuTe family com- mix-valence compounds with rock-salt structure must be pounds are d and f topological insulators with strong inter- topologically non-trivial. Especially, transport experiments action [77]. All AmN and PuTe family compounds showed, PuTe [79] has a big energy gap around 0.2 eV, have rock-salt crystal structure with space group Fm3m and this gap can be enhanced to 0.4 eV with pressure. Many of these f compounds host all kinds of magnetic phases, so they might open the opportunity to study QAH effect and dynamic Axion field. ones of tight-binding [70] and first-principles calculations [48]. The schematics of the difference among these results are shown in Fig. 5. We can see that the ARPES result indicates three surface states Σ1, 2,3 , but two surface states Σ1, 2

Figure 6 Band structure for AmN compound with U = 0 eV (a) and U = 2.5 eV (b) from the LDA + U method. The thickness of the band corresponds to the projected weight of the d character of Am. In Γ-X direction, one band with d character clearly comes down to cross with the valence bands which mainly having f character. (a) This part represents semi-metal without a band gap, but (b) represents a finite band gap. From Ref. [77]. www.pss-rapid.com

4 Summary and outlook In this review, we first introduced widely-used techniques within first-principles calculations including LDA and GGA, GW and MBJLDA, LDA + U, LDA + DMFT and LDA + Gutzwiller methods, because they play a crucial role on the field of topological insulators. Then the basic concepts of topological insulators and some useful methods to confirm the topological property are summarized. We classify topological insulators found to-date into three types as s–p, p–p and d–f based on the clear band inversion picture. For each type of topological insulators, we take several typical compounds as examples with talking about the electronic structure and the topological property. Though many topological insulators have been discovered, it is still important to find more with desired properties. First of all, a big band gap is important for the application of surface states of topological insulators. Up to now the biggest band gap is around 0.3 eV in Bi2Se3 compound. Secondly, the transport experiments to detect surface states are still very challenging [80–82]. One reason is that the quality of samples is not good enough with a low mobility. Another reason is that Dirac cone always coexists with some bulk carriers. In order to overcome this barrier, on the one hand, experimentalists are trying to improve the quality of samples. On the other hand, it is important to © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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find other new topological insulators with functional properties. In addition, it is interesting to study the cooperation of the topological property with other phases, such as, superconductivity, magnetism and so on. We hope that this review can provide some guidance in the search.

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Acknowledgements This work was supported by the Defense Advanced Research Projects Agency Microsystems Technology Office, MesoDynamic Architecture Program (MESO) through the contract number N66001-11-1-4105 and by the Army Research Office (No. W911NF-09-1-0508).

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Dedicated to the memory of Professor Shou-Cheng Zhang

APPENDIX S04 SYMMETRY IN A HUBBARD MODEL** CHEN NING YANG Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA and S.C. ZHANG IBM Research Division, Almaden Research Center, San Jose, CA 95120-6099, USA For a simple Hubbard model, using a particle-particle pairing operator 11 and a particle-hole pairing operator , , it is shown that one can write down two commuting sets of angular momenta operators J and J', both of which commute with the Hamiltonian. These considerations allow the introduction of quantum numbers j and j' , and lead to the fact that the system has S04 = (SU2 x SU 2)/Z2 symmetry. j is related to the existence of superconductivity for a state and j' to its magnetic properties.

In a recent paper1 it was found that a pairing operator 11 is useful for considering the Hamiltonian in a simple Hubbard model on an L x L x L lattice, where L = even. We shall extend such considerations in the present paper. All notations are the same as in Ref. 1. We introduce here a Hamiltonian H' and a momentum operator P' which are trivially different from the Hand P of Ref. 1, in order to bring out more symmetries of the system: H'

T'

=

=

T' + V' , -

2e

(1)

L (cos kx + cos ky + cos kz)(a; a" + bk b.,.) ,

(2)

k

V'

=

P'

=

2W~ (at a.- ~)(btb.- ~) ~(k

-

.

~ n)· fis the lowest eigenvalue of H' per site at fixed densities Pa and Pb· The function f has many symmetries. Because of Theorems 1 and 2, f(W,pa,Pb) = f(W,pb,Pa)

=

f(W, 1 - Pa, 1 - Pb) = f(W, 1 -Po, 1 - Pa) (23)

Because of (20),

(24)

These symmetries are illustrated in Fig. 2.

Pa

Fig. 2. Equi-f contours in Pa, Pb plane (schematic). Because of (23), these contours are reflection symmetrical with respect to the Pa = Pb axis and the Pa + Pb = 1 axis. Because of Theorem 8, these contours are convex. One can obtain the ( - W) contours from the ( W) contours by a rotation through 90" around the center of the square.

Theorem 8. f(W, Pa, Pb) as a function of Pa and Pb is continuous and concaves

upwards. Theorem 9. f(W, p 0 , Pb) as a function of W concaves downwards.

These two theorems can be proved using the methods of Ref. 5. Theorem 8 and Eq. (23) show that the minimum of f(W, p0 , Pb) for fixed W isf(W, 1/2, 112). This minimum value may be shared by fat other values of (p0 , Pb) than ( l/2, 1/2). Let the region of (p0 , Pb) where this is true be denoted by R , and call the states that have this minimum value off lowest states. (23) shows that R is reflection symmetrical with respect to the axis: Pa = Pb , and with respect to the axis: Pa + p6 = 1. Using Theorem 8 we can show

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Theorem 10. The region R in (Pm Pb) where f(W, Pm Pb) = f(W,)/2, l/2) is convex. Possible schematic shapes of R are illustrated in Fig. 3. Each of the lowest state belongs to a multiplet {j, ) ' }. Within that multiplet the leading state (i.e. where iz ~ j, J'z ~ j,) is also a lowest state. Hence it must be in the i z ~ 0, fz ~ 0 quadrant of R. Thus Theorem 11. All the lowest states on the boundary of R have j = IJz I, j' ~ IJ'z 1. Finally we remark that for the points Pa = 0 (or Pb = 0,) the system is devoid of a (or b) particles. Hence the value of f(W, 0, Pb) and f(W, Pm 0) can be easily evaluated. (23) then allows one to write down f(W, 1, Pb) and f(W, Pa, 1). Thus the value of f on the boundary of the square in Fig. 2 is known. We now define g ( W, Pa, Pb) to be highest eigenvalue of H ' per site. Equation (22) then shows that

More generally we define the free energy per site by F({J, W, Pa, Pb) ~ lim ( - M{J)- 1 ln (p.f.)

(26)

where (p.f.)

=

trace of block of exp (- {JH') belonging to given Pa, Pb ,

(27)

and the limit is for M -- oo. Then

(28)

The function F has many symmetries. Theorems 1 and 2 show that F({J, W, Pa. Pb)

=

F({J, W, Pb, Pa)

=

F({J, W, 1 - Pa. 1 - Pb)

= F({J, W, 1 - Pb, 1 - Pa) .

(29)

Equation (20) shows that

Equation (21) shows that (31)

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a

b

c

d

Fig. 3. Possible shapes for R. R is convex and is reflection symmetrical with respect to the Pa • p 6 , and the Pa + Pb = 1 axes. For case c there is particle-particle ODLRO at low temperatures in the open line segment. For case d there is particle-particle ODLRO at low temperatures inside of the region R. These cases exhibit superconductivity.

These two last equations together show that (32)

Acknowledgments

One of us (CNY) is supported in part by the National Science Foundation under grant number PHY 8908495. References l. C. N. Yang, Phys. Rev. Lett. 63 (1989) 2144. 2. C. N. Yang, Rev. Mod. Phys. 34 (1962) 694. 3. W. Kohn and D. Sherrington, Rev. Mod. Phys. 42 (1970) 1. 4. H. Shiba, Prog. Theor. Phys. 48 (1972) 21 71. 5. C. N. Yang and C. P. Yang, Phys. Rev. 147 (1966) 303.

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