Symmetries, Anomalies and Duality in Chern-Simons Matter Theories

Po-Shen Hsin

A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy

Recommended for Acceptance by the Department of Physics Adviser: Nathan Seiberg

This thesis investigates the properties of quantum field theory with Chern-Simons interaction in three spacetime dimension. We focus on their symmetries and anomalies. We find many theories exhibit the phenomenon of duality - different field theories describe the same long-distance physics, and we will explore its consequence.

We start by discussing Chern-Simons matter dualities with unitary gauge groups. The theories can couple to background gauge field for the global U(1) symmetry, and we produce new dualities by promoting the fields to be dynamical. We then continue to discuss theories with orthogonal and symplectic gauge groups and their dualities. For the orthogonal gauge algebra there can be discrete levels in addition to the ordinary Chern-Simons term, and the dualities require specific discrete levels as well as precise global forms of the gauge groups. We present several consistency tests for the dualities, such as consistency under deformation by the mass terms on both sides of the duality. When the matter fields are heavy the dualities reduce at long distance to the level-rank dualities between Chern-Simons theories, which we prove rigorously. We clarify the global form of the gauge groups, and we show the level-rank dualities hold generally only between spin topological quantum field theories.

Next, we apply the dualities to describe the symmetry that only emerges in the infrared.

For example, we argue that quantum electrodynamics (QED3) with two fermions has an emergent unitary O(4) symmetry. Similarly, we argue U(1)2 coupled to one scalar with the Wilson-Fisher interaction has an emergent SO(3) symmetry. We also investigate the microscopic symmetry and match its ‘t Hooft anomaly across the duality, thus providing another evidence for the duality.

Finally, we comment on the time-reversal symmetry T in three spacetime dimension. We ﬁnd examples where the square of T does not equal the fermion parity, but instead it

iii is modiﬁed by the Z2 magnetic symmetry. This occurs in QED3 with two fermions. We also clarify the dynamics of QED3 with fermions of higher charges using the conjectured dualities. The results are generalized to theories with SO(N) gauge group.

iv Acknowledgements

I would like to thank the Physics department of Princeton University for support. I would like to thank my parents, without whom my life would not be possible. I would like to thank my advisor Nathan Seiberg for discussion and guidance. I would also like to thank Francesco Benini, Clay C´ordova for fruitful collaborations and many helpful discussions. I would like to thank my dissertation committee. I would like to thank Herman Verlinde for reading the manuscript.

I would like to thank my friends and colleagues in the Physics Department and In- stitute of Advanced Study: Yuntao Bai, Ksenia Bulycheva, Laura Chang, Shai Chester, Will Coulton, Kolya Dedushenko, Kenan Diab, Yale Fan, Lin Fei, Tong Gao, Huan He, Luca Iliesiu, Jiming Ji, Jiaqi Jiang, Vladimir Kirilin, Dima Krotov, Hotat Lam, Aitor Lewkowycz, Aaron Levy, Jeongseog Lee, Jingjing Lin, Jingyu Luo, Zheng Ma, Lauren Mc- Gough, Alexey Milekhin, Kantaro Ohmori, Sarthak Parikh, Pavel Putrov, Shu-Heng Shao, Zach Sethna, Yu Shen, Siddharth Mishra Sharma, Joaquin Turiaci, Juven Wang, Jie Wang, Xin Xiang, Zhenbin Yang, Junyi Zhang and Yunqin Zheng. I would like to thank Cather- ine M. Brosowsky and Lisa Fleischer for dealing with the administration. I would like to thank Hsueh-Yung Lin for discussions about mathematics. I would also like to thank my two sisters, and my friends Yu-Hsuan Cheng, Chieh-Hsuan Kao and Ya-Ling Kao, for their friendship and support.

v To my parents.

vi Contents

Abstract...... iii

Acknowledgements...... v

1 Introduction and Overview1

2 Level-rank duality and Chern-Simons Matter Theories 10

2.1 Preliminaries...... 17

2.1.1 A brief summary of spinc for Chern-Simons theory...... 17

2.1.2 Some facts about U(N)K,L ...... 19

2.1.3 An almost trivial theory U(N)1 ...... 22

2.1.4 A useful fact...... 23

2.1.5 SU(N)K ...... 25

2.2 Simple explicit examples of level/rank duality...... 27

2.2.1 A simple Abelian example U(1)2 ←→ U(1)−2 ...... 28

2.2.2 A simple non-Abelian example SU(N)1 ←→ U(1)−N ...... 30

2.2.3 Gauging the previous example U(N)1,±N+1 ←→ U(1)−N∓1 ...... 31

2.3 Level/rank duality...... 32

2.3.1 The basic non-Abelian level/rank duality...... 32

vii 2.3.2 SU(N)K ←→ U(K)−N ...... 35

2.3.3 U(N)K,N+K ←→ U(K)−N,−(N+K) ...... 37

2.3.4 U(N)K,K−N ←→ U(K)−N,K−N ...... 38

2.4 Boson/fermion duality in Chern-Simons-Matter theories...... 39

2.5 New boson/boson and fermion/fermion dualities...... 45

2.5.1 Boson/boson dualities...... 45

2.5.2 Fermion/fermion dualities...... 46

2.5.3 Self-duality of QED with two fermions...... 47

2.6 Appendix A: Often used equations...... 51

3 Chern-Simons-matter dualities with SO and Sp gauge groups 53

3.1 Dualities between USp(2N) Chern-Simons-matter theories...... 57

3.1.1 RG ﬂows...... 60

3.1.2 Coupling to background gauge ﬁelds...... 61

3.1.3 Small values of the parameters...... 63

3.1.4 New fermion/fermion and boson/boson dualities...... 63

3.2 Dualities between SO(N) Chern-Simons-matter theories...... 65

3.2.1 Flows...... 66

3.2.2 Global symmetries...... 67

3.2.3 Small values of N and k ...... 72

3.2.4 The k = 1 case...... 74

3.2.5 The N = 1 case...... 75

3.2.6 The k = 2 case...... 75

3.2.7 The N = 2 case...... 76

viii 3.3 Relation to theories of high-spin gravity...... 77

3.4 Level-rank dualities with orthogonal and symplectic groups...... 78

3.4.1 Level-rank dualities of 3d TQFTs...... 78

3.4.2 Matching the symmetries...... 80

3.4.3 Level-rank dualities of spin-TQFTs...... 85

3.4.4 More non-spin level-rank dualities...... 88

3.5 T -invariant TQFTs from level-rank duality...... 89

3.6 Appendix A: Notations and useful facts about Chern-Simons theories.... 90

4 Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups 93

4.1 Chern-Simons Theories with Lie Algebra so(N)...... 104

4.1.1 Groups, Bundles, and Lagrangians...... 104

4.1.2 Ordinary Global Symmetries and Counterterms...... 110

4.1.3 One-form Global Symmetries...... 115

4.1.4 ’t Hooft Anomalies of the Global Symmetries...... 118

4.1.5 Chiral Algebras...... 120

4.2 Level-Rank Duality...... 122

4.2.1 Conformal Embeddings and Non-Spin Dualities...... 123

4.2.2 Level-Rank Duality for Spin Chern-Simons Theory...... 130

4.2.3 Consistency Checks...... 133

4.3 Chern-Simons Matter Duality...... 136

4.3.1 Fermion Path Integrals and Counterterms...... 136

4.3.2 Dualities with Fundamental Matter...... 143

ix 4.3.3 Phase Diagram of Adjoint QCD...... 148

4.4 Appendix A: Representation Theory of so(N)...... 155

4.5 Appendix B: Z2 Topological Gauge Theory in Three Dimensions...... 156

4.6 Appendix C: P in−(N) and O(N)1 from SO(N)...... 159

4.7 Appendix D: The Chern-Simons Action of O(N) for Odd N ...... 162

4.8 Appendix E: Derivation of Level-Rank Duality for Odd N or K ...... 165

4.8.1 Even N and odd K ...... 166

4.8.2 Odd N,K ...... 168

4.9 Appendix F: Low Rank Chiral Algebras and Level-Rank Duality...... 170

4.9.1 Chiral Algebras Related to so(2)...... 170

4.9.2 Chiral Algebras Related to so(4)...... 172

4.10 Appendix G: Projective Representations in SO(4)4 with CM 6= MC ..... 175

4.11 Appendix H: O(2)2,L as Family of Pfaﬃan States...... 177

4.12 Appendix I: Duality Via One-form Symmetry...... 179

5 Global symmetries, anomalies, and duality in (2 + 1)d 183

5.0.1 Global symmetries...... 186

5.0.2 Anomalies...... 189

5.0.3 Outline...... 195

5.1 ’t Hooft Anomalies and Matching...... 196

5.1.1 Global symmetry...... 197

5.1.2 Background ﬁelds...... 198

5.1.3 Symplectic gauge group...... 205

5.2 Quantum Global Symmetries from Special Dualities...... 206

x 5.2.1 U(1)k with one Φ...... 207

5.2.2 U(1) Nf with Nf Ψ...... 208 −N+ 2

5.2.3 SU(2)k with one Φ...... 209

5.2.4 SU(2)k with 2 Φ...... 209

5.2.5 SU(2) Nf with Nf Ψ...... 210 −N+ 2 5.2.6 Examples with Quantum SO(3) Symmetry and ’t Hooft anomaly matching...... 211

5.3 Example with Quantum O(4) Symmetry: QED with Two Fermions..... 212

5.3.1 QED3 with two fermions...... 213

5.3.2 Mass deformations...... 216

5.3.3 Coupling to a (3 + 1)d bulk...... 218

5.4 Example with Global SO(5) Symmetry...... 220

5.4.1 A family of CFTs with SO(5) global symmetry...... 220

5.4.2 Two families of CFTs with SO(3) × O(2) global symmetry...... 223

5.4.3 A family of RG ﬂows with O(4) global symmetry...... 226

5.4.4 Relation with a Gross-Neveu-Yukawa-like theory...... 228

5.5 Appendix A: Derivation of the Wess-Zumino term in the 3D S4 NLSM... 229

5.6 Appendix B: Comments on Self-Dual QED with Two Fermions...... 231

5.7 Appendix C: More ’t Hooft anomalies...... 235

6 Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)d 238

6.1 Introduction...... 238

6.1.1 T ...... 239

6.1.2 What is the Global Symmetry?...... 242

xi 6.1.3 Monopole Operators and Their Quantum Numbers...... 243

6.1.4 Summary of Models...... 245

6.2 QED3 with Nf Fermions of Charge One...... 249

6.2.1 Nf = 2: O(4) Unitary Symmetry...... 253

6.3 QED3 with Fermions of Even Charge...... 255

6.3.1 Infrared Behavior...... 257

6.4 SO(N)0 with Vector Fermions...... 265

6.5 SO(N)0 with Two-Index Symmetric Tensor Fermion...... 267

6.5.1 Time-Reversal Symmetry and its Anomaly...... 268

6.5.2 Time-Reversal Symmetry in the IR...... 270

6.5.3 Gauging the Magnetic Symmetry: Spin(N)0 + Tensor Fermion... 272

Bibliography 277

xii Chapter 1

Introduction and Overview

Chern-Simons gauge theory describes many condensed matter phenomena in two spatial dimensions, such as fractional quantum Hall eﬀect and non-abelian statistics (for a revew see e.g. [43, 48, 146]). It is speciﬁed by a gauge group G, and the action depends on dimensionless parameters that are often quantized depending on the gauge group [41]. An example is G = U(1), and the action can be written as

k Z udu (1.0.1) 4π

H where k is an integer called the level, and u is a U(1) gauge ﬁeld normalized by du ∈ 2πZ.

The theory is denoted as U(1)k. For even k the theory is a non-spin topological quantum field theory (TQFT) and its physics only depends on the topology [146]. For odd k the theory also depends on the spacetime spin structure, and the theory is a spin topological quantum field theory [41]. For a review on the quantization of the Chern-Simons theory and its relation with the two-dimensional conformal field theory, see e.g. [41,43,44,101,146].

The observables in the Chern-Simons theory are the Wilson lines labelled by the representations of the gauge group, and they are also the trajectories of quasiparticles. For

1 U(1) gauge group the lines are labelled by the electric charge Q, and braiding the lines of

0 2πi 0 charges Q, Q produces the phase exp k QQ , while rotating the quasiparticle on the line πi 2 with charge Q by changing the framing of the line produces the phase exp k Q . Thus the quasiparticles are not mutually local in general, and they can obey anyon statistics [99].

The anyons in U(1)k Chern-Simons theory obey abelian fusion rules i.e. fusing two Wilson lines produces a unique Wilson line. Another example is G = SU(2) with integral Chern-

1 k Simons level k, where the Wilson lines are labelled by the SU(2) spin j = 0, 2 , 1, ··· , 2 , and most of them obey non-abelian fusion rules i.e. fusing two Wilson lines does not produce a

k unique possibility. The Wilson lines with j = 0, 2 are abelian anyons. The set of abelian anyons forms an abelan group under fusion, and it generates the one-form symmetry of the Chen-Simons theory [54]. For example, SU(2)k Chern-Simons theory has Z2 one-form k symmetry generated by the line j = 2 , and for even k we can gauge this one-form symemtry to obtain SO(3) Chern-Simons theory. (For a discussion about gauging the one-form symmetry in Chern-Simons theory, see e.g. [35, 54, 101]). The anyon in Chern-Simons theory also has applications in quantum computation [51,85,106] and knot theory [146].

We can couple the Chern-Simons theory to matter fields in three-dimensional spacetime and produce a Chern-Simons matter theory. For example, Chern-Simons theory coupled to massless fermions can give rise to transition between phases distinguished by different topological order (i.e. the quasiparticles and their statics), which corresponds to different signs of the fermion mass. (For a review of the topological aspect of the fermion path integral, see e.g. [150]). The theory with matter contains various local operators, such as gauge invariant polynomials of the elementary matter fields or the non-perturbatve monopole operators. In particular, the Chern-Simons interaction modifies the quantum number of the monopole operators (see e.g. [27,36,69,122]).

2 It is important to identify the global symmetry of the theory and couple it to background gauge fields. By adding counterterms of the background fields and making the fields dynamical (i.e. including them in the path integral), we can produce new theories [54,149].

In general there can be an obstruction to coupling the system to background gauge fields. Instead, the theory can couple to the backgrounds in one dimension higher by living on the surface of a bulk, and the bulk characterizes the ‘t Hooft anomaly of the global symmetry. Two systems with different anomalies cannot be related by a perturbation that preserves the symmetry, and they are said to be in different symmetry protected topological (SPT) phases (e.g. [30,142]). Examples of nontrivial SPT phases include the topological insulator and topological superconductor (e.g. [30,115,150,151]), where the surface theory can be free fermions or Chern-Simons theories with time-reversal symmetry (e.g. [25,29,31,36,47,130]).

In recent years the phases of quantum matter have been explored both in high energy and condensed matter physics, and a useful tool is the conjectured duality of Chern-Simons matter field theories that provides different descriptions of the same infrared critical point (e.g. [2,3,22,35,36,62,67,75,83,88,95,121]). They generalize dualities studied in condensed matter physics (e.g. [112, 126, 141]), string theory (e.g. [4,5, 57]) and supersymmetric field theory (e.g. [7,8,58,76]). The conjectured dualities have survived stringent consistency tests such as matching global symmetries and anomalies [22,35, 62,88], and in the large N limit the dual theories describe the same higher spin gravity theory by holography [4,5,57]. The supersymmetric versions of these dualities are under more control for finite N, and more observables such as the partition function and the moduli space of vacua were computed and they match in the dual theories (e.g. [7,8]).

In this thesis we will explore Chern-Simons matter dualities and their applications. The discussions are based on the work [3,22,35,36,67]. In chapter two we discuss the dualities with unitary gauge groups. In chapter three and four we discuss the dualities with orthogonal and with symplectic gauge groups. Of particular interest is the bosonization duality

3 in (2+1) dimension, where a theory of scalar ﬁelds with the Wilson-Fisher interaction is dual to a theory of fermion ﬁelds:

SU(N)K with Nf φ ←→ U(K) Nf with Nf ψ −N+ 2

U(N)K with Nf φ ←→ SU(K) Nf with Nf ψ −N+ 2

U(N)K,K±N with Nf φ ←→ U(K) Nf Nf with Nf ψ −N+ 2 ,−N∓K+ 2

SO(N)K with Nf φ ←→ SO(K) Nf with Nf ψ −N+ 2

0 Spin(N)K with Nf φ ←→ O(K) Nf Nf with Nf ψ −N+ 2 ,−N+ 2

0 O(N)K,K with Nf φ ←→ Spin(K) Nf with Nf ψ −N+ 2

1 1 O(N)K,K−1+L with Nf φ ←→ O(K) Nf Nf with Nf ψ −N+ 2 ,−N+ 2 +1+L

Sp(N)K with Nf φ ←→ Sp(K) Nf with Nf ψ . (1.0.2) −N+ 2

In these dualities the boson or fermion ﬁelds φ, ψ are in the fundamental representation of the gauge groups (for orthogonal gauge algebra they are in the vector representation), and

r the Chern-Simons theory O(N)K,L has additional discrete levels r = 0, 1 and L = 0, 1 ··· , 7 to be explained in chapter four. The ﬁrst two dualities and the third one with U(N)K,K+N were ﬁrst conjectured in [2], and we provide the generalization with background gauge

field. For small enough Nf the dual theories are conjectured to describe the same critical point at long distance. For higher values of Nf the duality and its orientation reversal give rise to two bosonic descriptions for the same fermionic theory, and the fermionic theory is conjectured to flow to the two critical points described by the bosonic theories, with a symmetry breaking phase in between [35,88]. We provide many consistency tests for these dualities, such as deforming one duality with masses flow to another duality in the same family, and matching the local operators in the dual theories. Generalizations of the above

4 dualities with U, SU, SO and Sp gauge groups to theories that include both fundamental bosons and fermions are discussed in [20,73].

From theses bosonization dualities (1.0.2) we also obtain many boson-boson and fermion- fermion dualities. Some examples are as follows. For unitary gauge group, we ﬁnd the following theories

U(N)1 with one φ (1.0.3) for all N ≥ 1 ﬂow at long distance to a single free Dirac fermion. Similarly,

U(K) 1 with one ψ (1.0.4) 2 for all K ﬂow at long distance to the same Wilson-Fisher ﬁxed point with one complex scalar. For SO(N) gauge group, the theories

SO(N)1 with Nf φ (1.0.5)

for N ≥ Nf + 2 ﬂow at long distance to the same Nf free Majorana fermions. By gauging suitable Z2 symmetries (see chapter four for detail) we ﬁnd that

1 Spin(N)1 with Nf φ ←→ O(N)1,−1 with Nf φ , (1.0.6)

which flows at long distance to the same fixed point as (Z2)−N+Nf /2 with Nf Majorana fermions that couples to the Z2 gauge field by the Z2 transformation ψ → −ψ. Likewise,

Spin(K) with N ψ ←→ O(K)1 with N ψ , (1.0.7) −1+Nf /2 f Nf /2−1,Nf /2+1 f

which ﬂows at long distance to the same ﬁxed point as (Z2)K coupled to Nf real Wilson-

Fisher scalars φ by the Z2 transformation φ → −φ.

5 In the absence of matter these dualities become the level-rank dualities between Chern- Simons theories, and we prove them rigorously. Since they follow from the dualities with matter by giving the matter masses, this provides a nontrivial consitency check for the Chern-Simons matter dualities. Furthermore, we clarify the precise statement of level- rank dualities (i.e. between two equivalent theories) such as the global form of the gauge groups, and they are typically valid between spin TQFTs. As special cases of these precise level-rank dualities, we ﬁnd rich examples of time-reversal invariant spin TQFTs:

U(N)N,2N ←→ U(N)−N,−2N

SO(N)N ←→ SO(N)−N

1 1 O(N)N,N−1 ←→ O(N)−N,−N+1

1 1 O(N)N,N+3 ←→ O(N)−N,−N−3

Sp(N)N ←→ Sp(N)−N , (1.0.8) where if the theory is non-spin we need to multiply it with an invertible spin TQFT [50,122]

(they can be described by the Chern-Simons theory SO(M)1 for some integer M) to make the duality valid. They provide new surface states of topological insulator or superconductor in (3+1) dimensional bulk, generalizing the known examples in the literature that correspond to small values of N (e.g. [25,29,47,121,122]). The anomaly of the time-reversal symmetry in these theories are investigated in [19,31,36,62,130], see also chapter six.

In chapter four we also discuss the phase diagram of SO(N)K coupled to a fermion in the two-index tensor representation, with background gauge fields for the Z2 charge conjugation and the Z2 magnetic symmetry. We derive the necessary counterterms for the background fields, and find they are consistent throughout the proposed phase diagram, thus providing a nontrivial consistency check. By summing over the background gauge fields we obtain

6 the phase diagram of O(N)±1/2 and Spin(N) gauge theory with two-index tensor fermion. Similar phase diagrams for SU(N) and Sp(N) gauge groups are discussed in [62].

In chapter ﬁve we focus on the global symmetry of Chern-Simons matter theories and discuss some applications of the dualities. The global symmetry is often a discrete quotient of a naive symmetry, and this quotient results in nontrivial ‘t Hooft anomaly of the global symmetry. We compute this ‘t Hooft anomaly in Chern-Simons matter theories with unitary or symplectic gauge groups, and we provide a further consistency check for the dualities (1.0.2) by matching the ‘t Hooft anomaly in the dual theories. Using the Chern- Simons matter dualities we also ﬁnd many examples where the global symmetry manifests in one microscopic description is an enhanced symmetry in the other descriptions at the long distance critical point. Several interesting cases include: U(1)2 with one scalar has emergent SO(3) symmetry, and U(1)0 with two fermions has emergent O(4) unitary symmetry (for related discussions see e.g. [136,153]). We also study in detail the phase diagram ∼ of SU(2)K = Sp(1)K coupled with two scalars in the fundamental representation, which includes a potential critical point with SO(5) global symmetry.

In chapter six we discuss the time-reversal symmetry T in G = SO(N) gauge theory with fermions in the vector or two-index tensor representation. On operators formed by polynomials of the elementary fermion ﬁelds, the time-reversal symmetry satisﬁes

T 2 = (−1)F , (1.0.9) where F is the fermion parity. However, there are theories where the time-reversal symmetry is deformed in the sector with odd magnetic charge M

T 2 = (−1)F M . (1.0.10)

7 An example is quantum electrodynamics where the number of fermions with odd charge is 2 mod 4. When the gauge field is not dynamical, this is discussed in [92,139,151]. We revisit the example of quantum electrodynamics with two fermions of unit charge, which is argued to flow to a fixed point with O(4) unitary symmetry, and the time-reversal symmetry squares to the center of O(4). We also derive a duality that describes the long distance behavior of quantum electrodynamics with one fermion of charge two using the particle-vortex duality in [94,121,126,140]:

U(1)0 + ψ with q = 2 ←→ free Dirac fermion χ + U(1)2 , (1.0.11)

where U(1)2 as a spin TQFT is time-reversal invariant by the level-rank duality (1.0.8). We generalize the duality (1.0.11) to any even charge q ≥ 2, where we argue the long distance behavior of the theory is a Dirac fermion couples to U(1)2 via gauging a diagonal Zq/2 symmetry. For q = 4 i.e. quantum elctrodynamics with a fermion of charge four, we find the theory perturbed by a monopole operator that breaks the magnetic symmetry flows at long distance to the T-Pfaffian theory [25,29], which can be described by the Chern-Simons theory O(2)2,1 [35], and it is time-reversal invariant by the level-rank duality (1.0.8). This is consistent with the scenario of the dynamics proposed in [36,62].

We generalize the discussion of time-reversal symmetry to SO(N) gauge theory coupled to fermions in the vector or two-index tensor representation. For SO(N) gauge theory with Nf fermions in the vector representation, we ﬁnd the time-reversal symmetry does not commute with the charge conjugation of the O(Nf ) ﬂavor symmetry Cf in the sector with magnetic charge:

−1 TCf T = Cf M . (1.0.12)

F In particular we ﬁnd either T or Cf T must square to (−1) M. For T this occurs when

Nf = 2 mod 4, while for Cf T this occurs for Nf = 0 mod 4.

8 For SO(N) gauge theory with two-index tensor fermion, we ﬁnd the time-reversal symmetry T does not commute with the charge conjugation symmetry C:

TCT −1 = CM . (1.0.13)

This implies the anti-unitary symmetries T and CT cannot both obey the standard algebra (1.0.9), but one of them must square to (−1)F M. This occurs for T when N = 0 mod 4, while for CT this occurs when N = 2 mod 4. We investigate the long distance behavior of the theory using the conjectured phase diagram in [36, 62]. We gauge the magnetic symmetry to obtain Spin(N) gauge theory coupled to two-index tensor fermion, where both T and CT square to (−1)F , and we match the ‘t Hooft anomaly of these anti-unitary symmetries between the ultraviolet and the infrared with a suitable choice of the couplings. This provides another consistency test for the proposed phase diagram in [36,62].

9 Chapter 2

Level-rank duality and Chern-Simons Matter Theories

Level-rank duality is an interesting example of a surprising duality between two different theories based on two different set of variables. It is sufficiently simple and tractable that it can be established rigorously, and it seems related to other dualities that are harder to analyze.

One of the simplest contexts where this duality arises is in the theory of NK complex chiral fermions in two dimensions [105] (see also [108, 110, 119, 154]). This leads to the equivalence of the chiral algebras

SU(NK)1 SU(N)K ←→ , (2.0.1) SU(K)N

where the notation GL represents the chiral algebra of G with level L and in the right hand side of (2.0.1) we have a GKO coset [59]. In the context of the corresponding three- dimensional Chern-Simons theory (2.0.1) is represented as the duality between theories

10 based on the two Lagrangians

LSU(N)K ←→ LSU(NK)1/SU(K)N = L(SU(NK)1×SU(K)−N )/ZK , (2.0.2) where we used the Chern-Simons Lagrangian description of coset models of [101]. (In section 4.1 we will present another Lagrangian for this coset theory.)

Other versions of level/rank duality are

SU(N)±K ←→ U(K)∓N,∓N

U(N)K,K±N ←→ U(K)−N,−N∓K , (2.0.3)

where U(N)K,K+NK0 ≡ (SU(N)K × U(1)N(K+NK0))/ZN (see section 2) and N, K > 0.

Orientation-reversal exchanges the two dualities in the ﬁrst line of (2.0.3). The two dualities in the second line of (2.0.3) are mapped to themselves under orientation-reversal (the left hand side and the right hand side of the dualities are exchanged). Three of these dualities are well known (see e.g. the analysis of the Chern-Simons theory in [96, 103, 104] and in [42] and the discussion of the cohomology of the Grassmannian [147]), but as far as we know, the fourth duality

U(N)K,K−N ←→ U(K)−N,−N+K (2.0.4) is new.

Although the dualities (2.0.3) are common, they lead to several puzzles.

1. The two-dimensional central charges (and correspondingly the three-dimensional framing anomaly) do not match between the two sides of these dualities. What should we add to them in order to ﬁx it?

11 2. The chiral algebra of a standard two-dimensional rational conformal ﬁeld theory in-

cludes only integral-dimension operators. This is true in SU(N)K for all K, U(N)K,K

for even K, and U(N)K,K±N for odd K (see section 2). Correspondingly, the three- dimensional Chern-Simons descriptions of these theories do not depend on the choice of spin structure. This is not true in the other theories in (2.0.3). They are spin- Chern-Simons theories and depend on the choice of spin structure.1 How can a non- spin Chern-Simons theory be dual to a spin Chern-Simons theory?

3. Consider the cases where the two sides of the duality (2.0.3) do not depend on the choice of spin structure, i.e. the ﬁrst duality with even N and the second duality with odd N and K. Even here, the two-dimensional conformal weights of the representations and correspondingly the three-dimensional spins of the line observables do not quite match between the two sides of the duality. They match only modulo one-half. This does not occur in (2.0.2). But it is puzzling in (2.0.3).

Below we will resolve these puzzles. First, we will add to all the non-spin theories in (2.0.3) an almost trivial, purely gravitational sector consisting of a single non-trivial line observable of spin one-half. This sector was discussed in [121] and will be reviewed in section 2.3. (In the condensed matter literature this sector is often denoted as {1, f}.) We can also try to add this sector to the spin theories in (2.0.3), but since they already include such a transparent line, this added sector does not change them. After these additions all

1Ordinary, non-spin Chern-Simons theories are independent of a choice of spin structure and the spin theories depend on that choice. The spin theories include a line with half-integral spin (hence it is non- trivial), which is transparent, i.e. it has no braiding with all the lines in the theory. This line does not decouple from the rest of the theory, because it can be obtained by fusing other lines. We can consider such a transparent line also in the non-spin theories, but it decouples from the rest of the theory. It is intrinsically gravitational. A useful example to keep in mind is U(1)K Chern-Simons theory with the K Lagrangian 4π bdb. For even K this theory is not spin and does not depend on the choice of spin structure. For odd K this is a spin theory. Quantizing this theory with any K on a Riemann surface leads to Kh states, where h is the genus. But the expression for the number of independent line observables depends H H on K mod 2. For even K there are K lines ein b with the line eiK b being trivial. For odd K the line H eiK b is transparent and its spin is half-integral. Therefore, it is non-trivial and the theory with odd K has 2K distinct lines. For more details see, e.g. [121, 122].

12 the theories in (2.0.3) are spin theories. We will show that after this modiﬁcation of (2.0.3) the second and third puzzle above are resolved.

Next, we will couple these theories to appropriate background fields. These includes ordinary U(1) gauge fields B, C, etc. a spinc connection A (see section 2.1) and a metric g. In order for the dualities (2.0.3) to be valid with the background fields present one needs to take into account two facts. First, the map between the background fields in the left hand side and the right hand side of the duality will be non-trivial. This will account for a non-trivial map of the line observables. (Because of this non-trivial map one cannot simply factor out the decoupled sector with the transparent line; the duality map mixes the two sectors.) Second, as in [33], we will have to add appropriate finite counterterms constructed out of the background fields. These counterterms will ensure that the theories are mapped correctly and they will also account for the discrepancy in the framing anomaly mentioned in the first puzzle above.

Related dualities were found in N = 2 supersymmetric theories in three dimensions [7, 21, 58, 109]. These theories contain matter fields coupled to gauge fields with Chern- Simons terms. There is enormous evidence supporting these dualities, but they cannot be rigorously derived.2 These theories contain both fermions and bosons and they clearly depend on the choice of spin structure. Therefore, the second question above does not arise. However, when the matter fields are given masses and are integrated out, these theories lead to the topological theories (2.0.3) and we still need to add the appropriate, transparent, spin one-half line.

One approach to finding non-supersymmetric dualities is to study Chern-Simons theories coupled to matter in the fundamental representation with large N and large K with fixed N/K. In some cases two different theories, one of them is fermionic and the other is bosonic

2The new level/rank duality (2.0.4) has not been embedded in this context. But our discussion below can be repeated there and it leads to new supersymmetric dualities enjoying the same nontrivial tests.

13 were found [4,5, 57] to be dual to the same gravitational Vasiliev theory (see e.g. [132]). Another approach to finding such non-supersymmetric dualities with finite N and K is based on starting with a pair of dual supersymmetric theories and turning on a relevant operator that breaks supersymmetry. If the flow to the infra-red is smooth, we should find a non-supersymmetric duality [64, 71]. Motivated by this whole body work and the map between baryon operators and monopole operators in these theories [117] (see below) Aharony conjectured [2] three boson/fermion dualities and adding to it a new duality, which is motivated by (2.0.4) we will discuss:

Nf scalars with SU(N)K ←→ Nf fermions with U(K) Nf −N+ 2

Nf scalars with U(N)K ←→ Nf fermions with SU(K) Nf −N+ 2

Nf scalars with U(N)K,K±N ←→ Nf fermions with U(K) Nf Nf .(2.0.5) −N+ 2 ,−N∓K+ 2

Here the matter fields are in the fundamental representation of the gauge group and it is implicit that the scalars φ are at a |φ|4 fixed point. In addition, we can reverse the orientation of our manifold (apply time-reversal) and obtain four other dualities, which differ from (3.0.1) by an overall sign change of all the subscripts (the levels). Finally, the

N = K = Nf = 1 version of these dualities were analyzed and coupled to appropriate background ﬁelds in [83,102,121], thus providing further evidence that they are right.

One of the common tests of a duality is that the theories in the two sides of a duality must have the same global symmetry. Furthermore, the correlation functions of the currents of the global symmetry must be the same in the two dual theories. These statements apply to ordinary continuous global symmetries and to Poincare symmetry, where the associated current is the energy momentum tensor. A useful tool to study these correlation functions is based on coupling the system to classical background gauge ﬁelds for these symmetries. This can be done also for discrete global symmetries. Then, the partition functions as a

14 functionals of these background ﬁelds must match in the two dual theories. This matching guarantees that the correlation functions of the currents both at separated points and at coincident points are the same in the two dual theories.

It is often the case that a naive way of coupling the dual theories to background fields does not lead to the same contact terms. In that case, the proper map between the two dual theories involves also added appropriate counterterms in one side of the duality. These counterterms should be local, well-defined functionals of the background fields. In our case these will be Chern-Simons terms in the background fields [33]. And as with all Chern- Simons terms, their coefficients must be properly quantized.

These added background ﬁelds with their proper counterterms also allow us ﬁnd new dualities. More explicitly, we denote a duality between theories described by the Lagrangians

L1[B] and L2[B] that depend on the same background ﬁelds B (and include appropriate counterterms) as

L1[B] ←→ L2[B] . (2.0.6)

0 We can transform the Lagrangians to new Lagrangians that describe new theories L1[C]

0 and L2[C] and maintain the duality

0 0 L1[C] ←→ L2[C] . (2.0.7)

These operations include [149]

1 T : L[B] → L0[B] = L[B] + BdB 4π 1 S : L[B] → L0[C] = L[b] − bdC , (2.0.8) 2π where T adds a counterterm for the background field B and S promotes B to a dynamical field b and couples it to a new classical background field C. These satisfy S2 = (ST )3 = C,

15 where C acts as charge conjugation B → −B. On a spinc manifold odd powers of T - operation require a spinc connection (more in section 2.1). We can also perform orientation- reversal on the two sides of the duality and we can use other known dualities. Performing a sequence of such operations on a given duality (2.0.6) we can generate many new dualities (2.0.7).

One of the main points of this note will be to couple the dualities (2.0.1)-(3.0.1) to background ﬁelds with appropriate counterterms. This will allow us to resolve the puzzles mentioned above. It will also allow us to use the rigorously derived duality (2.0.1), (2.0.2) to derive a more precise version of the dualities (2.0.3), to relate the dualities (3.0.1), and to ﬁnd new Chern-Simons matter dualities.

We should mention our notations. We will use an equal sign between two Lagrangians

L1 = L2 when they are the same as functionals of the ﬁelds and we will use an arrow

L1 ←→ L2 when as functionals they are different (they might even be functionals of a different number of fields), but the theories described by the two Lagrangians are dual; i.e.

L1 and L2 lead to the same physics. All our background fields (except the metric g) will be denoted by upper case letters and all dynamical fields by lowercase letters. A will denote a background spinc connection (see below) and all other gauge fields will be standard gauge fields.

In section 2 we will review and extend some preliminary background material. We will review the notion of a spinc connection and some facts about U(N) Chern-Simons theories and in particular the almost trivial theory U(N)1. We will also discuss SU(N)K and will show how to couple it to background ﬁelds.

Sections 3 - 6 will present various dualities with increasing level of complexity and decreasing level of an explicit map between them.

16 Section 3 will discuss a number of dualities, which can be derived by an explicit and elementary change of variables. Section 4 will be devoted to the dualities (2.0.1) and . The ﬁrst of them is rigorously established, but there is no known explicit change of variables between the two dual theories. Then, given (2.0.1) we will derive a more precise version of (2.0.3).

Section 5 addresses the conjectural boson/fermion dualities (3.0.1) and section 6 will present several boson/boson and fermion/fermion dualities obtained from the conjectural boson/fermion dualities in section 5. Among other things we will use our conjectured dualities to ﬁnd a derivation of the self-duality of QED with two fermions [153]. That duality is closely related to the mirror symmetry of the N = 4 supersymmetric version of that theory [70] and it has many generalizations and extensions including to certain N = 2 dualities [6].

For convenience we collect many of the equations we use often in an appendix.

2.1 Preliminaries

2.1.1 A brief summary of spinc for Chern-Simons theory

A spinc-connection A is characterized by

Z dA 1 Z = w2(M) mod Z, (2.1.1) γ2 2π 2 γ2

for any two-cycle γ2. In particular 2A always has integral period and thus is a U(1) gauge ﬁeld. For more details about the application of spinc connections to this problem, see e.g. [91, 122]. There it was also related to the spin/charge relation of condensed matter physics.

17 We will use the gravitational Chern-Simons term

Z Z 1 Z CSgrav = π Ab(R) = Tr R ∧ R, (2.1.2) M=∂X X 192π X where X is a bulk four-manifold, whose boundary is our spacetime M. For a given M R different non-spin X can lead to different M CSgrav, which differ by πZ/8 and therefore R only exp(16i M CSgrav) is meaningful.

The framing anomaly of CSgrav will be important for matching the framing anomaly in R the dualities we discussed later. If the framing of spacetime is changed by n units, M CSgrav changes by nπ/24.

Consider background U(1) gauge fields B,C, a background spinc-connection A, and a background metric. Integrals over linear combinations with integer coefficients of the terms of the following form are well-defined on a spinc manifold:

1 BdC 2π 1 1 BdB + BdA 4π 2π 1 I[g, A] ≡ AdA + 2CS 4π grav

16CSgrav (2.1.3) and therefore they are valid counterterms.3 Finally, the relation to the Atiyah-Patodi-Singer

1 R η-invariant is η[g, A] = π I[g, A] mod 2Z. 3In fact, all these terms can be expressed as linear combinations of I[g, A] with appropriate values of A [122] 1 1 BdB + BdA = I[g, A + B] − I[g, A] + d(...) 4π 2π 1 BdC = I[g, A + B + C] − I[g, A + B] − I[g, A + C] + I[g, A] + d(...) 2π 16CSgrav = 9I[g, A] − I[g, 3A] . (2.1.4)

18 Although we will often couple our theories to a background spinc connection A, in many cases the resulting theory will not depend on the details of A; we will be able to absorb a shift A → A0 by a U(1) ﬁeld A0 −A in an appropriate shift of other dynamical or background ﬁelds. The explicit coupling to A will serve as a reminder that our theory does not depend on a choice of spin structure and will allow us to transform correctly line operators in dual theories (see below).

2.1.2 Some facts about U(N)K,L

Throughout this section the integer N will be positive, but the integers K and K0 can be positive or negative.

Deﬁne U(N)K,L as

SU(N)K × U(1)N(K+NK0) U(N)K,K+NK0 = , (2.1.5) ZN which is consistent only for K,K0 ∈ Z. This is manifest when we write the corresponding Chern-Simons Lagrangian

K 2i K0 Tr bdb − b3 + (Tr b)d(Tr b) , (2.1.6) 4π 3 4π

where b is a U(N) gauge ﬁeld. From here it is also clear that the U(N)K,K+NK0 Chern- Simons theory depends on the spin structure, if and only if K + K0 is odd.

0 We will be mostly interested in U(N)K,K (K = 0), which we will abbreviate as U(N)K

0 and U(N)K,K±N (K = ±1).

Next, we couple these theories to background ﬁelds in such a way that they do not depend on the spin structure. Speciﬁcally, we couple the non-spin theories to a background

c U(1) gauge ﬁeld C and the spin theories to a background spin -connection A. For U(N)K

19 we have

K 2i 1 L [b; C] ≡ Tr bdb − b3 + (Tr b)dC for even K U(N)K 4π 3 2π K 2i 1 L [b; A] ≡ Tr bdb − b3 + (Tr b)dA for odd K. (2.1.7) U(N)K 4π 3 2π

These can be combined as

K 2i 1 L [b; C + KA] ≡ Tr bdb − b3 + (Tr b)d (C + KA) , (2.1.8) U(N)K 4π 3 2π where the background ﬁeld C + KA is U(1) for even K and a spinc connection for odd K. It is important to stress the diﬀerence between the two expressions in (2.1.7), which is not manifest in (2.1.8). The even K theory in (2.1.7) is non-spin, while the odd K theory is a spin theory. Correspondingly, the odd K theory has a transparent line observable with spin one-half. Below we will also turn the even K theory into a spin theory by adding to it a decoupled transparent line with spin one-half.

An important example is the special case of N = 1

K 1 L [b; C + KA] ≡ bdb + bd (C + KA) . (2.1.9) U(1)K 4π 2π

It will be useful below that changing the orientation is the same as K → −K with C → −C, A → A. (If we view A as the electromagnetic ﬁeld, then our deﬁnition of orientation change is what is normally called CT ; i.e. time-reversal combined with charge conjugation.)

Similarly, for U(N)K,K±N

K 2i 1 1 L [b; C] ≡ Tr bdb − b3 ± (Tr b)d(Tr b) + (Tr b)dC for odd K U(N)K,K±N 4π 3 4π 2π

20 K 2i 1 1 L [b; A] ≡ Tr bdb − b3 ± (Tr b)d(Tr b) + (Tr b)dA for even K. U(N)K,K±N 4π 3 4π 2π (2.1.10)

These can be combined as

K 2i 1 L [b; C + (K ± 1)A] ≡ Tr bdb − b3 ± (Tr b)d(Tr b) U(N)K,K±N 4π 3 4π 1 + (Tr b)d (C + (K ± 1)A) , (2.1.11) 2π where the background ﬁeld C + (K ± 1)A is spinc for even K and U(1) for odd K. In the special case N = 1 this coincides with (2.1.9) with K → K ± 1. Note that (2.1.7)-(2.1.11) are consistent with (2.1.3).

Even though the Lagrangians (2.1.8),(2.1.11) depend on two background fields A and C, in fact the dependence is only on a certain linear combination of them. Therefore, for K even in (2.1.8) and K odd in (2.1.11) the Lagrangians depend only on an even multiple of A and as such the dependence on A can be absorbed in a redefinition of C. For K odd in (2.1.8) and K even in (2.1.11) A cannot be removed. But the dependence on the specific choice of A can be absorbed in C; the difference between the choices A and A0 can be absorbed in a shift of C by a multiple of A − A0, which is an ordinary U(1) field. (See the comment at the end of section 2.1.)

We can define monopole operators in the Chern-Simons theory by removing a point from spacetime and specifying the flux through a sphere around it. Because of the equations of motion that flux must be localized at points on the sphere, and we will take it to be at a single point. Therefore, the monopole operator can be thought of as creating a line with a trivial holonomy around it. In U(N)K with even K this line has integer spin and therefore it is completely trivial. We can say that this monopole operator is a trivial bosonic operator; i.e. it is like the identity operator. In U(N)K with odd K this operator is fermionic and

21 it creates a transparent line with half-integral spin. This reﬂects the fact that the even K theory is non-spin and the odd K theory is a spin Chern-Simons theory. Correspondingly, the even K theory is coupled to a U(1) background ﬁeld C and the odd K theory is coupled

c to a spin connection A. The situation with U(N)K,K±N is the opposite. Therefore the coupling to the background ﬁelds is also reversed, as in (2.1.10). Below we will couple these Chern-Simons theories to matter ﬁelds and then the monopole operators will be less trivial.

2.1.3 An almost trivial theory U(N)1

This is a slight generalization of Appendix B in [121], where more details can be found.

The special case U(N)1 is almost trivial. When quantizing this topological quantum ﬁeld theory on a Riemann surface the Hilbert space consists of a single state and the partition function on any three manifold is a phase. So this theory is dual to a classical ﬁeld theory

N L [A] ←→ −NI[g, A] = − AdA − 2NCS , (2.1.12) U(N)1 4π grav where the arrow means that the two Lagrangians describe dual theories. This expression makes manifest the framing anomaly of the theory. When the theory is quantized on a manifold with a boundary anomaly considerations force us to place degrees of freedom on the boundary. One choice is placing N chiral Dirac fermions there.

These properties of the theory can be used to show that

LU(M)1 [A] + LU(N)1 [A] ←→ LU(M+N)1 [A] , (2.1.13)

22 where we suppressed the dynamical gauge ﬁelds. We can extend the range of N and preserve this property by deﬁning

LU(−N)1 [A] ≡ LU(N)−1 [A] 1 1 L [x, y; A] ≡ L [x, y; A] ≡ xdx + xd(y + A) . (2.1.14) U(0)1 0 4π 2π

Below we will often use the Lagrangian L0, but we will not use U(N) with negative N.

Even though these theories are almost trivial, they do have a non-trivial line observable. The Wilson line in the fundamental representation of U(N) is a transparent line with spin one-half.

Because of (2.1.12) the theory based on the Lagrangian (2.1.14) is dual to a theory with vanishing Lagrangian. Yet, we will use L0[A] as a Lagrangian to denote the fact that the theory includes a transparent line with spin one-half.

2.1.4 A useful fact

We will often use the fact that the Chern-Simons theory based on

1 L = cdv (2.1.15) cv 2π

23 (with c and v U(1) gauge ﬁelds) is trivial [149].4 Speciﬁcally, in many cases we will encounter a Lagrangian like 1 L(v, ...) + cdv (2.1.17) 2π with L independent of c. Then, if the theory is non-spin, c acts as a Lagrange multiplier

L setting v to a pure gauge, which can be set to zero in L. For example, when L = 2π vdv + 1 2π vde (with e a linear combination with integer coeﬃcients of gauge ﬁelds), the fact that the integral over c sets v to zero can be seen explicitly by shifting c → c − Lv − e.

If L describes a spin theory, the situation is not that simple. For example, let L =

1 1 4π vdv + 2π vdA, which is consistent with (2.1.3). Now we cannot shift c in (2.1.17) to 1 1 eliminate the 4π vdv term. Indeed, the theory L + 2π cdv is precisely the theory (2.1.14), which has a line with spin a half. However, remembering that the theory is a spin theory, we can still set v to zero, but we should remember that the resulting theory should include that spin a half line. One way to see that is to add to L the theory L0 (2.1.14) and replace (2.1.17) with

1 1 1 1 1 L[v, c, x, y; A] = vdv + vdA + cdv + xdx + xd(y + A) . (2.1.18) 4π 2π 2π 4π 2π

As we explained above, this does not change the theory. Now, we can change variables

1 1 1 L[v, c + y, x − v, y + v; A] = cdv + xdx + xd(y + A) (2.1.19) 2π 4π 2π

4 This is the special case of N = 1, K = 0 of the Lagrangian for a ZN gauge theory of level K, whose Lagrangian is [16, 78, 90]

K N 1 L(Z ) [x, y; A] = xdx + xdy + xdA for K odd N K 4π 2π 2π K N 1 L(Z ) [x, y; B] = xdx + xdy + xdB for K even , (2.1.16) N K 4π 2π 2π where the theories with even K are non-spin and the theories with odd K are spin.

24 and see that we ﬁnd the same answer as with v to zero. This can be generalized to general spin theory.

2.1.5 SU(N)K

Using our U(N) theories (2.1.7),(2.1.10) with background U(1) field C we can find the Lagrangians for SU(N) theories by promoting C to a dynamical field, which we will denote by c. These impose the constraint that Tr b is trivial. We can also use this Lagrangians to couple the SU(N) theory to new U(1) background fields B

K 2i 1 L [b, c; B] ≡ Tr bdb − b3 + K (Tr b)d(Tr b) + cd (Tr b + B) SU(N)K 4π 3 4π 2π   0 K even   K = 1 K odd and positive . (2.1.20)    −1 K odd and negative

This Lagrangian remains consistent, if we shift the deﬁnition of K by an even (possibly

K dependent) integer 2n. But then we can redeﬁne c → c − n(Tr b − B) to shift K back

n to (2.1.20) and ﬁnd instead a counterterm 2π BdB. So the only freedom in the deﬁnition (2.1.20) is in the choice of this counterterm.5 It is important to stress that c implements

5Here we used (2.1.7),(2.1.10) and turned the background C into a dynamical ﬁeld. What would have happened if we did the same thing starting with the equations with background A? Gauging it amounts to replacing A → c + A with c a dynamical U(1) gauge ﬁeld:

K 2i + 1 1 1 Tr bdb − b3 + K (Tr b)d(Tr b) + cd(Tr b + B) + (Tr b)dA . (2.1.21) 4π 3 4π 2π 2π

Unlike (2.1.20) the theory based on (2.1.21) is a spin theory. Therefore, we can follow section 2.4 and as in the discussion around (2.1.18),(2.1.19) we can add the Lagrangian L0 (2.1.14) without aﬀecting the outcome and use a change of variables like (2.1.19) (with an appropriate shift of c by B) to ﬁnd that (2.1.21) plus L0 is the same as 1 1 L [b, c; B] + L [x, y; A] − AdB + BdB . (2.1.22) SU(N)K 0 2π 4π Therefore, gauging A in (2.1.7),(2.1.10) leads to the same theories as gauging C except that these are spin theories and we have to add some counterterms.

25 the constraint Tr b = −B, but we cannot simply replace the term with K in (2.1.20) by

K 4π BdB. (See section 2.4.) Note that with the deﬁnition (2.1.20) the theory with K is related to the theory with −K by a change of orientation combined with c → −c (without transforming b and B).

For trivial B the Lagrangian (2.1.20) describes the standard non-spin SU(N)K Chern- Simons theory. But the coupling of B leads to additional possibilities. If B is topologically nontrivial, then Tr b is nontrivial and correspondingly, the functional integral can receive its contribution from a nontrivial SU(N)/ZN bundle. Note that in an SU(N)/ZN theory we sum over such nontrivial bundles. Here, as our theory is an SU(N) gauge theory, we do not sum over such bundles. Instead, depending on B we have a given such bundle. We see that, as emphasized in [78], SU(N)/ZN bundles can appear as observables in the SU(N) theory. And here we control them using the background ﬁeld B in (2.1.20).

We will ﬁnd it useful to deﬁne

LbSU(N)K [b, c, x, y; B] K 2i 1 ≡ Tr bdb − b3 + (c + KA)d(Tr b + B) + L [x, y; A] + K BdB(2.1.23) , 4π 3 2π 0 4π

where L0[A] is the almost trivial theory (2.1.14). Note that it is consistent with (2.1.3). For

even K it is trivially the same as LSU(N)K [B] + L0[A] (simply shift c → c − KA). For odd K this is not true, but these two theories are still related. To see that, follow the discussion around (2.1.19) and write

1 2 1 2 LbSU(N) [b, c, x, y; B] = LSU(N) [b, c + (K − K )Tr b + (K + K )B − Ky; B] K K 2 2

+ L0[x + K(Tr b + B), y − K(Tr b + B); A] , (2.1.24)

26 2 where the change of variables is valid because K ± K are even. This can be written as the duality

LbSU(N)K [B] ←→ LSU(N)K [B] + L0[A] , (2.1.25) where we suppressed the dynamical ﬁelds.

We will often extend (2.1.23) and use the deﬁnitions

LbSU(N)K [B] ←→ LSU(N)K [B] + L0[A]

LbU(N)K [B + KA] ≡ LU(N)K [B + KA] + L0[A]

LbU(N)K,K±N [B + (K ± 1)A] ≡ LU(N)K,K±N [B + (K ± 1)A] + L0[A] . (2.1.26)

As we said above, L0[A] is dual to a theory with a vanishing Lagrangian, but we will use it to denote the fact that it has a transparent line with spin one-half. Therefore, our notation

is such that there is a diﬀerence between the theory based on LSU(N)K [B], which is not spin and the theory based on LbSU(N)K [B], which is spin. The latter theory has twice as many line observables because it includes also the transparent line of L0[A]. We should remind the reader that even though the presence of L0[A] leads to the transparent line, the theory still does not depend on the choice of spin structure.

For the U(N) theories in (2.1.26) the situation is diﬀerent. For K odd U(N)K is a spin

c theory and correspondingly, B + KA is a spin connection and LU(N)K [B + KA] already includes the transparent line. Therefore, in this case LbU(N)K [B + KA] ←→ LU(N)K [B +

KA]. Similarly, for K even LbU(N)K,K±N [B + (K ± 1)A] ←→ LU(N)K,K±N [B + (K ± 1)A].

2.2 Simple explicit examples of level/rank duality

Here we discuss three simple dualities, which can be derived by a straightforward change of variables. They demonstrate several points, which will be essential below.

27 2.2.1 A simple Abelian example U(1)2 ←→ U(1)−2

We start with

2 1 1 1 L [b; B] + L [c; A] = bdb + bdB − cdc + cdA . (2.2.1) U(1)2 U(1)−1 4π 2π 4π 2π

A simple change of variables leads to

LU(1)2 [b + c − B; B]+LU(1)−1 [c + 2b − B; A] 1 1 =L [b; B + 2A] + L [c; A] − BdB − BdA , (2.2.2) U(1)−2 U(1)1 4π 2π thus establishing the duality

U(1)2 × U(1)−1 ←→ U(1)−2 × U(1)1 , (2.2.3) which we can write as

1 1 L [B] + L [A] ←→ L [B + 2A] + L [A] − BdB − BdA , (2.2.4) U(1)2 U(1)−1 U(1)−2 U(1)1 4π 2π or, using (2.1.12) and the deﬁnition (2.1.26), as

1 1 LbU(1)2 [B] ←→ LbU(1)−2 [B + 2A] − BdB − BdA − 2I[g, A] , (2.2.5) 4π 2π

where we used LbU(1)±2 to remind ourselves that the theory includes a transparent line with spin one-half. As we commented at the end of section 2.1, neither side of the duality depends on the details of A; changing A → A0 can be absorbed in a redeﬁnition of the dynamical ﬁelds. This duality was derived in [122] and was used in [121]. It had appeared earlier

28 without the background ﬁelds and the essential counterterms in [47], where the theory of

LbU(1)2 [B] was denoted as U(1)2 × {1, f} and was referred to as the semion/fermion theory.

We would like to make several comments about this duality, which will be useful below.

1. In the change of variables (2.2.2) the dynamical fields b and c transform as a valid linear combination of the dynamical fields and the background fields B and A. This change of variables preserves the fact that they are U(1) fields and therefore the coefficient of A must be even (it vanishes).

2. The background fields B and A also transform, but their transformations are independent of the dynamical fields. Also, they remain U(1) and spinc fields respectively.

3. The duality involves a nontrivial counterterm of the background ﬁelds. These terms are consistent with (2.1.3).

4. The form of the duality (2.2.5) might suggest to write it, up to counterterms, as

U(1)2 ←→ U(1)−2. Below we will often use such shorthand notation, but it should be emphasized that it is misleading. This duality is valid only when we include a transparent spin one-half line on both sides of the duality. It couples to A with charge one. This is manifest in (2.2.4), (2.2.5).

5. In the left hand side of (2.2.5) the background ﬁeld B couples only to b and the background ﬁeld A couples only to the transparent line. This is not true in the right

hand side of the duality, where the dynamical U(1)−2, eb ﬁeld couples to B + 2A.

i H b 1 Explicitly, the line e of U(1)2 carries charge − 2 under B and is neutral under A. (This follows from the equation of motion 2db + dB = 0.) This should be contrasted with the basic line in the right hand side of the duality

H H ei eb = ei (−b+c) , (2.2.6)

29 1 which carries charge 2 under B and charge 1 under A. This reﬂects the change of variables in (2.2.2) and the map (2.2.6). Such a nontrivial mapping of the lines will be common below and it is accounted for by the nontrivial map of the background ﬁelds.

6. It is instructive to see how orientation-reversal (or time-reversal) acts on (2.2.5). We let it act on the background ﬁelds as A → A and B → −B − 2A and then (2.2.5) is mapped to itself (after moving the counterterms to the left hand side).

2.2.2 A simple non-Abelian example SU(N)1 ←→ U(1)−N

The second simple duality involves non-Abelian ﬁelds. Consider (2.1.23) for K = 1

LbSU(N)1 [b, c, x, y; B] 1 2i 1 1 = Tr bdb − b3 + (c + A)d(Tr b + B) + BdB + L [x, y; A] 4π 3 2π 4π 0 1 1 =L [b + c1; A] + L [c; B − NA] + AdB + BdB + L [x, y; A(2.2.7)] , U(N)1 U(1)−N 2π 4π 0 which can be written, using (2.1.12), as

1 1 LbSU(N)1 [B] ←→ LbU(1) [B − NA] + AdB + BdB − NI[g, A] . (2.2.8) −N 2π 4π

All the comments we made after (2.2.5) are clearly true also here. (In fact, for N = 2 the known relation SU(2)1 ←→ U(1)2 shows that the dualities (2.2.5) and (2.2.8) are the same.) In particular, the presence of a transparent spin one-half line and the dependence on A are essential. Below we will often abbreviate this duality in the misleading form

SU(N)1 ←→ U(1)−N , but we should remember to keep in mind these subtleties. In order

to remind ourselves about that, we prefer to use the notation LbSU(N)1 [B] as opposed to

30 simply using LSU(N)1 [B]. We would also like to point out that the two sides of the duality have a ZN one-form symmetry. In the left hand side it acts on SU(N)1 and in the right hand side it acts on U(1)−N .

2.2.3 Gauging the previous example U(N)1,±N+1 ←→ U(1)−N∓1

We start with (2.2.8) and turn B into a dynamical ﬁeld, which we denote by e and couple it to a U(1) background ﬁeld C in the two sides. This is the operation S in (2.0.8)[149]. The left hand side becomes

1 2i 1 1 1 Tr bdb − b3 + (c + A)d(Tr b + e) + ede − edC + L [x, y; A] . (2.2.9) 4π 3 2π 4π 2π 0

Integrating out c sets e = −Tr b and we have

1 2i 1 1 Tr bdb − b3 + (Tr b)d(Tr b) + (Tr b)dC + L [x, y; A] 4π 3 4π 2π 0

= LU(N)1,N+1 [b; C] + L0[x, y; A] . (2.2.10)

Repeating these operations in the right hand side of (2.2.8) we ﬁnd

N 1 1 1 1 − udu + ud(e − NA) − edC + edA + ede − NI[g, A] = 4π 2π 2π 2π 4π

LU(1)−N−1 [u; C − (N + 1)A] + LU(1)1 [e + u; A − C] − NI[g, A] . (2.2.11)

This establishes the duality

1 1 LbU(N) [C] ←→ LbU(1) [C − (N + 1)A] − CdC + CdA 1,N+1 −N−1 4π 2π − (N + 1)I[g, A] , (2.2.12)

31 or more brieﬂy U(N)1,N+1 ←→ U(1)−N−1.

In the special case N = 1 we ﬁnd the U(1)2 ←→ U(1)−2 duality of section 3.1, so this is a generalization of that duality. All the comments we mentioned after equations (2.2.5) about that duality are applicable here.

1 1 We can repeat the discussion in this subsection after adding − 2π BdB − 2π BdC to (2.2.8). In terms of (2.0.8) this is the transformation ST −2. Then, instead of (2.2.10) the left hand side becomes

1 2i 1 1 Tr bdb − b3 − (Tr b)d(Tr b) + (Tr b)dC + L [x, y; A] 4π 3 4π 2π 0

= LU(N)1,−N+1 [b; C] + L0[x, y; A] . (2.2.13)

And instead of (2.2.11) the right hand side becomes

LU(1)−N+1 [u; −C − (N − 1)A] + LU(1)−1 [e − u; A − C] − NI[g, A] . (2.2.14)

We derive the duality

1 1 LbU(N) [C] ←→ LbU(1) [−C − (N − 1)A] + CdC − CdA 1,−N+1 −N+1 4π 2π − (N − 1)I[g, A] , (2.2.15)

or more brieﬂy U(N)1,−N+1 ←→ U(1)−N+1.

2.3 Level/rank duality

2.3.1 The basic non-Abelian level/rank duality

In order to avoid cluttering the equations we take N,K to be positive integers.

32 The SU(N)K WZW model is known to be equivalent to the GKO coset [59]

SU(NK)1 SU(N)K ←→ . (2.3.1) SU(K)N

It was rigorously proven and further discussed in [108,110,119,154]. Using the rules of [101] for writing coset theories in Chern-Simons theory we can write this duality as

SU(NK)1 × SU(K)−N SU(N)K ←→ . (2.3.2) ZK

The Lagrangian for the left hand side is simply LSU(N)K [B]. The Lagrangian for the right hand side can be written as an SU(NK)1 × SU(K)−N Lagrangian and then mod out the gauge group by ZK by allowing additional bundles. Instead, we would like to follow our previous discussion and use U(NK) × U(K) gauge ﬁelds b and u.

We write

1 2i N 2i 1 h i h i Tr bdb − b3 − Tr udu − u3 + (Tr b) − N(Tr u) d (Tr b) − N(Tr u) 4π 3 4π 3 4π 1 h i 1 + cd (Tr b) − N(Tr u) + (Kc + Tr u)dB , 2π 2π (2.3.3)

where c is a U(1) gauge ﬁeld. The ﬁrst two terms are as in the Lagrangians for U(NK)1 and

U(K)−N . The third term, which we will further motivate below, is such the U(NK)×U(K) gauge theory is not a spin theory. First, let us ignore the term with B. The term with c correlates the gauge fields of U(1) ⊂ U(NK) and of U(1) ⊂ U(K), thus removing one degree of freedom. Also, it states that the flux of Tr b is a multiple of N, which is needed for the identification of this Lagrangian with the right hand side of (2.3.2). Next, we consider the term with B. Now, nontrivial B can lead to a shift of the fluxes of b (as in the SU(N) theory).

33 The Lagrangian (2.3.3) was designed such that it is invariant under the one-form gauge symmetry [54,78] b → b + ξ1, u → u + ξ1, c → c − ξ with ξ a U(1) gauge ﬁeld. (The third term in (2.3.3) and the coupling to B were motivated by this invariance.) This one-form gauge symmetry removes another degree of freedom and establishes that this Lagrangian describes the right hand side of (2.3.2).

The fact that the one-form gauge symmetry removes a degree of freedom can be made manifest by shifting b → b − c1 and u → u − c1 in (2.3.3) (or equivalently, ﬁxing the gauge c = 0) to ﬁnd

1 2i 3 N 2i 3 L SU(NK)1 [B] ≡ Tr bdb − b − Tr udu − u SU(K)N 4π 3 4π 3 1 h i h i 1 + (Tr b) − N(Tr u) d (Tr b) − N(Tr u) + (Tr u)dB . (2.3.4) 4π 2π

As a check that this Lagrangian indeed describes SU(NK)1 , we can write the gauge group in SU(K)N

(2.3.4) as U(NK) × U(K) = SU(NK) × SU(K) × U(1) × U(1)/ZNK × ZK . The two U(1) factors in the numerator have a k-matrix

  NK(1 + NK) −(NK)2   k =   , (2.3.5) −(NK)2 NK(−1 + NK) whose determinant is −(NK)2. The fact that it is negative shows that these two U(1) factors do not contribute to the framing anomaly. Also, this value of the determinant shows that this Abelian sector has (NK)2 representations. Therefore, the number of representations of this SU(NK)1 × SU(K)−N × U(1) × U(1)/ZNK × ZK theory is the same as the number of representations of SU(NK)1 × SU(K)−N /ZK .

34 Next, we would like to determine the coeﬃcient of a BdB counterterm in the duality.

SU(NK)1 The currents coupled to B in the SU(N)K and the theories are SU(K)N

1 1 K J = dc = + dB SU(N)K 2π 2π N K 1 1 K 2 J SU(NK)1 = dTr u = + K dB , (2.3.6) SU(K)N 2π 2π N where we used the equations of motion to express them in terms of the classical background dB. Therefore, in order to match them across the duality we should add a Chern-Simons counterterm and write the duality as

2 K − K LSU(N)K [B] ←→ L SU(NK)1 [B] + BdB . (2.3.7) SU(K)N 4π

We should emphasize that this duality is a duality between two non-spin-Chern-Simons theories. Neither side of the duality depends on the choice of spin structure and the theories couple only to a background U(1) gauge ﬁeld B; there is no need for a coupling to a spinc connection A.

2.3.2 SU(N)K ←→ U(K)−N

Next, we would like to add the eﬀect of the gravitational line in the duality (2.3.2),(2.3.7)

by adding L0[A] to the two sides. In the left hand side we now have LbSU(N)K [B](2.1.23). And in the right hand side we have the sum of the Lagrangian (2.3.4), the counterterm added in (2.3.7) and L0[A]

1 2i N 2i 1 h i h i Tr bdb − b3 − Tr udu − u3 + (Tr b) − N(Tr u) d (Tr b) − N(Tr u) 4π 3 4π 3 4π

35 1 1 1 − K2 + xdx + xd(y + A) + (Tr u)dB + K BdB . 4π 2π 2π 4π (2.3.8)

Substituting here b → b − (x + y − KB)1, u → u − (x + y − KB)1, x → −x + Tr b − NTr u + KB, y → y + x − Tr b + NTr u − KB it becomes

1 2i N 2i 1 1 Tr bdb − b3 − Tr udu − u3 + (Tr u)d(B − NA) + (Tr b)dA 4π 3 4π 3 2π 2π 1 1 K − xdx − xd(y + A) + BdA + K BdB 4π 2π 2π 4π K =L [A] + L [B − NA] − L [A] + BdA + K BdB . U(NK)1 U(K)−N 0 2π 4π (2.3.9)

We conclude that

K K LbSU(N) [B] ←→ LbU(K) [B − NA] + BdA + BdB − NKI[g, A] . (2.3.10) K −N 2π 4π

As a check, note that for K = 1 this agrees with (2.2.8), which was checked explicitly by a change of variables. Similarly, for N = 1 the two sides of the duality are almost trivial

K+K theories, where all the dynamical ﬁelds can be integrated out. This leads to 4π BdB in the two sides.

Finally, the orientation-reversal of (2.3.10) is

K K LbSU(N) [B] ←→ LbU(K) [−B + NA] − BdA − BdB + NKI[g, A] . (2.3.11) −K N 2π 4π

K K (Equivalently, this can be derived by adding to the two sides of (2.3.10) − 2π BdA− 4π BdB− 1 2π CdB, promoting B to a dynamical ﬁeld e, and then integrating it out. After C → −B, N ←→ K and some rearrangement we end up with (2.3.11))

36 2.3.3 U(N)K,N+K ←→ U(K)−N,−(N+K)

Given the duality (2.3.10) we can derive additional dualities by adding counterterms to the two sides and turning the background ﬁelds to dynamical ones, i.e. using the operations (2.0.8). An example of that was mentioned after (2.3.11).

1−K As another example, we add to the two sides of (2.3.10) the counterterm 4π BdB − 1 2π Bd(C + (K + 1)A) with another U(1) background ﬁeld C. Then we turn B into a dynamical ﬁeld e. The left hand side of (2.3.10) becomes

K 2i 1 1 1 Tr bdb − b3 + (c + KA)d(Tr b + e) + ede − ed(C + (K + 1)A) 4π 3 2π 4π 2π

+ L0[x, y; A] . (2.3.12)

Integrating out c sets e = −Tr b and (2.3.13) becomes

K 2i 1 1 Tr bdb − b3 + (Tr b)d(Tr b) + (Tr b)d(C + (K + 1)A) + L [x, y; A] 4π 3 4π 2π 0

= LU(N)K,N+K [b; C + (K + 1)A] + L0[x, y; A] . (2.3.13)

The right hand side of (2.3.10) becomes

N 2i 1 1 1 − Tr udu − u3 + (Tr u)d(e − NA) + ede − ed(C + A) − NKI[g, A] 4π 3 2π 4π 2π

= LU(K)−N,−N−K [u; C + (1 − N)A] + LU(1)1 [e + Tr u; −C − A] − NKI[g, A] .

(2.3.14)

This establishes the duality

LbU(N)K,N+K [C + (K + 1)A] ←→ LbU(K)−N,−N−K [C + (1 − N)A] 1 1 − CdC − CdA − (NK + 1)I[g, A] .(2.3.15) 4π 2π

37 As checks, note that it is consistent with (2.1.3) and that for K = 1 it coincides with (2.2.12). Also, under orientation-reversal combined with C → −C −2A the duality (2.3.15) is mapped to itself with N ←→ K.

2.3.4 U(N)K,K−N ←→ U(K)−N,K−N

Another duality is obtained by adding to the two sides of (2.3.10) the counterterm

K +1 1 − 4π BdB − 2π Bd(C + (K − 1)A) with another background ﬁeld C, and promoting B to be a dynamical ﬁeld e. After integrating out c and e the resulting duality is

1 LbU(N) [C + (K − 1)A] ←→ LbU(K) [−C − (N − 1)A] + CdC K,K−N −N,K−N 4π 1 − CdA − (NK − 1)I[g, A]. (2.3.16) 2π

We can set K = N in (2.3.16):

1 LbU(N) [C + (N − 1)A] ←→ LbU(N) [−C − (N − 1)A] + CdC N,0 −N,0 4π 1 − CdA − (N 2 − 1)I[g, A] . (2.3.17) 2π

Since U(N)N,0 = [SU(N)N × U(1)0]/ZN , the U(1) ﬁeld does not have a quadratic term in (2.3.17). Integrating over it constrains the background ﬁeld C + (N − 1)A.6 This leads to

6 Unlike the SU(N) theories discussed above, where the background ﬁeld leads to SU(N)/ZN bundles, here the partition function includes a sum over such twisted bundles and the background ﬁeld determines their relative phases.

38 the duality:7 SU(N) SU(N) N ←→ −N . (2.3.19) ZN ZN

Finally, as another check of all the dualities (2.3.10),(2.3.11),(2.3.15),(2.3.16), the coeﬃ- cient of I[g, A] accounts for the diﬀerence in the framing anomaly between the Chern-Simons gauge theories in the two sides.

2.4 Boson/fermion duality in Chern-Simons-Matter

theories

Our goal in this section is to write a more complete and precise version of (3.0.1)

Nf scalars with SU(N)K ←→ Nf fermions with U(K) Nf −N+ 2

Nf scalars with U(N)K ←→ Nf fermions with SU(K) Nf −N+ 2

Nf scalars with U(N)K,K±N ←→ Nf fermions with U(K) Nf Nf (2.4.1). −N+ 2 ,−N∓K+ 2

Again, we point out that the third duality in (2.4.1) with the bottom sign is new, as it is based on the new level/rank duality (2.0.4). In addition to (2.4.1) we have four similar dualities obtained from these by reversing the orientation. These diﬀer from (2.4.1) by a sign change of all the levels.

7 We could have started with (2.3.2) with N = K and mod it out by its ZN one-form symmetry to ﬁnd SU(N) SU(N) SU(N 2) N ←→ −N × 1 . (2.3.18) ZN ZN ZN

2 For even N these theories are spin and for odd N they are non-spin. In the latter case SU(N )1/ZN has only one representation and its framing anomaly is a multiple of 8. (For example, SU(9)1/Z3 ←→ E(8)1.) Therefore, for odd N (2.3.18) can be written as SU(N)N /ZN ←→ SU(N)−N /ZN without the need to add the transparent line.

39 In this section we write the precise Lagrangians of these theories with the background fields and their necessary counterterms. This will allow us to turn these background fields into dynamical fields and to derive some consistency conditions (like deriving some of these dualities from the others), and to find additional dualities.

Before we do that we should review some facts about the fermion determinant in three dimensions. Consider a three-dimensional fermion coupled to a background spinc connection A via the Dirac operatorD / A

iψD/ Aψ . (2.4.2)

Following [10,150] the fermion determinant is

− iπ η(A) Z(A) = | detD / A|e 2 , (2.4.3) where the sign in the exponent depends on the regularization and we pick it to be negative. If we give the fermion a positive mass and integrate it out, the low energy theory has no induced Chern-Simons term. And if the mass is negative, the low energy theory includes −I[g, A].

In the literature it is common to refer to the fermion coupled to A with this regularization as U(1) 1 and describe it as having a bare counterterm −I[g, A]/2. We will follow this − 2 terminology and will refer to this theory as U(1) 1 . But we will not add this improperly − 2 quantized Chern-Simons term. In more detail, although

eiπη(A) = eiI[g,A] , (2.4.4) we can take the square root of the left had side as the meaningful and gauge invariant expression eiπη(A)/2 (which appears in (2.4.3), but we cannot write the square root of the right hand side as eiI[g,A]/2, which is not meaningful.

40 In the various theories the gauge group is either SU(L) or U(L) with L being N in the scalar theories and K in the fermionic theories. As in section 2.4, we are going to represent the SU(L) theory in terms of a U(L) theory with a constraint. So our bosonic covariant derivative D includes a U(N) gauge field and the Dirac operatorD / includes A = u + A1 with u a U(K) gauge field and A a background spinc connection. Also, in the bosonic side of the duality we include a |φ|4 term to denote the fact that the theory is at the corresponding Wilson-Fisher fixed point. We will elaborate on this potential momentarily.

We claim that the dualities (2.4.1) are

2 4 |Dφ| − |φ| + LSU(N) [B] ←→ iψDψ/ + LU(K) [B − (N − Nf )A] b K b Nf −N K + BdA + K BdB − (N − N )KI[g, A] 2π 4π f 2 4 |Dφ| − |φ| + LU(N) [B + KA] ←→ iψDψ/ + LSU(K) [−B] b K b Nf −N N − N N−N − f BdA + f BdB − (N − N )KI[g, A] 2π 4π f 2 4 |Dφ| − |φ| + LbU(N)K,K±N [B + (K ± 1)A] ←→ iψDψ/

+ LU(K) [±B + (Nf − N + 1)A] b Nf −N,Nf −N∓K 1 1 − BdA ∓ BdB − ((N − N )K ± 1)I[g, A(2.4.5)] . 2π 4π f and their orientation reversed ones are

2 4 |Dφ| − |φ| + LbSU(N)−K [B] ←→ iψDψ/ + LbU(K)N [−B + NA] K − BdA − K BdB + NKI[g, A] 2π 4π 2 4 |Dφ| − |φ| + LbU(N)−K [−B − KA] ←→ iψDψ/ + LbSU(K)N [−B] N + BdA − N BdB + NKI[g, A] 2π 4π 2 4 |Dφ| − |φ| + LbU(N)−K,−K∓N [−B − (K ± 1)A] ←→ iψDψ/

+ LbU(K)N,N±K [∓B + (N − 1)A]

41 1 1 + BdA ± BdB + (NK ± 1) I[g, A] .(2.4.6) 2π 4π

For Nf = 0 there are no matter ﬁelds and the dualities (2.4.5),(2.4.6) go over to the four level/rank dualities (2.3.10),(2.3.11),(2.3.15),(2.3.16)

K K LbSU(N) [B] ←→ LbU(K) [B − NA] + BdA + BdB − NKI[g, A] K −N 2π 4π N N LbU(N) [B + KA] ←→ LbSU(K) [−B] − BdA + BdB − NKI[g, A] K −N 2π 4π 1 LbU(N) [B + (K ± 1)A] ←→ LbU(K) [±B + (1 − N)A] ∓ BdB K,K±N −N,−N∓K 4π 1 − BdA − (NK ± 1)I[g, A] (2.4.7) 2π and their orientation reversed versions. In fact, (2.4.5) is obtained from (2.4.7) by substituting N → N − Nf in the fermionic theory and (2.4.6) is obtained by simply changing the signs of the levels.

i I For Nf = 1 there is a single ﬂavor of N scalars φ (with i = 1, ..., N) and K fermions ψ

4 P i 2 (with I = 1, ..., K) and the |φ| interaction is | i φiφ | . The dualities refer to (assumed) P I nontrivial IR theories, where the coefficient of the mass terms I ψI ψ in the fermionic P i side and i φiφ in the bosonic side are tuned to the fixed point. We denote this fine-tuned value of the mass by zero.

ia Ia For Nf > 1 the scalars and the fermions also have a ﬂavor index φ and ψ (with

8 a = 1, ..., Nf ). Starting the bosonic theory at a free UV ﬁxed point, the SU(Nf )-invariant

2 2 b scalar potential can be a linear combination of Tr M, Tr M , and (Tr M) with Ma = P ib i φaiφ . We assume that this theory has a nontrivial IR fixed point with SU(Nf ) global P Ia symmetry with a single SU(Nf )-invariant relevant operator Ia ψaI ψ ←→ Tr M. The fixed point is achieved when the coefficient of this relevant operator is fine-tuned. Again, the duality refers to this IR theory.

8We thank O. Aharony and S. Minwalla for a useful discussion on the following points.

42 In section 4 we related the dualities (2.4.7) by starting with one of them, adding counterterms and promoting background fields to dynamical fields. Exactly the same manipulations can be performed in (2.4.5). In particular, assuming one of these dualities, say the first one, we can derive the others.

In the special case N = Nf = K = 1 these dualities go over to the dualities studied in [121]. The bottom sign in the last duality in (2.4.5) is our new duality, which becomes in this case

2 4 |Dφ| − |φ| + LbU(1)0 [B] ←→ iψDψ/ + LbU(1)1 [−B + A] + I[g, A − B] . (2.4.8)

As a check, it can be derived from gauging B in equation (2.10) in [121]. Note that the left hand side is manifestly orientation-reversal invariant but the right hand is not. Its orientation-reversal invariance will be one of the fermion/fermion dualities in section 6.

Assuming any of these dualities with fixed values of the integers (Nf ,N,K) we can deform the theory by a mass term for one of the flavors to derive a duality with fewer flavors.

N We turn on a mass term P ψ ψINf in the fermionic side and M f = P φ φiNf in the I Nf I Nf i Nf i bosonic side. Unlike the similar supersymmetric dualities, here we still need to ﬁne tune an

SU(Nf −1) invariant mass term in order to hit a nontrivial IR ﬁxed point. In the fermionic

P Ia P ia side it is aI ψaI ψ and in the bosonic side it is Tr M = ai φaiφ .

In the bosonic side for one sign of the mass square the scalar becomes massive and we

flow to the theory with (Nf − 1,N,K). For the other sign of the mass square the gauge group is Higgsed and we flow to the theory with (Nf −1,N −1,K). In the fermionic side of the duality these two flows correspond to different signs of the fermion mass. For positive mass there is no induced Chern-Simons term and for negative mass the Chern-Simons terms

are shifted by −LU(K)1 [u; A] − KI[g, A] (recall that the fermion couples to u + A1 with u

43 a U(K) gauge field). This has the effect of shifting the level of the existing Chern-Simons terms in the fermionic side. The dualities (2.4.5) are consistent with these flows.

This discussion shows that the duality (2.4.1) might be problematic for Nf > N. In that case we can turn on masses and flow to N = 0. Now the left hand side of the duality is an empty gapped theory, while the theory in the right hand side seems like it is non-trivial. In fact, it is conjectured in [88] that the duality in this case cannot describe a simple fixed point, but rather some of the dual descriptions are valid near different transition points.

It is instructive to understand the physical interpretation of the currents that the background fields B and A couple to in (2.4.5). The SU(L) theories are described in terms of U(L) fields u with a Lagrange multiplier that sets Tr u = −B. Therefore, in these theories B couples to the baryon number current. More precisely, in the fermionic SU(K) theory (the right hand side of the second duality in (2.4.5)) the baryon current couples to B +KA, which is a background U(1) field for even K and a spinc connection for odd K. Indeed, the baryons in this theory are bosons for even K and they are fermions for odd K. In the various U(L) theories in (2.4.5) there is no baryon number symmetry and B couples to the monopole charge. Depending on the levels of the U(L) Chern-Simons theory it is or is not a spin theory. (Recall that even when it is a spin theory, because of the coupling to A it is independent of the spin structure.) When it is a spin theory, its monopole operator is a fermion and it couples to a background spinc connection, and when it is not spin the monopole operator couples to a background U(1) field. It is straightforward to check that the levels in (2.4.5) and the couplings to A and B are such that the monopole and baryon operators are mapped correctly between the two sides of the duality. This was studied in [117] and was one of the motivations of [2].

44 2.5 New boson/boson and fermion/fermion dualities

Our goal in this section is to use the conjectured boson/fermion dualities (2.4.5), (2.4.6) to derive new boson/boson and fermion/fermion dualities. We will mainly limit ourselves to

Nf = 1 and only discuss an example of Nf = 2 in section 6.3.

2.5.1 Boson/boson dualities

We set K = 1 in the second duality of (2.4.5) and the second duality of (2.4.6) to ﬁnd a triality

2 4 iψD/ Aψ ←→ |Dφ| − |φ| + LbU(N)1 [A] + (N − 1)I[g, A]

2 4 ←→ |Dφ| − |φ| + LbU(N)−1 [−A] − NI[g, A] , (2.5.1) where the φ in the different Lagrangians are different fields. We see that all the interacting bosonic theories on the right hand side are dual to a theory of a single free fermion. Surprisingly, although their Lagrangians depend on N, their physics is independent of N. This amounts to an infinite set of dualities relating interacting bosonic theories.

The special case of N = 1 leads to

2 4 2 4 |Dφ| − |φ| + LbU(1)1 [A] ←→ |Dφ| − |φ| + LbU(1)−1 [−A] − I[g, A] , (2.5.2) which agrees with equation (2.6) of [121].

Given these dualities we can act on them with T and S of (2.0.8) and find new dualities. These are infinitely many purely bosonic theories, which are dual to a single fermion coupled to dynamical gauge fields, which in general have some Chern-Simons couplings.

45 For example, starting with the duality between the ﬁrst bosonic theory in (2.5.1) with general N and the second bosonic theory in (2.5.1) with N = 1 we can add to the two sides

1 1 of the duality the counterterms 2π CdA + 4π CdC + (1 − N)I[g, A] and then gauge A by substituting A → A + u to ﬁnd

2 4 2 4 1 1 N |Dφ| − |φ| +LbSU(N)1 [C] ←→ |Dbφ| − |φ| − bdb + bdu + udu 4π 2π 4π 1 1 1 1 − bdA + ud (C − NA) + CdC + CdA − NI[g, A] . (2.5.3) 2π 2π 4π 2π

In the special case N = 1 we can integrate out u and (2.5.3) becomes the well-known particle-vortex duality.

2.5.2 Fermion/fermion dualities

Next we look for fermion/fermion dualities. Set N = 1 in the first duality of (2.4.5) and the first duality of (2.4.6) to find the triality

2 4 K K |D−Bφ| − |φ| + L0[A] ←→ iψDψ/ + LbU(K)0 [B] − BdB + BdA 4π 2π K K ←→ iψDψ/ + LbU(K)1 [−B + A] + BdB − BdA + KI[g, A(2.5.4)] . 4π 2π

As in the previous subsection the fields ψ in these different dual Lagrangians are different fields. This leads to an infinite number of fermion/fermion dualities. The duality between the two fermionic theories with K = 1 in (2.5.4) agrees with equation (2.16) of [121] (up to a trivial use of charge conjugation).

Again, as in the previous section we consider the duality between the ﬁrst fermionic Lagrangian in (2.5.4) with K = 1 and the second one with arbitrary K. We add to the

K 1 1 1 two sides the counterterms − 4π BdB + 2π Bd(−C + KA) + 4π CdC + 2π CdA − KI[g, A] and

46 gauge B by denoting it by u to ﬁnd

1 K + 1 1 iψDψ/ + LbSU(K)1 [C] ←→ iψD/ b+Aψ + bdu − udu + ud (−C + (K + 1)A) 2π 4π 2π 1 1 + CdC + CdA − KI[g, A] . 4π 2π (2.5.5)

In the special case K = 1 the duality (2.5.5) becomes after a trivial change of variables the fermion/fermion duality in equation (2.14) of [121]

1 2 2 iψD/ ψ ←→ iψD/ ψ + bdu − udu + udA − I[g, A] . (2.5.6) A b+A 2π 4π 2π

2.5.3 Self-duality of QED with two fermions

In this section we are going to use the previous dualities to derive a fermion/fermion duality similar to that of [153]. Our derivation can be easily extended to lead to many other dualities.

We start by substituting A → A + X in (5.6.1) and A → A − X in its orientation reversed duality

1 2 1 iψD/ ψ ←→ iχD/ χ + adu − udu + ud(A + X) − I[g, A + X] X+A a 2π 4π 2π 1 1 2 1 iψD/ ψ ←→ iχD/ χ + ada − adu + udu − ud(A − X) + 2CS (2.5.7), −X+A a 4π 2π 4π 2π grav where a, appearing as b + A in (5.6.1), is a dynamical spinc connection. (Note the parity anomaly.) We denoted the fermions in the right hand side by χ to highlight that they are diﬀerent than the fermions in the left hand side ψ.

47 The direct product of the theories (2.5.7) is described by the sum of their Lagrangians

1 2 iψ D/ ψ1 + iψ D/ ψ2 ←→ iχ D/ χ1 + iχ D/ χ2 + a da − u du 1 A+X 2 A−X 1 a1 2 a2 4π 2 2 4π 1 1 2 1 1 + u du + (a du − a du ) + (u − u )dA 4π 2 2 2π 1 1 2 2 2π 1 2 1 1 + (u + u )dX − (A + X)d(A + X) , (2.5.8) 2π 1 2 4π where the indices 1 and 2 label the two sectors.

1 1 Next we gauge A by adding the counterterms 2π AdY − 4π Y dY + I[g, A] with Y is a background U(1) ﬁeld (note that it is consistent with (2.1.3)) and then promoting A to a dynamical spinc connection a. This gauging couples the two sectors labeled by 1 and 2. In the right hand side we integrate out most of the gauge ﬁelds and are left with a single

c dynamical spin connection ea

1 1 1 iψ D/ ψ1 + iψ D/ ψ2 + ada + adY − Y dY + 2CS 1 a+X 2 a−X 4π 2π 4π grav 1 1 1 ←→ iχ D/ χ1 + iχ D/ χ2 + ada + adX − XdX + 2CS (2.5.9). 1 ea−Y 2 ea+Y 4πe e 2πe 4π grav

The theory on the left hand side is QED coupled to two fermions ψ1 and ψ2. The

1 Chern-Simons term 4π ada arises from our conventions of the fermion determinant (see the discussion around (2.4.2)-(2.4.4)). We will return to it below. This theory has a manifest SU(2)X × U(1)Y global symmetry. X couples to the Cartan generator of SU(2)X and Y couples to the global U(1)Y symmetry associated with the connection a.

The theory on the right hand side is also QED coupled to two fermions χ1 and χ2. It also has a manifest SU(2)Y × U(1)X global symmetry. But here Y couples to the Cartan

Y X generator of SU(2) and X couples to the global U(1) associated with the connection ea. This means that the theory at the ﬁxed point must have a global SU(2)X × SU(2)Y = Spin(4) symmetry. Each dual description makes manifest only an SU(2) × U(1) subgroup

48 of it, but these are diﬀerent subgroups. Equivalently, each of the Lagrangians in (2.5.9) has a classical SU(2) × U(1) symmetry, but the quantum theory has an enhanced quantum SU(2) × SU(2) = Spin(4) symmetry.

The way the classical global U(1) symmetry is enhanced to a global SU(2) quantum symmetry is common in supersymmetric dualities [6, 70], and in fact, this particular enhancement occurs in the N = 4 supersymmetric version of this theory [70]. (See also [74] for a recent related discussion.)

Let us deform the theory on the left hand side of (2.5.9) by an SU(2) invariant mass

i term mψiψ and ﬁnd the low-energy theory

1 1 1 2 ada + adY − Y dY + 2CS ←→ − Y dY for m > 0 4π 2π 4π grav 4π 1 1 1 2 2 − ada + adY − Y dY − XdX − 2CS ←→ − XdX for m < 0 . 4π 2π 4π 4π grav 4π (2.5.10)

We ﬁnd the same low-energy theories when we deform the theory on the right hand side of

i i i (2.5.9) by −mχiχ . This means that the operators are mapped as ψiψ ←→ −χiχ .

We claim that our theory has a global SU(2)X × SU(2)Y global symmetry and here we coupled background ﬁelds only to the Cartan generators. As a check of our claims we note that the Lagrangians in (2.5.10) are normalized such that we can extend them by adding the other background ﬁelds for SU(2)X × SU(2)Y

Y Y U(1)−2 −→ SU(2)−1 for m > 0

X X U(1)−2 −→ SU(2)−1 for m < 0 . (2.5.11)

(In this particular case if X and Y become dynamical ﬁelds, there is no need to add degrees of freedom when moving from the left to the right as these theories are dual to each other.)

49 Next, we would like to compare our result (2.5.9) with [153]. In addition to ﬁtting this duality into a large web of dualities, i.e. deriving it by assuming other, well-motivated dualities, our analysis has three key diﬀerences relative to [153].

1. We have formulated the theory with spinc connections such that our formulation of

the theory does not depend on a choice of spin structure. In particular, our a and ea are dynamical spinc connections and X and Y are background U(1) ﬁelds, which in the quantum theory can be extended to SU(2) ﬁelds.

2. We are more precise about the deﬁnition of the fermion determinant (see the discussion around (2.4.2)-(2.4.4)). To relate to [153] we imprecisely describe each of our fermions that couples to some A as having an improperly quantized Chern-Simons

1 term − 8π AdA − CSgrav. This means that in order to compare with [153] we need to 1 1 add in the left hand side of the duality − 4π ada − 4π XdX − 2CSgrav and to add in the right hand side of the duality − 1 ada − 1 Y dY − 2CS . After doing that (2.5.9) is 4π e e 4π grav almost the same as the duality in [153]. To make them identical we also need to add

1 to the two sides of the duality the counterterms 4π (XdX + Y dY ).

1 3. The counterterms 4π (XdX + Y dY ) that we have to add to (2.5.9) in order to match with [153] are incompatible with (2.1.3). But what is worse is that if we add these counterterms and then deform the theory with fermion masses, the low-energy theory

X Y is U(1)±1 × U(1)∓1 (see equation (5) in [153]), which unlike (2.5.10), (5.3.6) cannot be extended to SU(2)X × SU(2)Y .

Finally, we have also established the Spin(4) global symmetry of the model, thus clarifying the relation to [123, 153] (once some counterterms are added). Note that we argued that the long distance behavior of the system is a non-trivial ﬁxed point with a Spin(4) global symmetry. But that ﬁxed point cannot be described by the strong coupling limit of a

50 continuum Lagrangian based on a non-linear sigma model, whose target space is S3. Such a continuum theory is non-renormalizable.

2.6 Appendix A: Often used equations

In this appendix we collect some useful equations.

The following terms are valid counterterms on spinc three-manifold (with B,C being U(1) gauge ﬁelds and A a spinc connection):

1 BdC 2π 1 1 BdB + BdA 4π 2π 1 I[g, A] ≡ AdA + 2CS 4π grav

16CSgrav. (2.6.1)

U(N)1 is dual to a classical ﬁeld theory

1 2i 1 L [b; A] = Tr [bdb − b3] + (Tr b)dA ←→ −NI[g, A] . (2.6.2) U(N)1 4π 3 2π

The Lagrangians for U(N) and SU(N) are

K 2i 1 L [b; C + KA] ≡ Tr bdb − b3 + (Tr b)d (C + KA) U(N)K 4π 3 2π K 2i 1 L [b; C + (K ± 1)A] ≡ Tr bdb − b3 ± (Tr b)d(Tr b) U(N)K,K±N 4π 3 4π 1 + (Tr b)d (C + (K ± 1)A) 2π K 2i 1 L [b, c; B] ≡ Tr bdb − b3 + K (Tr b)d(Tr b) + cd (Tr b + B) SU(N)K 4π 3 4π 2π

51   0 K even   K = 1 K odd and positive . (2.6.3)    −1 K odd and negative

And if we add to them the “trivial” theory, which includes the purely gravitational spin one-half line 1 1 L [x, y; A] ≡ xdx + xd(y + A) . (2.6.4) 0 4π 2π

We have

K 2i 3 1 K LbSU(N) [B] ≡ Tr bdb − b + (c + KA)d(Tr b + B) + L0[A] + BdB K 4π 3 2π 4π

←→ LSU(N)K [B] + L0[A]

LbU(N)K [B + KA] ≡ LU(N)K [B + KA] + L0[A]

LbU(N)K,K±N [B + (K ± 1)A] ≡ LU(N)K,K±N [B + (K ± 1)A] + L0[A] . (2.6.5)

52 Chapter 3

Chern-Simons-matter dualities with SO and Sp gauge groups

Our goal in this chapter is to extend the conjectural infrared dualities

Nf scalars with SU(N)k ←→ Nf fermions with U(k) Nf Nf −N+ 2 ,−N+ 2

Nf scalars with U(N)k,k ←→ Nf fermions with SU(k) Nf −N+ 2

Nf scalars with U(N)k,k±N ←→ Nf fermions with U(k) Nf Nf (3.0.1) −N+ 2 ,−N∓k+ 2

for Nf ≤ N, to orthogonal and symplectic gauge groups. We will conjecture the following IR dualities:1

Nf real scalars with SO(N)k ←→ Nf real fermions with SO(k) Nf −N+ 2

Nf scalars with USp(2N)k ←→ Nf fermions with USp(2k) Nf (3.0.2) −N+ 2 1For the symplectic groups our notation is USp(2N) = Sp(N), and in particular USp(2) = Sp(1) = SU(2). It is worth noting that for the orthogonal groups the level k, which is an integer, is normalized such k 2 3 that the Chern-Simons term is 2·4π Tr AdA + 3 A with a trace in the vector representation. As we will discuss below, the SO(N)k theory with even k is a conventional non-spin topological quantum ﬁeld theory (it does not require a spin structure), while the theory with odd k is spin.

53 The matter ﬁelds are in the fundamental representation of the gauge group. The SO dualities are conjectured to hold for Nf ≤ N −2 if k = 1, Nf ≤ N −1 if k = 2, and Nf ≤ N if k > 2. Notice that these dualities, as opposed to the ones considered before, involve real scalars and real (Majorana) fermions. The USp dualities are conjectured to hold for

Nf ≤ N. In the ’t Hooft limit of large N and k, with ﬁxed Nf and N/k, these dualities are supported by the same considerable evidence as their U(N) counter-parts (since the orthogonal and symplectic theories are just projections of the U(N) theories at leading order in 1/N).

As a particularly interesting special case, if we set k = 1 in the SO dualities, we conjecture that the SO(N)1 CS theory coupled to Nf scalar ﬁelds in the vector representation

(with any N ≥ Nf + 2) ﬂows to Nf free Majorana fermions.

In a companion paper [22] we will discuss a number of non-trivial ﬁxed points with enhanced global symmetry. Every one of them has a number of dual descriptions. The full global symmetry appears classically in some descriptions, but it only appears as a quantum enhanced low-energy symmetry in others. For example, taking special cases of (3.0.1), all the theories

one scalar with U(1)2 ←→ one fermion with U(1) 3 − 2 l l

one fermion with SU(2) 1 one scalar with SU(2)1 (3.0.3) − 2 lead to the same nontrivial ﬁxed point with a global SU(2) symmetry. The two descriptions at the bottom make the SU(2) global symmetry manifest at the classical level. (Actually, only SO(3) acts faithfully on gauge invariant operators.) However, the symmetry only appears quantum mechanically in the two descriptions at the top.

54 The duality between the two theories at the bottom of (3.0.3) is the simplest example of the USp dualities (3.0.2), thus providing a nontrivial check of them. Also the duality between the two theories at the top of (3.0.3) appears among our SO dualities for SO(2) = U(1) (3.0.2), but the SO duality acts on the operators in a diﬀerent way, that is related to the U(1) duality by a global SU(2) rotation. Thus, the enhanced global symmetry plays a crucial role in the consistency checks of the dualities (3.0.2).

If we set Nf = 0 in the suggested dualities (3.0.1) and (3.0.2), we ﬁnd simple dualities involving topological quantum ﬁeld theories (TQFTs):

SU(N)k ←→ U(k)−N,−N

U(N)k,k±N ←→ U(k)−N,−N∓k

SO(N)k ←→ SO(k)−N

USp(2N)k ←→ USp(2k)−N . (3.0.4)

These are level-rank dualities in pure Chern-Simons theory. They provide one of the main motivations for the dualities (3.0.1), (3.0.2), or conversely, they are a nontrivial check of them. Although there exists a large literature about such level-rank dualities, we could not find a precise version of them. In [67] a careful analysis derived the first two lines in (3.0.4) and clarified that they hold, in general, only when the theories are spin-Chern-Simons theories (see [67] for details). Below we will provide a similar proof of the orthogonal and symplectic dualities in (3.0.4) and will establish the need for the theories to be spin.2

The level-rank dualities can be used to show that several TQFTs, although not manifestly so, are time-reversal invariant at the quantum level. We provide a rich set of examples,

2However we will also ﬁnd that for special values of N, k the dualities (3.0.4) are valid for conventional TQFTs as well. In particular the ﬁrst SU/U duality is non-spin for N even and Nk = 0 mod 8, the SO duality is non-spin for N, k even and Nk = 0mod 16, and the USp duality is non-spin for Nk = 0 mod 4.

55 summarized in table 3.1. These theories represent new possible gapped boundary states of topological insulators and topological superconductors.

T -invariant theories N property framing anomaly 2 U(N)N,2N even N T -invariant spin theory (N + 1)/2 odd N need to add ψ 2 PSU(N)N even N T -invariant spin theory (N − 1)/2 odd N T -invariant non-spin theory 2 USp(2N)N even N T -invariant non-spin theory N odd N need to add ψ SO(N)N odd N T -invariant spin theory N = 0 mod 4 T -invariant non-spin theory N 2/4 N = 2 mod 4 need to add ψ

Table 3.1: Some TQFTs that are time-reversal invariant (up to an anomaly) at the quantum level. In each case we indicate whether the theories are spin or non-spin TQFTs, and in the latter case whether they need to be tensored with a trivial spin theory {1, ψ} to exhibit time-reversal symmetry.

In Section 2 we analyze in detail the Chern-Simons-matter dualities for USp(2N) gauge groups, and in Section 3 for SO(N) groups. In Section 4 we comment on the relation to high-spin gravity theories on AdS4. Section 5 gives a detailed description of level-rank dualities for orthogonal and symplectic groups, and Section 6 describes their implications for constructing new time-reversal-invariant TQFTs. Appendix A explains our notation and conventions.

After the completion of this work, we received [95] where the duality of free fermions to

SO(N)1 with scalars is worked out.

56 3.1 Dualities between USp(2N) Chern-Simons-matter

theories

In Section 5 we will derive and discuss certain dualities of spin-TQFTs that take the form of level-rank dualities:

USp(2N)k × SO(0)1 ←→ USp(2k)−N × SO(4kN)1 . (3.1.1)

In our conventions USp(2N) = SU(2N) ∩ Sp(2N, C), and further details are collected in Appendix A. We recall that a spin-TQFT—as opposed to a conventional topological quantum ﬁeld theory—can only be deﬁned on a manifold with a spin structure, and if multiple spin structures are possible, then the spin-TQFT will depend on the choice. A

1 spin-TQFT always has a transparent line operator of spin 2 . In (3.1.1) the USp factors are non-spin, while the SO factors are trivial spin-TQFTs (discussed e.g. in [122]), whose presence is important for the duality to work.

We can then add matter in the fundamental representation, bosonic on one side and fermionic on the other, and conjecture new boson/fermion dualities. This is done in such a way that renormalization group (RG) ﬂows in which all matter becomes massive are consistent with (3.1.1). We thus propose the following dualities between the low-energy limits of Scalar and Fermionic theories:

4 Theory S: A USp(2N)k theory coupled to Nf scalars with φ interactions

Theory F: A USp(2k) Nf theory coupled to Nf fermions −N+ 2 (3.1.2)

for Nf ≤ N. (The meaning of half-integer CS levels is the standard one, reviewed in

Appendix A.) More precisely, Theory S also includes the spin-topological sector SO(0)1

57 1 that makes it into a spin theory and provides a transparent line of spin 2 . Theory F is already manifestly spin, however to correctly reproduce its framing anomaly3 we include SO 4k(N − Nf ) 1. For completeness, let us rewrite the duality including its time-reversed version:

N USp(2N)k × SO(0)1 with Nf φi ↔ USp(2k) f × SO 4k(N − Nf ) 1 with Nf ψi −N+ 2

USp(2N)−k × SO(0)1 with Nf φi ↔ USp(2k) Nf × SO(4kN)−1 with Nf ψi N− 2 (3.1.3)

where φi are scalars and ψi are fermions.

Theory S contains Nf complex scalars in the fundamental 2N representation of

USp(2N). Since the representation is pseudo-real, we can rewrite them in terms of 4NNf

ab ij ∗ complex scalars subject to the reality condition ϕaiΩ Ωe = ϕbj, where a = 1, ··· , 2N is a fundamental index of USp(2N), i = 1, ··· , 2Nf is a fundamental index of USp(2Nf ), and

Ωab, Ωeij are the corresponding symplectic invariant tensors. This description makes the

4 USp(2Nf ) ﬂavor symmetry of the theory manifest. There is one quadratic term and two quartic terms one can write that preserve the USp(2Nf ) ﬂavor symmetry: in terms of the

ab antisymmetric meson matrix Mij = ϕaiΩ ϕbj they are

(2) X 2 (4) 2 (4) O = Tr (ΩeM) = |ϕai| , O1 = Tr (ΩeM) , O2 = Tr (ΩeMΩeM) . (3.1.4) ai

All these three terms are relevant in a high-energy Yang-Mills-Chern-Simons theory with these matter fields. We turn them on with generic coefficients and make a single fine- tuning (which we can interpret as the coefficient of the quadratic term). We conjecture

3In general, the two sides of the duality have different framing anomalies (see [146]). We fix that by adding a trivial spin-TQFT that has the difference in the anomaly. For our purposes this is the same as a gravitational Chern-Simons term with an appropriate coefficient. 4 For Nf = 1 there is only one independent quartic term.

58 that by doing that the long distance theory is at an isolated nontrivial fixed point. It is a generalization of the Wilson-Fisher fixed point, and it has a single USp(2Nf )-invariant relevant deformation which may be identified with O(2). Theory S is defined at such a fixed point.

Note that the Z2 center of the USp(2N) gauge group acts in exactly the same way as the Z2 center of the USp(2Nf ) global symmetry group. So local gauge-invariant operators actually sit in representations of USp(2Nf )/Z2 (e.g. for Nf = 1 they always have integer spin under SU(2)). However we can have non-local operators like a ϕai attached to a Wilson line, where the charge of the Wilson line under the center of USp(2N) is correlated with the charge of the operator that it ends on under the center of USp(2Nf ). In this sense the gauge × global symmetry group is USp(2N) × USp(2Nf ) /Z2.

Theory F contains Nf complex spinors in the fundamental 2k representation of

USp(2k), and again, rewriting them in terms of 4kNf Dirac spinors with a reality con-

ab ij c c dition ψaiΩ Ωe = ψbj (where ψ is the charge conjugate) makes the USp(2Nf ) ﬂavor symmetry manifest. At high energies the corresponding Yang-Mills-Chern-Simons theory has a single relevant USp(2Nf )-invariant operator, which is the quadratic mass term. We tune it to zero in the IR, and assume it is the only USp(2Nf )-invariant relevant operator there. The bare CS levels (see Appendix A) are (Nf − N) in (3.1.2) and N in (3.1.3).

Notice that while the spin-TQFTs involved in the level-rank duality (3.1.1) have a Z2 one-form global symmetry associated to the center of the group, such a symmetry is broken in the theories with matter because the latter transforms under the center [54]. In fact, the theories in (3.1.2) do not have any discrete global symmetries.

59 3.1.1 RG ﬂows

We cannot prove the dualities in (3.1.2), however we can perform some consistency checks. For instance, we can connect different dual pairs by RG flows triggered by mass deformations. In both Theories S and F, turning on a mass at high energies leads to turning on the unique relevant deformation of the low-energy conformal field theory (CFT). So turning

2 on a bosonic mass-squared mφ in Theory S should have the same eﬀect at low energies as turning on a fermion mass mψ in Theory F.

In Theory S, if we give a positive mass-squared to one of the complex scalars we simply reduce Nf by one unit. However, if we turn on a negative mass-squared, a complex scalar condenses Higgsing the gauge group to USp(2N − 2)k, in addition to reducing the number of flavors. In Theory F, when giving mass to one of the complex fermions, the phase of its partition function becomes either e−iπη(A) or 1 in the IR limit, depending on the sign of the mass. In both cases the number of flavors is reduced by one, however in the first case one can use the APS index theorem [14] s(see Appendix A) to rewrite the leftover regularization term e−iπη(A) in terms of a shift of the bare gauge and gravitational CS terms. Thus, tuning

5 the mass of the remaining (Nf −1) ﬂavors to zero, the RG ﬂow leads to the following pairs:

N −1 USp(2N)k × SO(0)1 with φi ↔ USp(2k) f × SO 4k(N − Nf + 1) 1 with ψi −N+ 2 N +1 USp(2N − 2)k × SO(0)1with φi ↔ USp(2k) f × SO 4k(N − Nf ) 1 with ψi −N+ 2 (3.1.5)

each with Nf − 1 matter ﬁelds. These are consistent with the proposed duality.

If Nf < N, by ﬂowing with diﬀerent combinations of positive and negative masses squared, in Theory S the gauge group can range between USp(2N)k and USp 2(N −Nf ) k,

5 Here we assume that the RG ﬂow still leads to a non-trivial CFT with a single USp(2Nf − 2)-invariant relevant operator.

60 while in Theory F the level can range between USp(2k)−N and USp(2k)−N+Nf . In all these cases we do not find any inconsistency in the duality. Starting with a dual pair with Nf = N, we can give negative mass-squared to all flavors and flow to a dual pair with Nf = N = 0.

In this case, Theory S is gapped and empty in the IR; Theory F is a USp(2k)0 Yang-Mills theory, which conﬁnes and has a single gapped ground state. Thus the duality is still valid.

On the other hand, consider the case Nf ≥ N + 1 and turn on generic masses for all the flavors. Theory F flows to a non-trivial topological theory. Let us compare with Theory S. Here generic negative mass-squared for all matter fields Higgses the gauge group completely. The IR theory could be gapped or could have massless Goldstone bosons, but since the gauge group is completely Higgsed, it cannot include a topological sector. Hence, the duality cannot be correct in this case. We conclude that none of the pairs with

Nf ≥ N + 1 can be dual.

3.1.2 Coupling to background gauge ﬁelds

Given that our system has a global USp(2Nf ) symmetry, we can couple it to background gauge ﬁelds for that symmetry. Our goal here is to identify the CS counterterms [33] for these ﬁelds that are needed for the duality.

Let us start with the scalar side of the duality. We start with a USp(2N)k CS theory for

the dynamical ﬁelds and we can also have USp(2Nf )ks for some integer ks for the classical ﬁelds. If we give masses to all scalars such that the gauge symmetry is not Higgsed, then

the low energy theory is purely topological. It is a USp(2N)k × USp(2Nf )ks CS theory, where the second factor is classical. In the fermionic side of the duality we start with

N USp(2k) f for the dynamical fields and USp(2Nf )kf for the classical fields. These −N+ 2 k mean that the bare CS levels for these two groups are −N + Nf and kf + 2 respectively. Repeating the mass deformation of the bosonic side we find at low energy a topological

61 USp(2k)−N as well as a CS counterterm for the classical ﬁelds USp(2Nf ) k . For this to kf − 2 match with the bosonic side we must choose6

k k = k + . (3.1.6) f s 2

Of course, we have the freedom to add the same CS counterterm on the two sides of the duality. This will add an arbitrary integer to ks and the same integer to kf .

We can repeat the same considerations with an opposite sign for the mass deformations. In the scalar theory, Higgsing occurs and the symmetry group is reduced to USp2(N − Nf ) k × USp(2Nf )ks+k, where the broken part of the gauge group is identified with the flavor group and this causes the shift of the CS counterterm for the classical fields. In the

k fermionic theory we ﬁnd USp(2k)−N+Nf × USp(2Nf ) . As a non-trivial check, equality kf + 2 on the two sides requires the very same relation (3.1.6).

As we discussed above, the global symmetry that acts faithfully on local operators is USp(2Nf )/Z2 and this puts restrictions on the CS counterterms. More precisely, in the bosonic side we would like the bare CS terms to be consistent for USp(2N)k × USp(2Nf )ks /Z2 and the Z2 quotient is consistent only for

Nk + Nf ks ∈ 2Z . (3.1.7) 6It is common in the literature to argue for ’t Hooft-like anomaly matching conditions restricting the level of CS terms for the global symmetry. These levels could be half-integral in theories with fermions and they are always integral in theories with bosons. Based on that, one might attempt to exclude many of these boson/fermion dualities. Instead, as in Appendix A, the fractional part of the CS terms always arise from the dynamics and the bare CS levels are always integral. In the cases where our theories have fermions leading to a fractional level, the bosonic dynamics in the other side of the duality must lead to equal fractional levels. Indeed, our analysis of the renormalization group ﬂow out of the ﬁxed point is consistent with this assertion. However, as we will now show, there do exist non-trivial ’t Hooft-like anomaly matching of the integer bare CS terms for the global symmetries.

62 In the fermionic side the bare CS terms are USp(2k)−N+Nf × USp(2Nf )ks+k /Z2, where we used (3.1.6). This is consistent for

− kN + 2kNf + ksNf ∈ 2Z . (3.1.8)

Fortunately, (3.1.7) is the same condition as (3.1.8). In the spirit of ’t Hooft anomaly matching, this is a non-trivial consistency check on our duality. The obstruction to the Z2 quotient is the same in the two sides of the duality.

When the condition (3.1.7) is not satisfied, we cannot mod out by the Z2 and fewer backgrounds of the gauge fields are allowed. In those cases it might still be possible to extend the USp(2Nf ) classical gauge fields to a (3 + 1)d bulk and consistently take the Z2 quotient there.

3.1.3 Small values of the parameters

It is instructive to look at the dualities (3.1.2) for small values of N, k and Nf . We already discussed that the case N = Nf = 0 is the statement that USp(2k)0 conﬁnes with a single vacuum. The case k = 0 is the statement that USp(2N)0 with Nf ≤ N complex scalars conﬁnes with a single vacuum: although we have no proof that this is true, it is surely plausible.

As discussed around (3.0.3), the case N = k = Nf = 1 can be derived from the dualities in [2,67], giving us more conﬁdence that the duality is correct.

3.1.4 New fermion/fermion and boson/boson dualities

Combining the dualities for symplectic groups in (3.1.2) with those for (special) unitary groups in [2,67], we can ﬁnd new fermion/fermion and boson/boson dualities.

63 For instance, we can take (3.1.2) with N = Nf = 1 and combine it with the ﬁrst duality of (5.5) in [67]. This gives us a fermion/fermion duality

USp(2k) 1 with 1 fermion in 2k ←→ U(k) 3 with 1 fermion in k . (3.1.9) − 2 − 2

(To be precise, the theory on the right should include a decoupled SO(2k)1 factor.) As we discussed, the duality with k = 1 can be derived from [2,67], but the other ones are new.

Note that for all k the theories in (3.1.9) have a global SU(2) symmetry, which is manifest in the LHS of (3.1.9). On the RHS of the duality we have a manifest U(1) monopole number symmetry and charge conjugation C, which does not commute with it

C because it maps the monopole number n to −n. Our duality suggests that this U(1) o Z2 classical symmetry is enhanced in the quantum theory to SU(2). The currents that extend the Abelian symmetry to SU(2) must carry monopole charge. We suggest that they are constructed out of the monopole operator and its conjugate in the U(k) theory. Since these carry charge, each of them should be dressed by a fermion to be gauge-invariant.

Similarly, from (3.1.2) with k = 1 and [2,67] we can obtain the boson/boson duality

USp(2N)1 with Nf scalars in 2N ←→ U(N)2 with Nf scalars in N (3.1.10)

with Nf ≤ N. Both sides should include SO(0)1 and be regarded as spin theories. The case

N = Nf = 1 was already found in [2,67], but the other ones are new. As above, the global symmetry of these theories is USp(2Nf ), which is manifest in the LHS, thus we conjecture that the manifest U(Nf ) and charge conjugation symmetries in the RHS are enhanced to

USp(2Nf ).

64 3.2 Dualities between SO(N) Chern-Simons-matter

theories

For orthogonal groups there exists a similar level-rank duality of spin-TQFTs (derived in Section 5):

SO(N)k × SO(0)1 ←→ SO(k)−N × SO(kN)1 . (3.2.1)

(For our conventions on CS terms see Appendix A and footnote 1.) Arguments similar to those of the previous section, and to those used for SU(N) and U(N) groups, suggest a duality between the low-energy limits of:

4 Theory S: An SO(N)k theory coupled to Nf real scalars with φ interactions

Theory F: An SO(k) Nf theory coupled to Nf real fermions. −N+ 2 (3.2.2)

As we explain below, the duality can only be true for Nf ≤ N − 2 if k = 1, for Nf ≤ N − 1 if k = 2, and for Nf ≤ N if k > 2. Moreover, one should remember that Theory S includes the trivial spin-topological sector SO(0)1, while Theory F includes SO k(N − Nf ) 1. The matter fields are all in the vector representation. There is considerable evidence for this duality at large N and k; at finite values of N and k we can check its consistency by mass flows, including flows to the level-rank dualities between pure Chern-Simons theories (3.2.1) described in Section 5.

The two low-energy theories are defined by tuning the masses to zero, and assuming that both sides flow to a fixed point, which has a single relevant operator consistent with the global symmetries of the high-energy theory. For finite values of N, k and Nf we do not know when this is true; additional operators could become relevant at low energies, which would prevent the two theories from flowing to the same fixed point, or the fixed point

65 may cease to exist (say because a mass gap develops, or because we spontaneously break some of the ﬂavor symmetries on one side but not the other). The limits on Nf above arise because we show that for larger values of Nf the duality cannot hold, but we conjecture that it does hold for the values mentioned above.

3.2.1 Flows

We can ﬂow from the duality with (N, k, Nf ) to the dualities with (N, k, Nf − 1) and with

(N − 1, k, Nf − 1) by adding a mass for a single flavor. Both the mass-squared deformation in Theory S, and the mass deformation in Theory F, are expected to flow to the same relevant operator in the low-energy CFT (which is in the symmetric tensor representation of the SO(Nf ) flavor group). Previously we mentioned the SO(Nf )-invariant component of this relevant operator, but we assume here that it also has a symmetric-traceless counter- part that is relevant. In theory F we can integrate out the massive flavor, schematically

1 shifting the Chern-Simons level of the remaining low-energy theory by ± 2 , depending on the sign of the mass. In theory S the same flavor, depending on the sign of its mass-squared, either becomes massive, or condenses and breaks the gauge group to SO(N − 1) (this is all similar to the U and USp cases). In this flow we fine-tune the mass of the remaining

(Nf −1) ﬂavors to zero. The ﬂow to the lower duality exists whenever no additional operator becomes relevant, and whenever the new IR CFT exists; under these assumptions it leads to a new duality between the lower theories S0 and F0.

Reversing this logic, if we have a non-trivial duality for some values of (N, k, Nf ), we

0 0 0 0 can assume that the duality holds for all (N , k, Nf ) with N + Nf − Nf ≥ N ≥ N and

0 Nf > Nf . If for some such value the duality fails but we still have a non-trivial theory with no extra relevant operators, then we expect the duality to fail also for the corresponding higher values (since otherwise we get a contradiction by first flowing to the “higher” IR CFT and then performing the mass flow).

66 When giving a mass to all Nf fermions, Theory F flows to a pure SO(k) Chern-Simons theory with a level between (−N) and (Nf −N), depending on the signs of the masses of the different flavors. Depending on the same signs, some flavors could condense in theory S, so that its gauge group is between SO(N)k and SO(N −Nf )k. The resulting theories are then dual by level-rank dualities of SO(N) spin-TQFTs (see Section 5), giving a consistency check on our dualities. Note that this assumes Nf < N.

For Nf = N and an appropriate sign of the mass deformation, the gauge theory in Theory S is completely broken, and for generic values of the masses all scalars become massive and the theory develops a mass gap. In Theory F for the same masses the low- energy Chern-Simons level vanishes and the fermions are massive, so again we expect a mass gap. For Nf > N with the same choice of signs for the masses for all the ﬂavors, we break the gauge group completely in Theory S, and generically all scalars are massive and we have a mass gap with a trivial theory at low energies. On the other hand, in Theory F for the same choice, all fermions become massive, but we end up with a non-trivial topological theory, so the duality necessarily breaks down for this case, as in the U(N) and USp(2N) cases.

3.2.2 Global symmetries

The UV Yang-Mills-Chern-Simons theories we start from have an O(Nf ) ﬂavor symmetry, as well as two discrete symmetries discussed below. The deﬁnition of the IR Theories

S and F involves ﬂows from these UV theories that preserve these SO(Nf ) and discrete symmetries. The SO(Nf ) symmetry allows for a single mass term which needs to be tuned, both on the scalar and on the fermion sides. As in (3.1.4), for Nf > 1 the UV description

4 of Theory S has two possible φ terms. In terms of Mij = φaiφaj (i, j are ﬂavor indices and

(4) 2 (4) 2 a is a color index) they are O1 = Tr (M) and O2 = Tr (M ) . We assume that for generic couplings of these operators they do not lead to any new relevant operator at low

67 energies.7 For some small values of N and k there are enhanced continuous symmetries, which we will discuss below.

In addition there are two discrete symmetries, that were discussed in detail in [8]. There is a global “charge conjugation” symmetry C, which acts on the matter ﬁelds as φ1i → −φ1i and all other φai are invariant. When C is gauged the gauge group changes from SO(N)

8 to O(N). The SO(N) vector indices may be contracted to form singlets either with δab

or with a1a2···aN . Operators that involve the latter contraction are odd under C, while all others are even. Since the product of two epsilon symbols may be replaced by a sum of products of δ’s, the symmetry is Z2.

In the fermionic SO(k) theory, the lowest-dimension operator charged under C is a baryon operator, involving k fermions contracted with an epsilon symbol. Classically the dimension of this operator is k, and in the quantum theory it has some anomalous dimension. Its Lorentz ×SO(Nf ) representation is a symmetric product of k spinors which are vectors of SO(Nf ). In the scalar SO(N) theory a similar contraction vanishes (for Nf < N) because of the statistics of the scalar operators. For Nf = 1 the lowest-dimension C-odd operator comes from choosing N diﬀerent derivative operators acting on the scalar, and then contracting them with an epsilon symbol. Similarly for Nf > 1 we need to choose

N diﬀerent combinations of derivatives and ﬂavor indices; when Nf = N we can contract

Nf diﬀerent scalars with no derivatives. In the large N limit with ﬁxed Nf , the classical dimension of the lightest C-odd operator scales as N 3/2 [125]. The Lorentz representation of these operators comes from the product of those of the derivative operators that we need to use.

7 For some values of N, k and Nf there are additional ﬁxed points where one or both quartic operators are tuned to zero, and that also have fermionic Gross-Neveu-like duals, but we will not discuss them here. 8For N even, this symmetry always exchanges the two spinor representations of Spin(N). They are in fact complex conjugate representations for N = 2mod 4, but not for N = 0mod 4.

68 Monopole operators in an SO(N) theory are characterized by having some quantized flux around them, which can be chosen to be in the Cartan algebra of SO(N) (this is a semiclassical characterization; in the quantum theory operators with different GNO charges can mix). The lowest one carries one unit of flux under a single SO(2) subgroup of SO(N). The flux breaks the SO(N) gauge group to O(2) × O(N − 2) /Z2. The smallest monopole charge (which we normalize to be 1) in the SO(N) theory is defined by requiring mutual locality with matter fields (or Wilson lines) in the vector representation of SO(N). In a Spin(N) theory we need to require mutual locality also with fields (or Wilson lines) in the spinor representation, and thus the minimal monopole charge is 2. This implies that the monopoles carry a Z2 global “magnetic” symmetry M, and operators that are allowed in SO(N) but not in Spin(N) are odd under M. Note that unlike in U(N) theories, the monopoles do not carry a U(1) global charge (except when the gauge group is SO(2)), but a Z2 charge.

This description of the monopole operator is not manifestly SO(N) invariant. Alterna- tively, we can deﬁne this operator by removing a point from our spacetime and specifying a non-trivial bundle on the sphere that surrounds it. Speciﬁcally, this monopole corresponds to having nontrivial second Stiefel-Whitney class w2 on that sphere. This makes it clear that the M charge is a Z2 charge.

Gauge-invariance requires that a monopole in the SO(N)k Theory S must come together with k ﬁelds charged under the SO(2) ⊂ SO(N) gauge group. Charged scalars in the

1 1 monopole background carry spin 2 [152], and their scaling dimension is shifted from 2 to 1. Thus the lightest monopole operator that is charged under M has classical dimension k, and lies in a Lorentz ×SO(Nf ) representation that is a symmetric product of k spinors that are fundamentals of SO(Nf ).

Charged fermions in a monopole background have integer spins, and in particular each fermion has a zero mode, and deﬁning the monopole operator requires quantizing these Nf

69 zero modes. After quantizing the zero modes, one has to add in the SO(k) Nf Theory −N+ 2 F N additional fermionic operators charged under SO(2), of various integer spins, in order to form a gauge-singlet.

The Lorentz ×SO(Nf ) representations of these fermionic monopole operators are identical to those of the baryons in Theory S that were described above, and their classical dimensions are also the same (since the fermion modes in the monopole background have the same Lorentz quantum numbers and dimensions as derivatives acting on scalars). In particular their classical dimension in the large N limit scales as N 3/2 [113] (and this statement is true also quantum mechanically [117]). Similarly, the lightest baryon operator in the fermionic theory has precisely the same quantum numbers and classical dimension as the lightest monopole operator in the scalar theory.

Above we were not careful about precisely which monopole we choose in the SO(2) theory, and how it transforms under charge conjugation; these issues were discussed in detail in [8], and we review the discussion here. In an SO(2) = U(1) theory, there are monopoles Vn that carry n units of the U(1)J magnetic charge (topological charge). Charge conjugation in this theory takes n to (−n), so we have one lightest monopole (V1+V−1) = V+ which is C-even, and another (V1 − V−1) = V− which is C-odd. We can choose the SO(N) monopole discussed above to correspond to either one of these operators in the SO(2) subgroup. However, since the precise gauge group that remains in the monopole background is O(2) × O(N − 2) /Z2, if we choose the C-even monopole operator in SO(2), we need to dress it by a C-even operator in SO(N −2), while if we choose the C-odd monopole in SO(2), it has to come with a C-odd operator in SO(N − 2), in order to be SO(N)-gauge-invariant. The monopoles we discussed above in SO(N) theories were singlets of SO(N − 2), so they are C-even and involve the monopole V+ in the O(2) subgroup. In order to form a C-odd monopole operator (which was called a monopole-baryon in [8]) we need to take V− and multiply it by a C-odd operator in O(N − 2), namely a product of (N − 2) matter ﬁelds

70 contracted with an epsilon symbol (in addition to the extra ﬁelds required for cancelling the SO(2) charge of the monopole). Repeating the same arguments as above, we ﬁnd that the lightest monopole-baryon-operator in both theories F and S has a classical dimension

3/2 scaling as N in the large N, k limit with ﬁxed Nf , and the operators also lie in identical

Lorentz ×SO(Nf ) representations in the two theories.

The arguments above strongly suggest that the duality exchanges monopoles and baryons, and takes the monopole-baryon operators to themselves, namely it exchanges the two Z2 global symmetries C and M. In fact, we can see that this must be the case by performing the mass ﬂow to the pure Chern-Simons theories, and noting (see Section 5) that the level-rank duality in these theories indeed exchanges C with M. So this gives a nice consistency check for the duality. The fact that the classical dimensions on both sides match (at least at large N) is somewhat surprising, since one would expect their dimension to receive quantum corrections (except for the monopole operator in the fermionic theory which was shown in [117] to receive no quantum corrections to its dimension in the ’t Hooft large N limit). This is all very similar to the duality between SU(N) and U(k) CS-matter theories, which also exchanges baryon number with monopole number [2,117].

By gauging C and/or M we can find related dualities involving O(N), Spin(N) and P in±(N) gauge theories. These theories can have additional labels, which are the coefficients of terms like w2w1 of the gauge bundle [8]. These can be thought of as CS terms of various discrete gauge fields or as discrete theta parameters analogous to those studied in [9,54,78]. We will not discuss them here.

In the N = 2 supersymmetric version of the Chern-Simons-matter dualities between

SO(N) gauge theories [8], the duality maps CSUSY to itself, while mapping MSUSY to

CSUSYMSUSY. Given the fact that one can ﬂow from the supersymmetric theories to the pure Chern-Simons theories, and perhaps also to the non-supersymmetric Chern-Simons- matter theories along the lines of [64, 71], this is confusing. The resolution is that in the

71 N = 2 supersymmetric Chern-Simons-matter theories there is also a complex gaugino field in the adjoint representation. It has (N − 2) zero modes in the monopole background, which are a vector of SO(N − 2), and their product is odd under CSUSY. Thus the minimal monopole in the supersymmetric theory, that carries those zero modes, has an opposite charge-conjugation transformation from the minimal monopole in the non-supersymmetric theory (bosonic or fermionic). So we have MSUSY = M, but CSUSY = CM (where C and M are defined as above in the non-supersymmetric theory). With this relation, the two dualities are consistent with the flows and identifications discussed above.

3.2.3 Small values of N and k

When the theories on both sides are non-Abelian it is diﬃcult to check the dualities. How- ever, for k = 1, 2, and for Nf = N − 1,N, we can (possibly after Higgsing) have Abelian or empty gauge groups, so we can test the dualities in more detail.

When k = 1 or N = 1 we have no gauge group on one of the two sides, and also no magnetic symmetry, though the charge conjugation symmetry remains and changes the sign of the matter ﬁelds. On the fermionic side for k = 1 we have Nf free real (Majorana) fermions. On the bosonic side for N = 1 we have an O(Nf )-invariant Wilson-Fisher ﬁxed point, which arises by turning on the quartic operator in the theory of Nf real scalars.

When k = 2 or N = 2 we have an Abelian gauge theory on one of the two sides. In this theory the magnetic Z2 symmetry M is enhanced to a U(1)J symmetry (equal to M modulo 2). As mentioned above, the charge conjugation symmetry does not commute with

U(1)J : it takes a monopole of U(1)J charge n to one of charge (−n). In addition, the UV theory in this case is equal to a U(1) Maxwell-Chern-Simons theory (since U(1)k0 with Nf

flavors is identical to SO(2)k0 with Nf flavors) which has, for Nf > 1, an enhanced SU(Nf ) flavor symmetry.

72 In the bosonic N = 2 theory, the mass operator (which we can write in the U(1) language

(2) † as O = Φi Φi in terms of complex ﬁelds Φi carrying U(1) charge 1) is SU(Nf )-invariant.

For Nf = 2, we have additional gauge-invariant quadratic operators: there is an operator

2 2 † † charged under the flavor SO(2) containing |Φ1| −|Φ2| and (Φ1Φ2 +Φ2Φ1) which is C-even, (2) † † and a flavor SO(2) singlet O2 = (Φ1Φ2 − Φ2Φ1) which is C-odd. So in flows from the

UV Yang-Mills-Chern-Simons theory that preserve both SO(Nf ) and C we do not need to consider either one of them, and we still require a single ﬁne-tuning at low energies. For

Nf > 2 there is only one quadratic SO(Nf )-invariant operator.

(2) 2 † † At quartic order there is an SU(Nf )-invariant C-even operator (O ) = Φi ΦjΦiΦj. † † For Nf > 1, though, there is another SO(Nf )-invariant C-even quartic operator Φi Φi ΦjΦj. When we view the theory as an SO(2) gauge theory, and in particular when we flow to it from higher SO(N) gauge theories, we only preserve the SO(Nf ) symmetry, so the latter operator is also turned on during the flow. We expect this extra operator to be irrelevant at the SU(Nf )-invariant fixed point of the U(1) theory, and if so then the SO(2) flow also reaches the same fixed point, at least for some values of the deformations from the UV theory. However, we do not know how to prove that this is the case, and there may be a different flow if the SU(Nf )-non-invariant operator is turned on with a large coefficient. We will assume below that we do end up at the same fixed point. For Nf = 2 this extra operator

(2) 2 (2) 2 is a linear combination of (O ) and (O2 ) , and there is another SO(2)-invariant C-odd

(2) (2) operator O O2 which will not be turned on in a C-invariant ﬂow (an additional operator coming from the SO(2)-charged operator squared is a linear combination of these).

In the fermionic k = 2 theory we have similar quadratic operators, and the quartic operators are irrelevant in the UV.

Thus, in both theories we expect the SO(Nf )-invariant C-invariant flow of the SO(2) theory to end up at the same fixed point as that of the SU(Nf )-invariant U(1) flow. We

73 can then use known facts about the latter ﬂow to learn about the status and implications of the SO(N) dualities.

Now let us discuss the dualities for various small values of k and N.

3.2.4 The k = 1 case

For k = 1 we have a duality between the theory of Nf free real (Majorana) fermions, and an SO(N)1 theory coupled to Nf real Wilson-Fisher scalars.

For N = 1 (implying also Nf = 1) this duality is obviously wrong, since the usual Wilson-Fisher ﬁxed point is not free (and thus the duality cannot hold also for higher

N = Nf which can ﬂow to this value).

For N = 2 we have on the bosonic side the ﬁxed point of U(1)1 coupled to Wilson-Fisher scalars. In this case we need Nf = 1, and then the U(L) dualities [2, 121], assuming they hold for L = 1, imply that this scalar theory is dual to a free complex fermion. So under this assumption the SO duality cannot be correct for (N, k, Nf ) = (2, 1, 1), and thus also for k = 1 and higher values of N with Nf = N − 1.

For higher values of N, with Nf ≤ N − 2, we cannot rule out the k = 1 duality by known results. Thus we conjecture an IR duality between:

4 Theory S: An SO(N)1 theory coupled to Nf real scalars with φ interactions

Theory F: Nf free Majorana fermions. (3.2.3)

As discussed above, the monopole operator of the scalar theory maps to the real fermion.

The lowest case is N = 3 and Nf = 1, where we have an SO(3)1 theory with a single scalar in the vector (adjoint) representation ﬂowing to a free Majorana fermion. Since Theory F

74 in this case has no magnetic symmetry, the duality implies that all baryons and monopole- baryons of Theory S decouple at low energies.

3.2.5 The N = 1 case

In this case, since we should have Nf = 1, the scalar theory is just a real Wilson-Fisher scalar. The dual fermionic theory has a real fermion coupled to an SO(k) 1 CS theory. − 2

The case k = 1 was already ruled out above. The case k = 2 is related to a U(1) 1 theory, − 2 which maps by the dualities of [2,121] to a complex Wilson-Fisher scalar. So, assuming the validity of the U(1) duality, the SO duality cannot hold in this case, and thus also for other cases with k = 2 and Nf = N.

For higher values of k and N = Nf = 1, the duality may be correct, namely the SO(k) 1 − 2 CS theory coupled to a single fermion may ﬂow to the ﬁxed point of a real Wilson-Fisher scalar. Again this implies that all baryonic operators of this theory decouple at low energies.

3.2.6 The k = 2 case

In this case we have a duality between the theory of Nf fermions coupled to SO(2) Nf = −N+ 2

U(1) Nf , and the theory of Nf real scalars coupled to SO(N)2. Such a duality implies −N+ 2 that the charge conjugation Z2 symmetry of the scalar theory is enhanced to U(1), and its

SO(Nf ) ﬂavor symmetry is enhanced to SU(Nf ), at low energies.

As discussed above we can rule out the cases with Nf = N, so the lowest case is N = 2 and Nf = 1. This case is interesting because, as discussed around (3.0.3), the two dual Abelian theories admit two more non-Abelian descriptions in which the full SU(2) global symmetry of the ﬁxed point is manifest in the UV.

Our duality maps Theory S, a U(1)2 CS theory coupled to a complex Wilson-Fisher scalar (for Nf = 1 there is no diﬀerence between the SO(2) and U(1) ﬂows) to Theory F,

75 a U(1) 3 CS theory coupled to a complex fermion. The very same two theories, viewed as − 2 U(1) theories, are also mapped to each other by the U(1) duality of [2, 67, 121]. However, interestingly enough, the operator mappings are not the same in the U(1) duality and the SO(2) duality: the U(1) ↔ U(1) duality preserves the magnetic symmetry and the charge conjugation, while the SO(2) ↔ SO(2) duality exchanges them. Fortunately, this perfectly ﬁts with the enhanced quantum SU(2) global symmetry. In each U(1) CS description there

C is a manifest U(1)J o Z2 magnetic and charge conjugation symmetry. The U(1) duality

C trivially maps the two copies of U(1)J o Z2 one into the other. The SO(2) duality, instead,

C maps U(1)J o Z2 in a nontrivial way, which follows from its embedding inside the global SU(2) symmetry: it is an SU(2) rotation.

For k = 2 and N > 2 we obtain more complicated dualities, which we cannot rule out.

The U(1) dualities map the fermionic SO(2) = U(1) theories to SU(N)1 theories coupled to Nf scalars, so we obtain boson-boson dualities between SU(N)1 and SO(N)2 theories coupled to Nf < N Wilson-Fisher scalars (which are complex and real, respectively).

3.2.7 The N = 2 case

For N = 2 we have an SO(2)k = U(1)k CS theory coupled to Nf charged scalars with

Wilson-Fisher couplings; here we can have Nf = 1 or Nf = 2. The dual for Nf = 1 is an

SO(k) 3 theory coupled to a real fermion, and for Nf = 2 an SO(k)−1 theory coupled to − 2 two real fermions. For k = 1 and k = 2 we have already discussed these theories above.

For k > 2 the dual theory is non-Abelian, and we cannot rule the duality out. Again, it implies that the charge conjugation symmetry of the fermionic theory should be enhanced to U(1) at low energies, and its SO(Nf ) ﬂavor symmetry to SU(Nf ).

The U(1) duality with Nf = 1 maps the same scalar theory to an SU(k) 1 theory − 2 coupled to a complex fermion, giving another fermion-fermion duality between SO(k) 3 − 2

76 and SU(k) 1 theories with one real/complex fermion ﬂavor. For Nf = 2 the U(1) duality − 2 breaks down, but the SO(N) duality of the previous paragraph may still be valid.

3.3 Relation to theories of high-spin gravity

The SO(N) theories with k = 0 and Nf = 1 were the ﬁrst ones to be suggested to be dual at large N to Vasiliev’s high-spin gravity theory on AdS4 [87, 124]; they are dual to the minimal Vasiliev theory, which has only even-spin excitations. There are two versions of this theory, diﬀering by a discrete parameter, that were argued to be dual to theories of N scalars and N fermions, respectively. This was later generalized to U(N) and SU(N) theories being dual to non-minimal Vasiliev theories that have excitations of all spins.

There is an obvious generalization of both dualities to higher Nf , with the SO(N) theory containing Nf (Nf + 1)/2 excitations of even spins, and Nf (Nf − 1)/2 excitations of odd spins. The Vasiliev theory is only known by its classical equations of motion, so a priori it is not known how to quantize it; the dual field theories with finite N can be viewed as giving a non-perturbative definition of this theory.

For the U(N) scalar/fermion theories, it was argued that a parameter θ0 in the Vasiliev theory is related to N/k when the U(N) Yang-Mills theory is replaced by U(N)k [57]. This parameter interpolates between the theory dual to parity-invariant scalars, and the one related to parity-invariant fermions, and this led to the conjectured duality between CS-scalar and CS-fermion theories. The same parameter exists also in the minimal Vasiliev theories, so it is natural to conjecture that turning it on in the minimal Vasiliev theories corresponds to having SO(N)k or USp(2N)k CS-matter theories. Again the fact that the Vasiliev theory has two interpretations, as a CS-scalar and as a CS-fermion theory, suggests that at least at large N the SO(N) and USp(2N) dualities that we discussed above are correct.

77 At leading order in large N, the orthogonal, symplectic and unitary theories are all the same, consistent with having the same classical equations of motion in the minimal and non-minimal Vasiliev theories (up to having a projection removing half of the fields in the SO and USp cases). However, the one-loop corrections should be different. In particular there should be a discrete parameter distinguishing the SO(2N) and USp(2N) theories, whose effect is to change the sign of all l-loop diagrams with odd values of l. This can be realized by inverting the signs of ~ and of Newton’s constant.9 There should also be discrete parameters on the gravity side distinguishing the different versions of the SO(N) theories, where one gauges some of the Z2 discrete symmetries.

3.4 Level-rank dualities with orthogonal and symplec-

tic groups

3.4.1 Level-rank dualities of 3d TQFTs

Level-rank dualities of 2d chiral algebras can be derived starting from systems of free fermions. For instance, consider a system of Nk free real (Majorana) fermions: writing them as ψ with a = 1, ··· ,N and a = 1, ··· , k one obtains the following conformal aea e embeddings of chiral algebras (see also [65,133]):

Spin(Nk)1 ⊃ Spin(N)k × Spin(k)N /Z2 N, k odd

Spin(Nk)1 ⊃ Spin(N)k × SO(k)N N even, k odd

Spin(Nk)1 ⊃ SO(N)k × SO(k)N N, k even . (3.4.1)

9Note that this is not the same as the case where only Newton’s constant is inverted, giving a theory in de Sitter space, as studied in [11, 12].

78 Here Spin is the standard Kac-Moody chiral algebra, while SO = Spin/Z2 is the extended chiral algebra [100, 101] obtained from Spin by adding a suitable Z2 generator of spectral

ﬂow (see below). The Z2 quotient in the ﬁrst line is the extension by the diagonal element. A series of equalities of chiral algebras follows:

Spin(Nk)1 Spin(N)k ←→ N, k odd Spin(k)N

Spin(Nk)1 Spin(N)k ←→ N even, k odd SO(k)N Spin(Nk)1 SO(N)k ←→ N odd, k even Spin(k)N Spin(Nk)1 SO(N)k ←→ N, k even . (3.4.2) SO(k)N

On the right hand sides we have GKO cosets [59]. Moving from two-dimensional chiral algebras to three-dimensional Chern-Simons theories [101], one obtains dualities between the following Lagrangian theories:

Spin(Nk)1 × Spin(k)−N Spin(N)k ←→ N, k odd Z2

Spin(N)k ←→ Spin(Nk)1 × SO(k)−N N even, k odd Spin(Nk) × Spin(k) SO(N) ←→ 1 −N N odd, k even k B Spin(Nk)1 × SO(k)−N SO(N)k ←→ N, k even . (3.4.3) Z2

On the right hand sides, the Lagrangian is the one corresponding to the Lie algebra of the numerator, while the gauge group is the result of the quotient. On the third line,

B = Z2 × Z2 for k = 0mod 4 and B = Z4 for k = 2mod 4. We stress that these level-rank dualities of 3d TQFTs (or equivalently of 2d chiral algebras) can be rigorously proven. They have also been analyzed in [96,103].

79 Similarly, one can start with a system of 4Nk 2d real fermions, and writing them as 4Nk complex fermions with a symplectic Majorana condition one obtains the conformal embedding

Spin(4Nk)1 ⊃ USp(2N)k × USp(2k)N . (3.4.4)

This leads to the duality of chiral algebras

Spin(4Nk)1 USp(2N)k ←→ , (3.4.5) USp(2k)N and in terms of three-dimensional Chern-Simons theories one has the duality

Spin(4Nk)1 × USp(2k)−N USp(2N)k ←→ (3.4.6) Z2 between 3d TQFTs.

3.4.2 Matching the symmetries

Before going on with the analysis of those dualities, let us fix some notations for orthogonal and symplectic chiral algebras. The center B(G) of the simply-connected group G associated to a Lie algebra g acts on the affine Lie algebra Gk as an outer automorphism, and its action is generated by elements σi of Gk via spectral flow. In the corresponding 3d CS theory, B(G) appears as a one-form symmetry [54] generated by the lines σi. Such lines can act in two different ways on the other lines of the theory: either by “fusion” (or spectral flow), if they are placed parallel to the lines they act upon, or by “monodromy”, if they are wound on a small circle around the other lines.

Whenever the generator of spectral ﬂow has integer dimension, it can be included in the chiral algebra to give an extended chiral algebra; equivalently, in 3d the one-form symmetry can be gauged to give a Gk/H CS TQFT, where H is a subgroup of B(G).

80 When the generator has half-integer dimension, the chiral algebra can be augmented to a

Z2-graded chiral algebra (with half-integer dimensions) that depends on the spin structure; equivalently, in 3d the one-form symmetry can be gauged but the Gk/H CS theory is a spin-TQFT. In our case, the center of USp(2N) is Z2 while the center of Spin(N) is Z2 for

N odd, Z2×Z2 for N = 0 mod 4 and Z4 for N = 2mod 4. Let us identify the corresponding generators of spectral ﬂow.

10 In USp(2N)k, the Z2 spectral ﬂow is generated by

σ :[λ0, λ1, ··· , λN ] → [λN , λN−1, ··· , λ0] . (3.4.7)

kN The generator is given by σ = (0, ··· , 0, k) with dimension h(σ) = 4 . The action of σ via monodromy is

c Qσ[λ] = (−1) [λ] , (3.4.8)

PN where the congruence class c of a representation [λ] is given by c = j odd λjmod 2. In par-

Nk ticular the self-parity of σ is (−1) . For Nk = 0mod 4, one can consider the P USp(2N)k ≡

USp(2N)k/Z2 CS theory; for Nk = 2mod 4, one can consider the P USp(2N)k spin-CS theory.

In Spin(N)k with N odd, the Z2 spectral ﬂow is generated by

σ :[λ0, λ1, λ2 ··· , λr] → [λ1, λ0, λ2, ··· , λr] . (3.4.9)

k The generator is given by σ = (k, 0, ··· , 0) with dimension h(σ) = 2 . The action via monodromy is

λr Qσ [λ] = (−1) [λ] . (3.4.10)

10 We indicate a highest weight representation by its Dynkin labels (λ1, ··· , λr) or by its extended Dynkin labels [λ0, ··· , λr], where r is the rank. In sp(2N), λr refers to the long root. In so(2n + 1), λr refers to the short root, while in so(2n), λr−1 and λr refer to the two roots at the “bifurcated tail” of the Dynkin diagram.

81 For N even we have the two spectral ﬂow operations

  [λr−1, λr, λr−2, ··· , λ1, λ0] for N = 2mod 4 js :[λ0, λ1, ··· , λr−1, λr] → (3.4.11)  [λr, λr−1, λr−2, ··· , λ1, λ0] for N = 0mod 4 and

σ :[λ0, λ1, ··· , λr−1, λr] → [λ1, λ0, λ2, ··· , λr−2, λr, λr−1] . (3.4.12)

Nk They are generated by js = (0, ··· , 0, k) with dimension h(js) = 16 and σ = (k, 0, ··· , 0) k 2 with dimension h(σ) = 2 . For N = 2mod 4 the group structure is Z4: js = σ and 4 2 js = σ = 1. The action via monodromy is

N 2 c Qjs [λ] = i [λ] (3.4.13)

N −2 P 2 where the congruence class is c = j odd 2λj + λr − λr−1 mod 4. For N = 0mod 4 the

2 2 group structure is Z2 × Z2: js = σ = 1 and jsσ = σjs. The action via monodromy is

  (−1)cc [λ] for N = 0mod 8 Qjs [λ] = (3.4.14)  (−1)cs [λ] for N = 4mod 8

N −3 N −3 P 2 P 2 where cs = j odd λj + λrmod 2 and cc = j odd +λr−1mod 2. The action of σ via monodromy is

λr+λr−1 Qσ[λ] = (−1) [λ] (3.4.15) in both N even cases. The one-form symmetry generated by σ can always be gauged to obtain the SO(N)k CS theory: it is a TQFT for k even, and a spin-TQFT for k odd. Only for N, k both even there is another generator js that survives the quotient, thus only in this case SO(N)k has a Z2 one-form symmetry.

82 Let us also discuss what type of conventional zero-form symmetries the Chern-Simons theories can have. In Spin(N)k and SO(N)k with N even, one deﬁnes a “charge conjugation” Z2 symmetry C that transforms representations as

C : λr−1 ↔ λr . (3.4.16)

In SO(N)k gauging this symmetry gives O(N)k. Counterterms for the classical gauge field of C lead to additional parameters in the O(N)k theory [8]. In SO(N)k with k even, we define a “magnetic” Z2 symmetry M that exchanges the two representations of the extended chiral algebra resulting from a fixed point of the Z2 spectral flow of Spin(N)k [100, 101]. From the 3d point of view, the magnetic quantum number of a monopole operator is the second Stiefel-Whitney class w2 of the SO(N) bundle around its location. This symmetry is gauged when going from SO(N)k to Spin(N)k.

Whenever a three-dimensional TQFT has a Z2 one-form global symmetry generated by 1 σ with spin h(σ) = 2 mod 1, we can deﬁne a quantum zero-form symmetry Kσ acting on the lines in the following way:

  [λ] if Qσ[λ] = [λ] Kσ[λ] = (3.4.17)  σ · [λ] if Qσ[λ] = −[λ] where by σ · [λ] we mean fusion. This deﬁnition guarantees that K preserves the fusion rules and that K[λ] has the same spin as [λ].

Having settled the basic deﬁnitions, we can analyze the precise mapping of symmetries between the dual theories in (3.4.3) and (3.4.6).

Consider

Spin(Nk)1 × Spin(k)−N Spin(N)k ←→ N, k odd , (3.4.18) Z2

83 where the quotient is generated by σ ⊗ σ (which has ﬁxed points). Both sides have a Z2 one-form global symmetry, and the map of generators is11 σ ↔ σ ⊗ 1 ∼ 1 ⊗ σ. Both sides have Z2 zero-form symmetry. On the RHS it is the quantum symmetry Kσ. On the LHS it is the magnetic symmetry MZ2 associated to the ﬁxed points of the Z2 quotient. The map of generators is Kσ ↔ MZ2 .

Consider

Spin(N)k ←→ Spin(Nk)1 × SO(k)−N N even, k odd . (3.4.19)

Both sides have a Z2 × Z2 one-form global symmetry for N = 0mod 4, and Z4 for N =

2mod 4. The map of generators is js ↔ js ⊗ 1, σ ↔ σ ⊗ 1. Both sides have a Z2 × Z2 zero-form symmetry, and the map of generators is CKσ ↔ 1 ⊗ M, Kσ ↔ C ⊗ 1.

Consider

Spin(Nk) × Spin(k) SO(N) ←→ 1 −N N odd, k even . (3.4.20) k B

For k = 2mod 4, B = Z4 is generated by js ⊗ js, while for k = 0mod 4, B = Z2 × Z2 is generated by js ⊗ js and σ ⊗ σ (the quotient has no ﬁxed points). Both sides have no zero-form symmetry (on the RHS, all generators of the numerator are projected out by the quotient). The zero-form symmetry is Z2 and the map of generators is M ↔ CKσ (on the RHS, all other generators do not commute with the quotient action and are thus broken).

Consider

Spin(Nk)1 × SO(k)−N SO(N)k ←→ N, k even , (3.4.21) Z2 11Here and in the following, ∼ means identiﬁcation by the quotient.

84 where the quotient is generated by js ⊗js (with no ﬁxed points). On both sides the one-form global symmetry is Z2 and the map of generators is

  Nk  js ⊗ 1 ∼ 1 ⊗ js 4 even js ↔ (3.4.22)  Nk  σjs ⊗ 1 ∼ σ ⊗ js 4 odd.

On the RHS, all other generators are projected out by the quotient. The zero-form symmetry is Z2 × Z2, and the map of generators is C ↔ 1 ⊗ M, M ↔ 1 ⊗ C.

Finally, consider

Spin(4Nk)1 × USp(2k)−N USp(2N)k ←→ , (3.4.23) Z2

where the quotient is generated by js ⊗σ (with no ﬁxed points). On both sides the one-form symmetry is Z2 and the map of generators is

  js ⊗ 1 ∼ 1 ⊗ σ Nk even σ ↔ (3.4.24)  σjs ⊗ 1 ∼ σ ⊗ σ Nk odd.

The zero-form symmetry is Z2 for Nk = 2mod 4 and nothing otherwise. The map of generators is Kσ ↔ 1 ⊗ Kσ.

3.4.3 Level-rank dualities of spin-TQFTs

So far we have discussed level-rank dualities between TQFTs. We can obtain simpler dualities if we consider spin-TQFTs. As we explain below, we obtain the following:

SO(N)k × SO(0)1 ←→ SO(k)−N × SO(Nk)1 (3.4.25)

USp(2N)k × SO(0)1 ←→ USp(2k)−N × SO(4Nk)1 . (3.4.26)

85 1 We recall that SO(N)1 is a trivial spin-TQFT with two transparent lines of spins 0, 2 N and with framing anomaly c = 2 (see e.g. [122]). Before deriving (3.4.25) and (3.4.26), let us discuss the symmetries and their map, starting with the orthogonal case (3.4.25). As we discussed after (3.4.15), SO(N)k has a

Z2 one-form global symmetry for N, k both even, and not otherwise. Thus the one-form symmetries match. Moreover, SO(N)k has a charge conjugation Z2 zero-form symmetry

12 C for N even, and a magnetic Z2 symmetry M for k even. Those two symmetries are exchanged in the duality (3.4.25),

C ←→ M , (3.4.27) as it also follows from the derivation of the duality that we give below.

In the symplectic case (3.4.26), on both sides there is a Z2 one-form global symmetry generated by σ, and a quantum zero-form symmetry Kσ for Nk = 2mod 4.

Note that in the CS theories with matter in the fundamental representation discussed in the main text, both for gauge group SO and USp, the possible one-form global symmetry is broken by the presence of matter [54].

Next, we derive the dualities (3.4.25) and (3.4.26). The simplest case is to start with

Spin(N)k ↔ Spin(Nk)1 × SO(k)−N with N even, k odd. We can gauge the Z2 one-form symmetry generated by σ ↔ σ ⊗ 1 to directly obtain (3.4.25). Since k is odd, SO(N)k is a spin theory and hence adding SO(0)1 does not change it. Another simple case is to start with Spin(N)k ↔ Spin(Nk)1 × Spin(k)−N /Z2 with N, k odd. The quotient is by σ ⊗ σ, and it preserves the Z2 generator σ ↔ σ ⊗ 1 ∼ 1 ⊗ σ. If we gauge the latter one-form

12 For k even, the Z2 quotient Spin(N)k/Z2 = SO(N)k has ﬁxed points, but not for k odd when SO(N)k is spin.

86 symmetry as well, we obtain

Spin(Nk)1 Spin(k)−N SO(N)k ←→ × , (3.4.28) Z2 Z2 which is precisely (3.4.25).

To get the other cases, we make use of the following identity of spin-TQFTs:

Spin(2L)1 × SO(0)1 ←→ SO(2L)1 × (Z2)−L , (3.4.29)

discussed in [122]. Starting from USp(2N)k ↔ Spin(4Nk)1×USp(2k)−N /Z2, we multiply both sides by SO(0)1 making them into spin theories, then apply (3.4.29) and obtain

SO(4Nk)1 × (Z2)−2Nk × USp(2k)−N USp(2N)k × SO(0)1 ←→ . (3.4.30) Z2

The theory (Z2)−2Nk is a TQFT with Z2 × Z2 one-form symmetry. The Z2 quotient is Nk generated by pairing σ in USp(2k)−N , whose spin is h = − 4 mod 1, with a generator in

(Z2)−2Nk that has opposite spin. In the quotient SO(4Nk)1 remains as a spectator. The product of USp(2k)−N by (Z2)−2Nk gives four times as many ﬁelds, however the freely- acting quotient by Z2 reduces to the original ones (one can check that the surviving states have the same dimensions as the original ones). Thus one has a simple duality of TQFTs: (Z2)−2Nk × USp(2k)−N /Z2 ↔ USp(2k)−N . This leads to the duality in (3.4.26). Exactly the same reasoning can be applied to the case SO(N)k ↔ Spin(Nk)1 ×SO(k)−N /Z2 with

N, k even. We multiply both sides by SO(0)1, then we apply (3.4.29), and ﬁnally observe that the quotient can be “unfolded” by the simple duality: (Z2)−2nm ×SO(2m)−2n /Z2 ↔

SO(2m)−2n. Here we set N = 2n and k = 2m. This leads to (3.4.25).

87 The last case is SO(N)k ↔ Spin(Nk)1 × Spin(k)−N /B, with N odd, k even. After multiplication by SO(0)1 and application of (3.4.29), we obtain

(Z2)− Nk × Spin(k)−N SO(N) × SO(0) ←→ SO(Nk) × 2 . (3.4.31) k 1 1 B

(js) (σ) This case is a little bit more intricate. For Nk = 0mod 4, B = Z2 × Z2 . The first factor Nk is obtained by pairing js in Spin(k)−N , whose spin is h(js) = − 16 mod 1, with a generator in (Z2) Nk that has the opposite spin. Therefore the first quotient is non-spin. The second − 2 factor is generated by W1,0⊗σ (in the notation of [122]) and it is a spin quotient. We can first

(js) (σ) use (Z2) Nk to unfold the quotient by 2 , and be left with Spin(k)−N / 2 = SO(k)−N . − 2 Z Z

This reproduces (3.4.25). For Nk = 2mod 4, B = Z4. Its generator is obtained by pairing js in Spin(k)−N with a generator in (Z2) Nk that has the opposite spin. This quotient − 2 has no ﬁxed points, and its eﬀect is to restrict to the lines with c = 0mod 4 with respect

1 to js, times the transparent lines 0, 2 . This is precisely the eﬀect of the spin quotient (σ) Spin(k)−N /Z2 for N odd. Again we end up with (3.4.25).

3.4.4 More non-spin level-rank dualities

For special values of N, k the spin dualities (3.4.25)-(3.4.26) can be upgraded to non-spin dualities. This uses the fact when N = 0 mod 8, the spin duality (3.4.29) can be replaced by the non-spin duality

L Spin(16L)1 ←→ (Z2)0 × (E8)1 . (3.4.32)

88 We will omit the trivial TQFT (E8)1. For even N, k and Nk = 0 mod 16, repeating the derivation of (3.4.25) with (3.4.32) replacing (3.4.29) gives the non-spin duality

SO(N)k ←→ SO(k)−N . (3.4.33)

Similarly, for Nk = 0 mod 4, repeating the derivation of (3.4.26) with (3.4.32) gives the non-spin duality

USp(2N)k ←→ USp(2k)−N . (3.4.34)

A special case of (3.4.33) is SO(8L)2 ↔ SO(2)−8L, which can be rewritten as

SU(8L)1 ↔ U(1)−8L using some relation in Appendix A. In turn this can be used in the arguments of [67] to show the non-spin duality

SU(N)k ←→ U(k)−N (3.4.35) for even N and Nk = 0mod 8.

In the next section we will make use of the non-spin level-rank dualities (3.4.33) and (3.4.34) to ﬁnd new T -invariant TQFTs.

3.5 T -invariant TQFTs from level-rank duality

It is of considerable interest to find topological field theories that are time-reversal invariant (T -invariant) up to an anomaly, because they can lead to gapped boundary states of topological insulators or topological superconductors. Some known examples are based on Chern-Simons theories with various product groups and possibly appropriate quotients [25,29,47,93,94,122,137,140]. It turns out that level-rank duality is a powerful tool to find

89 new examples.13 Speciﬁcally, a level-rank duality that exchanges N ↔ k, when applied to a theory with N = k, shows that such a theory is T -invariant quantum mechanically.

Let us examine a few examples. First, from the dualities of spin-TQFTs (4.15), (4.19) in [67], and (3.4.25), (3.4.26) here, we ﬁnd spin-TQFTs that are T -invariant, up to an anomaly. In some cases the theory involved is already a spin theory. In some other cases the theory is not spin but the level-rank duality, and therefore T -invariance, only holds after we tensor with a trivial spin theory, which we denote by ψ. Second, from the dualities of non-spin TQFTs (4.18) in [67] and (3.4.33), (3.4.34) here we ﬁnd conventional TQFTs that are T -invariant, up to an anomaly. This leads to the examples in table 3.1. The special case SO(3)3 appears in the literature as SU(2)6/Z2 [47]. The special case U(1)2 is known as the “fermion/semion theory” [47] and was discussed recently in [121,122].

3.6 Appendix A: Notations and useful facts about

Chern-Simons theories

Let us start by reviewing some facts about the fermion determinant [10, 150] and our notation. A (2 + 1)d fermion coupled to a gauge ﬁeld A and transforming in a complex or pseudo-real representation r has partition function

Zψ = |Zψ| exp − iπη(A)/2 , (3.6.1) where η(A) is the eta-invariant of the Dirac operator and the sign in the exponent is a matter of convention. For a fermion in a real representation, the phase of the partition function is exp − iπη(A)/4 instead.

13We thank E. Witten for a useful discussion about this point.

90 By the Atiyah-Patodi-Singer index theorem [14],

Z Z Z iπη(A) F/2π e = exp 2πi Ab(R)Trr e = exp 2ixr CS(A) + 2i(dim r) CSgrav X (3.6.2) where X is a bulk four-manifold that bounds the three-manifold, xr is the Dynkin index of

14 R R the representation r, and CSgrav = π X Ab(R) is the gravitational Chern-Simons term. R −in CSgrav In particular e is the partition function of the almost trivial spin-TQFT SO(n)1.

The Lagrangian of a Chern-Simons-matter theory can include a bare Chern-Simons term kbare. This must be properly normalized. For example, if the gauge group is SU(N) we have kbare ∈ Z, while in SU(N)/ZN we must have kbare ∈ NZ. In (3.0.1), (3.0.2) and below we deﬁne the level k as the sum of the bare value and the possibly fractional value δk that comes from (3.6.1):

k = kbare − δk , (3.6.3)

1 where δk = xr for complex or pseudo-real representations, and δk = 2 xr for real representations.

Let us collect some useful formulas. First, in our notation:

SO(2)k = U(1)k , Spin(2)k = U(1)4k

SO(3)k = SU(2)2k/Z2 , Spin(3)k = SU(2)2k . (3.6.4)

Second, chiral algebras at small level satisfy special relations:

SO(N)2 ↔ SU(N)1 . (3.6.5)

14 1 For SU(N), the fundamental has xr = 2 and the adjoint has xr = N. For SO(N), the vector has 1 xr = 1 and the adjoint has xr = N − 2. For USp(2N), the fundamental has xr = 2 and the adjoint has xr = N + 1.

91 Finally, the theories of a U(1) or Z2 gauge ﬁeld with an action expressed in terms of the eta-invariant are dual to simple spin-TQFTs:

SU(N)1 × SO(0)1 ↔ S = −Nπ η U(1) gauge ﬁeld Nπ Spin(N) × SO(0) ↔ S = − η gauge ﬁeld , (3.6.6) 1 1 2 Z2

where SO(1)1 and Spin(1)1 are the spin and non-spin Ising TQFT, respectively [122]. These dualities are proven in [67] and [122].

92 Chapter 4

Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups

In this chapter we study various gauge theories with a gauge group based on the Lie algebra so(N). These include SO(N), Spin(N), O(N), and P in±(N) gauge theories. (There are also other such groups that we will not study here; e.g. SO(N)/Z2.) Depending on the gauge group the Lagrangians of these theories can include various Chern-Simons couplings and discrete θ-parameters, which can also be viewed as more subtle Chern-Simons terms. We will discuss them and show how they aﬀect the behavior of the theory. We will also couple these gauge theories to bosonic or fermionic matter ﬁelds in various representations.

The novelty in our analysis will be the careful attention to the global aspects of the gauge group and the dependence on these discrete θ-parameters. This understanding will lead to

93 a rich spectrum of dualities including level-rank dualities between topological quantum ﬁeld theories (TQFTs), and various dualities between Chern-Simons-Matter (CSM) theories.

Beyond the intrinsic interest in these theories, there are several motivations for our investigations.

Consider for example the Spin(N) gauge theory with matter fields in the vector representation. This theory has a Z2 one-form global symmetry [54, 78] associated with the center of the gauge group. This means that we can explore the behavior of Wilson loops in a spinor representation of Spin(N) and learn about confinement. This has been done in many papers and our discussion here will follow the line of investigation of [9, 54, 78] in 4d and of [8,62,88] in 3 d. Specifically, we will focus on the one-form global symmetry as a clear diagnostic of confinement.

Our second motivation is associated with the role of global symmetries. Consider for example the SO(N) theories. They have a Z2 charge conjugation symmetry C and a Z2 magnetic symmetry M, which we will discuss in detail below. In addition, depending on the matter fields, the Chern-Simons couplings and the value of N they can have time-reversal symmetry and one-form global symmetry. (More subtle phenomena associated with time- reversal symmetry will be discussed in [36].) We will couple these global symmetries to background gauge fields and analyze their t’ Hooft anomalies, and Chern-Simons counterterms [33,34]. Once these counterterms are understood, it is straightforward to gauge these symmetries, i.e. to promote the background fields to dynamical fields.

Another motivation for studying these theories is the presence of discrete θ-parameters. It is often the case that the conﬁguration space of a gauge theory breaks into distinct sectors labelled by ν with a well deﬁned partition function in each sector Zν. The total partition function X Z = aνZν (4.0.1) ν

94 depends on the coefficients aν. These can be viewed as partition functions of some other theory that couples to our gauge theory. In the special case where all the coefficients aν are phases, aν are the partition functions of invertible topological quantum field theories [50].

But we can also have situations where some of the coeﬃcients aν vanish; i.e. some sectors are absent in the sum. Examples of that were presented in [49,120]. Here we will see more examples of this phenomenon.

As is clear from these points, these systems are concrete examples of the interplay between topology and symmetries. The recent interest in this interplay was formalized in [17,46]. Our systems are an explicit laboratory allowing exploration of these phenomena.

We will start our discussion with a review of known facts about the various groups that we will study, SO(N), Spin(N), O(N), and P in±(N). We will also review how gauge theories based on these groups are constructed. We will be particularly interested in three cases.

Spin(N) gauge theories are special because they do not involve a sum over topological sectors as in (4.0.1). In the other extreme, O(N) gauge theories include all the bundles of interest to us. In addition to the regular Chern-Simons level K ∈ Z, these theories are labeled by discrete parameters associated with distinct consistent ways to sum over these

r bundles as in (4.0.1). We will denote them as O(N)K,L with r = 0, 1 and L = 0, 1, ..., 7.

(The label r = 0, 1 was denoted as O(N)± in [8].)

SO(N) gauge theories are the ones with the largest (zero-form) discrete symmetry.

They have a charge conjugation Z2 and a magnetic Z2 symmetries. We will couple them to background gauge fields BC and BM. When these gauge fields are made dynamical the SO(N) gauge theory becomes the gauge theory of a different group.

We will then turn to a careful analysis of level-rank duality in topological ﬁeld theories based on Chern-Simons theories of these gauge groups. For unitary gauge groups the level-

95 rank duality relations are [42,67,96,103,104,147]

SU(N)K ←→ U(K)−N,−N , (4.0.2)

U(N)K,K±N ←→ U(K)−N,−N∓K ,

1 where U(N)K,K0 = (SU(N)K × U(1)NK0 )/ZN . For orthogonal groups they are [3, 65, 96, 103,133]

SO(N)K ←→ SO(K)−N . (4.0.3)

Most level-rank dualities in (4.0.2) and (4.0.3) are valid only when the theories involved are spin-TQFTs [3, 67].2 If the theory is a spin-TQFT as it stands, no change is needed. But if it is not, we should make it into a spin-TQFT. In the unitary theories (4.0.2) this amounts to tensoring with an almost trivial TQFT – {1, Ψ}, which can be described by

U(L)1 with any L. We can think of it as a theory with two lines, a trivial one and a complex fermion Ψ. (The fermion has to be complex and it should be charged so that the spin/charge relation is satisﬁed [122].) In the orthogonal theories (4.0.3) the analogous almost trivial theory is {1, ψ} with real ψ, which can be described by SO(L)1 with any L.

It is important to couple the TQFTs in (4.0.2),(4.0.3) to background fields. Moreover, these background fields need specific counterterms for the duality to be valid [3,22,67]. One such background field is the metric and the necessary counterterm is a gravitational Chern- Simons term. It was discussed in these papers and therefore, for simplicity, we will suppress

1Throughout most of this paper we label TQFTs by the corresponding Chern-Simons gauge group and its level. A quotient as in this expression is interpreted from the 2d RCFT as an extension of the chiral algebra [100] and from the 3d Chern-Simons theory as a quotient of the gauge group [101]. More abstractly, it can be interpreted as gauging a one-form global symmetry of the TQFT [54,78]. This quotient is referred to in the condensed matter literature as “anyon condensation” [15]. Occasionally we will meet a TQFT that is easier to describe not by the Chern-Simons gauge group, but by starting with another TQFT and performing such a quotient by a “magnetic one-form symmetry.” Such TQFTs can be described by a Chern-Simons group, but that description is more complicated. We will alert the reader whenever we use this more abstract notation. 2 In some cases a duality of spin TQFTs can be promoted to a related duality of non-spin TQFTs.

96 it in most of the discussion below. Another important background is a U(1) ﬁeld coupled to (4.0.2). A careful analysis of this ﬁeld and its counterterms led to an easy derivation of any two of the dualities in (4.0.2) by assuming the third one [67].

One of the goals of this work is to couple the orthogonal dualities (4.0.3) to background ﬁelds BC and BM and to determine the necessary counterterms to make the dualities valid. We will ﬁnd

C M C M C M M C SO(N)K [B ,B ] + (K − 1)f[B ] + (N − 1)f[B ] + f[B + B ] ←→ SO(K)−N [B ,B ] . (4.0.4)

C M Here SO(N)K [B ,B ] denotes the action for SO(N)K coupled to the two background Z2 gauge fields BC, BM.3 The coupling to BC, which is the background gauge field for the charge conjugation symmetry, means that the dynamical gauge field is actually an O(N)

C gauge field constrained to satisfy w1 = B . (See more details below.) f[B] is a specific counterterm for a Z2 gauge field B, which is related to the η invariant of a massive fermion coupled to B. Its coefficient is an integer modulo 8. We will discuss f[B] in detail below. Note that the background fields BC, BM are exchanged under the duality. This reflects the fact that the duality exchanges the global symmetries C ←→ M [3].

The result (4.0.4) makes the duality relation more complete. It also enables us to derive many other dualities by gauging the Z2 ×Z2 symmetry (or a subgroup of it); i.e. by making BC, or BM, or BC + BM, or both of them dynamical. For example, we will derive 4

0 O(N)K,K ←→ Spin(K)−N , (4.0.5)

3 There are two different conventions for Zn gauge fields. The first is where they are viewed as Zn connections with periods 0, ··· n − 1. The second is where they are viewed as constrained U(1) gauge fields with periods 0, 2π/n, ··· , 2π(n − 1)/n. In general, we use the first convention unless otherwise explicitly indicated. 4It is straightforward to use our techniques to find many additional dualities, for instance, involving P in±(N) gauge groups or O(N) theories with other levels. The resulting dual theories are more complicated and we will not discuss them here.

97 1 1 O(N)K,K−1+L ←→ O(K)−N,−N+1+L . where the additional superscript and subscript of the orthogonal groups are speciﬁc terms in the Lagrangian, which we will discuss in detail below. (The integer L ∼ L+8 is arbitrary.)5

An important special case that will be used repeatedly throughout the following is N = 1 which yields6

(Z2)K ←→ Spin(K)−1 . (4.0.7)

The Z2 level K appearing above is an integer deﬁned modulo eight. (See appendix 4.5 for a detailed discussion.) In particular, the theory (Z2)1 is the (time-reversal of) the Ising TQFT. ∼ ∼ Our conventions are such that SO(2)K = U(1)K and Spin(2)K = U(1)4K . Therefore, we ﬁnd from (4.0.3) and (4.0.5) the interesting special cases

∼ SO(2)K = U(1)K ←→ SO(K)−2 ,

∼ 0 Spin(2)K = U(1)4K ←→ O(K)−2,−2 , (4.0.8)

1 O(2)K,L ←→ O(K)−2,L−K , which simplify for K = 2 to

U(1)2 ←→ U(1)−2 ,

U(1)8 ←→ O(2)−2,−2 , (4.0.9)

5 1 1 Note as a particular consequence that the theories O(N)N,N−1 and O(N)N,N+3 are time-reversal r invariant quantum mechanically (the time-reversal symmetry ﬂips the signs of all levels r, K, L in O(N)K,L):

1 1 1 1 O(N)N,N−1 ↔ O(N)−N,−N+1 ,O(N)N,N+3 ↔ O(N)−N,−N−3 . (4.0.6)

The special case O(2)2,1 is equivalent to the T-Pfaﬃan spin-TQFT [25, 29]. See appendix 4.11 for details. 6For O(N) with N ≤ 2 the level encoded by the superscript does not exist and hence we drop it from our notation throughout [30, 143].

98 O(2)2,L ←→ O(2)−2,L−2 .

Armed with this understanding of the background fields and their counterterms we can reexamine the CSM dualities of [3, 95] between bosons and fermions in the vector representation. As in the discussion of dualities of TQFTs above, we couple them to the background fields BC and BM (which means, e.g. that the gauge fields are constrained O(N) fields).

It is important to clarify the treatment of the fermions. They are coupled to O(N) gauge fields and integrating over them leads to a phase involving the η-invariant (see below). When the fermions are given positive or negative masses this phase becomes a local functional of the gauge fields, but when the fermions are massless the phase of the functional integral is typically non-local. As is customary, we label the massless theory by the “effective level,” which is the average of the integral levels for positive and negative fermion masses. This effective level can be fractional.

Taking into account the above discussion, we ﬁx conventions such that if no bare counterterm is present in the Lagrangian, the theory of Nf vector fermions is denoted as

C M C SO(N)Nf /2[B ,B ] + (Nf /2)f[B ] with Nf ψ . (4.0.10)

Note, this does not mean that we have a term with a fractional coeﬃcient in the Lagrangian. Then we can add to this theory additional, properly-quantized, bare Chern-Simons terms for the dynamical and classical gauge ﬁelds.

Using these conventions we determine the necessary counterterms in the SO(N) boson- fermion duality:

C M C SO(N)−K+Nf /2[B ,B ] + (Nf /2)f[B ] with Nf ψ ←→ (4.0.11)

99 M C M C M C SO(K)N [B ,B ] + (N − 1)f[B ] + (K − 1)f[B ] + f[B + B ] with Nf φ .

Here the scalars have a φ4 interaction and the duality is valid only in the infrared.

In addition to making this duality more precise, we can now make the background ﬁelds dynamical; i.e. gauge them, and ﬁnd new dualities. Some examples are

0 O(N)K,K with Nf φ ←→ Spin(K)−N+Nf /2 with Nf ψ ,

Spin(N) with N φ ←→ O(K)0 with N ψ , (4.0.12) K f −N+Nf /2,−N+Nf /2 f O(N)1 with N φ ←→ O(K)1 with N ψ . K,K−1+L f −N+Nf /2,−N+Nf /2+1+L f

In the above, Nf is required to satisfy the following constraint for the duality to describe a transition point [3]: Nf ≤ N − 2 if K = 1, Nf ≤ N − 1 if K = 2, and Nf ≤ N if

K > 2. For other values of Nf below some theory-dependent number N? the dualities are conjectured to be valid near diﬀerent phase transition points with a symmetry-breaking phase in between [88] (see ﬁgure 4.1).

Some interesting special cases of these dualities are highlighted in section 5.0.3 below. ∼ For instance with K = 3 and Nf = 1 we can use Spin(3) = SU(2) to ﬁnd

0 O(N)3,3 with vector φ ←→ SU(2)−N+1/2 with adjoint ψ , (4.0.13) which was used in [62], with particular interest in N = 1 and N = 2.

Essential to our discussion of the dualities (4.0.12) is the fact that the discrete θ- parameters can be changed by integrating out massive fermions. For a single Majorana fermion λ coupled to a Z2 gauge field, the level L for the gauge field is shifted by one as we transition from negative to positive mass. This means that we can describe the massless theory as in the discussion above (4.0.10) by saying that the effective level is one-half as

100

푆푝푖푛(푁)퐾 with 푁푓 휓 in vector for 2퐾 < 푁푓 < 푁⋆

0 푁 0 푁푓 푂 ( 푓 + 퐾) + 푁 휙̂ 푂 ( − 퐾) + 푁푓 휙 2 푓 2 푁,푁 −푁,−푁

푚휓 ≪ 0 small |푚휓| 푚휓 ≫ 0

푆푝푖푛(푁) 푆푝푖푛(푁)퐾+푁 /2 퐾−푁푓/2 푆푂(푁 ) 푓 푓 + 푁Γ 푁푓 푁푓 푊푍 ↕ ↕ S(푂( +퐾)×푂( −퐾)) 2 2 0 0 푁푓 푁푓 푂 ( − 퐾) 푂 ( + 퐾) 2 푁,푁 ℤ2 strings 2 −푁,−푁

Figure 4.1: The phase diagram of Spin(N) gauge theory coupled to fermions in the vector representation. For large |mψ| the infrared is a topological field theory, which is visible semiclassically. For small |mψ| the theory spontaneously breaks the flavor symmetry and is described by a sigma model with Grassmannian target. There is a Wess-Zumino term, and stable strings, signaling confinement. The transition from the semiclassical phase to the quantum phase is weakly coupled in the dual bosonic variables (φ or φb). This diagram completes the discussion in [88] by specifying the discrete θ-parameters needed in the various gauge theories, and provides a non-trivial consistency check on that proposal. shown below.

mλ < 0 mλ = 0 mλ > 0 (4.0.14)

(Z2)L (Z2)L+1/2 (Z2)L+1

r This is the notation adopted in (4.0.12) for expressing the second subscript level of O(N)K,L, where we view the ﬁrst Stiefel-Whitney class w1 as a Z2 gauge ﬁeld.

Next we will move to an analysis of the suggested phase diagram of orthogonal gauge groups with fermions in symmetric and anti-symmetric tensor representations [62]. We add to the discussion in [62] the necessary background ﬁelds and their counterterms in each

101 phase. This leads to highly non-trivial tests of the proposal. Again a crucial role is played by the shifts in the counterterms generated by integrating out massive fermions. In addition to the shifts described by (4.0.14), we also find that the C charge of monopole operators in the SO(N) theory changes when we transition a two-index tensor fermion from negative to positive mass. This also means that in the O(N) theory, the superscript level (controlling R the discrete theta term exp(iπ X w1 ∪ w2)) jumps by one across such a transition. As in all our other cases, once these counterterms are set we can easily turn the background fields in the SO(N) phase diagram into dynamical fields and make similar predic- tions for Spin(N) and O(N) gauge theories. It is important that these are not logically independent proposals. They follow from the suggested phase diagram of SO(N) theories.7 Therefore, in addition to being interesting in their own right, their consistency also gives further evidence to the original suggestion. An example phase diagram that we find is shown in figure 4.2.

In section 4.1 we review some facts about Chern-Simons theory with Lie algebra so(N). In section 4.2 and appendix 4.8 we derive the level-rank dualities for SO Chern-Simons theories coupled to the gauge ﬁelds for the Z2 symmetries. In section 4.3 we discuss Chern- Simons matter dualities and the phase diagram of QCD with tensor fermion, and conjecture new boson-boson and fermion-fermion dualities. 7We can also add a pair of adjoint gauginos to the Chern-Simons theories to make them N = 2 supersymmetric. After taking into account the appropriate level shifts we ﬁnd the following N = 2 dualities:

SU(N)K+N ↔ U(K)−K−N,−N ,

U(N)K+N,K+N ↔ U(K)−K−N,−K−N ,

U(N)K+N,K−N ↔ U(K)−K−N,K−N ,

Sp(N)K+N+1 ↔ Sp(K)−N−K−1 , (4.0.15)

SO(N)K+N−2 ↔ SO(K)−K−N+2 , 1 Spin(N)K+N−2 ↔ O(K)−K−N+2,−K−N+1 , 0 0 O(N)K+N−2,K+N−2+L ↔ O(K)−K−N+2,−K−N+2+L .

These agree with existing results [7,8, 58, 76]. (The second subscript level of the orthogonal group O(N) was previously ignored.)

102

푁 푆푝푖푛(푁) with adjoint 휆 for 0 ≤ 퐾 < − 2 퐾 2 1 1 − 푁 − 2 2 푁 − 2 2 푂 ( − 퐾) 푂 ( + 퐾) 3푁+2퐾−2 3푁+2퐾+4 3푁−2퐾−2 3푁−2퐾+4 2 , 2 − ,− 4 4 4 4 + symmetric S + symmetric Ŝ

SUSY

푚휆 ≪ 0 푚휆 ≫ 0 푁 − 2 1 푂 ( − 퐾) 푆푝푖푛(푁) 푁−2 푁−2 푁 푆푝푖푛(푁) 푁−2 퐾− 2 +퐾, +퐾+2 퐾+ 2 2 2 2 ↕ ↕ massless Goldstino ↕ 1 푁 − 2 0 푁 − 2 푁 − 2 0 푂 ( − 퐾) 푂 ( + 퐾) 푂 ( + 퐾) 푁−2 푁 2 푁,푁 2 − +퐾,− +퐾−2 2 −푁,−푁 2 2

Figure 4.2: The phase diagram of Spin(N) gauge theory coupled to an adjoint fermion. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate transitions from the semiclassical phase to a quantum phase. Each transition is weakly coupled in a dual theory, which covers part of the phase diagram. The dual theory has symmetric tensor fermions (S and Sb). Across these transitions the superscript level of the TQFT jumps, i.e. O(N)0 and O(N)1 are exchanged. The fractional levels represent the eﬀective contributions from the massless fermions. At a special value of the mass in the quantum phase, the ultraviolet theory is supersymmetric. This symmetry is spontaneously broken leading to a massless Goldstino [148]. This phase diagram is obtained from that of the SO(N) theory proposed in [62], by tracking the global symmetries and counterterms and then gauging.

103 There are several appendices. In appendix 4.4 we summarize some facts about the representation theory of so(N). In appendix 4.5 we discuss the spin topological Z2 gauge theories and their classiﬁcation, as well as the generalization to other unitary symmetry groups. In appendix 4.6 we give more detail about the relations between P in−(N),O(N)1 and SO(N) gauge theories. In appendix 4.7 we derive the relation between the O(N) and SO(N) Chern-Simons actions for odd N. In appendix 4.9 we present explicit examples of extended and orbifold chiral algebras. In appendix 4.10 we give examples of Wilson lines in Chern-Simons theory that transform in projective representations of the global symmetry. In appendix 4.11 we discuss the relationship between O(2)2,L and Pfaﬃan theories. In appendix 4.12 we present dualities of Chern-Simons theory related to the one-form symmetry.

4.1 Chern-Simons Theories with Lie Algebra so(N)

4.1.1 Groups, Bundles, and Lagrangians

In this section we review aspects of these Chern-Simons theories. A discussion of the relevant group theory may be found in [23,84,89,98,131], while aspects of their bundles are described in [84,89].

Gauge Groups

The various global forms of the gauge group can be understood starting from SO(N), the smallest group of interest, and performing extensions by elements of order two.8 One possible extension is to include reﬂections, leading to the orthogonal group O(N). Another

8 For even N one may also consider the group SO(N)/Z2, which we do not discuss here.

104 extension allows a 2π rotation to act non-trivially thus permitting spinor representations. This leads to the group Spin(N).

The two extensions may be combined leading to the two groups P in±(N). Both groups are simply connected and contain as their identity component the subgroup Spin(N). The diﬀerence between them is whether the reﬂection r along a single axis squares to the identity or to 2π rotation s:

P in+(N): r2 = 1 , P in(N)− : r2 = s . (4.1.1)

The groups P in±(N) may thus be presented explicitly as semidirect products of Spin(N) with cyclic groups generated by r

+ ∼ − ∼ Spin(N) o Z4 P in (N) = Spin(N) o Z2 , P in (N) = . (4.1.2) Z2

A summary of these extensions is illustrated in the diagram below, where each row and column is exact.

s s Z2 Z2 (4.1.3)

± r Spin(N) / P in (N) / Z2

r SO(N) / O(N) / Z2

When N is odd, a simultaneous reﬂection along all axes commutes with SO(N) and we

∼ ± can further simplify O(N) = SO(N) × Z2. The analogous relation for the groups P in (N) is [131]   Spin(N) × Z2 N = ±1 (mod 4) P in±(N) ∼= , (4.1.4)  (Spin(N) × Z4)/Z2 N = ∓1 (mod 4)

105 where the signs are correlated in each line above. Note that in contrast to (4.1.2), the cyclic factors in the products above are generated by reﬂection along all axes. For N even, there are no analogous simpliﬁcations in the groups.

For even N, the reﬂection that extends SO(N) to O(N) may be understood as an outer automorphism of SO(N), and may be seen from the symmetry of the Dynkin diagram. We refer to this Z2 symmetry as charge conjugation C. It acts on the representations of SO(N) by exchanging the Dynkin labels associated with the two permuted nodes.9 Later, we will see that this outer automorphism gives rise to a global symmetry of the related quantum ﬁeld theories.

Bundles

Next let us discuss the possible topological types of gauge bundles that exist for the various gauge groups deﬁned above. The topology of the bundles may be parameterized in terms

i of the Stiefel-Whitney characteristic classes wi ∈ H (X, Z2), where X is the spacetime three-manifold and i = 1, 2. The allowed classes of bundles for each possible gauge group are indicated in table 4.1 below [84, 89]. Note that the group O(N) has the largest set of

Group w1 w2 SO(N) 0 unrestricted O(N) unrestricted unrestricted Spin(N) 0 0 P in+(N) unrestricted 0 − P in (N) unrestricted w1 ∪ w1

Table 4.1: Allowed bundle topology for various gauge groups. possible gauge bundles, while for other groups the topology of the bundles are restricted.

For O(N) bundles, the classes wi may be deﬁned as follows:

9In the Lie algebra so(N) these are the Dynkin labels associated with the two distinct spinor representations.

106 • w1 is a Z2 gauge ﬁeld for the subgroup of O(N) that reﬂects a single axis. It is the obstruction to restricting the structure group of the bundle to SO(N).

+ • w2 is the obstruction to lifting the structure group of the O(N) bundle to a P in (N) bundle.

∼ When N is odd, the product structure O(N) = SO(N) × Z2 implies that every O(N) bundle deﬁnes an SO(N) bundle. The Stiefel-Whitney class w2(SO(N)) is the obstruction to lifting this SO(N) bundle to a Spin(N) bundle.10 Using the formula (4.1.4) one may determine that

  w2(SO(N)) N = +1 (mod 4) , w2(O(N)) = (4.1.5)  w2(SO(N)) + w1 ∪ w1 N = −1 (mod 4) , where we have used the fact that w1 ∪w1 is the obstruction to lifting w1 to a Z4 cohomology class.11 When N is even there is no analogous formula since in that case an O(N) bundle does not in general deﬁne an SO(N) bundle. For uniformity throughout the following we use the Stiefel-Whitney classes of O(N) bundles unless otherwise explicitly indicated.

Lagrangians

We now turn to a discussion of the possible Lagrangians for Chern-Simons theories based on the gauge groups discussed above.

In each possible theory there is a continuous ﬁeld variable A that is a connection valued in the Lie algebra so(N). We may include in the action a Chern-Simons term CS(A) with

10 Here and below we denote by wi(G) is the i-th Stiefel-Whitney class of a principle bundle with structure group G. 11A simple way to see the necessity of this condition is to note that for any degree one cohomology classes x, y we have x ∪ y = −y ∪ x. Thus, if x = y this vanishes unless the coeﬃcients are Z2. A more detailed discussion may be found in [66, 127].

107 a level K K Z 2 K · CS(A) = Tr A ∧ dA + A ∧ A ∧ A , (4.1.6) 8π X 3 where in the above the trace is deﬁned in the vector representation. Our normalization is such that the level must be quantized in integral units, K ∈ Z. If the level K is odd, then the action (4.1.6) depends on a choice of spin structure on X. If K is even, no spin structure is required. For any gauge group G, we denote the Chern-Simons theory with level K by GK .

In Chern-Simons gauge theory we perform a path integral over the connection A and the Chern-Simons action exp(iKCS(A)) tells us the weight assigned to each ﬁeld conﬁguration. In addition we must also sum over the possible topological types of bundles allowed for the gauge groups, and there are choices for how to weight the various bundles in the sum. The additional coupling constants that parameterize the possible weights are sometimes referred to as discrete θ-parameters. As we describe below, they may also be thought of as Chern-Simons interactions for discrete groups.

It is clearest to present the discussion starting from the gauge group O(N) which admits bundles with all possible Stiefel-Whitney classes. There are two distinct discrete θ-parameters to specify.

R • There is a possible weight exp(iπ X w1 ∪ w2). Since the characteristic classes are Z2- 12 valued this evaluates to ±1 for each possible bundle. We may introduce a Z2-valued level p that speciﬁes whether this interaction is present (p = 1) or absent (p = 0).13 We indicate its value in a superscript O(N)p.

• Another possible coupling depends only on the characteristic class w1. Viewing w1

as a Z2 gauge ﬁeld, the possible additional couplings arise from local eﬀective actions 12In the special case of O(2) this coupling is trivial [143]. We therefore omit the superscript in this case. 13 0 1 In [8] our O(N) was denoted by O(N)+ and our O(N) was called O(N)−.

108 depending on this gauge ﬁeld. Since we are interested in spin TQFTs we permit these actions to depend on the spin structure of the underlying three-manifold X.

Such local actions have been completely determined [63, 81, 82, 135], and admit a Z8 classiﬁcation.

We indicate the minimal local action by f[B] where B is any Z2 gauge ﬁeld. The properties of these actions are discussed in detail in appendix 4.5.14 The relationship

of these Z2 gauge theories to the fermion path integrals is crucial in our discussion of Chern-Simons matter duality in section 4.3.

In general, we indicate the additional level by a subscript valued mod 8 (e.g. (Z2)L).

For Chern-Simons theory with gauge group G we indicate this Z8-valued level by a second subscript.

In summary, our complete list for possible quantum O(N) Chern-Simons theories and their coupling constants is

p O(N)K,L ,K ∈ Z ,L ∈ Z8 , p ∈ Z2 . (4.1.7)

This classiﬁcation of discrete levels may be repeated for any gauge group G, and in particular the Z8-valued level exists for any group that admits non-vanishing ﬁrst Stiefel-Whitney class

± w1. Thus for instance, the groups P in (N) may be used to deﬁne Chern-Simons:

± P in (N)K,L ,K ∈ Z ,L ∈ Z8 . (4.1.8)

One of our main results will be explicit level-rank dualities for these gauge theories, including the required maps on the discrete θ-parameters.

14 R The special case 4f[B] has an elementary action π X B ∪ B ∪ B, and represents a bosonic Dijkgraaf- 3 Witten theory [41] for Z2 gauge group classiﬁed by H (BZ2,U(1)) = Z2.

109 It is important to stress that the gauge theories we describe here are in general spin topological ﬁeld theories. That is, they depend explicitly on a choice of spin structure S on the three-manifold X. In particular, all such theories contain a transparent line ψ of spin 1/2.

p When K is even and L = 0 (mod 4) the theories O(N)K,L may in addition be deﬁned without the choice of spin structure. However, both the level-rank dualities and Chern- Simons matter dualities of interest to us only make sense as spin theories. For instance, the latter involve dynamical fermions where the spin structure is required. We can promote

p O(N)K,L at bosonic values of the levels to a spin theory by tensoring with {1, ψ}: an almost trivial theory containing two transparent lines, the identity and the spinor. In particular, the existence of the line ψ restores the dependence on spin structure. In the following,

p we use the notation O(N)K,L to denote the spin version of the theory (i.e. tensoring with {1, ψ} implicit) unless explicitly indicated. We adopt the same convention for other global forms of the gauge group.

4.1.2 Ordinary Global Symmetries and Counterterms

Let us describe the global symmetries of these gauge theories. There may be ordinary (zero-from) global symmetries, as well as one-form global symmetries and we describe them in turn.

The ordinary global symmetries may be understood by starting with the smallest group

SO(N), which has the largest zero-form symmetry. The symmetries of SO(N)K depend on the parity of N and K.

If N is even, the Lie algebra so(N) has an outer automorphism C discussed in section

4.1.1. This deﬁnes a charge conjugation symmetry C of SO(N)K , which acts non-trivially on

110 the lines in the topological ﬁeld theory.15 Equivalently, one may view this as acting directly on the SO(N) connection. Expanding in a standard Lie algebra basis of antisymmetric matrices the transformation is

  [ij]  A i, j 6= 1 , C(A[ij]) = (4.1.9)  [ij] −A i = 1 , or j = 1 .

If N is odd, the transformation (4.1.9) can be achieved by conjugation by an element of SO(N) and hence does not permute representations. Therefore, for odd N charge conjugation is not a true symmetry of the topological field theory SO(N)K . Nevertheless, when N is odd we still find it useful to discuss a symmetry C which we will identify with (−1)F where F is the fermion number. As we will see, this identification is natural from the point of view of level-rank duality.

Similarly, for K even we may deﬁne a magnetic global symmetry M of SO(N)K that permutes the lines. Like charge conjugation, the magnetic symmetry is a Z2 global symmetry. To measure the magnetic charge, one integrates the second Stiefel-Whitney class w2 of the gauge bundle over a two-cycle.

To understand how M acts on lines in the TQFT, it is convenient to view the theory

SO(N)K as Spin(N)K /Z2 where the quotient means that we gauge a Z2 one-form global symmetry [54, 78]. The generating line of this one-form global symmetry gives rise in

SO(N)K to a transparent line transforming in the K-th symmetric traceless power of the vector representation (i.e. Dynkin indices (K, 0, ··· , 0)), which has spin K/2. In the case of K even this transparent line is a boson, and there are pairs of line defects that are related by fusion with the transparent line. Those lines of Spin(N)K that are ﬁxed points under fusion with the one-form symmetry generator are doubled in the spectrum of SO(N)K .

15See Appendix 4.4 for a discussion of the representation theory of so(N).

111 The two lines in a given doublet are exchanged under the action of the symmetry M [101]. An equivalent point of view on the magnetic symmetry M via chiral algebras is discussed in section 4.1.5 below.

When K is odd one may similarly deﬁne a conserved charge M measured by w2. How- ever the associated symmetry acts trivially on lines since pairs related by fusion with the transparent line may be distinguished by their spin. The meaning of this magnetic symmetry is the following. For odd level the theory SO(N)K depends on the spin structure S. We can probe this dependence by shifting S by a Z2 gauge ﬁeld B. Under this transformation the minimally quantized Chern-Simons action responds as [72]

Z CS(A, S + B) = CS(A, S) + π w2 ∪ B. (4.1.10) X

Thus, for odd level, B may be interpreted as a background gauge ﬁeld for M. More physically, at odd level the transparent line is a spinor and the magnetic charge is identiﬁed with fermion number mod 2.

We summarize the identiﬁcations between the charge conjugation and magnetic symmetry in table 4.2.

N = 0 (mod 2) N = 1 (mod 2) K = 0 (mod 2) C, M, (−1)F M, C = (−1)F K = 1 (mod 2) C, M = (−1)F C = M = (−1)F

F Table 4.2: Identiﬁcations between the zero-form global symmetries C, M, (−1) of SO(N)K .

Depending on N and K there may be additional ordinary global symmetries. For instance, if N and K are even and NK is an odd multiple of 8 there is a Z2 quantum zero- form symmetry that permutes the anyons [3] deﬁned by fusing the anyons charged under

112 the Z2 one-form symmetry with the generator of the one-form symmetry. In the following we focus on the symmetries C and M.

In the theory SO(N)K with even N and K the symmetries C and M both square to one

16 and commute, and hence the zero-form symmetry is Z2 × Z2. We may couple our system

C M to background gauge fields B and B for these Z2 symmetries. Additionally we may add to the action local counterterms for these gauge fields. The possible local counterterms have been determined and admit a Z8 × Z8 × Z4 classification [135]. Each Z8 factor is the

C M level for the individual gauge ﬁelds B and B (i.e. if it is gauged one obtains (Z2)L), while the Z4 controls a counterterm that depends on both backgrounds. We express the counterterm action as

C M C M Scounterterm = xf[B ] + yf[B ] + zf[B + B ] , (4.1.11) where the parameters enjoy the identiﬁcation

x ∼ x + 8 , y ∼ y + 8 , (x, y, z) ∼ (x + 4, y + 4, z + 4) . (4.1.12)

More details on these counterterns are presented in appendix 4.5.

Starting from the case of SO(N)K we can gauge any of the ordinary global symmetries with specified values of the counterterms. In this way we can produce all the gauge groups discussed in section 4.1.1 together with specified discrete θ-parameters, as well as other models. The zero-form symmetries of the resulting theories are the quotient of Z2 × Z2 by the subgroup that is made dynamical. We illustrate this in figure 4.3.

As an illustrative example of this process, consider starting with SO(N)K and gauging

R M M the magnetic symmetry M. This inserts in the action a term π X w2 ∪ B , with B now M a dynamical Z2 gauge ﬁeld. The resulting sum over B constrains w2 to vanish and implies 16Interestingly, the symmetries are represented projectively on some lines. See Appendix 4.10 for details.

113 Spin(N) 9 M C

C&M+(x,y,z)=(−2,−2, 2) CM M & P in−(N) o SO(N) / O(N)1 / P in+(N) 8

C M % O(N)0

Figure 4.3: A map of possible gauge groups obtained by starting with SO(N) and gauging symmetries (Z2 levels suppressed). In Appendix 4.6 we discuss some details of this map. that the gauge group is Spin(N). If we include the counterterm Lf[BM] then we construct the theory Spin(N)K,L deﬁned in (4.3.34). This corresponds to the choice y + z = L in (4.1.11).

Analogously, starting from SO(N)K and gauging either C or the product CM with ﬁxed

0 1 counterterms produces O(N)K,L and O(N)K,L respectively.

In the special case where N and K are both odd the identiﬁcation of global symmetries discussed around table 4.2 implies that M=C and hence gauging them should produce equivalent ﬁeld theories. To make this precise, recall that for odd N we have O(N) ∼=

SO(N) × Z2. The relationship between the O(N) Chern-Simons action (with our choice of charge conjugation (4.1.9)) and that of the factorized SO(N) × Z2 variables is derived in appendix 4.7

Z CS(O(N)) = CS(SO(N)) + π w2(SO(N)) ∪ w1 + (N − 1)f[w1] . (4.1.13) X

Therefore, for N and K both odd, we have the following equivalence of spin Chern-Simons theories17

0 O(N)K,K−NK ←→ Spin(N)K . (4.1.14)

17 The equivalence only holds as spin TQFTs. In particular for Spin(N)K we must promote it to a spin theory by tensoring with {1, ψ}.

114 No analogous relationship holds for even N or K.

4.1.3 One-form Global Symmetries

Now let us turn to the one-form global symmetry of these models. A starting point to understanding these symmetries is via the center of the gauge groups shown in table 4.3 [23,89,98,131].

Group N = 0 (mod 4) N = 1 (mod 4) N = 2 (mod 4) N = 3 (mod 4)

SO(N) Z2 1 Z2 1

O(N) Z2 Z2 Z2 Z2

Spin(N) Z2 × Z2 Z2 Z4 Z2 + P in (N) Z2 Z2 × Z2 Z2 Z4 − P in (N) Z2 Z4 Z2 Z2 × Z2

Table 4.3: Centers of the Gauge Groups (N > 2).

In general, each element of the center of the gauge group may give rise to a one-form global symmetry. For each Wilson line in a representation R of the gauge group, the charge of the line is the phase deﬁned by the action of the associated element of the center in the representation R. In order for this deﬁnition to truly give rise to a global symmetry, it is necessary that there exists a line in the spectrum of the theory that can measure this phase by linking. This is the case in all examples in table 4.3 except for SO(N)K and O(N)K . The line generating the one-form symmetry associated to the center of the gauge group has Dynkin indices (0, ··· , 0,K). If K is odd this representation is a spinorial and excluded from the spectrum. If K is even it generates a Z2 global symmetry.

For SO(N)K , and O(N)K the center symmetry at even level exhausts the one-form global symmetry. However for other gauge groups there may exist additional quantum

115 one-form symmetries that are not manifest at the level of the classical action. Let us ﬁrst consider the case of vanishing Z2 discrete θ-parameter and subsequently generalize. A practical way to understand the additional symmetries is as bonus symmetries that arise from gauging ordinary global symmetries.

In general, if we begin with any three-dimensional quantum ﬁeld theory and gauge a

Zr global symmetry, the resulting theory has an emergent Zr one-form global symmetry whose generating line is the Wilson line of the Zr gauge theory. The inverse process also exists. Namely, gauging the emergent Zr one-form global symmetry in the resulting theory produces an emergent zero-form global symmetry generated by the Wilson surface of the

18 Zr two-form gauge theory, and the result is the original theory.

We can apply this logic to the gauge groups appearing in ﬁgure 4.3, starting from the one-form symmetry of SO(N)K . For instance if N and K are even we deduce that

0 1 ± O(N)K,0,O(N)K,0 and P in (N)K,0 must have quantum Z2 one-form global symmetries.

The exact form of the resulting one-form global symmetry group is either Z2 × Z2 or Z4 depending on the level. Some useful explicit examples are (N even):

   2 × 2 K = 0 (mod 4) 0 Z Z one-form symmetry O(N)K,0 = (4.1.15)  Z4 K = 2 (mod 4) ,    4 N + K = 0 (mod 4) 1 Z one-form symmetry O(N)K,0 =  Z2 × Z2 N + K = 2 (mod 4) .

We can also easily add the Z2 discrete θ-parameter to the discussion. Instead of viewing the θ-parameter as an added Lf(w1) in the Lagrangian, as we did above, we use (4.1.15).

The theory (Z2)L has one-form global symmetry Z2 if L is odd, Z2 × Z2 if L = 0 mod 4,

18 We can also gauge the Zr zero-form symmetry with different counterterms for the Zr gauge field to produce different theories, and each of them has the dual Zr one-form symmetry. Gauging this one-form symmetry takes us back to the original theory.

116 and Z4 if L = 2 mod 4. Then if G is any gauge group discussed in section 4.1.1, we can express the models with discrete θ-parameter as

GK,0 × (Z2)L GK,L = . (4.1.16) Z2

In this expression we deviate from our standard notation of labeling the TQFT by the gauge group of the Chern-Simons theory. Instead, here the quotient means that we gauge a

Z2 one-form global symmetry of the TQFT in the numerator. This Z2 one-form symmetry is generated by a product of a Wilson line in GK,0 and the electric Wilson line in the (Z2)L factor. The remaining one-form symmetry after gauging is spanned by the subset of Abelian anyons in GK,0 × (Z2)L that are uncharged under the gauged Z2, with the identiﬁcation of

19 lines that diﬀer by fusion with the Z2 generator.

In any theory with one-form global symmetry we may activate background two-form gauge fields. For instance, consider any gauge group that admits bundles with non-trivial w1. If we insert a line defect, then the integral of w1 around a cycle linking the line measures a one-form charge. We may refine our observables by coupling this global symmetry to R background fields by adding to the action π X w1 ∪ B2 where B2 is a Z2 two-form gauge field.

19 In the case where the generating line of Z2 has half-integral spin, we do not identify lines that differ by fusion with the one-form symmetry generator since they have different spin and can be distinguished. However, the difference between any two such lines is the transparent spin one-half line. Since the transparent line is local with respect to all lines in the theory, it does not generate a non-trivial one-form global symmetry and thus does not affect the resulting one-form symmetry in the quotient. Further discussion of the spectrum of lines after gauging a one-form symmetry is presented in section 4.1.5.

117 4.1.4 ’t Hooft Anomalies of the Global Symmetries

The global symmetries described in the previous sections participate in ’t Hooft anomalies.

We present the anomalies of SO(N)K with N and K even; those of other models may be deduced by gauging.

As described in section 4.1.2, there is a Z2 × Z2 global symmetry generated by M and C. These zero-form symmetries do not have any anomalies amongst themselves. Indeed, they may clearly be gauged.

Next consider the possible anomalies involving only one-form symmetries. The generating line of the Z2 one-form global symmetry has spin NK/16. Since we are considering spin theories, this symmetry may be gauged if the spin is half-integral. Thus in general there is an anomaly, which may be represented by a classical action on a four-manifold Y with boundary X. This action is written in terms of the background two-form gauge ﬁeld

B2 as [78] πNK Z P(B ) 2 , (4.1.17) 4 Y 2

2 4 where P : H (Y, Z2) → H (Y, Z4) is the Pontryagin square, and we have used the fact that on a spin four-manifold Y , P(B2) is divisible by two.

Finally, let us describe the mixed anomalies between the zero-form and one-form global symmetries. These are encoded in the following four-dimensional action:

Z K C C N M M C M π B2 ∪ B ∪ B + B ∪ B + B ∪ B . (4.1.18) Y 2 2

To derive the formula (4.1.18), we must consider the one-form symmetry after gauging any of the ordinary global symmetries. For instance, consider gauging M with trivial

C background B2 and B . Since the anomaly action (4.1.18) vanishes, there is no obstruction to this process. Now suppose that we turn on a background two-form B2. In this case

118 (4.1.18) is generally non-zero if N = 2 (mod 4). This means that the allowed class of background B2 is restricted.

j j+1 To deduce the allowed backgrounds, let β : H (Y, Z2) → H (Y, Z2) be the Bockstein homomorphism. Then using the fact that β obeys a Leibniz rule we can reexpress the

M anomaly involving B2 and B as

Z Z Z πN M M πN M πN M B2 ∪ B ∪ B = B2 ∪ β(B ) = β(B2) ∪ B . (4.1.19) 2 Y 2 Y 2 Y

Thus the allowed class of two-form backgrounds are those where β(B2) vanishes. The

Bockstein map is the obstruction to lifting the class B2 from a Z2 gauge field to a Z4 gauge field. Therefore, we deduce that after gauging BM our topological field theory can naturally couple to a Z4 two-form gauge field.

We will now show that this Z4 is the one-form global symmetry after gauging and that the Z4 two-form gauge ﬁeld is a background ﬁeld for this symmetry. When we promote

M B to a dynamical ﬁeld there is an emergent Z2 one-form global symmetry. This emergent

0 symmetry couples to a background ﬁeld B2 by modifying the action to include a term

R 0 M 0 π X B2 ∪ B . Let us consider the behavior of the action, including the coupling to B2, under gauge transformation of BM. Since BM is now a dynamical ﬁeld the total action must be invariant under any such transformation, i.e. the anomalous transformation of the action under BM gauge transformations must cancel. This is achieved by imposing the following relation (the symbol δ is the coboundary in cohomology)

0 δB2 = β(B2) . (4.1.20)

0 Note that in theories without a mixed anomaly (4.1.19), we require that B2 is closed. The modiﬁcation (4.1.20) cancels the anomalous variation of the action and ensures that

119 BM can be consistently gauged. Observe also that the left-hand-side of (4.1.20) is trivial in cohomology ensuring that B2 may be extended to a class with Z4 coeﬃcients. Let Be2 be any

0 Z4 cochain extending B2. We construct a Z4 cocycle by the combination Be2 + 2B2, which

0 is closed by (4.1.20). In particular, the Z2 emergent one-form symmetry that couples to B2 is the Z2 subgroup of this Z4. Therefore we conclude that the one-form global symmetry

20 after gauging is extended to Z4.

This matches with the expected one-form global symmetry. Indeed when the anomaly is non-trivial i.e. N = 2 (mod 4), the one-form symmetry of Spin(N)K is Z4 (see table 4.3).

Similarly, one can deduce the other terms in (4.1.18) by gauging either C or CM and

0 1 using the one-form symmetries of O(N)K,0,O(N)K,0 given in (4.1.15). This interplay between mixed ‘t Hooft anomalies and extensions of the global symmetry is analogous to the discussion in [53] involving CP symmetry and the one-form symmetry.

4.1.5 Chiral Algebras

The properties of the Chern-Simons theories described in previous sections are also encoded in their associated chiral algebras. We discuss examples in section 4.2.3 and appendix 4.9.

The most familiar case is that of the simply connected group associated to the algebra so(N), i.e. Spin(N)K . In this case the associated chiral algebra is the (enveloping algebra of the) Kac-Moody algebra of currents.

To obtain the chiral algebra of SO(N)K from that of Spin(N)K we must extend the chiral algebra [100]. This is the chiral algebra description of gauging a one-form global symmetry in the associated Chern-Simons theory [101]. Let a denote the generator of the one-form global symmetry, and assume that a has integer spin. In the Chern-Simons theory gauging a has the following eﬀects on the spectrum of lines [15,100,101]:

20A similar analysis shows that if we gauge the one-form symmetry, the mixed anomaly (4.1.19) forces the magnetic symmetry of SO(N)/Z2 to be extended to Z4.

120 • We exclude from the spectrum all lines that carry non-trivial charge under a.

• We identify lines b and a · b that diﬀer by fusion with a.

• Lines b that are stabilized under fusion (i.e. b = a · b) are taken as several distinct lines in the spectrum of the gauged theory.

The associated operation on chiral algebras is extension. The chiral algebra is enlarged to include the current associated to the line a, forcing all allowed modules to be local with respect to these additional currents. The modules of the extended chiral algebra are thus enlarged to include the action of modes of a. Finally, those modules of the original theory, which are stabilized under the action of the a modes may now be assigned a phase under the a action and hence give rise to several distinct modules of the extended chiral algebra.

If a has half-integral spin, we still exclude all lines that carry non-trivial charge under a, but we do not identify b with a · b since their spins diﬀer by a half integer, and hence they are always distinct.

We may apply this general procedure to obtain the chiral algebra of SO(N)K . The extending representation is the transparent line of spin K/2 in the K-th symmetric traceless power of the vector representation (Dynkin labels (K, 0, ··· , 0)). The magnetic symmetry M acts to exchange the lines that are doubled under this extension.

To obtain the chiral algebras for other gauge groups we can apply the chiral algebra analogue of gauging a zero-form global symmetry. This is an orbifold [101]. The zero-form symmetries C and M act on the SO(N)K chiral algebra as automorphisms. In particular, for even K the magnetic symmetry M permutes the two distinct modules in each pair b± satisfying b · a = b. By suitably introducing twisted sectors [40] for either C or CM we

0 1 0 obtain the chiral algebra of O(N)K,0 and O(N)K,0. The chiral algebras of O(N)K,L and

1 O(N)K,L can then be constructed as in (4.1.16).

121 4.2 Level-Rank Duality

In this section we derive level-rank dualities for Chern-Simons theories with gauge group based on the Lie algebra so(N).

One way to phrase our result is to consider SO(N)K coupled to background Z2×Z2 gauge ﬁelds BC and BM for the global symmetry C and M. The level-rank duality then states how the correlation functions in the presence of background ﬁelds are related including the

C M required map on counterterms for these gauge fields. Let us denote by SO(N)K [B ,B ], the SO(N)K topological field theory coupled to background fields. Then, our explicit duality is

C M M C C M C M SO(N)K [B ,B ] ←→ SO(K)−N [B ,B ]−(K−1)f[B ]−(N−1)f[B ]−f[B +B ] . (4.2.1) A special case of this result, with K = N = 2 so that both theories are Abelian, was discussed in [122].

Note crucially that under level-rank duality the symmetries C and M (and hence also their background ﬁelds) are exchanged [3]. We explain this feature in more detail in section 4.2.1 below.

Starting from this result we may add any desired counterterms and gauge the global symmetries to obtain a host of other level-rank dualities. Several examples are

0 O(N)K,K ←→ Spin(K)−N , (4.2.2)

1 1 O(N)K,K−1+L ←→ O(K)−N,−N+1+L .

Note that these are dualities among spin TQFTs, thus keeping with our earlier convention, we add the transparent spin 1/2 line ψ to the spectrum if required. Without this addition the dualities are in general false. Additionally, we observe that the second duality in (4.2.2)

122 holds for arbitrary L. This is possible since the symmetry CM is mapped to itself under level-rank duality.

We claim that the dualities (4.2.1) and (4.2.2) hold for all N and K. In order to make sense of this for either N or K odd, where some of the global symmetries do not act on

SO(N)K we use our general discussion in section 4.1.2 to deﬁne C and M. This implicitly

0 1 deﬁnes our conventions for O(N)K,L and O(N)K,L for odd N (where the group is a product

O(N) = SO(N) × Z2) by starting from SO(N)K with counterterms and gauging.

In the remainder of this section we prove the dualities (4.2.2). This also establishes the level-rank duality (4.2.1) of SO(N)K coupled to background ﬁelds.

4.2.1 Conformal Embeddings and Non-Spin Dualities

Our strategy for deriving the result (4.2.2) will be to bootstrap our way up starting from the level-rank dualities that arise for non-spin theories, and then subsequently generalize to the spin theories of interest. Thus in this section, unlike others, we discuss non-spin TQFTs and their associated chiral algebras (i.e. we do not tensor with the {1, ψ} sector).

The basic tool is the conformal embedding of non-spin chiral algebras. Consider NK

α real two-dimensional fermions ζa where α = 1, ··· ,N and a = 1, ··· K. These generate the current algebra Spin(NK)1. By forming singlets out of the a or α indices, we can generate various currents. The ﬁrst are

[αβ] ab α β α β J = δ ζa ζb ,J[ab] = δαβζa ζb , (4.2.3)

which generate Kac-Moody algebras for Spin(N)K , and Spin(K)N respectively. Other objects of interest are:

Λ(α1α2···αK ) = εa1a2···aK ζα1 ζα2 ··· ζαK , Λ = ε ζα1 ζα2 ··· ζαN . (4.2.4) a1 a2 aK (a1a2···aN ) α1α2···αN a1 a2 aN

123 If K is even, Λ(α1α2···αK ) has integer dimension and can be used to extend the chiral algebra

from Spin(N)K to SO(N)K . Similarly if N is even Λ(a1a2···aN ) can be used to extend the chiral algebra to SO(K)N . (See section 4.1.5.)

Thus we deduce the following conformal embeddings of chiral algebras (see also [65,133]):

SO(N)K × SO(K)N ⊂ Spin(NK)1 N even,K even ,

Spin(N)K × SO(K)N ⊂ Spin(NK)1 N even,K odd ,

Spin(N)K × Spin(K)N ⊂ Spin(NK)1 N odd,K odd , (4.2.5)

Note that in the above embeddings, the subalgebras on the left are not in general maximal, i.e. depending on N and K, there may be strictly larger chiral algebras that embed inside

Spin(NK)1. In particular, the centers on the left and right of (4.2.5) need not agree.

A related point is that in general there are modules of the subalgebras that do not occur in the decomposition of any module of Spin(NK)1. One might attempt to remedy this by quotienting the groups on the left by elements of their center, but in general there is no quotient that imposes the required selection rules.21 A consequence of this is that in general there is no map between the simple currents on the left and right of the conformal embeddings (4.2.5). Indeed, since there are more modules of the subalgebra, the simple currents of the subalgebra need not act faithfully on the subset of modules that arise via decomposition of modules of the larger algebra. Similarly, the simple currents of the larger algebra may fail to be local with respect to modules of the subalgebra that do not arise from the decomposition.22

21 ∼ For instance, consider a primary of spins (1, 1, 1, 1) for the group Spin(4)4 × Spin(4)4 = SU(2)4 × SU(2)4 × SU(2)4 × SU(2)4 ⊂ Spin(16)1. There is no quotient that can remove this representation, and it never occurs in a module of Spin(16)1. 22This corrects some misleading comments about the centers of conformal embeddings in [3].

124 Although the centers in the conformal embeddings (4.2.5) are not in general matched, it is crucial for the derivation of level-rank duality that each factor of the subalgebra acts faithfully. Thus, every module of one of the factors occurs in some decomposition of a module of Spin(NK)1.

For our application, we aim to gauge some of the zero-form symmetries in (4.2.5). Thus, we must understand their action in more detail. For concreteness, we focus on the case N and K both even where C and M act as true symmetries of the topological ﬁeld theory that permute anyons. The required generalizations of level-rank duality for odd N or K are discussed in Appendix 4.8.

Let us explain why C and M are exchanged via level-rank duality. We will do this by direct inspection of the representations of the chiral algebras.

The representations of the group SO(N) are labelled by Young tableau, Y, built from the vector representation. Such a tableau is speciﬁed by a tuple (l1, ··· , lN/2) of row lengths

(see Appendix 4.4 for conventions). The li for i < N/2 are non-negative integers. The ﬁnal

23 element lN/2 is an integer of either sign accounting for the two chiralities of spinors.

When we discuss modules of the chiral algebra SO(N)K we must take into account the fact that the chiral algebra of SO(N)K is larger than that of Spin(N)K (see section 4.1.5). The extending representation is a current in the K-th symmetric power of the vector representation, which is described by a tableau with (l1, l2, ··· , lN/2) = (K, 0, ··· , 0).

This means that as modules of SO(N)K there is the following spectral ﬂow identiﬁcation [100,101]:

(l1, l2, ··· , lN/2−1, lN/2) ∼ (K − l1, l2, ··· , lN/2−1, −lN/2) . (4.2.6)

Using this identiﬁcation we may restrict to those primaries with l1 ≤ K/2.

23 When drawing such a tableau one typically takes the ﬁnal row length to be lN/2 and adds an extra sign label.

125 We can also explicitly see the primaries of representations that are doubled due to the extension. A representation is fixed under spectral flow if l1 = K/2 and lN/2 = 0. Each such representation gives rise to two modules of the chiral algebra SO(N)K . They are permuted by the magnetic symmetry M. The charge conjugation symmetry C is also visible, it simply acts on the final label by lN/2 → −lN/2.

Next let us inspect the modules of the chiral algebra Spin(NK)1. Since NK is even, there are four distinct representations: 1, ζ, σ±. We ask how these modules decompose under the embedded chiral algebra SO(N)K × SO(K)N . The current generating Spin(NK)1 is a general bilinear in the fermions ζ. As a representation of SO(N) × SO(K) it decomposes as:24

α β a α a α J = ζ ζ = α β , ⊕ , a b ⊕ 1 , ⊕ , 1 . (4.2.7) Spin(NK)1 a b b β b β

The last two terms in the direct sum above are simply the currents of the embedded chiral algebra identiﬁed in (4.2.3). The ﬁrst two terms are new primaries that appear in the decomposition of the identity under the subalgebra SO(N)K × SO(K)N . Notice that these primaries are labelled by pairs of tableaux each with an even number of boxes that are related by transposition. This remains true for all other primaries obtained from this decomposition. Indeed, the current algebra primaries of each of the subalgebra factors do

α not involve derivatives of the fermions ζa and hence are subject to Fermi statistics. This implies that whenever two upper indices α, β are symmetrized, the corresponding lower indices a, b must be anti-symmetrized and vice versa.

24Here means the symmetric tensor with the trace removed.

126 Based on this logic, we can readily deduce properties of decompositions of modules of

Spin(NK)1 under SO(N)K × SO(K)N . We have

M T 1Spin(NK)1 −→ (Yeven, Yeven) , (4.2.8)

M T ζSpin(NK)1 −→ (Yodd, Yodd) , where the subscript on the tableau indicates the parity of the number of boxes. Similarly, we can also deduce properties of the decomposition of the modules σ+ and σ− = σ+ · ζ. In fact, at least one of them contains the simple current of SO(N)K or SO(K)N [65,133], and thus the decompositions can be obtained by fusion with the right-hand-sides of (4.2.8).

The fact that Young tableaux are paired with their transpose in the decomposition of modules of Spin(NK)1 clariﬁes the map of global symmetries C ↔ M under level-rank duality. The basic point is simply that if Y is any tableau which is acted on non-trivially

T by C then M acts non-trivially on Y . Indeed, C acts on a tableau if l1 < K/2 and lN/2 6= 0.

T T When this is so, the transposed Young tableau has l1 = N/2 and lK/2 = 0. In particular such a representation is stabilized by the spectral ﬂow (4.2.6) and hence is acted on by M.

We are now equipped to discuss gauging the ordinary global symmetries in the chiral algebra embeddings. Consider the symmetry that acts with C on the ﬁrst factor and M on the second factor in (4.2.5). It is achieved by an inner automorphism of Spin(NK)1,

α which acts on the fermions ζa by giving some of them periodic boundary conditions on the cylinder (compare to the deﬁnition (4.1.9))25

1 α>1 periodic: ζa , antiperiodic: ζa . (4.2.9) 25Recall that when we study the RCFT on a cylinder the representations (4.2.8) arise in the NS sector α where the fermions ζa are antiperiodic.

127 Since we have mapped the symmetries between the two sides of the conformal embedding we can gauge to produce new embeddings. Gauging the symmetry (C, M) is equivalent to an inner-automorphism orbifold of Spin(NK)1 with the twists (4.2.9). This leads to the conformal embedding:

0 O(N)K,0 × Spin(K)N ⊂ Spin(K)1 × Spin(NK − K)1 . (4.2.10)

Similarly we can gauge the symmetry (CM, CM). The related inner automorphism of

Spin(NK)1 now acts on the fermions as

1 α>1 α>1 1 periodic: ζa>1 , ζ1 , antiperiodic: ζa>1 , ζ1 . (4.2.11)

Therefore we deduce the conformal embedding26

1 1 O(N)K,0 × O(K)N,0 ⊂ Spin(N + K − 2)1 × Spin(NK − K − N + 2)1 . (4.2.12)

The embeddings (4.2.10)-(4.2.12) allow us to obtain dualities among non-spin Chern- Simons theories. First we have the following equivalences of chiral algebras27

0 ∼ Spin(K)1 × Spin(NK − K)1 O(N)K,0 = , (4.2.13) Spin(K)N 1 ∼ Spin(N + K − 2)1 × Spin(NK − N − K − 2)1 O(N)K,0 = 1 , O(K)N,0 where on the right-hand side above the chiral algebras are GKO cosets. Here we make use of the fact mentioned above that in the conformal embeddings (4.2.5) each factor of the subalgebra acts faithfully. Thus every module of the left-hand side of (4.2.13) is an allowed

26 0 1 1 The special case K = 2 or N = 2 coincides with (4.2.10) since O(2)K ↔ O(2)K , O(N)2 ↔ Spin(N)2. This uses the fact that M and C are not outer automorphisms of SO(2)K and SO(N)2 respectively. 27The case N = 2 of the ﬁrst equation was studied in [56].

128 representation of the coset. In particular this means that the simple currents must match between the left and right of (4.2.13), since if they did not, some modules would necessarily be forbidden.

From (4.2.13) we obtain the Chern-Simons dualities

Spin(K) × Spin(NK − K) × Spin(K) O(N)0 ←→ −N 1 1 , (4.2.14) K,0 Z 1 O(K) × Spin(N + K − 2)1 × Spin(NK − N − K − 2)1 O(N)1 ←→ −N,0 , K,0 Z0 where the common one-form symmetries Z and Z0 are

    Z2 × Z2 K = 0 (mod 4) , Z4 N + K = 0 (mod 4) , Z = Z0 =   Z4 K = 2 (mod 4) , Z2 × Z2 N + K = 2 (mod 4) . (4.2.15)

In these equations the Z2 × Z2 quotients use the two generators of the one-form symmetry

χ and j from each factor in the numerator of (4.2.14). In the Z4 case the quotient uses the generator j (with χ = j2) from each factor.28 The consistency of the quotient depends on the spin of these generators being integral. These spins are the sum of the contributions from each factor in (4.2.14). (In checking this we use the fact that the spins of χ and j in

NK the ﬁrst factors are 0 and − 16 .) We also require that χ and j have trivial mutual braiding so that there is no anomaly [54].

We stress that these dualities, and the associated chiral algebra isomorphisms can be rigorously proven.

28 The Abelian anyon χ is the line in Spin(K)−N transforming in the N-th symmetric power of the vector representation, i.e. Dynkin indices (N, 0, ··· , 0). Similarly j transforms as a power of the spinor representation with Dynkin indices (0, ··· , 0,N).

129 4.2.2 Level-Rank Duality for Spin Chern-Simons Theory

We now promote the non-spin dualities (4.2.14) to spin dualities and simplify. Thus in this section we restore our convention that all theories are spin TQFTs.

The ﬁrst step is to use the duality Spin(L)1 ↔ (Z2)−L discussed in appendix 4.5. This gives

Spin(K) × ( ) × ( ) O(N)0 ←→ −N Z2 −K Z2 −NK+K (4.2.16) K,0 Z 1 O(K)−N,0 × (Z2)−(N+K−2) × (Z2)−NK+N+K−2 O(N)1 ←→ . K,0 Z0

When the common one-form symmetry is Z4 the quotient is generated by a product of the Wilson line j in the continuous factor and the basic magnetic line in each Z2 factor.

When the common one-form symmetry is Z2 × Z2, one Z2 is generated by a product of the

Wilson line j in the continuous factor and the basic magnetic line in each Z2 factor. The quotient by this Z2 may be viewed as a quotient on the gauge group in the numerator of

(4.2.16). However, the second Z2 is generated by a product of Wilson lines, χ from the continuous group and the electric Wilson line in the two Z2 groups. The quotient by this

Z2 is not a quotient of the gauge group, but of the TQFT and thus the above deviates from our convention of labeling TQFTs by the Chern-Simons gauge groups. (See the discussion around (4.1.16).)

When j generates a Z2 one-form symmetry, we have (see Appendix 4.12 for proof)

Spin(K)−N × (Z2)−NK Spin(K)−N ←→ , (4.2.17) Z2 1 1 O(K)−N,0 × (Z2)−NK O(K)−N,0 ←→ , Z2

130 where the quotient on (Z2)−NK uses the basic magnetic line. When j generates a Z4 one- form symmetry (with j2 = χ) we instead ﬁnd

Spin(K)−N × (Z4)−2NK Spin(K)−N ←→ , (4.2.18) Z4 1 1 O(K)−N,0 × (Z4)−2NK O(K)−N,0 ←→ , Z4

where (Z4)M is the Abelian Chern-Simons theory (M/4π)udu+(4/2π)udv with U(1) gauge H fields u, v, and the quotient on (Z4)−2NK uses the basic magnetic line (i.e. exp(i v)). (Note that the Z4 level is that defined by the Abelian Chern-Simons theory, which differs by a factor of two from our conventions for the Z2 level. In order to remind us of this difference we change the fonts, e.g. Z2 vs. Z4).

We next use the following Abelian Chern-Simons dualities:

(Z2)4M × (Z2)2L ←→ (Z2)4M−2L × (Z2)2L even L (4.2.19) (Z4)8M × (Z2)2L ←→ (Z2)4M−2L × (Z2)2L odd L, (4.2.20) Z2

where the quotient uses the product of the charge two magnetic line in (Z4)8M and the basic Wilson line in (Z2)2L. These may be established by changing variables. To prove the ﬁrst, write

2M L 2 2 2M − L L 2 2 ada+ bdb+ adx+ bdy = a0da0 + b0db0 + a0dx0 + b0dy0 , (4.2.21) 4π 4π 2π 2π 4π 4π 2π 2π where a0 = a, b0 = b + a, x0 = x − y − (L/2)b, y0 = y. To prove the second line of (4.2.19), proceed as

2M − L L 2 2 8M L 4 2 ada+ bdb+ adx+ bdy = a0da0 + b0db0 + a0dx0 + b0dy0 , (4.2.22) 4π 4π 2π 2π 4π 4π 2π 2π

131 0 0 0 0 0 0 0 0 where a = 2a , b = b + 2a , x = x − y + La , y = y − La . The Z2 quotient is generated by the line exp −iL H a + 2i H x = exp 2i H x0 − 2i H y0 = exp 2i H x0 + i H b0.

Consider the ﬁrst duality in (4.2.16). We can simplify the right hand side as follows.

Suppose L = −K/2 is even, we tensor the ﬁrst duality in (4.2.17) with (Z2)2L and take the

Z2 quotient generated by the product of χ and the basic Wilson line on the left. Using the change of variables (4.2.19) for the the numerator on the right makes the quotient diagonal and we ﬁnd the right hand side of (4.2.16) for the ﬁrst duality. The left hand side is thus dual to [Spin(K)−N × (Z2)2L]/Z2.

Similarly for odd L = −K/2 we tensor the first duality in (4.2.18) with (Z2)2L and take the Z2 quotient generated by the product of χ and the basic Wilson line on the left. Using the change of variables (4.2.19) for the numerator on the right makes the quotient diagonal and we find the right hand side of (4.2.16) for the first duality. The left hand side is again dual to [Spin(K)−N × (Z2)2L]/Z2.

Therefore we prove that for all even N,K (rearranging (Z2)2L),

0 O(N)K,K ←→ Spin(K)−N . (4.2.23)

By repeating the same steps with L = −(N + K − 2)/2 we similarly establish

1 1 O(N)K,K−1 ←→ O(K)−N,−N+1 . (4.2.24)

From the dualities (4.2.23) and (4.2.24) we conclude the level-rank duality map of SO spin Chern-Simons theories coupled to the backgrounds for C, M symmetries stated in (4.2.1).

132 4.2.3 Consistency Checks

In this section we present a few simple consistency checks of the level-rank dualities stated in equation (4.2.2). In appendix 4.9 we also discuss various low-rank examples of chiral algebras involved in the dualities.

0 Conformal Dimensions in O(N)K,K and Spin(K)−N

Consider the duality with even N. The lowest dimension twisted operator in the chi-

0 0 ral algebra O(N)K,K is a product of the twist operator of O(N)K and the spinor of

Spin(K)−1 ↔ (Z2)K . On the other side of duality the lowest dimension twisted operator in Spin(K)N is the spinor of Dynkin label (0, ··· , 0, 1). We will compute their dimensions and demonstrate that they agree.

0 The conformal dimension of the twist operator σ of O(N)K can be computed as follows.

Denote the Kac-Moody current by Jµν, the C symmetry changes the sign of J1i with i = 2, ··· N. We evaluate the one-point function of the Sugawara stress tensor in the presence of the twist operator by expanding J1i in half-integral modes:

1 X m m0 −m−1 −m0−1 hT (z)itwisted = limh Jij Jij z w w→z 2(N + K − 2) m,m0∈Z,i,j=2,···N,i

where we used the central term in the current commutator Kmδm+m0,0, thus for integral modes it vanishes after summing over positive and negative integers, while for half-integral

133 hσ(0)σ(0) modes it gives a non-trivial contribution. Since T (z)σ(0)|0i ∼ z2 |0i + ··· this deter- (N−1)K mines the conformal weight of the twist operator σ to be 16(N+K−2) . The same result was obtained in [55].

Using the fact that the spinor of Spin(K)−1 has spin −K/16 we see that

(N − 1)K K −K(K − 1) h[σO(N)0 ] = − = . (4.2.26) K,K 16(N + K − 2) 16 16(K + N − 2)

This agrees with the conformal weight of the spinor representation of Spin(K)N , which can be computed from the Casimir (see e.g. [39] and appendix 4.4). Therefore we ﬁnd the lines have the same spin as expected from the duality.

Counting the Lines

As another consistency check, we count the number of lines in each theory and show that they match in the level-rank dualities stated in equation (4.2.2). We take even N, K > 2.

The number of lines in Spin(N)K with even N,K is the number of non-negative integral PN/2−2 solutions for the aﬃne Dynkin labels λ0 +λ1 +λ N +λ N +2 λi = K where λ0 is the 2 −1 2 i=2 label for the extended node in the aﬃne Dynkin diagram. The tensor representations have

Dynkin labels λN/2−1, λN/2 both even or both odd, otherwise the representation is spinorial.

The number of tensor and spinor representations are given by

(N + K − 2)/2 (N + K − 4)/2 (N + K − 2)/2 N = 4 + ,N = 4 . tensor (K − 2)/2 K/2 spinor (K − 2)/2 (4.2.27)

As discussed in section 4.1.5, the chiral algebra of SO(N)K is the chiral algebra of

Spin(N)K extended by the representation of Dynkin label (K, 0, ··· , 0). The spinor representations are projected out and the tensor representations are identiﬁed with each other

134 or doubled. For even N,K the number of lines is

(N + K − 4)/2 (N + K − 4)/2 (N + K − 4)/2 (N + K − 4)/2 + + 2 + 2 , (N − 2)/2 (K − 2)/2 N/2 K/2 (4.2.28)

(N+K−4)/2 which has 2 K/2 doubled representations permuted by the magnetic symmetry M.

0 The chiral algebra O(N)K,0 for even N,K can be obtained from SO(N)K by an orbifold with the symmetry C, which acts on the Dynkin labels as λN/2−1 ↔ λN/2. The number of

0 untwisted and twisted primaries in the chiral algebra of O(N)K can be derived by [40]:

(N + K − 2)/2 (N + K − 4)/2 (N + K − 2)/2 N = 4 + ,N = 4 . untwisted (N − 2)/2 N/2 twisted (N − 2)/2 (4.2.29)

1 The chiral algebra O(N)K,0 for even N,K can be obtained from SO(N)K by orbifold with the symmetry CM, which acts as λN/2−1 ↔ λN/2, and permutes the two primaries in every doubled representations. The number of untwisted and twisted primaries in the

1 chiral algebra of O(N)K,0 can be derived by [40]:

(N + K)/2 (N + K − 4)/2 (N + K − 4)/2 N = + + , (4.2.30) untwisted K/2 (N − 2)/2 (K − 2)/2 (N + K − 4)/2 (N + K − 4)/2 N = 2 + 2 . twisted (N − 2)/2 (K − 2)/2

Using the above formulas, one can verify that the number of lines match in the dualities

SO(N)K ←→ SO(K)−N ,

0 O(N)K,K ←→ Spin(K)−N , (4.2.31)

1 1 O(N)K,K−1 ←→ O(K)−N,−N+1 .

135 0 0 1 Note that for even K and as spin theories, O(N)K,K0 (or O(N)K,K0 ) has the same number of

0 1 0 lines as O(N)K,0 (or O(N)K,0). Meanwhile for odd K , the number of untwisted primaries is

0 doubled by the presence of the lines 1, χ of Spin(K )−1, but the number of twisted primaries remains the same. In particular in the last two dualities of (4.2.31) the number of primaries matches for the untwisted/twisted sectors respectively in the dual chiral algebras.

4.3 Chern-Simons Matter Duality

In this section we derive new Chern-Simons matter dualities. We extend the previously conjectured dualities in [3,95] by coupling them to background ﬁelds for the global symmetries. Gauging the global symmetries we ﬁnd new dualities. We similarly extend the phase diagrams of [62].

4.3.1 Fermion Path Integrals and Counterterms

In order to utilize level-rank dualities in the context of Chern-Simons matter theories, we must understand how the counterterms we have studied in the previous sections may be generated by integrating out massive degrees of freedom. For a review of many of the elements described below see [150].

We first consider a real (Majorana) fermion λ coupled to an SO(N) gauge field A and transforming in representation R. Since R is real, λ may be given an SO(N) invariant mass m. The phase of the fermion partition function depends on the sign of m. We fix

136 conventions such that the phase is29

iπ Z [A]| = |Z | exp η(A) ,Z [A]| = |Z | . (4.3.1) λ m>0 λ 2 λ m<0 λ

In particular, in the massless theory we take the phase to be the average between these,

iπ that is exp 4 η(A) . In these formulas, η(A) is the eta-invariant deﬁned by a regularized sum of eigenvalues of the Dirac operator.30

The APS index theorem [13] relates the eta invariant to the minimally quantized spin

Chern-Simons action CS(A) as well as a gravitational Chern-Simons term CSgrav.

iπ Z Z exp η(A) = exp icR CS(A) + idim(R) CSgrav . (4.3.2) 2 X X

In the above cR is the quadratic Casimir of the representation R, and the gravitational Chern-Simons terms is deﬁned by

Z 1 Z CSgrav = tr(R ∧ R) , (4.3.3) X 192π Y where Y is a four-manifold with boundary X and R is the curvature two-form. In practice we will use formula (4.3.2) for vectors and two index tensors for which:

cR = 1 , cR = N − 2 , cR = N + 2 . (4.3.4)

Note that the partition function of the massive fermion (of either sign of m) defines a local effective action of the gauge field A. By contrast the massless fermion gives a non-local

29Note that these conventions imply a choice of scheme for the Fermion path integral. A diﬀerent choice of scheme would shift the Chern-Simons level for both positive and negative mass by an integer k. The invariant scheme independent statement is that the diﬀerence between positive and negative mass is a level cR Chern-Simons term. 30 P −s For instance using zeta function regularization we have: η(A) = lims→0 k sign(αk)|αk| , where αk are the eigenvalues of the Dirac operator coupled to A.

137 phase. In general, when describing the level of a Chern-Simons matter theory, we follow the convention that the theory is labelled by an eﬀective level, which is the average of the level at m > 0 and m < 0.

Similar analysis applies to the case of complex fermions. For instance, a massive complex fermion of charge q coupled to a U(1) gauge ﬁeld gives a contribution to the eﬀective level of

2 km>0 − km<0 = q . (4.3.5)

As in the case of real fermions, one typically labels the massless theory by the eﬀective level, which receives a contribution q2/2 from such a fermion.

Now let us extend our discussion to a single real Majorana fermion λ coupled to a

Z2 gauge field B. Again we fix conventions such that for negative mass the phase of the partition function is trivial, whereas for positive mass it is non-trivial and defined by the

iπ eta invariant. In particular, the massless theory has phase exp 4 η(B) . The analog of the APS index theorem is [13]

iπ Z exp η(B) = exp if[B] + i CSgrav . (4.3.6) 2 X

Here f[B] is the basic Z2 action discussed in section 4.1.1 and Appendix 4.5. We again label theories by an eﬀective level, which includes the bare coupling (an integer mod 8) as well as a fraction arising from fermions that are odd under the Z2 symmetry. With these conventions, (4.3.6) implies that integrating out a massive fermion shifts the Z2 level by sign(m)/2

One simple way to check (4.3.6) is to consider two Majorana fermions, which we combine into a single complex fermion. If both fermions are odd under the Z2, the eﬀective action for m > 0 is simply the square of (4.3.6). Thus, neglecting gravitational counterterms, the phase of the partition function for positive mass is exp(2if[B]). Since the level is even,

138 we can represent this eﬀective action by Abelian Chern-Simons theory following (4.5.4) and compare to (4.3.5). We see that, for positive mass, a charge q complex fermion yields

2 (Z2)2q2 . This is the expected answer since, when q is even, 2q = 0 (mod) 8 and hence

2 trivial for a Z2 gauge ﬁeld, while if q is odd 2q = 2 (mod) 8.

In the following applications we will be interested in how the levels for the Z2 ×Z2 global symmetry defined by C and M are changed by integrating out massive degrees of freedom. We define the action of charge conjugation to be compatible with the action (4.1.9) on gauge fields. In particular, C acting on a field (either scalar or fermion) reflects the first index in a vector representation, and acts trivially on other indices:

0 C(ρi1i2···i` ) = (−1)#1 sρi1i2···i` . (4.3.7)

Let xR indicate the number of components of an SO(N) representation that are charged under C. For the representations of interest to us:

xR = 1 , xR = N − 1 , xR = N − 1 . (4.3.8)

We take a parameterization of the counterterms in the presence of massless ﬁelds to be

xf[BC] + yf[BM] + zf[BC + BM] . (4.3.9)

Upon activating a mass m for a fermion in representation R these counterterms shift. Since the magnetic symmetry M is non-perturbative and all level shifts are one-loop exact, the coefficients y and z above are unmodified. The total effect is therefore

sign(m)x (x, y, z) −→ x + R , y, z . (4.3.10) 2

139 To further apply our results to theories with more general tensor matter, it is necessary to investigate the properties of monopole operators in more detail. In particular we will argue that the charge of a monopole operator V under the charge conjugation symmetry C may be changed by transition through a point where a fermion becomes massless. ∼ We ﬁrst consider the theory of SO(2)K+1/2 = U(1)K+1/2 coupled to a single complex fermion ψ of charge one. Since the Chern-Simons level jumps by one between positive and negative fermion mass, the gauge charge of the bare classical monopole diﬀers by one as well. Thus, if V represents the gauge invariant monopole operator for negative mass, the operator for positive mass must be dressed by an additional fermion. Therefore, the monopole operator for positive mass is V ψ.

This analysis carries over straightforwardly to SO(N)K coupled to fermions in the vector representation. The only essential diﬀerence is that in this case, the monopole charge is only conserved mod 2. Across a vector fermion transition, the M charge is unchanged, the electric charge is shifted, and the gauge invariant monopole acquires an additional fermion across the transition.

If we now consider SO(N) Chern-Simons theory coupled to a fermion ψ[i,j] transforming in the adjoint representation we ﬁnd additional eﬀects. We examine a GNO monopole embedded in the SO(2) subgroup of SO(N) which rotates the two-plane spanned by the last two indices. We refer to this operator as V [N,N−1]. Note that this is the minimally charged monopole. Across a massless transition for ψ[i,j], the electric charge of V [N,N−1] is changed by N − 2. Therefore across the transition the gauge invariant monopole operator jumps as

V [N,N−1] −→ V [N,N−1] ψ[1,N−1] + iψ[1,N] ψ[2,N−1] + iψ[2,N] ··· ψ[N−2,N−1] + iψ[N−2,N] . (4.3.11)

140 Notice also that depending on N and K there may in fact be no gauge invariant monopole

cR operator in the spectrum. Speciﬁcally, when N is even and K − 2 is odd the monopole operator is odd under the center of SO(N) and this charge cannot be screened by any other

cR ﬁelds in the model. Meanwhile, for N odd, or K − 2 even, there is a gauge invariant local monopole operator.

Assuming that a gauge invariant monopole operator exists, let us now act on it with the charge conjugation symmetry C. As in (4.1.9) we take the operator C to act as a minus sign on the index 1. From (4.3.11) we therefore see that the C charge of the monopole

(i,j) operator diﬀers for mψ < 0 and mψ > 0. An identical analysis applies to the case of ψ in a symmetric traceless tensor representation.

We can phrase this result in terms of the discrete θ-parameters of the section 4.1.1. Consider the SO(N) gauge theory in the presence of a background C gauge ﬁeld BC. The monopole charge is measured by the second Stiefel-Whitney class w2. The fact that the monopole charge diﬀers across a tensor transition means that the positive and negative mass

R C theories diﬀer by the coupling π X w2 ∪B . This means that the theory with vanishing mψ can be described imprecisely by saying that the coupling is ±1/2. This coupling is shifted when a tensor ﬁeld is given a mass by sign(mψ)/2, so that it is properly quantized after the massive fermions have been integrated out.

This eﬀect described above is even more dramatic in the O(N) gauge theory, where

C R B = w1 is dynamical. In that case the diﬀerent integral values of the coupling π X w2 ∪w1 label the theories O(N)0 and O(N)1. Thus we see that these two gauge theories are related through a transition of a massless tensor fermion. We denote the theory at the origin by O(N)±1/2. In particular this implies that O(N) Chern-Simons theory coupled to an odd number of fermions in a two index tensor representation has a parity anomaly.

One immediate application of this analysis is that we can rephrase the discussion in [3] about the relationship between the non-supersymmetric Chern-Simons matter dualities

141 reviewed in section 4.3.2 and the supersymmetric N = 2 dualities described in [8]. As discussed in section 4.2, level-rank duality of SO(N)K exchanges the Z2 symmetries C and

M. This map of symmetries also holds in the boson-fermion dualities for SO(N)K coupled to vector matter.

By contrast, in N = 2 supersymmetric theories there are also symmetries Csusy and

Msusy however under the duality the map of symmetries is instead

Csusy ←→ Csusy , Msusy ←→ CsusyMsusy . (4.3.12)

To explain the difference in the symmetry map under duality, note that the N = 2 theory has a pair of gauginos transforming in the adjoint representation under SO(N). The non- supersymmetric duality can be obtained from the supersymmetric one by giving mass to these gauginos and flowing to the infrared. Therefore, according to the discussion above, the supersymmetric theory differs from the non-supersymmetric theory by the coupling

R C π X w2 ∪ B . In particular, this means that the symmetries of the supersymmetric and non-supersymmetric theories are related as

Csusy −→ CM , Msusy −→ M . (4.3.13)

In fact, the superymmetric theory, deﬁned by the massless gauginos, has eﬀective cou-

R C pling π X w2 ∪ B . If the gauginos are given mass (of either sign) the coeﬃcient of this term shifts to become trivial. But at the massless point it still has non-trivial eﬀects on the C charges of monopole operators.

142 4.3.2 Dualities with Fundamental Matter

We can apply our reﬁned understanding of level-rank duality of orthogonal gauge theories to obtain dualities of Chern-Simons theories coupled to vector matter.

Let us recall brieﬂy the dualities described in [3]. They concern SO(N) Chern-Simons theories coupled to real scalars (φ) or Majorana fermions (ψ) in vector representations. Explicitly the duality is

SO(N)K with Nf φ ←→ SO(K)−N+Nf /2 with Nf ψ , (4.3.14) where in the above, the scalars are subject to a quartic potential and both theories are tuned to ﬂow to a transition point in the infrared.

Level-rank duality provides an essential consistency check on (4.3.14). Indeed, the obvious mass terms in the UV description of the theories flow to relevant operators at the fixed point that may be used to give mass to any of the scalar or fermi fields. In particular, if we gap all the matter fields we find level-rank dual topological field theories.

We can also turn this logic around, and use the consistency of the duality (4.3.14) to provide a check on our result (4.2.1) for level-rank duality coupled to background ﬁelds.

C Consider the theory with scalars, now coupled to background Z2 gauge fields B and BM for the charge conjugation and magnetic symmetry. As is common, the validity of the duality (4.3.14) in the presence of background fields may require us to add counterterms for these gauge fields [33,34]. Without loss of generality, we couple the fermionic theory to background gauge fields without counterterms. Then, the bosonic theory as a function of background fields is

C M C M C M (SO(N)K with Nf φ)[B ,B ] + αf[B ] + βf[B ] + γf[B + B ] . (4.3.15)

143 Here, the coefficients α, β, γ parameterize our ignorance. We could use (4.2.1) to fix them. Instead, we will constrain them by demanding consistency of the duality (4.3.14) in the presence of background fields.

Note that the parameters α, β, γ are part of the ultraviolet deﬁnition of the theory, and cannot depend on symmetry breaking patterns, i.e. they depend only on N and K. If we now give negative mass squared to L scalars and positive mass squared to Nf − L we arrive in the infrared at the topological ﬁeld theory

C M C M C M SO(N − L)K [B ,B ] + αf[B ] + βf[B ] + γf[B + B ] . (4.3.16)

On the fermion side of the duality, we can give positive mass to L of the vectors and negative mass to Nf − L. The gauge group is unmodiﬁed, but the Chern-Simons level is shifted according the discussion in 4.3.1. Using (4.3.10) we ﬁnd

M C M SO(K)−N+L[B ,B ] + Lf[B ] . (4.3.17)

We now demand that (4.3.16) and (4.3.17) are dual for all L. From this we determine:

α(N,K) = α(N−L, K) , β(N,K)−L = β(N−L, K) , γ(N,K) = γ(N−L, K) . (4.3.18) We can also apply spacetime parity to (4.3.16) and (4.3.17) and again demand a consistent duality. This implies that α(N,K) = β(K,N) and further generates the recursion relations:

α(N,K)−L = α(N,K−L) , β(N,K) = β(N,K−L) , γ(N,K) = γ(N,K−L) . (4.3.19)

144 From this we conclude that

α(N,K) = K − p , β(N,K) = N − p , γ(N,K) = q , (4.3.20) for some N and K independent integers p and q. This matches our general result (4.2.1) for level-rank duality, provided we ﬁx p = q = 1.

Dualities for Spin(N) and O(N)

Now that we have ﬁxed the exact counterterms required for the duality (4.3.14) to hold in the presence of background ﬁelds, we can immediately derive new results. Gauging C, M, and CM produces three new dualities:31

0 O(N)K,K with Nf φ ←→ Spin(K)−N+Nf /2 with Nf ψ ,

Spin(N) with N φ ←→ O(K)0 with N ψ , (4.3.21) K f −N+Nf /2,−N+Nf /2 f O(N)1 with N φ ←→ O(K)1 with N ψ . K,K−1+L f −N+Nf /2,−N+Nf /2+1+L f

Note that as a special case when there is no matter (Nf = 0), the above reduce to the level-rank dualities of section 4.2.

Some special cases of these are worth mentioning. For K = 1 we ﬁnd

0 O(N)1,1 with Nf φ ←→ Nf free Majorana fermions + decoupled (Z2)0 ,

Spin(N)1 with Nf φ ←→ (Z2)−N+Nf /2 with Nf ψ ,

1 O(N)1,L with Nf φ ←→ (Z2)−N+Nf /2+1+L with Nf ψ , (4.3.22)

31 For odd N the group O(N) is a product Z2×SO(N) and therefore its representation theory is factorized. In particular there is a vector representation of SO(N) that is even under the Z2 factor as well as one that is odd under the Z2 factor. Our dualities hold for the representation that is odd under the Z2 factor.

145 where in the second and last dualities the fermions are odd under the Z2 gauge ﬁeld. The

first duality gives infinitely many scalar theories that describe the same Nf free Majorana fermions with the decoupled (Z2)0, which is time-reversal invariant (similar dualities appeared in [3,95]). The theories on the right-hand-side of the second and third dualities are also essentially free. Specifically, they are free fermions coupled to a discrete gauge theory, which reduces the local operators to the Z2 invariant sector. Comparing the second duality with the last duality for L = −1 also gives infinite many boson/boson dualities

1 Spin(N)1 with Nf φ ←→ O(N)1,−1 with Nf φ , (4.3.23)

which describe the same ﬁxed point as that of (Z2)−N+Nf /2 coupled to Nf ψ.

Similarly, for N = 1 we ﬁnd

(Z2)K with Nf φ ←→ Spin(K)−1+Nf /2 with Nf ψ ,

N real WF scalars + decoupled ( ) ←→ O(K)0 with N ψ , f Z2 0 −1+Nf /2,−1+Nf /2 f ( ) with N φ ←→ O(K)1 with N ψ(4.3.24) , Z2 K−1+L f −1+Nf /2,Nf /2+L f

where in the first and last dualities the scalars are odd under the Z2 gauge field. The second duality gives infinitely many fermionic theories that describe the same Nf real

Wilson-Fisher scalars with the decoupled (Z2)0. Comparing the ﬁrst duality with the last duality for L = 1 also gives inﬁnite many fermion/fermion dualities

Spin(K) with N ψ ←→ O(K)1 with N ψ , (4.3.25) −1+Nf /2 f Nf /2−1,Nf /2+1 f

which describe the same ﬁxed point as that of (Z2)K coupled to Nf φ.

146 ∼ For K = 2 we use Spin(2)L = U(1)4L to ﬁnd

0 O(N)2,2 with Nf φ ←→ U(1)−4N+2Nf with Nf ψ ,

Spin(N)2 with Nf φ ←→ O(2)−N+Nf /2,−N+Nf /2 with Nf ψ , (4.3.26)

1 O(N)2,1+L with Nf φ ←→ O(2)−N+Nf /2,−N+Nf /2+1+L with Nf ψ ,

where the fermion in the ﬁrst duality has charge 2. These dualities are valid only for Nf < N [3]. For other values the dualities are still valid but with a more subtle interpretation [36, 62, 88]. Comparing the second duality with the last duality for L = −1 gives the boson/boson duality

1 Spin(N)2 with Nf φ ←→ O(N)2,0 with Nf φ , (4.3.27)

which describes the same theory as O(2)−N+Nf /2,−N+Nf /2 with Nf ψ.

Similarly, for N = 2 we ﬁnd

O(2)K,K with Nf φ ←→ Spin(K)−2+Nf /2 with Nf ψ ,

U(1) with N φ ←→ O(K)0 with N ψ , 4K f −2+Nf /2,−2+Nf /2 f O(2) with N φ ←→ O(K)1 with N ψ , (4.3.28) K,K−1+L f −2+Nf /2,−1+Nf /2+L f where the scalar in the second duality has charge 2. Comparing the ﬁrst duality with the last duality for L = 1 gives the fermion/fermion duality

Spin(K) with N ψ ←→ O(K)1 with N ψ , (4.3.29) 2−Nf /2 f 2−Nf /2,−Nf /2 f

which describes the same theory as O(2)−K,−K with Nf φ.

147 If we specialize to N = K = 2 and Nf = 1 they become

O(2)2,2 with φ ←→ U(1)−6 with ψ ,

U(1)8 with φ ←→ O(2)−3/2,−3/2 with ψ , (4.3.30)

O(2)2,1+L with φ ←→ O(2)−3/2,−1/2+L with ψ , where the matter in the U(1) theories has charge 2. They can be summarized into

O(2)2,0 with φ ←→ U(1)8 with φ of charge two x x   y y U(1)−6 × (Z2)−2 O(2)−3/2,−3/2 with ψ ←→ with ψ of U(1) charge two .(4.3.31) Z2

Here we also deviate from our standard notation and the quotient is not of the gauge group, but denotes a gauging of the one-form global symmetry. Speciﬁcally, the quotient uses the

Wilson line of (Z2)−2.

4.3.3 Phase Diagram of Adjoint QCD

As a ﬁnal application of our level-rank duality result (4.2.1), we consider the phase diagram of orthogonal gauge theories coupled to fermionic matter in a two-index tensor representation discussed in [62]. We focus on the case of small Chern-Simons level, i.e. the range:

N SO(N) + adjoint λ 0 ≤ K < − 2 , (4.3.32) K 2 N SO(N) + symmetric S 0 ≤ K < . (4.3.33) K 2

148 The problem is to determine the infrared behavior of these theories as a function of the mass m of the fermionic matter. A complete solution to this problem was conjectured in [62] and we review it below.

There are two obvious phases where |m| is large and the fermions may be integrated out semi-classically. These are gapped TQFTs described by SO(N) Chern-Simons theory with level depending on the sign of the mass.

As |m| is reduced, a new quantum phase appears. This quantum phase is also gapped and described by an SO Chern-Simons theory but with a diﬀerent gauge group and value of the level.

The passage from a semiclassical phase to the quantum phase proceeds through a transition that is weakly coupled in dual variables. The dual theory also has a two-index tensor fermion. However, this tensor has the opposite symmetry to the original matter deﬁning the model. Thus, in the adjoint theory, the transition from the semiclassical to quantum phase is governed by a symmetric tensor transition, while in the theory with symmetric matter it is governed by an adjoint transition.

As we dial further into the quantum regime, we find that the adjoint theory becomes N = 1 supersymmetric at a special value of the bare mass. Based on index calculations in [148], it is expected that this theory spontaneously breaks supersymmetry and therefore in addition to the TQFT we also find a massless goldstino. In the symmetric tensor theory, no such effect occurs.

These phase diagrams are illustrated in ﬁgure 4.4. As is evident from the explicit levels shown there, level-rank duality is crucial for the consistency of the full picture.

We can use our improved understanding of level-rank duality in the presence of background gauge ﬁelds to provide a new consistency check of this conjectured picture. As we have discussed in section 4.2, level-rank duality shifts the values of the Z2 × Z2 countert-

149

푁 푆푂(푁) with adjoint 휆 for 0 ≤ 퐾 < − 2 퐾 2 ℤ2 × ℤ2 symmetry (퐶, 푀)

푁−2 푁−2 ̂ 푆푂 ( − 퐾) + symmetric S 푆푂 ( + 퐾) 3푁−2퐾−2 + symmetric S 2 3푁+2퐾−2 2 − 4 4 (퐶푀, 퐶) (푀, 퐶푀)

SUSY

푚휆 ≪ 0 푚휆 ≫ 0 푁 − 2 푆푂 ( − 퐾) massless Goldstino 푆푂(푁) 푁−2 푁−2 푆푂(푁) 푁−2 퐾− 2 +퐾 퐾+ 2 2 2 (퐶, 푀) (퐶푀, 퐶) (퐶푀, 푀) ↕ ↕ ↕ 푁 − 2 푁 − 2 푁 − 2 푆푂 ( − 퐾) 푆푂 ( + 퐾) 푆푂 ( + 퐾) 푁−2 2 푁 2 − +퐾 2 −푁 (푀, 퐶) 2 (퐶, 퐶푀) (푀, 퐶푀)

푁 푆푂(푁) with 푆 symmetric for 0 ≤ 퐾 < 퐾 2 ℤ2 × ℤ2 symmetry (퐶, 푀)

푁+2 푁+2 ̂ 푆푂 ( − 퐾) + adjoint 휆 푆푂 ( + 퐾) 3푁−2퐾+2+ adjoint λ 2 3푁+2퐾+2 2 − 4 4 (푀, 퐶푀) (퐶푀, 퐶)

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0 푁 + 2 푆푂(푁) 푁+2 푆푂 ( − 퐾) 푆푂(푁) 푁+2 퐾− 푁+2 퐾+ 2 2 +퐾 2 2 ↕ (퐶, 푀) (퐶푀, 퐶) ↕ (퐶푀, 푀) ↕ 푁 + 2 푁 + 2 푁 + 2 푆푂 ( − 퐾) 푆푂 ( + 퐾) 푆푂 ( + 퐾) 2 푁 푁+2 2 −푁 2 − +퐾 (푀, 퐶) 2 (퐶, 퐶푀) (푀, 퐶푀)

Figure 4.4: The phase diagrams of SO(N) gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears. These ﬁgures are identical to those in [62] except that we now add the map of the Z2 × Z2 global symmetry.

150 erms. Additionally the various tensor matter transitions change the counterterms as well

R C as shift the X w2 ∪ B counterterm as discussed in section 4.3.1. The fact that all these discrete counterterms close on a consistent picture is a striking test of the validity of this phase diagram.

One way to present the information encoded by the counterterm consistency described above is to promote some of the background ﬁelds to be dynamical and hence obtain the phase diagram for another gauge group. We illustrate the results for the Spin(N) and O(N) theories in ﬁgures 4.5, 4.6, and 4.7 below.

In these ﬁgures it is convenient to introduce a notation analogous to the Z2 level for theories with gauge group Spin(N):

Spin(N)K × Spin(L)−1 Spin(N)K × (Z2)L Spin](N)K,L ≡ ←→ . (4.3.34) Z2 Z2

The ﬁrst expression is deﬁned in terms of a Chern-Simons theory based on the group

Spin(N)×Spin(L) . The third expression uses the duality (4.0.7). Here the 2 quotient is not Z2 Z simply a quotient of the gauge group in the numerator. It is given by gauging the diagonal one-form symmetry of the Z2 subgroup in the center of Spin(N), whose quotient results in

SO(N), and the Z2 one-form symmetry generated by the Wilson line of (Z2)L. Note that this quotient exists for all L. An example of an allowed line in this quotient is the product of a spinor in Spin(N) and an ’t Hooft line in (Z2)L. For L = 0 the theory Spin](N)K,0 is the same as the standard Spin(N)K Chern-Simons theory.

151

SUSY

푁 푆푝푖푛(푁) with 푆 symmetric for 0 ≤ 퐾 < 퐾 2

1 1 − 푁+2 푁+2 2 푂 ( + 퐾)2 + λ̂ 푂 ( − 퐾) 3푁+2퐾+2 3푁+2퐾 + adjoint 휆 3푁−2퐾+2 3푁−2퐾 adjoint 2 , 2 − ,− 4 4 4 4

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0

푁 + 2 1 푆푝푖푛(푁) 푁+2 푂 ( − 퐾) 푆푝푖푛(푁) 푁+2 퐾− 푁+2 푁 퐾+ 2 2 +퐾, +퐾 2 2 2 ↕ ↕ ↕ 0 0 푁 + 2 푁 + 2 1 푁 + 2 푂 ( − 퐾) 푂 ( + 퐾) 푂 ( + 퐾) 2 푁,푁 푁+2 푁 2 −푁,−푁 2 − +퐾,− +퐾 2 2

Figure 4.5: The phase diagrams of Spin(N) gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. Across these tensor transitions O(L)0 and O(L)1 are exchanged. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears.

152 1 푁 푂(푁)2 with adjoint 휆 for 0 ≤ 퐾 < − 2 퐾,퐿 2 1 − 푁 − 2 푁 − 2 2 푆푝푖̃푛 ( − 퐾) 푂 ( + 퐾) 2 3푁+2퐾−2 1 3푁−2퐾−2 3푁+2퐾−6 ,퐿−퐾− 2 − ,퐿− 4 2 4 4 + symmetric S + symmetric Ŝ

SUSY

푚휆 ≪ 0 푚휆 ≫ 0

푁 − 2 ̃ 0 푆푝푖푛 ( − 퐾) 1 푂(푁) 푁−2 푁−1 푁−2 1 푂(푁) 푁−2 푁−1 퐾− ,퐿− 2 +퐾,퐿−퐾− 퐾+ ,퐿+ 2 2 2 2 2 2 massless Goldstino ↕ ↕ ↕ 푁 − 2 푁 − 2 0 푁 − 2 1 푆푝푖̃푛 ( − 퐾) 푂 ( + 퐾) 1 푂 ( + 퐾) 5 2 푁,퐿−퐾− 푁−2 푁 1 2 −푁,퐿−푁−퐾+ 2 2 − +퐾,퐿− + 2 2 2 2

1 푁 푂(푁)2 with 푆 symmetric for 0 ≤ 퐾 < 퐾,퐿 2

1 − ̃ 푁+2 푁+2 2 ̂ 푆푝푖푛 ( − 퐾)3푁+2퐾+2 3 + adjoint 휆 푂 ( + 퐾) 3푁−2퐾+2 3푁+2퐾−2+ adjoint λ 2 ,퐿−퐾+ 2 − ,퐿− 4 2 4 4

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0

푁 + 2 ̃ 1 0 푆푝푖푛 ( − 퐾) 푂(푁) 푁+2 푁−1 푂(푁) 푁+2 푁−1 푁+2 3 퐾+ ,퐿+ 퐾− ,퐿− 2 +퐾,퐿−퐾+ 2 2 2 2 2 2 ↕ ↕ ↕ 0 푁 + 2 1 푁 + 2 푁 + 2 푂 ( + 퐾) 푆푝푖̃푛 ( − 퐾) 1 3 푂 ( + 퐾) 2 −푁,퐿−푁−퐾+ 2 푁,퐿−퐾+ 푁+2 푁 1 2 2 2 − +퐾,퐿− + 2 2 2

1 Figure 4.6: The phase diagrams of O(N) 2 gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. Across these tensor transitions O(L)0 and O(L)1 are exchanged. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears. The notation Spin](N)K,L is explained around (4.3.34).

153 1 − 푁 푂(푁) 2 with adjoint 휆 for 0 ≤ 퐾 < − 2 퐾,퐿 2 1 푁 − 2 푁 − 2 2 푆푝푖푛̃ ( + 퐾) 푂 ( − 퐾) 3푁+2퐾−2 3푁−2퐾−6 3푁−2퐾−2 1 ,퐿+ 2 − ,퐿−퐾+ 2 4 4 4 2 + symmetric S + symmetric Ŝ

SUSY

푚휆 ≪ 0 푚휆 ≫ 0

푁 − 2 0 1 푂 ( − 퐾) 0 푂(푁) 푁−2 푁−1 푁−2 푁 1 푂(푁) 푁−2 푁−1 퐾− ,퐿− 2 +퐾,퐿+ − 퐾+ ,퐿+ 2 2 2 2 2 2 2 massless Goldstino ↕ ↕ ↕ 푁 − 2 1 푁 − 2 푁 − 2 푆푝푖̃푛 ( + 퐾) 푂 ( − 퐾) 푆푝푖푛̃ ( + 퐾) 1 5 푁−2 1 2 −푁,퐿−퐾+ 2 푁,퐿+푁−퐾− 2 − +퐾,퐿−퐾+ 2 2 2 2

1 − 푁 푂(푁) 2 with 푆 symmetric for 0 ≤ 퐾 < 퐾,퐿 2

1 푁+2 푁+2 ̃ ̂ 2 푆푝푖푛 ( + 퐾) 3푁−2퐾+2 3+ adjoint λ 푂 ( − 퐾) 3푁+2퐾+2 3푁−2퐾−2 + adjoint 2 − ,퐿−퐾− 2 ,퐿+ 4 2 4 4 휆

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0

푁 + 2 0 0 1 푂 ( − 퐾) 푂(푁) 푁+2 푁−1 푂(푁) 푁+2 푁−1 푁+2 푁 1 퐾+ ,퐿+ 퐾− ,퐿− 2 +퐾,퐿+ − 2 2 2 2 2 2 2 ↕ ↕ ↕ 1 푁 + 2 푁 + 2 푁 + 2 푆푝푖̃푛 ( + 퐾) 3 푂 ( − 퐾) 푆푝푖푛̃ ( + 퐾) 2 −푁,퐿−퐾− 1 푁+2 3 2 2 푁,퐿+푁−퐾− 2 − +퐾,퐿−퐾− 2 2 2

− 1 Figure 4.7: The phase diagrams of O(N) 2 gauge theory coupled tensor fermions. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. This proceeds through a tensor transition described by a dual theory, which covers part of the phase diagram. Across these tensor transitions O(L)0 and O(L)1 are exchanged. At a special value of the mass in the quantum phase of the adjoint theory, a massless goldstino appears. The notation Spin](N)K,L is explained around (4.3.34).

154 4.4 Appendix A: Representation Theory of so(N)

The Young tableaux is deﬁned from Dynkin labels λi by non-increasing row lengths li, i = 1, ··· , n where n is the rank.

n−1 λn X λn so(2n + 1) : l = + λ for 1 ≤ i ≤ n − 1; l = , (4.4.1) i 2 j n 2 j=i n−1 λn − λn−1 X λn − λn−1 so(2n): l = + λ for 1 ≤ i ≤ n − 1; l = .(4.4.2) i 2 j n 2 j=i

Pn The number of boxes is r = i=1 li. Note for so(2n) the last row length comes with a sign while other row lengths are non-negative.

Tensor representations are deﬁned by even λn for so(2n + 1) and even (λn − λn−1) for so(2n), which correspond to Young tableaux with integral row lengths. Otherwise they are spinor representations (we also refer to them as spinorial representations).

The conformal weight of so(N)K representation with row lengths {li} can be computed from the Casimir [39]

n ! 1 X h = Nr + l (l − 2i) . (4.4.3) 2(N + K − 2) i i i=1

For example, the fundamental spinor representation of Dynkin labels (0, ··· , 0, 1) cor-

1 1 1 responds to the Young tableaux with row lengths ( 2 , 2 , ··· , 2 ), with conformal weight N(N−1) 16(N+K−2) for all N.

155 4.5 Appendix B: Z2 Topological Gauge Theory in Three Dimensions

In this section we describe Z2 classical gauge theories in three spacetime dimensions. We are interested in theories that depend on the spin structure of the underlying three-manifold X. These are also known as fermionic symmetry protected topological phases (SPT) with

Z2 unitary symmetry. Such theories have been completely determined [63,81,82,135], and admit a Z8 classiﬁcation.

A practical way to produce these local actions is via SO(L)1 spin Chern-Simons theory.

Since these theories depend on the spin structure, they can couple to a Z2 gauge ﬁeld B by shifting the spin structure with the background B. Equivalently, from the relation (4.1.10) they couple to B by the magnetic symmetry M, and we deﬁne the action fL[B] via the normalized partition function

Z(SO(L)1)[B] e−ifL[B] = . (4.5.1) Z(SO(L)1)[0]

Although we have deﬁned the action by a functional integral, it is known that the result is local since SO(L)1 is an invertible ﬁeld theory [50].

There are two signiﬁcant properties that are useful in manipulating these counterterms.

First, the index L behaves as a level (i.e. fL[B] is linear in L). This follows from the simple duality

0 0 SO(L)1 × SO(L )1 ←→ SO(L + L )1 . (4.5.2)

We use this linearity to simplify notation and write fL[B] = Lf[B] where f[B] is the minimal non-trivial action.

156 A second useful property is that for even level L = 2n the action 2nf[B] is simply related to Abelian Chern-Simons theory. Explicitly32

Z 2i Z in Z exp(i2nf[B]) = [Da] exp adB + BdB , (4.5.4) 2π X 4π X where in the above, the U(1) field a is dynamical and enforces the constraint that B is a Z2 gauge field [16, 90]. (For a discussion of these theories see also [78, 122].) Note that n = 1 above is the minimal allowed level in U(1) spin Chern-Simons theory. Thus Lf[B] for odd L cannot be expressed using continuum U(1) actions. Observe that by redefining a → a+B in (4.5.4) we change the counterterm by 8f[B], and hence we have the identification

8f[B] ∼ 0 . (4.5.5)

The special cases Lf[B] for L = 0, 4 mod 8 are the bosonic Dijkgraaf-Witten theories [41]

3 for Z2 gauge group classiﬁed by H (BZ2,U(1)) = Z2. They are independent of the spin R structure. The particular case L = 4 has the action π X B ∪ B ∪ B.

If we promote the Z2 background gauge ﬁeld B to be a dynamical ﬁeld b we have

Lf[b] ←→ Spin(L)−1 × SO(L)1 . (4.5.6)

We will denote the left hand side by (Z2)L (with L = 0 mod 4 deﬁned to be bosonic). It is a spin TQFT for L 6= 0 mod 4. In particular, (Z2)L for odd L has anyons that obey the 1 L fusion rule of the Ising TQFT with spins 0, 2 , − 16 mod 1, tensored with {1, ψ} where ψ is 32 ∼ It is suﬃcient to show the relation for 2f[B]. Using SO(2)1 = U(1)1 and w2 = c1 mod 2 for SO(2) bundle, we can express SO(2)1[B] as

1 da du ada + + 2 B. (4.5.3) 4π 2π 2π

1 2 Integrating out a gives the Abelian Chern-Simons theory − 4π BdB + 2π Bdu − 2CSgrav, where the gravitational Chern-Simons term is cancelled by SO(2)1[0].

157 the transparent spin one-half line. The Wilson line of (Z2)L corresponds to the product of the line in the vector representation of Spin(L) and ψ, and it has integral spin. The basic magnetic line of (Z2)L corresponds to the line in the fundamental spinor representation L of Spin(L) and it has spin − 16 .(Z2)L has zero framing anomaly. The chiral algebra corresponding to (Z2)L can be read oﬀ from (4.5.6) (also from (4.5.4) for even L).

This construction of SPT phases may be generalized to any discrete group G that contains a Z2 subgroup. Consider a G bundle on X. A gauge ﬁeld γ may be viewed as

1 a map to the classifying space γ : X → BG. For every choice ρ ∈ H (BG, Z2), we then

∗ 1 obtain a Z2 gauge ﬁeld γ ρ ∈ H (X, Z2). Consider

∗ ∗ Z(SO(L) )[γ ρ] e−iLf[γ ρ] = 1 , (4.5.7) Z(SO(L)1)[0]

1 where as before Z(SO(L)1)[B] with B ∈ H (X, Z2) is the partition function of SO(L)1 coupled to B by the magnetic symmetry M (equivalently it couples via shifting the spin structure by B and using (4.1.10)). f[γ∗ρ] is a local topological action of the classical G gauge theory. For discrete symmetry G it can be expressed as the Arf invariant of the symmetry defects [81,82]. Other descriptions include [24,52] and the references there.

We can use the above construction, together with fermionic SPT phases that can be produced from Abelian Chern-Simons actions to produce a complete list of the fermionic

SPT phases for Z2n unitary symmetry. We will reproduce the classiﬁcations in [63] and [135] obtained from diﬀerent methods.

All fermionic SPT phases for Z2n unitary symmetry can be expressed as follows (denote

1 the background gauge ﬁeld by B ∈ H (X, Z2n))

L0 Z 2n Z Lf[(nB)] + BdB + adB , (4.5.8) 4π X 2π X

158 where in the above a is a dynamical field that constrains the U(1) field B to be a Z2n gauge field.

The parameters have the identiﬁcation

L ∼ L + 8,L0 ∼ L0 + 4n, (L, L0) ∼ (L + 2,L0 − n2) . (4.5.9) where the last equation uses (4.5.4). Thus the parameter space is L = 0, 1 and L0 = 0...4n−1 for n > 1, while L = 0, ...7,L0 = 0 for n = 1. For even n, 4n is multiple of 8 and the

0 2 classiﬁcation is Z4n × Z2 generated by (L, L ) = (1, 1) of order 4n and (1, −n /2) of order

0 2. For odd n > 1, the classiﬁcation is Z8n generated by (L, L ) = (1, 1). For n = 1 the Z8 classiﬁcation is generated by (L, L0) = (1, 0).

For unitary symmetry Z2n+1 there is no Z2 subgroup and thus no coupling with w2(SO(L)). Therefore all fermionic SPT phases for unitary symmetry Z2n+1, can be

33 constructed from Abelian Chern-Simons actions, and have a Z2n+1 classiﬁcation.

4.6 Appendix C: P in−(N) and O(N)1 from SO(N)

Since an SO(N) bundle is an O(N) bundle with w1 = 0, we can write the SO(N) Lagrangian as

Z Z Z L[SO(N)] ←→ L[O(N)] + π w1(O(N)) ∪ B2 , (4.6.1) X X X

33 Consider the Abelian Chern-Simons theory (ZN )K × SO(0)1 with odd N K N 1 1 B0dB0 + B0du + xdx + xdy . (4.5.10) 4π 2π 4π 2π

N+1 0 0 0 The change of variables u → u + 2 B − y, z → x + NB , y → y − NB produces the identiﬁcation K ∼ K + N. For even N there is no such identiﬁcation.

159 where the dynamical Z2 two-form gauge ﬁeld B2 constrains w1(O(N)) = 0. Thus turning on background BC for C changes the Lagrangian to34

Z Z Z C C L[SO(N),B ] ←→ L[O(N)] + π w1(O(N)) + B ∪ B2 , (4.6.2) X X X such that making BC dynamical without any additonal counterterm produces the O(N)

C theory. The dynamical multiplier B2 implies that an SO(N) bundle with background B

C produces O(N) bundle with ﬁxed ﬁrst Stiefel-Whitney class w1(O(N)) = B . In addition, we can include the coupling with background BM for M

Z L[SO(N),BC,BM] ←→ X Z Z Z C M L[O(N)] + π w1(O(N)) + B ∪ B2 + π w2(O(N)) ∪ B . (4.6.3) X X X

We can gauge the Z2 symmetries by promoting the gauge ﬁelds to be dynamical. In this way we produce ﬁgure 4.3.

• Gauging C, M by promoting BC,BM to be dynamical and adding a counterterm

C C M Sct[B ] that depends only on B . The multiplier B imposes w2(O(N)) = 0, which changes the gauge group to P in+(N), and the counterterm becomes

+ Sct[w1(P in (N))].

• Gauging C, M by promoting BC,BM to be dynamical and adding the counterterm

C R M C C C Sct[B ]+π X B ∪B ∪B . The equation of motion for B2 imposes B = w1(O(N)), M and the dynamical multiplier B imposes w2(O(N)) + w1(O(N)) ∪ w1(O(N)) = 0. Thus the theory becomes a P in−(N) gauge theory, and the other counterterm becomes

− Sct[w1(P in (N))].

34 In this section, as throughout the paper, wi(G) refers to the Stiefel-Whitney classes of a principle bundle with structure group G.

160 • Gauging the diagonal CM. We take BC = BM (denoted by B) and add some coun-

terterm Sct[B]. Then we promote B to be dynamical. The dynamical multiplier B2

imposes B = w1(O(N)), and the theory becomes

Z Z L[O(N)] + π w1(O(N)) ∪ w2(O(N)) + Sct[w1(O(N))] . (4.6.4) X X

Thus it produces an O(N) gauge theory with discrete topological term w1∪w2, namely an O(N)1 theory.

We can also gauge the one-form symmetry in the P in+(N) theory. We will focus on one- form symmetries dual to the gauged C, M symmetries. Denote their background two-form

0 00 + Z2 gauge ﬁelds by B2,B2 , they couple to the P in (N) theory by

Z Z C M C 0 M 00 L[SO(N),B ,B ] + π B ∪ B2 + B ∪ B2 , (4.6.5) X X where BC,BM are dynamical. As in the discussion below (4.6.2), the presence of the dynamical ﬁelds BC and BM means that the bundles are P in+(N) bundles with non-trivial Stiefel-Whitney classes.

0 00 • We can gauge both Z2 one-form symmetries by promoting both B2,B2 to be dynamical. This imposes BC = BM = 0 and we recover the SO(N) gauge theory.

0 00 • We can gauge either one-form symmetry by promoting either B2 or B2 to be dynamical. This produces the Spin(N) or O(N)0 theory.

0 00 • Gauging the diagonal one-form symmetry by promoting the diagonal B2 = B2 to be dynamical. This imposes BC = BM, and we recover the O(N)1 theory.

161 4.7 Appendix D: The Chern-Simons Action of O(N)

for Odd N

As discussed in section 4.1.1, the Lagrangian for O(N) = SO(N) o Z2 can have discrete couplings involving w1 of the O(N) bundle. We view the Stiefel-Whitney class w1 as a Z2 gauge field for the charge conjugation C of SO(N). These discrete couplings depend on how the Z2 is embedded in O(N). For even N there is no natural choice, and our convention is specified by (4.1.9). If we change our convention and define C to flip the signs of more indices, the Z2 levels are modified. Formulas for the change in the action can be derived following the logic below.

When N is odd, the orthogonal group factorizes O(N) = SO(N) × Z2, and thus every

O(N) bundle is separately an SO(N) bundle and a Z2 bundle. One natural choice of Z2 subgroup is this Z2 factor, which commutes with SO(N). However, to be uniform with our treatment of even N we continue to use the convention (4.1.9), which corresponds to a Z2 that is embedded non-trivially in each factor of SO(N) × Z2.

Let us derive the relation between the Lagrangians for these two choices of Z2 subgroup. Explicitly the formula we would like to prove is:

Z CS(O(N)) = CS(SO(N)) + π w2(SO(N)) ∪ w1 + (N − 1)f[w1] , (4.7.1) X where on the left the Z2 with gauge ﬁeld w1 is non-trivially embedded in SO(N) × Z2 as deﬁned by the charge conjugation (4.1.9), but on the right it is factorized from SO(N).

162 To derive (4.7.1) observe that the charge conjugation transformation (4.1.9) can be expressed as eπJ where J is the element of the Lie algebra

  0      0 1         −1 0  J =   . (4.7.2)    0 1       −1 0      ···

Therefore the dynamical O(N) gauge ﬁeld may be expressed as a sum of an SO(N) gauge

C ﬁeld together with a Z2 gauge ﬁeld B which also sources J

C AO(N) = (ASO(N),B ) . (4.7.3)

C For practical manipulations it is useful to view B as a U(1) gauge ﬁeld constrained to Z2 by a Lagrange multiplier ﬁeld.

We now evaluate the Chern-Simons action (4.1.6). To evaluate the terms involving the field BC it suffices to consider the case where the SO(N) gauge field is of the form

  0      0 a   1       −a1 0  ASO(N) =   (4.7.4)    0 a2       −a 0   2    ···

163 Evaluating the O(N) Chern-Simons action on (4.7.3) we then ﬁnd (below BC is normalized to have periods 0 and π):

Z (N−1)/2 Z X dai N − 1 CS(O(N)) = CS(SO(N)) + BC + BCdBC . (4.7.5) 2π 8π X i=1 X

P(N−1)/2 dai We recognize the sum over ﬂuxes i=1 2π is the expression for the Stiefel-Whitney class w2(SO(N)). Meanwhile, using the relationship between Abelian Chern-Simons theory and

C Z2 gauge theory discussed in appendix 4.5, we see that the ﬁnal term above is (N −1)f[B ].

As a consistency check, integrating out one Majorana fermion ψ of positive mass in the vector representation of O(N) generates the topological term (4.3.6):

Z CS(O(N)) + f[w1] + CSgrav . (4.7.6) X

On the other hand, up to an SO(N) gauge transformation, charge conjugation acts on the fermion in the same way as ﬂipping the sign of ψi for all SO(N) vector indices i. The latter symmetry is the Z2 subgroup of O(N) that commutes with SO(N). Note also that for positive vs negative mass, the charge of the monopole under this Z2 factor subgroup of O(N) is changed as in section 4.3.1. Thus, in the factorized variables, integrating out the same fermion generates the topological terms

Z Z CS(SO(N)) + π w2(SO(N)) ∪ w1 + Nf[w1] + CSgrav . (4.7.7) X X

Equating the two expressions (4.7.6),(4.7.7) reproduces the relation (4.7.1).

164 4.8 Appendix E: Derivation of Level-Rank Duality for

Odd N or K

In this appendix we derive the counterterm in (4.2.1) when N or K is odd. As in section 4.2, we derive this result, by proving the spin Chern-Simons dualities (4.2.23)-(4.2.24) for N or K odd:

0 O(N)K,K ←→ Spin(K)−N

1 1 O(N)K,K−1 ←→ O(K)−N,−N+1 . (4.8.1)

We will use various formulas from section 4.1.2 for the Chern-Simons action of SO(N)K with odd N coupled to the backgrounds BC,BM for the symmetries C, M. The Chern- Simons action can be expressed as

C M M C C SO(N)K [B ,B ] = SO(N)K [0,B + KB ] + (K − 1)(N − 1)f[B ]

− (N − 1)f[BM] + (N − 1)f[BC + BM] odd N. (4.8.2)

0 C From (4.8.2) we can obtain the O(N)K,0 theory by promoting B to be dynamical with BM = 0. Thus

   SO(N)K × (Z2)K(N−1) odd N, even K O(N)0 = (4.8.3) K,0  Spin(N)K × (Z2)K(N−1) /Z2 odd N, odd K.

165 1 C M 35 Similarly, we can obtain the O(N)K theory by promoting B = B to be dynamical

  Spin(N) × ( ) / odd N, even K 1  K Z2 (K−2)(N−1) Z2 O(N)K,0 = (4.8.4)  SO(N)K × (Z2)(K−2)(N−1) odd N, odd K.

In both (4.8.3) and (4.8.4), the Z2 quotients use the product of χ in the continuous factor and the electric Wilson line of the Z2 gauge theories in the numerator, and thus the theories are not based on quotients of the gauge group that appears in the numerator. (See the discussion around (4.1.16).)

4.8.1 Even N and odd K

We start with the conformal embedding with NK real fermions

Spin(N)K × SO(K)N ⊂ Spin(NK)1 , (4.8.5) where in the above the relation between the centers of the left and right chiral algebras is subject to the same discussion as that following (4.2.5).

Consider gauging (C, M) on the left hand side (and an outer automorphism equivalent to C on the right). C transforms the currents of Spin(N)K by changing the sign of J1i with i = 2, ··· ,N. To be compatible with the transformation the NK indices are partitioned into K and NK as in section 4.2.1. Therefore we obtain the conformal embedding

+ P in (N)K × Spin(K)N ⊂ Spin(K)1 × Spin(NK − K)1 . (4.8.6)

35 0 1 From (4.8.3) and (4.8.4) one can also ﬁnd the one-form global symmetry of O(N)K,0,O(N)K,0 for odd 0 1 N. In O(N)K,0 it is Z2 × Z2 for odd N and even K, while Z2 for odd N and K. In O(N)K,0 it is Z2 for odd N and even K, Z2 × Z2 for odd K and N = 1 mod 4, and Z4 for odd K and N = 3 mod 4.

166 The conformal embedding implies the duality of chiral algebras

+ Spin(K)1 × Spin(NK − K)1 P in (N)K ←→ , (4.8.7) Spin(K)N where we again use the fact that each factor in the subalgebra in (4.8.6) acts faithfully. (See the discussion around (4.2.13).) This implies the non-spin Chern-Simons duality

+ Spin(K)−N × Spin(K)1 × Spin(NK − K)1 P in (N)K ←→ . (4.8.8) Z2

We denote the generators of the Z2 × Z2 simple currents on the left-hand side by χ, jC . 1 They have spin of 2 and 0 respectively. Gauging the Z2 one-form symmetry generated by 0 χ results in the theory O(N)K , while gauging the Z2 one-form symmetry generated by jC results in the theory Spin(N)K . On the right-hand side χ maps to (1, 1, χ) and jC maps to (1, χ, χ). (See also footnote8.)

Taking the Z2 quotient in (4.8.8) generated by χ on the left and (1, 1, χ) on the right produces the spin duality

0 Spin(K)−N × Spin(K)1 O(N)K,0 ←→ , (4.8.9) Z2

0 which implies the spin Chern-Simons duality O(N)K,0 ↔ Spin](K)−N,−K . By adding a

(Z2)K theory on both sides and gauging a one-form symmetry as in appendix 4.12, the above implies

0 O(N)K,K ←→ Spin(K)−N . (4.8.10)

0 For the special case K = 1 the spin duality (4.8.9) reproduces O(N)1,0 ↔ Spin(1)1 with

Spin(1)N ≡ (Z2)0 for even N.

167 + Consider next the Z2 quotient in (4.8.8), which on the left brings P in (N)K to

Spin(N)K , and on the right it is generated by (1, χ, χ). We ﬁnd

Spin(N)K ←→ SO(K)−N × Spin(NK)1 . (4.8.11)

Comparing the right-hand side with (4.8.3), which deﬁnes the O(K) Chern-Simons theory for odd K, gives the spin Chern-Simons duality

0 Spin(N)K ←→ O(K)−N,−N . (4.8.12)

+ Similarly consider the Z2 quotient in (4.8.8), which on the left brings P in (N)K to

1 O(N)K , and on the right it is generated by (1, χ, 1) :

1 Spin(K)−N × Spin(NK − K)1 O(N)K ←→ . (4.8.13) Z2

Comparing with (4.8.4) gives the spin Chern-Simons duality

1 1 O(N)K,K−1 ←→ O(K)−N,−N+1 . (4.8.14)

4.8.2 Odd N,K

We start from the conformal embedding for odd N,K

Spin(N)K × Spin(K)N ⊂ Spin(NK)1 . (4.8.15)

168 Each factor of the subalgebra above acts faithfully and thus we obtain the following duality of chiral algebras

Spin(NK)1 Spin(N)K ←→ . (4.8.16) Spin(K)N

The corresponding non-spin Chern-Simons duality is

Spin(K)−N × Spin(NK)1 Spin(N)K ←→ . (4.8.17) Z2

Promoting to spin theories and comparing the right-hand side with (4.8.3) gives the spin Chern-Simons duality

0 Spin(N)K ←→ O(K)−N,−N . (4.8.18)

Note also that since N and K are both odd we also have the duality (4.1.14). Combining with the above, this implies

0 0 O(N)K,K−NK ←→ O(K)−N,−N . (4.8.19)

Taking the Z2 quotient in (4.8.17) generated by χ of Spin(N)K , Spin(K)−N gives

SO(N)K ←→ SO(K)−N . (4.8.20)

Comparing with (4.8.4) produces the spin Chern-Simons duality

1 1 O(N)K,K−1 ←→ O(K)−N,−N+1 . (4.8.21)

Therefore for N or K odd we establish the same dualities (4.8.1), and (4.2.1) follows.

169 Note as an application of (4.2.1) and (4.8.2), we can express the Chern-Simons action of SO(n)k with even n and odd k coupled to backgrounds for symmetries C, M as follows

C M C M SO(n)k[B ,B ] = SO(n)k[B , 0] − (nk + k − 2) f[B ]

− (k − 2)f[BC] + (k − 2)f[BC + BM] even n, odd k . (4.8.22)

4.9 Appendix F: Low Rank Chiral Algebras and

Level-Rank Duality

In this appendix we present some details of familiar chiral algebras and their relation to level-rank duality.

4.9.1 Chiral Algebras Related to so(2)

Let us describe the chiral algebras related to the Lie algebra so(2).36 We consider

+ SO(2)k = U(1)k , Spin(2)k = U(1)4k ,O(2)k,0 , P in (2)k ↔ O(2)4k,0 . (4.9.1)

+ We start with P in (2)k,0, its chiral algebra is the charge conjugation orbifold of

Spin(2)k = U(1)4k that takes charge Q to −Q mod 4k. This chiral algebra was constructed in [40]. The spectrum is built from U(1)4k as:

• the identity in U(1)4k is split into the invariant part 1 and j = i∂φ. They have conformal dimensions 0,1.

1 • the original primary of charge 2k splits into a symmetric part φ2k and an antisym-

2 metric part φ2k. They have conformal dimensions k/2.

36 − For simplicity we will not discuss P in (2)k.

170 • the original primary of charge q is identiﬁed with charge 4k − q, thus we have the

2 states φq with q = 1, 2, ··· , 2k − 1. The conformal dimension is q /(8k).

• the twisted sectors of the original identity and primary of charge 2k. Denote them

1 2 1 1 2 9 by σ , σ of conformal dimension 16 and τ , τ of conformal dimension 16 .

The modular S matrix is summarized in table 4.4[40].

1 j φa φ σa τ a 2k q √ √ 1 1 1 1 2 √2k √2k j 1 1 1 2 − 2√k − 2√k b q a+b a+b φ2k 1 1 1 2(−1) (−1) 2k (−1) 2k q0 πqq0 φ 0 2 2 2(−1) 4 cos 0 0 q √ √ √ 4k √ √ b a+b σ √2k −√2k (−1) √2k 0 δa,b √2k −δa,b√ 2k b a+b τ 2k − 2k (−1) 2k 0 −δa,b 2k δa,b 2k

√ + Table 4.4: Modular S matrix for P in (2)k, normalized by S1,1 = 1/(4 k). The labels are a, b = 1, 2, q, q0 = 1, ··· 2k − 1.

The fusion rules can be obtained from the modular S matrix (see [40]). In particular

1 2 there are Z2 × Z2 simple currents 1, j, φ2k, φ2k. They have conformal weights 0, 1, k/2, k/2 respectively. These currents are the Abelian anyons generating the one-form global symmetry. In particular by extending the chiral algebra by these currents we can obtain the other algebras in (4.9.1).

+ • Extending the chiral algebra P in (2)k by j. From the modular S matrix we ﬁnd that

a a 1 2 σ , τ are projected out, and from the fusion rules we ﬁnd that φ2k, φ2k are identiﬁed.

Thus the resulting chiral algebra is U(1)4k = Spin(2)k.

+ 1 • Extending the chiral algebras P in (2)k by φ2k. From the modular S matrix we ﬁnd

2 2 that σ , τ and φq with odd q are projected out.

1 For even k the extending primary φ2k has integral conformal weight, and from the

2 fusion rules we find that j, φ2k are identified, φq, φ2k−q are identified (for q 6= k), and

171 1 1 ± 1 1 φk, σ , τ are doubled. Thus the spectrum in the new theory is 1, j, φk , σ±, τ± and

φ2, φ4 ··· φk−2 i.e. k/2 + 7 primaries. Thus the chiral algebra is O(2)k,0.

1 For odd k the extending primary φ2k has half-integral conformal weight thus there is

1 no doubling and the primaries diﬀered by fusing with φ2k are not identiﬁed. Thus the

1 2 1 1 spectrum in the new theory is 1, j, φ2k, φ2k, σ , τ and φ2, φ4, ··· , φ2k−2 together k + 5

primaries. The resulting chiral algebra is O(2)k,0. This corresponds to a spin TQFT.

+ 2 • Extending the chiral algebras P in (2)k with φ2k. Since the modular S matrix respects

1 2 1 2 1 2 φ2k ↔ φ2k, σ ↔ σ , τ ↔ τ , we find that the extended chiral algebra is also O(2)k,0. This is consistent with the fact that the discrete θ-parameter differentiating O(N)1 from O(N)0 does not exist for N = 2, and further agrees with the classification of bosonic topological gauge theory with O(2) gauge group [30,41,143].

4.9.2 Chiral Algebras Related to so(4)

Next we discuss chiral algebras related to so(4). The simplest is Spin(4)k which is simply the product SU(2)k × SU(2)k.

+ The chiral algebra P in (4)k,0 is the charge conjugation orbifold of Spin(4)k that acts on representations by (j1, j2) ↔ (j2, j1) where ja are SU(2) × SU(2) spins. The spectrum is (see e.g. [26]):

2 • Oﬀ-diagonal ﬁelds (j1, j2) with j1 6= j2. There are [(k + 1) − (k + 1)]/2 = k(k + 1)/2 of them. Their conformal weight is

j (j + 1) j (j + 1) h = 1 1 + 2 2 . (4.9.2) k + 2 k + 2

• Diagonal ﬁelds j1 = j2 = j denoted by (j, s) where s = 0, 1 denotes two states in the (two dimensional) untwisted sector. There are (k + 1) of them. Their conformal

172 weight is 2j(j + 1) h = . (4.9.3) k + 2

• Twisted diagonal ﬁelds j1 = j2 = j denoted by ([j, s), where s = 0, 1 denotes two states in the (two-dimensional) twisted sector. Their conformal weight is

j(j + 1) 3k s h = + + . (4.9.4) 2(k + 2) 16(k + 2) 2

The modular S matrix is given in [26]. We list some of the entries

SU(2) SU(2) SU(2) SU(2) S(i,j),(i0,j0) = Si,i0 Sj,j0 + Si,j0 Sj,i0 ,

SU(2) SU(2) S(i,j),(i0,s) = Si,i0 Sj,i0 ,

S(i,j),([i0,s) = 0 , (4.9.5) 2 1 SU(2) S 0 0 = S 0 , (i,s),(i ,s ) 2 i,i 1 s SU(2) S = (−1) S 0 . (i,s),(\i0,s0) 2 i,i

Denote by 1, J, σ and Jσ the Z2 × Z2 simple currents (j = 0, s = 0), (j = 0, s = 1), (j = k k k k 2 , s = 0) and (j = 2 , s = 1) of conformal weights 0, 1, 2 , 2 respectively. In particular, + extending the chiral algebra P in (4)k,0 with J produces Spin(4)k by projecting out the twisted sectors, identifying s = 0, 1 for the untwisted diagonal fields and doubling the off-diagonal fields.

+ 0 Consider extending the chiral algebra P in (4)k,0 with σ. The resulting theory is O(4)k,0. For odd k there are no ﬁxed modules under fusion with σ, and therefore the spectrum is the

+ subset of primaries in P in (4)k,0 that are mutually local with σ. Namely, we exclude the oﬀ-diagonal ﬁelds (j1, j2) for j1, j2 not both SU(2) tensor or both spinor, and the diagonal

173 ﬁeld (\j0, s) for SU(2) spinor j0. For even k we need to take into account the identiﬁcation

0 and doubling, and the spectrum of O(4)k,0 theory is

k k k k • Oﬀ-diagonal ﬁelds (j1, j2) ⊕ (j2, j1) ∼ ( 2 − j1, 2 − j2) ⊕ ( 2 − j2, 2 − j1) with j1, j2 both k either SU(2) tensor or spinor, and j1 6= j2 or 2 −j2. There are doubled representations k k k [(j, 2 − j) ⊕ ( 2 − j, j)]± for j < 4 .

k k k • Diagonal ﬁelds (j, s) ⊕ ( 2 − j, s) for j < 4 and the doubled representation ( 4 , s)±.

[ k • Twisted diagonal ﬁelds (j, s)± with SU(2) tensor j = 0, 1, ··· 2 (all of them are doubled).

+ Instead, consider extending the chiral algebra P in (4)k with σJ. The resulting theory

1 is O(4)k,0. For odd k the spectrum is the original spectrum without the primaries that are projected out, namely without the off-diagonal fields (j1, j2) for j1, j2 not both SU(2) tensor or both spinor, and without the diagonal field (\j0, s) for SU(2) tensor j0. For even

1 k we need to take into account the identiﬁcation and doubling, and the spectrum of O(4)k theory is

0 • Oﬀ-diagonal ﬁelds the same as that of O(4)k.

k 1 k • Diagonal ﬁelds (j, s = 0) ⊕ ( 2 − j, s = 1) with j = 0, 2 , 1, ··· 2 .

[ 1 3 k−1 • Twisted diagonal ﬁelds (j, s)± with SU(2) spinor j = 2 , 2 , ··· 2 (all of them are doubled).

Note that unlike the previous section, which discussed O(2)k chiral algebras, here we see

0 1 that O(4)k,O(4)k have diﬀerent twisted sectors and are distinct theories.

As a check, the spins match in the spin Chern-Simons duality

1 1 O(4)4,3 ←→ O(4)−4,−3 . (4.9.6)

174 1 Here, the Z2 quotient in O(4)4,3 is generated by the product of J and the basic Wilson line in (Z2)3. The quotient pairs the twisted sector with the magnetic lines of (Z2)3. In particular the lowest-dimension twisted-sector primary matches in the two theories: the \ 3 conformal weight of (1/2, 0) is 16 , which is cancelled up to an integer by the basic magnetic line of (Z2)3. Thus this primary maps to itself under the duality (4.9.6).

4.10 Appendix G: Projective Representations in

SO(4)4 with CM 6= MC

The anyons of SO(4)4 can be labelled by spin (j1, j2) of Spin(4) = SU(2) × SU(2), with the extending representation σ = (2, 2). It acts as by fusion as σ · (j1, j2) = (2 − j1, 2 − j2). Fixed points under fusion with σ lead to a pair of distinct representations of the extended chiral algebra, denoted by (j1, j2)±. Meanwhile, for (j1, j2) 6= σ · (j1, j2) the representation of the extended chiral algebra is (j1, j2) ⊕ σ · (j1, j2).

The resulting list of anyons of SO(4)4, together with their conformal weights h =

1 6 (j1(j1 + 1) + j2(j2 + 1)) , are summarized in table 4.5.

(j1, j2) (0, 0) (0, 1) (1, 0) (1, 1)+ (1, 1)− (0, 2) (1/2, 1/2) (1/2, 3/2) h 0 1/3 1/3 2/3 2/3 1 1/4 3/4

Table 4.5: Anyons and their conformal dimensions for the theory SO(4)4.

The chiral algebra has Z2 × Z2 anyon permutation symmetry C, M deﬁned by

C : (0, 1) ←→ (1, 0)

M : (1, 1)+ ←→ (1, 1)− . (4.10.1)

175 In addition the theory has time-reversal symmetry that permutes the anyons as (0, 1) ↔

1 1 1 3 (1, 1)+, (1, 0) ↔ (1, 1)− and ( 2 , 2 ) ↔ ( 2 , 2 ). We will focus on the unitary symmetries C, M. Consider anyons that are not permuted by C, M. M assigns ±1 to the two states in the representation (j1, j2) ⊕ σ · (j1, j2) respectively. C is inherited from the symmetry that exchanges (j1, j2) as Spin(4) representations. C may not preserve the value assigned by M.

This happens when σ · (j1, j2) = C(j1, j2) = (j2, j1), namely the anyon (j1, j2) is invariant under C only up to fusion with the extending representation σ. This occurs for the primaries

1 3 2 (j1, j2) = (0, 2), ( 2 , 2 ), and they realize the symmetry projectively with (CM) = −1. The value of (CM)2 for each anyon can be changed by a sign that preserves the fusion rules and anyon permutations. The sign comes from braiding the anyon with the Abelian

2 1 1 1 3 anyon (0, 2), which is not permuted by C, M. Thus the values of (CM) for ( 2 , 2 ), ( 2 , 2 ) can be +1, −1 or −1, +1, while the values for (0, 0), (0, 2) stay unchanged.

Consider the following two orbits of three-punctured spheres under the permutation C, M:

{|(0, 1), (0, 1), (0, 0)i, |(1, 0), (1, 0), (0, 0)i} and

{|(0, 1), (0, 1), (0, 2)i, |(1, 0), (1, 0), (2, 0)i} .

Since (CM)2 = +1 for (0, 0) and (CM)2 = −1 for (0, 2), the two orbits of three-punctured spheres cannot both be in a linear representation of C, M. Thus, the three-punctured spheres in general are in projective representation of C, M.37

This does not occur in every orbit of three-punctured spheres (in particular, the sphere without anyons is in the trivial linear representation of the symmetry), and it does not imply

37 We thank M. Barkeshli and M. Cheng for pointing out the same result can be derived from the method of [17] for SO(4)4 as an abstract TQFT.

176 the symmetries C, M have an ‘t Hooft anomaly. In fact gauging C, M without counterterm

+ produces the well-deﬁned P in (4)4 theory.

4.11 Appendix H: O(2)2,L as Family of Pfaﬃan States

In this appendix we show that the Pfaﬃan and T-Pfaﬃan theories are special cases of the spin Chern-Simons theory O(2)2,L for odd L. For time-reversal invariant theories, we also analyze the mixed anomaly between the time-reversal and U(1) symmetries.

The theories O(2)2,L as spin Chern-Simons theories can be expressed as

U(1)8 × Spin(L)−1 O(2)2,L ←→ , (4.11.1) Z2

where we used O(2)2,0 ↔ U(1)8 and (4.5.6). For odd L the theory has 12 lines, with non-Abelian fusion algebra that can be read oﬀ from the right hand side.

The Moore-Read Pfaﬃan theory [97] can be expressed as the spin Chern-Simons theory [122]

SU(2) × U(1) × U(1) 2 −4 8 , (4.11.2) Z2 × Z2 where the ﬁrst Z2 quotient acts on SU(2)2 × U(1)−4 and the second quotient acts on

U(1)−4 × U(1)8. To simplify the quotients we can use the duality U(1)−4 × U(1)8 ↔

U(1)4 × U(1)−8 (which can be proven by a change of variables as in [122]). In the new variables, the ﬁrst Z2 quotient acts diagonally on SU(2)2 × U(1)4 × U(1)−8, while the second Z2 quotient acts on U(1)4 and turns it into the trivial spin TQFT U(1)1. Thus the quotient simpliﬁes to

SU(2)2 × U(1)−8 Moore-Read = × U(1)1 . (4.11.3) Z2 177 ∼ Since SU(2)2 = Spin(3)1, the Moore-Read theory is equivalent to O(2)−2,−3 ↔ O(2)2,7 up to a gravitational Chern-Simons term, where we used the second duality in (4.0.5). The

1 1 1 1 1 1 5 5 spins of the 12 lines in the theory modulo integers are {0, 0, 2 , 2 , ± 4 , ± 4 } and { 8 , 8 , 8 , 8 }, and thus the theory is not time-reversal invariant. The time-reversal of the Moore-Read

Pfaﬃan theory is also called the anti-Pfaﬃan theory, and it is dual to O(2)2,3.

38 Both O(2)2,1 and O(2)2,5 are time-reversal invariant by the level-rank duality (4.0.5):

O(2)2,1 ←→ O(2)−2,−1

O(2)2,5 ←→ O(2)−2,−5 . (4.11.4)

The theory O(2)2,1 is equivalent to the T-Pfaffian theory [25, 29], which can be expressed as U(1)8 × Ising /Z2. The theory O(2)2,5 has the same fusion algebra as O(2)2,1, but 5 1 the spin of the twist field in the Ising anyons is different (spin − 16 instead of − 16 ). In particular, both O(2)2,1 and O(2)2,5 have Z4 one-form symmetry. They are related by

O(2)2,1 × (Z2)4 O(2)2,5 ←→ , (4.11.5) Z2 where the quotient on (Z2)4 uses the Wilson line (of integral spin). In particular, this means that the theory is not deﬁned by a quotient of the gauge group in the numerator.

(See the discussion around (4.1.16).) Note (Z2)4 ↔ U(1)2 ×U(1)−2 is the semion-antisemion ∼ theory, and it is time-reversal invariant. Since O(2)2,5 = O(2)2,−3 and (Z2)−3 ↔ Spin(3)1 =

SU(2)2, the spin Chern-Simons theories O(2)2,5 and U(2)2,4 are dual, where U(2)2,4 is time- reversal invariant by the level-rank duality (4.0.2).

We can couple the theories O(2)2,1,O(2)2,5 to a background U(1) gauge ﬁeld A and investigate the mixed anomaly between the U(1) and the time-reversal symmetries. The

38 In fact, from the spins of lines one can conclude that O(2)2,L for other integers L 6= 1 mod 4 are not time-reversal invariant.

178 O(2) Chern-Simons theory does not have U(1) magnetic symmetry, but it can couple to U(1) symmetry using its one-form symmetry.39

Both O(2)2,1 and O(2)2,5 have Z4 one-form symmetry, and the generating line has spin 1 4 . Consider coupling the dualities (4.11.4) to the two-form background Z4 gauge ﬁeld B2 (with the period in (2π/4)Z) for the one-form symmetry:

8 Z 8 Z O(2)2,1[B2] + B2B2 ←→ O(2)−2,−1[B2] − B2B2 4π bulk 4π bulk 8 Z 8 Z O(2)2,5[B2] + B2B2 ←→ O(2)−2,−5[B2] − B2B2 , (4.11.6) 4π bulk 4π bulk

8 R where the bulk terms ± 4π B2B2 are equivalent on a closed four-manifold as expected from the anomaly matching for the dualities. By substituting the value for the background

1 1 R B2 = 4 dA we ﬁnd the mixed anomaly 8π dAdA between the time-reversal symmetry and the U(1) symmetry.40 This is the same parity anomaly as that of one massless Dirac fermion. The lines can carry fractional U(1) charges determined by their charges under the one-form symmetry.

4.12 Appendix I: Duality Via One-form Symmetry

Consider a TQFT T with an Abelian anyon a of spin L/(2N) mod 1 that generates a

41 ZN one-form symmetry (we assume NL is even for simplicity). Then we can prove the 39 This is an example of coupling a TQFT to an ordinary global symmetry using the Abelian anyons as in [17] for discrete symmetries. Here we couple O(2)2,L to the continuous U(1) symmetry that does not permute anyons. The O(2)2,L theory also has an intrinsic Z2 magnetic symmetry that permutes anyons (it is not the Z2 subgroup of the U(1) symmetry). 40 1 The analysis of the mixed anomaly can be generalized to the theories SO(N)N , O(N)N,N−1 and 1 O(N)N,N+3, which are time-reversal invariant by the level-rank dualities (4.0.3),(4.0.5). When N = 2 mod 4 they have one-form symmetry with non-trivial ‘t Hooft anomaly. 41 For example, T can be the Chern-Simons theory with gauge algebra so(N), with the one-form symmetry discussed in section 4.1.

179 following duality42

T × (Z ) Even L : T ←→ N 0 ZN T × (Z ) Odd L : T ←→ N N , (4.12.1) ZN

k N where (ZN )k denotes the Abelian Chern-Simons theory 4π xdx + 2π xdy with U(1) gauge

ﬁelds x, y. The ZN quotient on the (ZN )0 and (ZN )N theories are generated by the line H L H H L−1 H exp i y − i 2 x and exp i y − i 2 x with even and odd L respectively. (Note that L is deﬁned mod 2N.)

To prove (4.12.1) denote the ZN two-form gauge ﬁeld of the gauged one-form symmetry on the right by B2 and couple it to (ZN )0 and (ZN )N for even and odd L as

Z N N L NL Z Even L : xdy + B2(y − x) − B2B2 X=∂M4 2π 2π 2 4π M4 Z N N N L − 1 NL Z Odd L : xdx + xdy + B2(y − x) − B2B2(4.12.2), X=∂M4 4π 2π 2π 2 4π M4 and the one-form gauge transformation is

L Even L : B → B − dλ, x → x + λ, y → y − λ 2 2 2 L + 1 Odd L : B → B − dλ, x → x + λ, y → y − λ . (4.12.3) 2 2 2

In (4.12.2) the last terms B2B2 can be ignored, since they will be cancelled by the ‘t

Hooft anomaly of the ZN one-form symmetry in T . In this step, we use the fact that the ‘t Hooft anomaly of a one-form symmetry is given by the self-braiding of the generating line [54,61,78].

42 Notice the intentional change of font below. The theory (Z2)L is not the same as the theory (ZN )L when N = 2. They are related by (4.5.4) (i.e. (Z2)L ↔ (Z2)2L).

180 To simplify (4.12.2) we can perform the one-form gauge transformation with λ = −x, which change the gauge ﬁelds (B2, x, y) to (B2+dx, 0, y+Lx/2) for even L and (B2+dx, 0, y+ (L + 1)x/2) for odd L. Then (4.12.2) becomes R N(B + dx)y/(2π) where y = y + Lx/2 X 2 e e for even L and ye = y + (L + 1)x/2 for odd L. Integrating out ye constrains B2 to be a trivial two-form ZN gauge ﬁeld and removes the quotient. Therefore the theory on the right of (4.12.1) is the original theory T .

A simple example illustrating the duality (4.12.1) is the relationship between the SU(N) Chern-Simons theory, and U(N) Chern-Simons theory coupled to a U(1) multiplier that constrains the U(N) gauge ﬁeld [67,78]. Consider SU(N)K × (ZN )NK with the Lagrangian

K 2 NK N Tr bdb + b3 + xdx + xdy , (4.12.4) 4π 3 4π 2π where b is an SU(N) gauge ﬁeld, and x, y are U(1) gauge ﬁelds. For simplicity we will take

K to be even. Next we perform a ZN quotient such that b, x are not properly quantized but the U(N) gauge ﬁeld u = b + 1N x is well-deﬁned. The Lagrangian in the new variables is

K 2 1 Tr udu + u3 + (Tr u)dy , (4.12.5) 4π 3 2π

which is the U(N)K,K Chern-Simons action constrained by a U(1) multiplier y, and we recognize it as the SU(N)K Chern-Simons action [67]. Thus we ﬁnd that for even K,

SU(N)K × (ZN )NK SU(N)K ←→ , (4.12.6) ZN

where the quotient on (ZN )NK leads to selection rule on Wilson line of x but not for y, H H ∼ namely it is generated by exp(i y + iK x). For even K we can use (ZN )NK = (ZN )0

K by the redeﬁnition y → y − 2 x, then the quotient on (ZN )0 is generated by the line

181 H K H exp(i y + i 2 x). This agrees with the duality (4.12.1), where the generating line of the

ZN one-form symmetry in SU(N)K has spin −K/(2N) and thus L = −K.

182 Chapter 5

Global symmetries, anomalies, and duality in (2 + 1)d

The main boson/fermion dualities that we will study are [2,67]

SU(N)k with Nf scalars ←→ U(k) Nf with Nf fermions −N+ 2

U(N)k with Nf scalars ←→ SU(k) Nf with Nf fermions −N+ 2

U(N)k,k±N with Nf scalars ←→ U(k) Nf Nf with Nf fermions (5.0.1) −N+ 2 ,−N∓k+ 2

conjectured to hold for Nf ≤ N (our notation is U(N)k ≡ U(N)k,k), and [3]

SO(N)k with Nf real scalars ←→ SO(k) Nf with Nf real fermions −N+ 2

USp(2N)k with Nf scalars ←→ USp(2k) Nf with Nf fermions (5.0.2) −N+ 2

for Nf ≤ N in the USp case, and Nf ≤ N − 2 if k = 1, Nf ≤ N − 1 if k = 2, and Nf ≤ N if k > 2 in the SO case.

183 1 with a fermion 1 with a scalar S 2 with a fermion S 2 with a scalar

య భ ଶ ଵ ܷ ିమ ܷ ܷ ିమ ܷ Global 2 symmetry Global SO 3 symmetry

ܱ / duality

ܷܵ ܷ / or / duality / duality / duality ܷ ܷ ܱܵ ܱܵ ܷܵ ܷ ܵܲ ܵܲ

Same IR fixed point Global SO 3 symmetry

Figure 5.1: Four UV theories, related by dualities, ﬂow to the same IR ﬁxed point. The two UV theories with SU(2) gauge symmetry have a global SO(3) symmetry, and therefore the IR theory also has that global symmetry. The two UV theories with U(1) gauge symmetry have only O(2) global symmetry. The duality implies that they have an enhanced quantum SO(3) global symmetry in the IR.

In the fermionic theories the only interactions are gauge interactions. On the contrary, the scalar theories also have generic quartic potential terms compatible with a given global symmetry. Hence, it is important to specify what symmetry we impose, as different choices in general lead to different fixed points. We will discuss it in more details below.

Some special examples of the dualities in (5.0.1) and (5.0.2) lead to

U(1)0 with a scalar ←→ a scalar

U(1)1 with a scalar ←→ a free fermion

U(1)2 with a scalar ←→ SU(2)1 with a scalar. (5.0.3) | {z } enhanced SO(3) global symmetry

184 The ﬁrst duality is the celebrated particle/vortex duality of [37, 111]. The second duality maps an interacting bosonic theory to a free fermion [121]. The theory in the third duality has a quantum SO(3) global symmetry [3] (see ﬁgure 5.1). In all these cases the monopole

k operator of U(1)k in the theory on the left side of the duality, whose spin is 2 , is an important operator in the theory on the right side. It is the scalar in the ﬁrst case, it is the free fermion in the second case, and it is the new current of the enhanced SO(3) symmetry in the third case.1

All these dualities are IR dualities. We start at short distances with a renormalizable Lagrangian and impose some global symmetry on its terms. Then, we scan the relevant deformations that are consistent with the global symmetry. These are typically mass terms, but there are also others. For generic values of these parameters the low-energy theory is gapped. As these parameters are varied there could be phase transitions between different phases and the phase transition points occur at fine-tuned values of the scanned parameters. We will assume that, as we vary these parameters, the phase transitions can be second order. Then the long-distance physics is described by a fixed point of the renormalization group, which is a continuum conformal field theory. The statement of the IR duality is about this fixed point and its neighborhood. If, on the other hand, the IR theory is always gapped with possible first-order transitions between phases, the statement of the duality is significantly weaker and it applies only to the gapped phases.

1 One might wonder whether the theory of U(1)3 with a scalar, which has a monopole operator of spin 3 and a global U(1) symmetry, could have N = 2 supersymmetry in the IR. It has a dual description as 2 ∼ SO(3)− 3 = SU(2)−3/Z2 with a fermion in the adjoint [3], which seems to have N = 1 supersymmetry. How- 2 ∼ ever, this supersymmetric theory is expected to be gapped [148] with a low energy SO(3)−1 = SU(2)−2/Z2 trivial TQFT. As we vary the fermion mass, we can ﬁnd a transition to another gapped phase with a TQFT ∼ SO(3)−2 = SU(2)−4/Z2 ↔ SU(3)−1 ↔ U(1)3. The duality statement could mean that the theory at this transition point is dual to the U(1)3 theory with a scalar. However, since we needed to change the fermion mass from the supersymmetric point, we broke supersymmetry explicitly and there is no reason to believe that the IR theory at the transition point is supersymmetric. Alternatively, if the supersymmetric N = 1 SO(3) 3 theory is actually gapless, it could be dual to the U(1)3 theory with a scalar, in which case it will − 2 also have enhanced N = 2 supersymmetry.

185 5.0.1 Global symmetries

Our starting point is to identify the correct global symmetry of a quantum ﬁeld theory. For the moment we ignore discrete symmetries like time reversal T and higher-form global symmetries [54,78]. We will discuss them later.

We should distinguish between the global symmetry of the UV theory GUV and the global symmetry of the IR theory GIR. Although there might be elements in GUV that do not act on the IR degrees of freedom, we should still pay attention to them in the IR. The IR eﬀective action might contain topological local counterterms for background gauge ﬁelds coupled to those elements.

Conversely, there could be new elements in GIR that are not present in the UV. These lead to an accidental or quantum symmetry in the IR. These symmetries are approximate and are violated by higher-dimension operators in the IR theory. Examples of such quantum symmetries are common in (1 + 1)d ﬁeld theories and have played an important role in supersymmetric dualities, in particular in (2 + 1)d mirror symmetry [38,70].

A noteworthy simple example [3] is summarized in figure 5.1, where four different UV theories, some with GUV = O(2) and some with GUV = SO(3), flow to the very same IR fixed point with SO(3) global symmetry (we will discuss this example in Section 3).

These considerations are extremely important in the context of duality. Two dual theories TA and TB that ﬂow to the same IR ﬁxed point must have the same global symmetries.

UV UV In some cases the UV symmetries are the same GA = GB . But it is also common that

UV UV the UV symmetries are diﬀerent GA 6= GB , and yet they are enhanced to the same IR symmetry GIR. Again, the example in ﬁgure 5.1 demonstrates it and gives interesting consistency checks on the various dualities. We will see several examples of that in Section 3.

186 When we discuss the global symmetry G (either GUV or GIR) we should make sure that it acts faithfully on the operators. Speciﬁcally, we will see many examples where all the local gauge-invariant operators in the theory transform in certain representations of the naive global symmetry group Gnaive, but the true global symmetry G—which acts faithfully—is a quotient G = Gnaive/C by an appropriate C.

A key tool in the analysis of a quantum field theory is its coupling to background gauge fields for the global symmetry. If we misidentify the global symmetry and couple the system to background Gnaive gauge fields, we miss important observables. In particular, if all the local operators transform trivially under C ⊂ Gnaive we can couple the system to

G = Gnaive/C bundles, which are not G bundles.

For example, consider the SU(2)1 theory with a scalar in figure 5.1. The naive global symmetry is Gnaive = SU(2). However, in this case all gauge-invariant operators in the theory are in integer isospin representations of this group and therefore the true global symmetry is G = Gnaive/Z2 = SO(3). This means that the system can be coupled to additional background fields—SO(3) gauge fields, which are not SU(2) gauge fields. The response to such more subtle backgrounds leads to interesting observables, which give us more diagnostics of the theory.

More explicitly, we can couple the matter ﬁelds to

SU(2)dyn × SU(2)naive /Z2 (5.0.4)

gauge fields. This is consistent because the matter fields do not sense the Z2 quotient. The expression (5.0.4) means that when the classical fields are ordinary SU(2)naive gauge fields, the dynamical gauge fields are ordinary SU(2)dyn gauge fields. However, when the classical fields are nontrivial SO(3) gauge fields (i.e. SO(3) fields with nontrivial second Stiefel-

187 Whitney class w2), also the dynamical fields are in SO(3) bundles. This demonstrates that by using SO(3) background fields we can probe more twisted sectors of the dynamical fields.

Below we will see many generalizations of this example. We will encounter dynamical

fields b for a gauge group Gdyn and background fields B for the true global symmetry of the model G = Gnaive/C for some C. As in the example (5.0.4), the dynamical and classical fields can be combined to a gauge field B with group

Gdyn × Gnaive /C . (5.0.5)

If the classical ﬁelds B are in Gnaive bundles, the dynamical ﬁelds b are in Gdyn bundles.

But when B are in nontrivial G = Gnaive/C bundles, also the dynamical ﬁelds b are in

Gdyn/C bundles rather than in Gdyn bundles. The consistency of the theory under gauge transformations in (5.0.5) and possible anomalies in these transformations will be extremely important below.

We should point out that the authors of [79, 80] have examined such anomalies for discrete groups from a diﬀerent perspective.

We will be particularly interested in the theories in (5.0.1) and (5.0.2), so let us discuss their UV symmetry GUV. In the fermionic case that is the actual UV symmetry of the theory, while in the bosonic case that is the symmetry that we impose on the quartic

2 C potential. The naive UV global symmetry Gnaive is U(Nf ) o Z2 (where the second factor

C is charge conjugation) in the theories with SU gauge group, SU(Nf ) × U(1)M o Z2 (where

M C the second factor is the magnetic symmetry) for U gauge group, O(Nf ) × Z2 × Z2 for

3 SO gauge group, and USp(2Nf ) for USp gauge group. However, we will ﬁnd that the

2In this discussion we mostly neglect time-reversal symmetry T. 3 C C In this case Z2 is an element of O(N) not connected to the identity. When N,Nf are both odd, Z2 is already contained in O(Nf ) (up to a gauge transformation), and should not be listed as an independent symmetry.

188 faithfully-acting symmetry GUV is (we will not discuss the SO case here):

Theory Global Symmetry GUV

C SU(N)r with Nf scalars or fermions U(Nf )/ZN o Z2

C U(N)k with Nf scalars U(Nf )/Zk o Z2

C U(k) Nf with Nf fermions U(Nf )/ZN o Z2 −N+ 2

USp(2N)r with Nf scalars or fermions USp(2Nf )/Z2 (5.0.6)

Nf where r is an integer in the theory with scalars and an integer plus 2 in the theory with 2πi/N fermions. Here by U(Nf )/ZN we mean the quotient by e 1. In the special cases of U(N)0

N with Nf scalars and U(k) f with Nf fermions, the global symmetry is SU(Nf )/ZNf × 2 C 4 C U(1)M o Z2 which is isomorphic to U(Nf )/ZNf o Z2 . One should be careful at small values of the ranks. For instance, SU(2)r with Nf fermions has USp(2Nf )/Z2 symmetry as manifest in the USp(2)r description, while the symmetry of SU(2)r with Nf scalars depends on what we impose on the quartic potential. This will be analyzed in Section 3.

5.0.2 Anomalies

It is often the case that the global symmetry G has ’t Hooft anomalies. This means that the correlation functions at separated points are G invariant, but the contact terms in correlation functions cannot be taken to be G invariant. Related to that is the fact that the system with nontrivial background gauge ﬁelds for G is not invariant under G gauge transformations. Often, this lack of G gauge invariance of background ﬁelds can be avoided by coupling the system to a higher-dimensional bulk theory with appropriate bulk terms.

4 More generally U(Nf )/ZN is isomorphic to U(Nf )/ZN+Nf . Representing U(Nf )/ZN as g ∈ 2πi/N −2πi/N 2πi/N SU(Nf ), u ∈ U(1) with (g, u) ∼ e f g, e f u ∼ g, e u , the isomorphism is (g, u) → g, v = N 2πi − 2πi 2πi N+Nf Nf Nf N+Nf u . The identiﬁcations map to (g, v) ∼ e g, e v ∼ g, e v that represent U(Nf )/ZN+Nf . See also footnote8.

189 Let us discuss it more explicitly. Since we denote the classical gauge fields by uppercase letters, A, B, etc., we will denote the coefficients of their Chern-Simons counterterms [33,34] by K.5 They should be distinguished from the Chern-Simons coefficients of dynamical fields a, b, etc., which we denote by lower case k. It is important that k and K should be properly normalized as (2 + 1)d terms. As we will see below, it is often the case that the proper normalization of these coefficients involves a nontrivial relation between K and k.

It might happen that imposing the entire symmetry G there is no consistent value of K. In that case we say that G has an ’t Hooft anomaly and we have two options. First, we consider only a subgroup or a multiple cover of G and turn on background fields only for that group. Alternatively, we allow gauge fields for the entire global symmetry group G, but extend them to a (3 + 1)d bulk. In this case the partition function has a dependence on how the background fields are extended to the bulk. It is important, however, that the dynamical gauge fields are not extended to the bulk.

We will not present a general analysis of such anomalies. Instead, we will ﬁrst mention two well known examples. Then make some general comments, and later in the body of the paper we will discuss more sophisticated examples.

A well known typical example in which we can preserve only a subgroup Gb ⊂ G is the time-reversal anomaly of (2 + 1)d free fermions. Here G includes a global U(1) symmetry and time reversal, but they have a mixed anomaly. One common option is to preserve

Gb = U(1), but not time reversal. Alternatively, in the topological insulator we extend the background U(1) gauge ﬁeld to the bulk and we turn on a (3 + 1)d θ-parameter equal to π [45,114], such that the entire global symmetry G is preserved.

In this case the bulk term with θ = π is time-reversal invariant on a closed four- manifold, but not when the manifold has a boundary: a time-reversal transformation shifts

5We will use uppercase N in the gauge group of dynamical ﬁelds and Chern-Simons levels of dynamical ﬁelds depending on N and Nf . We hope that this will not cause confusion.

190 the Lagrangian by a U(1)1 Chern-Simons term. This is an anomaly in time reversal. The fermion theory on the boundary has exactly the opposite anomaly, such that they cancel each other and the combined (3 + 1)d theory is anomaly free.

Another well known example, where we can preserve a multiple cover Gnaive of the global symmetry G, is the following. Consider a quantum mechanical particle moving on S2 with a Wess-Zumino term with coeﬃcient k. (This is the problem of a charged particle on S2 with magnetic ﬂux k.) The global symmetry of the problem is G = SO(3), but as we will soon review, for odd k this symmetry is anomalous.

One way to represent the theory uses two complex degrees of freedom zi with a potential forcing |z1|2 + |z2|2 = 1. This system has an O(4) global symmetry. Next we introduce a dynamical U(1) gauge ﬁeld b coupled to the phase rotation of zi. The resulting theory is

1 the CP model whose target space is a sphere. We can add to the theory the analog of a 1 Chern-Simons term, which is simply a coupling kb. In terms of the eﬀective CP model this is a Wess-Zumino term with coeﬃcient k [116]. The spectrum of the theory is well known: it is ⊕jHj, where Hj is the isospin j representation of SU(2) and the sum over j runs over

k k j = 2 , 2 + 1, ...

i Naively, the global symmetry is Gnaive = SU(2) which rotates z . However, the global ∼ symmetry that acts faithfully is G = SU(2)/Z2 = SO(3). To see that, note that the coordinates zi are coupled to a U(1) gauge ﬁeld b, can be further coupled to an SU(2) classical ﬁeld B, but then b and B combine into a

U(2) = U(1)dyn × SU(2)naive /Z2 (5.0.7)

gauge ﬁeld B. The expression (5.0.7) shows that the element in the center of SU(2)naive is not a global symmetry transformation—it acts as a gauge transformation. Hence, the global

191 symmetry that B couples to is really SO(3). Indeed, all gauge-invariant local operators are in SO(3) representations.

For even k the Hilbert space includes integer j representations and represents SO(3) faithfully. In this case there is no anomaly. But for odd k all the states in the Hilbert space have half-integer j and the global symmetry acts projectively—it represents the double cover Gnaive.

What should we do about this anomaly? One option is to say that the global SO(3) symmetry acts projectively, or equivalently, the global symmetry is SU(2). A more interesting option is to introduce a (1 + 1)d bulk Mf2 (with boundary the original timeline), and add a bulk term that depends on the SO(3) gauge ﬁeld B.

Explicitly, the original degrees of freedom zi couple to a U(2) gauge ﬁeld B. Therefore,

k the CS term kb should be written as 2 Tr B. Although this is properly normalized as a CS term for a U(1) gauge field b, for odd k it is not properly normalized for a U(2) gauge field B. However, we can extend B to the (1 + 1)d bulk and replace the ill-defined contribution

i k R Tr B to the functional integral e 2 by the gauge-invariant expression

ik Z exp Tr FB , (5.0.8) 2 Mf2

where FB is the ﬁeld strength of B.

We should check whether (5.0.8) depends on the bulk values of the ﬁelds. A standard

0 way to do that is to replace the bulk Mf2 by another bulk Mf2 with the same boundary, and to consider the integral in (5.0.8) over the closed manifold M2 constructed by gluing

0 Mf2 with the orientation-reversal of Mf2:

k R R i Tr FB ik db e 2 M2 = e M2 . (5.0.9)

192 If this is always 1 then (5.0.8) is independent of the bulk fields. Had b been an ordinary U(1) gauge field, this would have been 1 for every b in the bulk. But since the bulk involves nontrivial SO(3) bundles, the gauge field b can have half-integral periods (it is a spinc connection) and (5.0.9) can be ±1, thus showing that it depends on the bulk values of b. However, this does not mean that the term (5.0.8) is not a valid term. In fact its dependence on b is completely fixed in terms of the SO(3) gauge field B:

k R R i Tr FB ikπ w2(B) e 2 M2 = e M2 , (5.0.10)

2 where w2 ∈ H (M2, Z2) is the second Stiefel-Whitney class of the SO(3) bundle.

In other words, the bulk term (5.0.8) can be viewed as a (1+1)d discrete θ-parameter for the background SO(3) gauge field, which is independent of the bulk values of the dynamical field b as long as we keep the background fixed. (It does depend on the boundary values of b.) The addition of such a bulk term, which depends on the background SO(3) fields, is familiar in the famous Haldane chain.6

The perspective on this phenomenon that we will use below is the following. The boundary theory—in this example a particle in the background of an odd-charge magnetic monopole—is anomalous and its action is not well-defined in (0 + 1)d in the presence of G background fields. To make it well-defined, we extend the background fields to a (1 + 1)d bulk, making sure that there is no dependence on the extension of the dynamical gauge R fields at fixed background. Then the bulk term w2(B) is well-defined mod 2 on a closed two-manifold, and it captures the dependence of the partition function on the extension R of the background field B. On a manifold with boundary the definition of w2(B) mod 2 depends on additional data. It is anomalous. This anomaly is exactly canceled by the anomaly in the boundary theory, such that the combined system is well defined.

6We thank E. Witten for a useful discussion about the Haldane chain.

193 Below we will see higher-dimensional generalizations of these examples. Using the notation discussed around (5.0.5), the dynamical fields b will typically have Chern-Simons couplings k while the background fields B will have Chern-Simons couplings K. In addition, for U(1) factors in the two groups there can be mixed Chern-Simons couplings. The way to properly define these couplings is by writing them as (3+1)d bulk terms of the form

θTr FB ∧ FB or θTr FB ∧ Tr FB with various θ’s, where FB are the ﬁeld strengths of the gauge ﬁelds B. In addition, we will also encounter discrete θ-parameters, like those in [9].

In this form we have a well deﬁned expression for gauge ﬁelds of (Gdyn × Gnaive)/C.

As in the quantum mechanical example of a particle on S2, it is crucial that these bulk terms must be independent of the bulk values of b at fixed B. This guarantees that b is a dynamical field living on the boundary. If the bulk terms are also independent of the bulk values of B, we say that the global symmetry G is anomaly free. Instead, if there is a dependence on the bulk values of B, the global symmetry suffers from ’t Hooft anomalies.

As in the same quantum mechanical example, we can check the independence of the bulk values of b and characterize the dependence on the bulk values of B by considering the bulk terms on a closed four-manifold M4. Then the integrals of the various F ∧ F terms of B should be expressed in terms of characteristic classes of B. These characteristic classes characterize the ’t Hooft anomalies.

Some of these characteristic classes are related to various discrete θ-parameters. We have already seen such a discrete θ-parameter in (5.0.10). Below we will encounter the discrete

θ-parameter of [9], which is associated with the Pontryagin square operation P(w2)[28,144]. As in [54,78], these can be represented by a two-form ﬁeld B with a (3+1)d coupling B ∧B. This coupling is gauge invariant on a manifold without boundary. But when a boundary is present, this term has an anomaly. The anomaly is cancelled by having an appropriate boundary theory, which has the opposite anomaly. For a (0+1)d boundary we have already seen that around (5.0.10), while below we will see examples with a (2 + 1)d boundary.

194 It is well known that in (3 + 1) dimensions, ’t Hooft anomaly matching conditions lead to powerful consistency constraints on the IR behavior of a theory and on its possible dual

UV UV descriptions. Consider first the simpler case of GA = GB . Then the ’t Hooft anomaly, which is the obstruction on the theory to be purely (2 + 1)-dimensional, must be the same on the two sides of the duality. In other words, if we need to couple the theory to a (3 + 1)d bulk and add some bulk terms with coefficients θ, these bulk terms should be the same in the two dual theories. Such θ-parameters can be ordinary or discrete ones. More precisely, θ should be the same, but the boundary counterterms KA and KB in the two theories can be different, provided they are properly quantized. This condition is the same as the celebrated ’t Hooft anomaly matching.

UV UV In the more interesting case that GA 6= GB , we can use the constraint in the UV by UV T UV coupling background ﬁelds to the common subgroup GA GB . Their θ must be the same

IR IR on the two sides of the duality. The IR theory can then be coupled to GA = GB gauge ﬁelds and this analysis also allows us to determine the value of θ for these ﬁelds. Again, we will see examples of that below.

5.0.3 Outline

In Section 2 we check ’t Hooft anomaly matching in the dualities (5.0.1)-(5.0.2). This is both an example of our methods and a nontrivial new test of those dualities.

In Section 3 we focus on some interesting special cases of the dualities with gauge group

U(1) ∼= SO(2) and SU(2) ∼= USp(2), either in the fermionic or the bosonic side. Such theories participate in more than one duality in (5.0.1)-(5.0.2). This leads to new tests of the dualities and to deeper insights into their dynamics. We also use those special cases to analyze theories with a surprising quantum SO(3) global symmetry in the IR, as in ﬁgure 5.1.

195 In Section 4 we follow [67, 153] and consider in detail a fermion/fermion duality that leads to an enhanced O(4) global symmetry. We extend previous discussions of this system by paying close attention to the global structure of the global symmetry and to the counterterms. This allows us to ﬁnd the precise anomaly in O(4) and time-reversal, and to restore those symmetries by adding appropriate bulk terms.

In Section 5 we analyze the phase diagram of systems with global SO(5) symmetry and clarify some possible confusions about various ﬁxed points with that global symmetry.

Appendix A derives the induced Wess-Zumino term in the model of Section 5, while Ap- pendix B describes carefully the duality of [32] paying attention to the proper quantization of CS couplings, to the spin/charge relation, to the global structure of the symmetry group, and to the bulk terms. In Appendix C we discuss more examples of ’t Hooft anomalies.

5.1 ’t Hooft Anomalies and Matching

We start by determining the ’t Hooft anomalies in the following theories:

SU(N)k with Nf Φ ←→ U(k) Nf with Nf Ψ −N+ 2

U(N)k with Nf Φ ←→ SU(k) Nf with Nf Ψ . (5.1.1) −N+ 2

The dualities are valid only for Nf ≤ N, but we will determine the symmetries and anomalies for generic integer values of N, k, Nf . Here and in the following, to be concise, we indicate complex scalars as Φ, real scalars as φ, complex fermions as Ψ and real fermions as ψ.

C All four theories have a naive global symmetry SU(Nf ) × U(1) o Z2 , where the last factor is charge conjugation. In the theories with SU gauge group, the ﬁrst two factors combine into a manifest U(Nf ) acting on the scalars or fermions. In the theories with

196 U gauge group, SU(Nf ) acts on the scalars or fermions, while the Abelian factor is the magnetic U(1)M , whose charge is the monopole number. However the faithfully-acting

C symmetry G is a quotient thereof, which as we will soon see is U(Nf )/ZN o Z2 in the C 7 ﬁrst line of (5.1.1) and U(Nf )/Zk o Z2 in the second line, as summarized in (5.0.6). For

Nf ≤ N this is a check of the dualities.

There might be an obstruction—an ’t Hooft anomaly—to turning on background gauge ﬁelds for G. We will show that the obstruction is the same on the two sides of the dualities, thus providing a nontrivial check of them.

5.1.1 Global symmetry

The ﬁrst step is to identify the global symmetry that acts faithfully on the four theories in (5.1.1). To do that, we analyze the local gauge-invariant operators.

Let us start with SU(N)k with Nf scalars. There is a U(Nf ) symmetry that acts on the scalars in the fundamental representation, but only U(Nf )/ZN acts faithfully on gauge invariants. In the absence of a magnetic symmetry, monopole operators do not change this result (since GNO ﬂux conﬁgurations [60] are continuously connected to the vacuum).

C There is also a charge-conjugation symmetry Z2 that exchanges the fundamental with the C antifundamental representation, therefore the symmetry is U(Nf )/ZN oZ2 . By the same C argument, SU(k) Nf with Nf fermions has U(Nf )/Zk o Z2 symmetry. −N+ 2

Next consider U(k) Nf with Nf fermions. The bare CS level is −(N − Nf ). There −N+ 2 is an SU(Nf ) symmetry that acts on the fermions in the fundamental representation and a U(1)M magnetic symmetry, whose charge is the monopole number. A monopole conﬁg- uration of monopole number QM has gauge charge (Nf − N)QM under U(1) ⊂ U(k). To

7 In the special cases of U(k)Nf /2 with Nf Ψ and U(N)0 with Nf Φ the global symmetry is SU(Nf )/ZNf × C C U(1)M o Z2 , which is isomorphic to U(Nf )/ZNf o Z2 (see footnote4). The scalar theory is also time- reversal invariant. More care has to be used in the case of SU(2) gauge group, as explained in Section 3.

197 form gauge invariants we dress the monopole with fermionic ﬁelds and their conjugates, and the net number of fundamentals minus antifundamentals is (N − Nf )QM . Therefore the operators are in SU(Nf ) representations of Nf -ality NQM mod Nf . We can then combine

SU(Nf ) with an N-fold multiple cover of U(1)M to form U(Nf ), and gauge-invariant local operators give representations of U(Nf )/ZN . Going to the multiple cover is natural from the point of view of the duality, since a monopole of charge 1 under U(1)M is mapped to a

8 baryon of charge N under U(1) ⊂ U(Nf )[117]. Charge conjugation acts both on SU(Nf )

C and U(1)M , therefore the full symmetry is U(Nf )/ZN o Z2 . In the case that N = 0 the C symmetry is U(Nf )/ZNf o Z2 .

By the same argument, U(N)k with Nf scalars has a faithfully-acting symmetry

C C T U(Nf )/Zk o Z2 . When k = 0 the symmetry is U(Nf )/ZNf o Z2 × Z2 .

5.1.2 Background ﬁelds

Now we turn on a background for the SU(Nf ) × U(1) symmetry of the four theories in (5.1.1), which can always be done, and analyze under what conditions the background gauge ﬁelds can be extended to U(Nf )/ZN or U(Nf )/Zk bundles.

Consider SU(N)k with Nf scalars. Turning on background gauge ﬁelds with generic CS counterterms we obtain the theory

SU(N)k × SU(Nf )L × U(1)J with Φ in (N, Nf , 1) . (5.1.2) ZN × ZNf

The ZN quotient acts anti-diagonally on SU(N) and the Abelian factor by a phase rotation

2πi/N 2πi/Nf e , while ZNf acts anti-diagonally on SU(Nf ) and the Abelian factor by e . The

8 Instead of going to the multiple cover and then take the ZN quotient, we can represent the symmetry −2πi/Nf 2πiN/Nf group as SU(Nf ) × U(1)M /ZNf where ZNf is generated by g = e 1, e . This ZNf has a N /d Zd subgroup generated by g f , where d = gcd(N,Nf ), which acts only on SU(Nf ). Therefore, only the representations of SU(Nf )/Zd appear.

198 quantization conditions on CS counterterms are

2 2 L ∈ Z ,J − Nk ∈ N Z ,J − Nf L ∈ Nf Z ,J ∈ NNf Z . (5.1.3)

The ﬁrst condition comes from the SU(Nf ) factor. The second and third conditions come from the separate quotients by ZN and ZNf , respectively. The last condition ensures that the generators of ZN and ZNf have trivial braiding and one can take the simultaneous quotient.

The equations in (5.1.3) have solutions in L, J, if and only if k = 0 mod gcd(N,Nf ). If this is not the case, there is an ’t Hooft anomaly and the theory with background is not consistent in (2 + 1)d. One can make sense of the theory on the boundary of a (3 + 1)d bulk, but then there is an unavoidable dependence on how the classical background ﬁelds are extended to the bulk. We will express the anomaly below.

Now consider U(k) Nf with Nf fermions. With background gauge ﬁelds we obtain −N+ 2

N k U(k) f × SU(Nf )L+ with Ψ in (k, Nf ), magnetic U(1)Kf and /ZNf . (5.1.4) −N+ 2 2

We stress that the magnetic U(1) is coupled to U(k) by a mixed CS term. The quotient by

ZNf acts on SU(Nf ) and the two Abelian factors. We have chosen to parametrize the CS counterterms in a way that matches the dual description (5.1.2) when the duality is valid.

9 J−Nk Then the topological symmetry U(1)Kf has CS counterterm Kf = N 2 . To see that, we mass deform the scalar theory by ±|Φ|2 and the fermionic theory by ∓ΨΨ. The two resulting topological theories are identiﬁed, exploiting level-rank duality on the dynamical ﬁelds.10 The map of CS counterterms for the U(1) global symmetry was already discussed in [67].

9 It can be interpreted as the ZN quotient of U(1)J−Nk. 10We cannot use level-rank duality on the background ﬁelds, which are not integrated over in the path- integral.

199 The quantization condition for SU(Nf ) k with k fermions is L ∈ , which reproduces L+ 2 Z the ﬁrst condition in (5.1.3). The topological factor U(1)Kf determines the condition Kf ∈

Z, which reproduces the second one in (5.1.3). To understand the ZNf quotient, consider the Abelian factors:

k(N − N ) k K L = − f ada + adB + f BdB , (5.1.5) Abelian 4π b b 2πb 4π where we have indicated as ba1k the Abelian factor in U(k). The equations of motion are as follows (neglecting the matter contribution):

k(Nf − N) dba + k dB = 0

k dba + Kf dB = 0 . (5.1.6)

We are after a ZNf one-form symmetry—then the matter contribution is canceled by a rotation in the center of SU(Nf ). An integer linear combination of the equations in (5.1.6) gives N k da + J dB = 0, which describes a one-form symmetry, if and only if J ∈ f b N ZNf

NNf Z. This reproduces the fourth condition in (5.1.3). The generator of the one-form symmetry is the line

I J I J + N k W = k a + B with spin SW = f mod 1 . (5.1.7) Nf b Nf 2 NNf 2Nf

To perform the ZNf quotient in the fermionic theory we combine WNf with the ZNf generator of SU(Nf ). The spin of the latter is −(L + k)/2Nf mod 1 (since the bare CS counterterm of SU(Nf ) is L + k). The ZNf quotient is well-deﬁned if its generator has integer or half-integer total spin,

! J + Nf k L + k 2 2 − ∈ Z , (5.1.8) 2Nf 2Nf

200 which reproduces the third condition in (5.1.3). Thus the ’t Hooft anomaly is the same on the two sides of the duality (5.1.1).

The discussion in the other two cases is similar. Consider SU(k) Nf with Nf fermions −N+ 2 ﬁrst. Turning on background gauge ﬁelds we have

SU(k) Nf × SU(Nf )L+ k × U(1) kNf −N+ 2 2 J+ 2 with Ψ in (k, Nf , 1) . (5.1.9) Zk × ZNf

Taking into account the bare CS levels, the quantization conditions are

2 2 L ∈ Z ,J + kN ∈ k Z ,J − Nf L ∈ Nf Z ,J ∈ kNf Z . (5.1.10)

They have solutions, if and only if N = 0 mod gcd(k, Nf ), otherwise there is an ’t Hooft anomaly.

Next consider U(N)k with Nf scalars. With background gauge ﬁelds we have

U(N)k × SU(Nf )L with Φ in (N, Nf ), magnetic U(1)Ks and /ZNf . (5.1.11)

The CS counterterms are chosen to match with those in (5.1.9) when the duality is valid,

J+kN with Ks = k2 . The SU(Nf ) and U(1) factors give the quantization conditions L ∈ Z and Ks ∈ Z, respectively. An integer linear combination of the equations of motion for the Abelian factors is (J/k) dB = 0 (where B is the U(1)Ks background gauge ﬁeld) which describes a ZNf one-form symmetry, if and only if J ∈ kNf Z. Such a one-form symmetry J H 2 is generated by the line WN = B with spin J/2N mod 1. This has to be combined f kNf f 1 with the generator in SU(Nf )L, and the condition that the total spin be in 2 Z reproduces the third condition in (5.1.10). Thus, all conditions in (5.1.10) are reproduced and the anomaly matches across the duality.

201 We should emphasize again that if we are only interested in the naive global symmetry group Gnaive = SU(Nf ) × U(1), which does not act faithfully, there is no problem turning on background gauge fields. The issue is only in considering gauge fields of the quotient group. In that case we can attach the system to a bulk, extend the fields to the bulk and replace the Chern-Simons terms by F ∧ F type terms there. Then the point is that the resulting theory depends on the extension. From this perspective, the ’t Hooft anomaly matching is the statement that we can use the same bulk with the same background fields there and attach to it either of the two dual theories on the boundary.

Consider the theory SU(N)k with Nf scalars in (5.1.2). To express the dependence on the bulk ﬁelds, we proceed as follows. A U(Nf )/ZN bundle can be represented by

NNf two correlated bundles, PSU(Nf ) and U(1)/ZD, where we set D = lcm(N,Nf ) = d and

(Nf ) 2 d = gcd(N,Nf ). We deﬁne w2 ∈ H (M4, ZNf ) as the second Stiefel-Whitney class of the

PSU(Nf ) bundle, and Fe = DF (in terms of the U(1) ﬁeld strength F ) as the well-deﬁned

ﬁeld strength of the U(1)/ZD bundle. Then the correlation between the two bundles is expressed by the fact that

Fe Nf (N) N (Nf ) = w + w mod D (5.1.12) 2π d 2 d 2

(N) 2 for some class w2 ∈ H (M4, ZN ). Such a class is the obstruction to lift a U(Nf )/ZN bundle to a U(Nf ) bundle.

Now consider a general bundle for the group in (5.1.2). The PSU(N) bundle associated to the dynamical ﬁelds is correlated with the U(Nf )/ZN bundle such that their Stiefel- (N) Whitney classes are equal: w2 PSU(N) = w2 . Therefore the dependence on the bulk

ﬁelds is completely ﬁxed by the classical U(Nf )/ZN background. Such a dependence is

202 described by

Z (N) (Nf ) 2 k P(w2 ) L P(w2 ) J Fe Sanom = 2π − − + 2 2 . (5.1.13) M4 N 2 Nf 2 D 8π

The integral is on a closed spin four-manifold M4, and P is the Pontryagin square operation

(N) 4 [28, 144] such that P(w2 )/2 ∈ H (M4, ZN ), etc. (for more details see [9] and references therein). We say that eiSanom captures the phase dependence of the partition function on the bulk extension of the U(Nf )/ZN bundle, in the sense that given two diﬀerent extensions

iSanom one can glue them into a closed manifold M4 and then e is the relative phase of the two partition functions.

If we choose J ∈ DZ, then we can substitute the square of (5.1.12) into (5.1.13) to obtain11

Z (N) (Nf ) J − Nk P(w2 ) J − Nf L P(w2 ) J (N) (Nf ) Sanom = 2π 2 + 2 + w2 ∪ w2 , (5.1.14) M4 N 2 Nf 2 NNf which is well-deﬁned modulo 2π. From this expression it is clear that if we can solve the constraints in (5.1.3), then eiSanom = 1 and there is no anomaly. On the other hand, it is always possible to make a suitable choice of L, J such that Sanom reduces to

Z (N) (k mod d) P(w2 ) Sanom = −2π . (5.1.15) N M4 2

We can regard this as a minimal expression for the anomaly.

As we have shown, the anomaly in U(k) Nf with Nf fermions is the same as in −N+ 2 (5.1.13). However one has to remember that the U(1) in (5.1.13) is an N-fold multiple cover of U(1)M . The special case N = 0 is discussed in Appendix C. The other two cases are similar, with an obvious substitution of parameters, and are presented in Appendix C.

11 (N) (Nf ) If J 6∈ DZ then (5.1.13) contains more information than w2 and w2 .

203 Although we checked the anomaly matching separately for the two dualities, in fact they are related by performing S,T operations on the U(1) symmetry [67, 149]. Since the operations add equal terms on both sides, the change in the bulk dependence on both sides must be equal, and thus the anomaly must still match. The anomaly also matches for other dualities obtained from them by S,T operations, such as the last two dualities in (5.0.1).

In general, the anomaly is characterized by bulk terms that are meaningful on closed manifolds, but anomalous when there is a boundary.12 This is true for the anomaly (5.1.13) where P(w2) is meaningful only on a closed manifold, and it is also true for the two examples discussed in Section 1.2.

Although we do not need it for the dualities, it is nice to demonstrate our general analysis of the anomaly by specializing it to a U(1) gauge theory of scalars with k = 0. Ignoring charge conjugation, the global symmetry is SU(Nf )/ZNf × U(1)M . The scalars are coupled to a U(Nf ) gauge ﬁeld B, where the U(1) ⊂ U(Nf ) gauge ﬁeld b is dynamical.

More precisely, b satisﬁes Nf b = Tr B. Therefore, the coupling to the magnetic U(1)M

1 background field BM is the ill-defined expression (Tr B)dBM that needs to be moved 2πNf to the bulk. This highlights that the global symmetry suffers from ’t Hooft anomalies, which are characterized by the bulk term

Z 1 dBM Sanom = 2π w2 PSU(Nf ) ∪ . (5.1.16) Nf 2π

This discussion is analogous to a similar example in [53]. See Appendix C for more details.

12We thank Dan Freed for a useful discussion about this point.

204 5.1.3 Symplectic gauge group

We conclude this section by brieﬂy analyzing the ’t Hooft anomalies in the two theories

USp(2N)k with Nf Φ ←→ USp(2k) Nf with Nf Ψ . (5.1.17) −N+ 2

Again, the dualities are valid only for Nf ≤ N, but we will study these theories for generic integer values of N, k, Nf . Since there is no magnetic symmetry, the faithfully-acting symmetry G is the one acting on gauge invariants constructed out of the scalars or fermions, which is USp(2Nf )/Z2 in both cases.

Coupling the two theories to a generic background, we obtain

USp(2N)k × USp(2Nf )L with Φ in (2N, 2Nf ) ←→ Z2 USp(2k) Nf × USp(2Nf )L+ k −N+ 2 2 with Ψ in (2k, 2Nf ) . (5.1.18) Z2

Recall that the scalars and fermions are in a pseudo-real representation, therefore they are subject to a symplectic reality condition. The CS counterterms are chosen in such a way that they match when the theories are dual. The quantization conditions are

Nk + Nf L ∈ 2Z (5.1.19) together with L ∈ Z in both theories. This provides ’t Hooft anomaly matching for the duality [3].

When Nk is odd and Nf is even, (5.1.19) cannot be solved and we have an ‘t Hooft anomaly. The anomaly is captured by the bulk term

Z P(w2) Sanom = π , (5.1.20) M4 2

205 where w2 is the second Stiefel-Whitney class of the USp(2Nf )/Z2 bundle. Given two

iSanom diﬀerent extensions of the bundle, e = ±1 (evaluated on their gluing M4) is the relative sign of the two partition functions.

5.2 Quantum Global Symmetries from Special Duali-

ties

Infrared dualities provide alternative descriptions of the same IR physics. It might happen that one description, say TA, makes a symmetry transformation manifest all along its RG

ﬂow, while the same symmetry is not present in the other description, say TB. Then, duality predicts that TB develops the symmetry quantum mechanically in the IR, because of strong coupling. In this section we survey various dualities at our disposal [2,3, 67, 121] and examine in what cases they predict a quantum enhancement of the global symmetry in the IR.

The theories we consider have Nf scalars or fermions in the fundamental representation.

C For gauge group U they have a naive global symmetry SU(Nf ) × U(1)M o Z2 , for gauge

C M C group SU have symmetry U(Nf )oZ2 , for gauge group SO have O(Nf )×Z2 ×Z2 , and for gauge group USp have USp(2Nf ). In addition, they might have time-reversal symmetry depending on the CS level. We have analyzed in Section 5.1 how Chern-Simons interactions determine the faithfully-acting subgroup, and the result for large enough values of N is summarized in (5.0.6).

The special cases U(1) ∼= SO(2) and SU(2) ∼= USp(2) need special attention. Fermionic

C theories with SO(2) gauge group have SU(Nf ) × U(1)M o Z2 naive symmetry (as seen in the U(1) language) and fermionic theories with SU(2) gauge group have USp(2Nf ) (as seen in the USp(2) language).

206 For scalar theories there are two subtleties to take into account. First, when using these theories in dualities Nf is restricted (Nf ≤ N in SU/U and USp dualities, while Nf ≤ N −2 for k = 1, Nf ≤ N − 1 for k = 2, and Nf ≤ N for k > 2 in SO dualities). Second, in the scalar theories we turn on quartic couplings in the UV and we must analyze their global symmetries. ∼ The U(1) = SO(2) theory with one scalar (Nf = 1) has only a single gauge-invariant

† 2 quartic coupling (ΦΦ ) . The theory preserves a U(1)M global symmetry, not present in generic SO(N) theories. The theory with Nf = 2 scalars has a single gauge-invariant SU(2)-

i † 2 invariant quartic coupling (Φ Φi ) . However, this theory does not participate in the SU/U dualities. The SO dualities use this theory for k > 2, but they require only SO(2) ⊂ SU(2) invariance (in addition to the U(1)M global symmetry). There are two quartic couplings

i † 2 i i † † that respect that symmetry, (Φ Φi ) and Φ Φ ΦjΦj, and the SO(2)k theories with those two couplings have SO duals. ∼ The USp(2) = SU(2) theory with one scalar (Nf = 1) has a single gauge-invariant

a † 2 quartic coupling (Φ Φa) which preserves a global SO(3) symmetry. The same theory with

Nf = 2 has several gauge-invariant quartic couplings. One of them is SO(5) invariant,

ai † 2 (Φ Φai) . However, this theory does not participate in the USp dualities. There are two distinct gauge-invariant quartic couplings that preserve an SO(3) × O(2) ⊂ SO(5) global

ai † bj † symmetry, the previous one and Φ ΦajΦ Φbi. These two couplings are assumed to be present in the theories with SU/U duals.

In the following, we analyze in detail these low-rank cases.

5.2.1 U(1)k with one Φ

∼ We exploit that U(1)k with 1 Φ = SO(2)k with 1 φ. The SO duality requires Nf = 1 if k = 2 and Nf ≤ 2 if k > 2. The SU/U duality requires Nf = 1. Therefore consider Nf = 1

207 ∼ and k ≥ 2. There is only one quartic term in the U(1)k = SO(2)k scalar theory and the following ﬁxed points are all the same:

∼ SU(k) 1 with 1 Ψ ←→ U(1)k with 1 Φ = SO(2)k with 1 φ − 2 x U(k + 1) 1 1 with 1 Ψ ←→  − 2 , 2 +k y

U(k − 1) 1 1 with 1 Ψ ←→ SO(k) 3 with 1 ψ . (5.2.1) − 2 , 2 −k − 2

C ∼ In the generic case the ﬁxed point has U(1)oZ2 = O(2) symmetry, which is a quantum symmetry in the fermionic SO(k) 3 theory. In the special case k = 2 the symmetry becomes − 2

SO(3), which is visible in the SU(2) 1 fermionic theory while it is a quantum symmetry in − 2 all other descriptions. This case is precisely the one in ﬁgure 5.1, indeed the scalar theory is the third example in (5.0.3).

5.2.2 U(1) Nf with Nf Ψ −N+ 2 ∼ We exploit that U(1) Nf with Nf Ψ = SO(2) Nf with Nf ψ. The SO duality requires −N+ 2 −N+ 2

Nf ≤ N − 1. Then the following ﬁxed points coincide:

∼ SU(N)1 with Nf Φ ←→ U(1) with Nf Ψ = SO(2) with Nf ψ Nf Nf −N+ 2 −N+ 2 x  U(N + 1)1,−N with Nf Φ ←→ y

U(N − 1)1,N with Nf Φ ←→ SO(N)2 with Nf φ . (5.2.2)

C In the generic case there is a U(Nf )/ZN o Z2 symmetry, which is a quantum symmetry in the SO(N)2 bosonic description. In the special case N = 2 and Nf = 1, the ﬁxed point coincides with (5.2.1) with k = 2 (this case is the one in ﬁgure 5.1 and in the third line

208 of (5.0.3)). The symmetry becomes SO(3), which is visible in the SU(2)1 bosonic theory while it is a quantum symmetry in the other descriptions.

5.2.3 SU(2)k with one Φ

∼ We exploit SU(2)k = USp(2)k. In the case Nf = 1 both the SU/U and USp dualities are valid. The scalar theory has only one quartic gauge invariant, thus the two dualities share the same ﬁxed point:

SU(2)k with 1 Φ ←→ USp(2k) 1 with 1 Ψ − 2 x  y

U(k) 3 with 1 Ψ (5.2.3) − 2

The two theories in the ﬁrst row have manifest SO(3) symmetry. The theory in the second

C row only has U(1)M oZ2 classically visible, thus it has enhanced quantum SO(3) symmetry. In the special case k = 1, the ﬁxed point coincides with (5.2.1) with k = 2 (as in ﬁgure 5.1 and third line of (5.0.3)).

5.2.4 SU(2)k with 2 Φ

We could write the theory as USp(2)k with 2 Φ, which has N = 1 and Nf = 2, however the

USp duality requires Nf ≤ N and so it is not valid. The SU/U duality, instead, is valid.

C ∼ In such a duality the scalar theory has two quartic terms, singlets under U(2)/Z2 o Z2 =

2 SO(3) × O(2). One term can be written as O1, where O1 is the quadratic gauge-invariant SO(5)-invariant operator (the subscript indicates the SO(5) representation). The other term is one of the components of O14 (a symmetric traceless rank-2 tensor of SO(5)) with the choice of coupling λ14 ∝ diag(−3, −3, 2, 2, 2), which breaks SO(5) to SO(3)×O(2). (As

209 we discuss later in Section 5, different signs of λ14 could lead to two distinct fixed points. Here we choose the sign that produces the fixed point involved in the SU/U duality.) Tuning

λ14 = 0 in the scalar theory gives a different fixed point with SO(5) symmetry. The flows are summarized as follows:

λ1,λ14 SU(2)k with 2 Φ −→ CFT with SO(3) × O(2) ←− U(k)−1 with 2 Ψ   λ1y CFT with SO(5) (5.2.4)

This example, discussed at length in Section 5, does not develop quantum symmetries.

5.2.5 SU(2) Nf with Nf Ψ −N+ 2

Both SU/U and USp dualities require Nf ≤ N. The two dualities have common fermionic theory and thus the ﬁxed points are the same:

SU(2) Nf with Nf Ψ ←→ USp(2N)1 with Nf Φ −N+ 2 x  y

U(N)2 with Nf Φ (5.2.5)

The ﬁxed point has USp(2Nf )/Z2 symmetry, which is a quantum symmetry in the bosonic

U(N)2 theory. When N = Nf = 1, the ﬁxed point coincides with (5.2.1) with k = 2 (as in ﬁgure 5.1 and the third line of (5.0.3)

210 5.2.6 Examples with Quantum SO(3) Symmetry and ’t Hooft

anomaly matching

Consider the examples with enhanced SO(3) symmetry, speciﬁcally the family of CFTs in

(5.2.3) parametrized by k and the family in (5.2.5) with Nf = 1 parametrized by N. We have already checked in Section 5.1 that the ’t Hooft anomaly for the manifest symmetry matches across the various dualities. In the case of quantum symmetries, some description does not have the full symmetry GIR manifest in the UV, and therefore we can only couple the UV theory to a background for the subgroup GUV ⊂ GIR. Still, we can check that the CS counterterms for GUV, when forced to be embeddable in GIR, present the same obstruction as the ’t Hooft anomaly for GIR. This in general provides a check of the quantum enhancement. In the examples with enhanced SO(3) symmetry, the ’t Hooft anomaly vanishes in all descriptions.

Combining the dualities in (5.2.1) and (5.2.2) we obtain six dual descriptions for the CFT with SO(3) global symmetry that appeared in the third line of (5.0.3) and in ﬁgure 5.1:

U(1)2 with 1 Φ ←→ SU(2)1 with 1 Φ ←→ U(3)1,−2 with 1 Φ x x x y y y

U(1) 3 with 1 Ψ ←→ SU(2) 1 with 1 Ψ ←→ U(3) 1 5 with 1 Ψ (5.2.6) − 2 − 2 − 2 , 2

The ﬁrst two columns are special cases of the discussion above (and had already been considered in [3]). The two theories in the last column can be coupled to a U(1) background for the maximal torus of SO(3) with Lagrangians

1 3 U(3) with 1 Φ → L = |D Φ|2 − |Φ|4 + Tr ada − 2i a3 − (Tr a)d(Tr a) 1,−2 a 4π 3 4π

211 1 1 + (Tr a)dB + Ks−3 BdB 2π 4π 2 3 (Tr a)dB 1 Ks+3 U(3)− 1 , 5 with 1 Ψ → L = iΨ/DaΨ + (Tr a)d(Tr a) + + BdB 2 2 4π 2π 4π 2 (5.2.7)

∼ where the parameter Ks is identiﬁed with the level of the SU(2)Ks /Z2 = SO(3)Ks/2 background in the upper middle description in (5.2.6). The needed CS counterterms have been computed in [67]. In all six cases, the CS counterterms are well-deﬁned for Ks + 1 ∈ 2Z, providing a check of the dualities.

5.3 Example with Quantum O(4) Symmetry: QED

with Two Fermions

In this section we consider three-dimensional QED, i.e. U(1)0, with two fermions of unit charge. As ﬁrst observed in [153], this model enjoys self-duality. The analysis of [67] paid more attention to global aspects of the gauge and global symmetries and to the Chern- Simons counterterms. Here we continue that analysis and discuss in detail the global symmetry and its anomalies. In particular, we will show that the IR behavior of this model has a global O(4) symmetry and time-reversal invariance T, but these symmetries have ’t Hooft anomalies. As in previous sections, various subgroups or multiple covers of this symmetry are anomaly free and can be preserved in a purely (2 + 1)d model. We also add bulk terms to restore the full global symmetry.

212 5.3.1 QED3 with two fermions

We consider a pair of dual UV theories ﬂowing to the same IR ﬁxed point. As in [67], we start with a purely (2 + 1)d setting and study13

1 1 1 iΨ D/ Ψ1 + iΨ D/ Ψ2 + ada + adY − Y dY + 2CS 1 a+X 2 a−X 4π 2π 4π grav 1 1 1 ←→ iχ D/ χ1 + iχ D/ χ2 + ada + adX − XdX + 2CS (5.3.1) 1 ea+Y 2 ea−Y 4πe e 2πe 4π grav

c where a, ea are dynamical U(1) gauge ﬁelds (more precisely they are spin connections [122]) while Ψ1,2, χ1,2 are complex fermions.

We would like to identify the global symmetry of the model. The UV theory in the left side of (5.3.1) has a global SU(2)X × O(2)Y symmetry. The explicit background ﬁeld

X X Y Y Y X couples to U(1) ⊂ SU(2) and Y couples to U(1) ⊂ O(2) . The Z2 ⊂ O(2) transformation CY acts as charge conjugation: CY (Y ) = −Y , CY (a) = −a, CY (X) = X,

Y j X C (Ψi) = ijΨ , hence it commutes with SU(2) . Similarly, the UV theory in the right side has a SU(2)Y × O(2)X symmetry which includes a CX transformation. We will soon see that they do not act faithfully.

Before we identify the global symmetry of the IR theory, we should ﬁnd the precise global symmetry of the UV theories (5.3.1). First we study how local operators transform under SU(2)X × U(1)Y in the left side of the duality (5.3.1). A gauge-invariant polynomial

i X X made out of Ψ , Ψi and derivatives has even U(1) charge corresponding to SU(2) isospin jX ∈ Z, and it is neutral under U(1)Y . A monopole of a has U(1)Y charge QY = 1 and

U(1)a charge 1. In order to make it gauge invariant, we must multiply it by a fermion,

13 R 1 R CSgrav is a gravitational Chern-Simons term deﬁned as M=∂X CSgrav = 192π X Tr R ∧ R. In this section we also use that the partition function of U(N)1 is reproduced by the classical Lagrangian −2NCSgrav. See [121, 122] for details.

213 X 1 14 thus making the operator have j = 2 . More generally, it is easy to see that all gauge invariant operators have 2jX + QY ∈ 2Z.

As in the previous sections, this means that the dynamical U(1)a and the classical

X − Y X − Y SU(2) × P in (2) should be taken to be U(1)a × SU(2) × P in (2) /Z2 and the X − Y global symmetry that acts faithfully is SU(2) × P in (2) /Z2.

A similar argument can be used in the right hand side of (5.3.1) showing that the global

Y − X symmetry there is SU(2) ×P in (2) /Z2. The duality (5.3.1) means that the IR theory ∼ X Y 15 should have the union of these two symmetries SO(4) = SU(2) × SU(2) /Z2. Also,

C the duality means that the theory is invariant under a transformation Z2 that exchanges ∼ C X ↔ Y , thus the global symmetry is really O(4) = SO(4) o Z2 . The Lagrangians in (5.3.1) use only U(1)X × U(1)Y gauge ﬁelds and in terms of these the global symmetry is

X Y C O(2) × O(2) /Z2 o Z2 .

In addition, in the absence of background ﬁelds (i.e. as long as we consider correlators at separate points) the theory is clearly time-reversal invariant:16

1 1 1 T iΨ D/ Ψ1 + iΨ D/ Ψ2 + ada + adY − Y dY + 2CS 1 a+X 2 a−X 4π 2π 4π grav 1 1 2 1 = iΨ D/ Ψ1 + iΨ D/ Ψ2 + ada + adY + XdX + Y dY + 2CS (5.3.2), 1 a+X 2 a−X 4π 2π 4π 4π grav where T(a) = a, T(X) = X, T(Y ) = −Y . Of course, we can combine this transformation with CY and/or with an element of SU(2)X . With a background, the theory is time-

2 reversal invariant up to the anomalous shift 4π (XdX + Y dY ). This anomaly should not 14It is easy to see that the basic monopole operators can have spin zero. More generally, our theory satisfies the spin/charge relation with a the only spinc connection. Therefore, all gauge-invariant local operators must have integer spin. 15The additional conserved currents in the IR are provided by monopole operators of magnetic charge ±2 dressed by two fermion fields with flavor indices contracted, schematically M(−2)ΨΨ and their conjugates. The fermions are contracted symmetrically with respect to Lorentz indices to give spin one, then by Fermi statistics flavor indices are antisymmetric giving a flavor singlet. 16When applying time reversal, one should be careful about the η-invariant from the regularization of the fermion path-integral [121].

214 be surprising. The U(1)X symmetry is embedded into SU(2)X and in terms of that, the functional integral over Ψ leads to an η-invariant (that can be described imprecisely as

X SU(2) 1 ) which has a time-reversal anomaly. Note that in the other side of the duality − 2 this transformation must act as T(ea) = −ea.

X Y Next, we would like to examine whether the Z2 quotient of U(1) ×U(1) is anomalous X Y or not. Since we should take the quotient U(1)a × U(1) × U(1) /Z2 (and similarly with U(1) ), this means that the ﬂuxes of a, a, X, Y are no longer properly quantized, but a±X, ea e c ea ± Y are properly quantized spin connections and X ± Y are properly quantized U(1) gauge ﬁelds. A simple way to implement it is to change variables a → a − X, ea → ea − Y c in (5.3.1) such that a, ea become ordinary spin connections:

2Y dY ada ad(Y − X) (X − Y )d(X − Y ) − + iΨ D/ Ψ1+iΨ D/ Ψ2 + + + + 2CS 4π 1 a 2 a−2X 4π 2π 4π grav x y 2XdX ada ad(X − Y ) (X − Y )d(X − Y ) − + iχ D/ χ1+iχ D/ χ2 + e e + e + + 2CS . 4π 1 ea 2 ea−2Y 4π 2π 4π grav (5.3.3)

2 2 Except for the ﬁrst term in each side, namely − 4π Y dY and − 4π XdX, all the terms are properly normalized Chern-Simons terms under the quotient gauge group.

The existence of these terms means that the two dual UV theories (5.3.1) have an

’t Hooft anomaly preventing us from taking the Z2 quotient.

2 We can change this conclusion by adding appropriate counterterms, e.g. 4π XdX, to the two sides of the duality (5.3.1) or equivalently (5.3.3). Denoting the Lagrangians in these equations by L0(X,Y ) ←→ L0(Y,X), we set

2 L (X,Y ) = L (X,Y ) + XdX . (5.3.4) 1 0 4π

215 This removes the ﬁrst term in the right side of (5.3.3) and makes also the left side consistent

X Y with the quotient. Then, we can place the UV theory in U(1) ×U(1) /Z2 backgrounds.

X In the left side of the duality this term represents adding SU(2)1 while in the right side this interpretation is meaningful only in the IR theory. After this shift, the IR theory can be placed in nontrivial SO(4) backgrounds. However, now the IR duality symmetry, which exchanges X ↔ Y , is anomalous:

2 L (X,Y ) ←→ L (Y,X) + (XdX − Y dY ) , (5.3.5) 1 1 4π

C X Y i.e. under the Z2 transformation the IR theory is shifted by SU(2)−1 × SU(2)1 .

To summarize, the global symmetry that acts faithfully is O(4), but we cannot couple the system to background O(4) gauge fields. Starting with (5.3.1) we can couple it to P in±(4) background fields,17 or starting with (5.3.4) we can couple it to SO(4) background fields.

5.3.2 Mass deformations

We can check the duality (5.3.1) by deforming both sides with fermion bilinear operators in either the singlet or vector representation of the SU(2) ﬂavor symmetry factors.

i The deformation by the SO(4)-singlet mass term mΨiΨ was discussed in [67]. The theory ﬂows to the Lagrangians

2 − Y dY i.e.U(1)Y ⊂ SU(2)Y for m > 0 , 4π −2 −1 2 − XdX i.e.U(1)X ⊂ SU(2)X for m < 0 . (5.3.6) 4π −2 −1 17Since in the IR there are no operators transforming in spinor representations of Spin(4), we can extend O(4) to both P in±(4).

216 i This makes it clear that mΨiΨ is odd under Z2 ⊂ O(4). The result of the deformation is consistent with the magnetic symmetry U(1)Y , U(1)X on two sides of the duality being enhanced to SU(2)Y , SU(2)X respectively.

X 18 1 2 The SU(2) triplet mass term m(Ψ1Ψ − Ψ2Ψ ) is in the (3, 3) representation of SU(2)X × SU(2)Y . In fact, the duality (5.3.1) can be derived by combining two fermion/fermion dualities involving a single fermion (e.g. see Section 6.3 of [67]), and from there one ﬁnds that the SU(2)X triplet mass term maps to the SU(2)Y triplet mass term

1 2 m(χ1χ − χ2χ ). The representation (3, 3) is completed by monopole operators M(−2)ΨΨ and their conjugates, where now Lorentz indices are antisymmetric while ﬂavor indices are symmetric. The triplet mass term explicitly breaks the symmetry to U(1)X × U(1)Y . Deforming the CFT (5.3.1) by this mass term leads to the low energy Lagrangians

1 1 a d(Y + X) − (XdX + Y dY ) for m > 0 , 2π 4π 1 1 a d(Y − X) − (XdX + Y dY ) for m < 0 . (5.3.7) 2π 4π

We see that the theory is not gapped: the photon a is massless and its dual is the Goldstone boson of a spontaneously broken global symmetry. From (5.3.7) we see that the unbroken symmetry is a diagonal mixture of U(1)X × U(1)Y . Under both deformations (5.3.6) and (5.3.7) we ﬁnd consistency of the duality.

We could entertain the possibility that the symmetry of the CFT be SO(5) ⊃ O(4).

i j That would imply that at the ﬁxed point the O(4) invariant operator ΨiΨ ΨjΨ sit in

i j the same representation 14 of SO(5) as Ψi(σ3)jΨ , and share the same dimension. As we

i j just discussed, we can assume that the operator Ψi(σ3)jΨ , which is relevant in the UV, is relevant in the IR as well: this leads to a coherent picture. This would imply that also the 4-Fermi interaction is relevant in the IR, and since it is irrelevant in the UV, it would be

18We thank D. Gaiotto for a useful discussion about this deformation.

217 a dangerously-irrelevant operator. Then, in order to reach the putative CFT with SO(5)

i j symmetry, one would need to tune the irrelevant operator ΨiΨ ΨjΨ in the UV. The theory we have been discussing in this section—QED with two fermions—does not have such a tuning, and therefore it would not reach the SO(5) ﬁxed point even if the latter existed.

5.3.3 Coupling to a (3 + 1)d bulk

We have seen that the IR behavior of the UV theories (5.3.1) has an O(4) global symmetry and time-reversal T. But these symmetries suffer from an ’t Hooft anomaly. We cannot couple them to background gauge fields for these symmetries. We saw that depending on the choice of counterterms we can have either P in±(4) or SO(4) background fields, but we cannot have O(4) background fields and in either case we do not have time-reversal symmetry.

However, we can couple our (2+1)d system to a (3+1)d bulk and try to add background gauge ﬁelds in the bulk such that the full global symmetry is realized.

∼ X Y C Let us start with the O(4) = SU(2) ×SU(2) /Z2oZ2 symmetry. The bulk couplings of these gauge ﬁelds are characterized by two θ-parameters, θX and θY . Because of the Z2 quotient, they are subject to the periodicity

(θX , θY ) ∼ (θX + 2π, θY + 2π) ∼ (θX + 4π, θY ) (5.3.8)

and the semidirect product restricts to (θX , θY ) ∼ (θY , θX ).

Consider a bulk term SB with (θX = −2π, θY = 0). For a closed four-manifold with X and Y being P in±(4) gauge ﬁelds, this bulk term is trivial. When X and Y are O(4) gauge

iSB ﬁelds the partition function e is ±1 and depends only on w2 of the gauge ﬁelds. (More precisely, the sign is determined by the Pontryagin square P(w2)/2.) This means that even

218 for O(4) gauge ﬁelds the partition function is independent of most of the details of X and Y in the bulk.

C The Z2 transformation, which exchanges X and Y , shifts the bulk term SB by the term

(θX = 2π, θY = −2π). On a closed four-manifold this shift has no eﬀect on the answers. But in the presence of a boundary it shifts the boundary Lagrangian by the Chern-Simons terms of SU(2)1 × SU(2)−1 /Z2. In other words, in the presence of a boundary the bulk

C term SB has an anomaly under Z2 .

Starting with the boundary theory (5.3.1) we add the boundary term in (5.3.4) and the bulk term SB. Naively, this did not change anything. The bulk term might be thought of as

X an SU(2)−1 boundary Chern-Simons term and therefore it seems like it removes the term added in (5.3.4). However, because of the quotients this conclusion is too fast. Instead, the

C bulk term is meaningful for SO(4) ﬁelds and has an anomaly under Z2 . The boundary term we added in (5.3.4) made the boundary theory meaningful for SO(4) ﬁelds and created an

C anomaly under Z2 . Together, we have a theory with a bulk and a boundary with the full O(4) symmetry.

Now that we have achieved an O(4) symmetry we can try to add additional terms to restore time-reversal symmetry. We would like to add a bulk O(4) term that even with

C a boundary does not have a Z2 anomaly, but such that it compensates the anomaly in

0 time reversal (5.3.2). Clearly, we need to add a bulk term SB with (θX = π, θY = π).

C Without a boundary this term is T and Z2 invariant. With a boundary it does not have

C an anomaly under Z2 but it has a T anomaly which exactly cancels that of the boundary theory (5.3.2). Note that the time-reversal anomaly (5.3.2) was not modiﬁed by adding the boundary term in (5.3.4) and the bulk term SB with (θX = −2π, θY = 0). These two terms almost completely cancel each other. To summarize, the theory with the added boundary

0 term in (5.3.4) and a bulk term SB + SB with (θX = −π, θY = π) has the full symmetry of the problem.

219 We should make a ﬁnal important comment. As we said above, the bulk term SB with

(θX = −2π, θY = 0) leads to dependence only on some topological information of the bulk

0 ﬁelds. Instead, the bulk term SB with (θX = π, θY = π) depends on more details of the bulk ﬁelds.

5.4 Example with Global SO(5) Symmetry

In this section we would like to study in some detail the theory

∼ USp(2)k = SU(2)k with two scalars , (5.4.1)

and the relation with its SU/U dual U(k)−1 with two fermions. Since Nf = 2, there are various quartic terms we can include in the potential, and depending on the choice we reach diﬀerent IR ﬁxed points. We will use mass deformations to check the duality, and exploit the ‘t Hooft anomaly matching for general SU/U dualities discussed in Section 5.1.

5.4.1 A family of CFTs with SO(5) global symmetry

Let us ﬁrst consider USp(2)k with two Φ. As a USp theory, it has maximal global symmetry ∼ SO(5) = USp(4)/Z2. We can classify relevant deformations accordingly. We describe the scalars through complex ﬁelds ϕai with a = 1, 2, i = 1,..., 4 subject to the reality condition

ab ij ∗ ϕai Ω = ϕbj (where Ω is the USp(4) symplectic invariant tensor). The quadratic gauge

ab invariants are collected into the antisymmetric matrix Mij = ϕaiϕbj , which decomposes under SO(5) as 1 O = −Tr ΩM, O = M − Ω O . (5.4.2) 1 5 4 1

Here the subscript is the SO(5) representation and we suppress the indices. Given the decomposition (1 + 5) ⊗S (1 + 5) = 1 ⊕ 5 ⊕ 1 ⊕ 14, in principle there are four quartic gauge

220 2 2 invariants: O1, O1O5, Tr ΩO5ΩO5 ≡ O5, and O14 constructed as

1 (O ) = (O ) (O ) + Ω Ω − 2Ω Ω + 2Ω Ω O2 . (5.4.3) 14 ijkl 5 ij 5 kl 20 ij kl ik jl il jk 5

2 2 However, since the gauge group has rank one, it turns out that O1 = 4O5 and so there is only one quartic singlet.

2 In USp(2)k with two Φ we insist on SO(5) global symmetry: we turn on O1 and O1 (k) with a fine-tuning on O1 and we assume that it flows to a nontrivial fixed point T0 . Such a fixed point has SO(5) global symmetry. We can couple the theory to SO(5) background gauge fields as USp(2) × USp(4) k L with Φ in (2, 4) , (5.4.4) Z2 with some CS counterterm with level L. The conditions to have a well-defined (2 + 1)d action are L ∈ Z and k − 2L ∈ 2Z. As we discussed in Section 2.3, for k even the equations can be solved, but for k odd they cannot and there is an ‘t Hooft anomaly. The anomaly is captured by the bulk term

Z P(w2) Sanom = k π . (5.4.5) M4 2

iSanom For a closed manifold M4 this is trivial for k even, in the sense that e = 1, but it is

±1 for k odd. For M4 with a boundary this term is anomalous for odd k and corrects the anomaly in the boundary theory.

(k) We can consider relevant deformations of T0 that preserve SO(5), see figure 5.2, by turning on µ1O1, which is an equal mass for all scalars. With positive mass-squared µ1 > 0 we flow to the TQFT SU(2)k. With negative mass-squared µ1 < 0 some of the scalars condense Higgsing SU(2)dyn completely. To figure out the breaking pattern, notice that we start with 8 real fields, one is an overall scale, 3 are eaten by the Higgs mechanism, so

221 2\Ta Tw/ 2 5 > 0 ܷܵ݌ ௞ Ȱ ૚૝ ( ) ܱܵ ߣ 2 w/ 2 w/ 2 \TaT \TaT 3\T ×a௞T 2 3 × 2 3 × 2 ି ௞ ିଵ ࣮? 2 ܷܵ Ȱ ܷ ݇ Ȳ ܱܵ ܱ ܱܵ ܱ ܱܵ ܱ < 0 ௞ ׽ ܷ ݇ ିଶ / duality ܷܵ ߣ૚૝ ܷܵ ܷ ଶ \TaT < 0 ௞5 > 0 ॺ ସ ࣮଴ 2 ૚ ߤ૚ ॺ௞ ܱܵ ߤ ௞ ׽ ܷ ݇ ିଶ ܷܵ ଵ > 0 ॺ ߣ૚૝ ( ) \TaT 3 ×௞ 2 ࣮ା ܱܵ ܱ

Figure 5.2: RG diagram and fixed points of SU(2)k with two scalars. The blue and brown lines are RG flows that preserve the global symmetry of the UV fixed point they emanate from. The green lines are deformations that break the SO(5) global symmetry. The solid green and brown lines separate between phases with different IR behavior (S1, S2, and gapped with a TQFT), with an S4 theory (with a Wess-Zumino term with coefficient k) along the brown solid line. There is no phase transition along the dotted brown line, as the IR physics is gapped there.

at most there can be 4 massless ﬁelds. However the largest subgroup of SO(5) is SO(4), so there must be at least 4 Goldstone bosons. Thus the only possible breaking pattern

is SO(5) → SO(4), leaving an S4 ∼= SO(5)/SO(4) sigma model in the IR. More directly,

1 2 the potential V = µ1O1 + 2 O1 has minima at O1 = −µ1. By the algebraic relation 2 1 2 O5 = 4 O1 we ﬁnd that also O5 condenses, and it spontaneously breaks the symmetry to 4 SO(4), producing the Goldstone modes living on S . Alternatively, the equation O1 = −µ1 describes an S7, which is an SU(2) Hopf ﬁbration over S4, therefore gauging SU(2) leaves the S4 NLSM.

As was shown in [116] (we review it in Appendix A), the S4 NLSM has a Wess-Zumino

interaction term kSWZ that originates from the level k Chern-Simons term in the UV. The

222 Wess-Zumino term can be written as

Z ∗ k SWZ = 2πk σ (ω4) (5.4.6) Mf4

where the integral is over a four-manifold with boundary, σ are the NLSM fields, and ω4 is the volume form of S4 normalized to total volume 1. We can couple the theory to SO(5) background gauge fields (gauging in general dimension was discussed in [68]). From our derivation in Appendix A it is clear that for odd k, the WZ action depends on how the SO(5) background fields are extended to the bulk, and the dependence is captured by the very same term (5.4.5).19 This is ’t Hooft anomaly matching along the RG flow.

Let us stress that the far IR limit of the S4 NLSM is given by 4 free real scalar ﬁelds. At higher energies there are irrelevant interactions that turn it into the S4 NLSM with WZ term. As we show in Appendix A, at the same scales there are also other irrelevant (higher-derivative) interactions that break time reversal T (for k > 0). Therefore such an S4 NLSM has only SO(5) global symmetry, as the UV theory.

5.4.2 Two families of CFTs with SO(3) × O(2) global symmetry

Let us now consider SU(2)k with two Φ (i.e. the same gauge group and matter content as before), but imposing only SU(2) × U(1) symmetry on the quartic terms, as it is the case in the SU/U dualities. Then there is another quartic deformation we can add in the UV:

jk il Oe(14) = (O14)ijkl η η , (5.4.7)

where η is a U(2)-invariant tensor. This operator is contained in O14 from the decomposition

14 → 10 ⊕ 50 ⊕ 31 ⊕ 3−1 ⊕ 12 ⊕ 1−2 under USp(4) → SU(2) × U(1). It turns out that the

19 We thank Todadri Senthil for pointing out to us the relevance of P(w2) in this context and for mentioning [136].

223 preserved symmetry acting faithfully is SO(3)×O(2).20 If we want to avoid the appearance of directions in ﬁeld space where the potential is unbounded from below, the absolute value

2 of the coeﬃcient λ14 of Oe(14) should not be too large compared to that of O1 (which is positive). Since at λ14 = 0 there is a phase transition with enhanced SO(5) symmetry, we

(k) expect two different RG flows for λ14 ≷ 0 that can lead to two families T± of fixed points with SO(3) × O(2) global symmetry.

According to (5.1.3), the theory SU(2)k with two Φ for odd k has an anomaly when ∼ coupled to U(2)/Z2 = SO(3) × SO(2) backgrounds. This anomaly directly follows from (5.4.5), if we restrict SO(5) backgrounds to SO(3) × SO(2).

(k) We can learn about the properties of the fixed points T± , if they exist, by looking at (k) the RG diagram in figure 5.2. In particular, we can predict what T± flow to, if we deform them by O1, which is the only other relevant deformation invariant under the symmetries

(notice that O5 does not contain SO(3) × O(2) singlets). This should be the same as

(k) ﬁrst deforming T0 by O1, as in Section 5.1, and then by Oe(14). The TQFT SU(2)k is not aﬀected by Oe(14), because SO(5) only acts on massive particles and thus Oe(14) is

4 P5 2 decoupled. In the S NLSM we use coordinates ρ1,...,5 with I=1 ρI = 1. Then we have

2 2 2 2 2 Oe(14) = −3(ρ1 + ρ2) + 2(ρ3 + ρ4 + ρ5). If we deform the potential by Oe(14) with positive coefficient, we flow to an S1 NLSM, while a negative coefficient leads to an S2 NLSM. We

(k) 1 conclude that, for µ1 < 0, the µ1O1 deformation of T+ gives an S NLSM, while the µ1O1

(k) 2 deformation of T− gives an S NLSM. Notice that when the NLSM maps are restricted to an equatorial S1 or S2, the WZ term (5.4.6) vanishes.

20 The reduced symmetry is O(3) × O(2) /Z2 embedded into SO(5), which includes charge conjugation C C Z2 . Equivalently, it is U(2)/Z2 o Z2 embedded into USp(4)/Z2 as follows: a 2 × 2 unitary matrix T T 0 0 −1 is mapped to 0 T ∗ , the quotient is by −1, and charge conjugation is mapped to Ω = 1 0 ∈ USp(4). 0 1 C The U(2)-invariant tensor is then η = 1 0 , odd under Z2 .

224 (k) We can provide two diﬀerent descriptions of T+ through the SU/U duality

SU(2)k with 2 Φ ←→ U(k)−1 with 2 Ψ , (5.4.8)

2 where the theory on the left has O1 and Oe(14) quartic couplings both with positive coeﬃ-

21 (k) cient. This gives evidence that the ﬁxed points T+ exist for all k > 0. One can check that the SO(3)×O(2) invariant mass deformation of U(k)−1 with two Ψ gives U(k)−2 (dual

1 to SU(2)k) for negative fermion mass, and U(k)0 (whose low energy limit is the S NLSM) for positive mass. Moreover, as discussed in Section 5.1, the fermionic theory correctly reproduces the ’t Hooft anomaly.

(k) What about T− ? For k = 1 a natural candidate for a dual description is

(1) T− : U(1)2 with 2 Φ . (5.4.9)

C ∼ This theory has U(2)/Z2 o Z2 = SO(3) × O(2) global symmetry acting faithfully, and it reproduces the anomaly of SU(2)1 with 2 Φ. Besides, the theory has unique SO(3) × O(2) invariant quadratic and quartic terms. Upon invariant positive mass-squared deformation it flows to U(1)2, which is dual to SU(2)1. For negative mass-squared, the minima of the potential form an S3 with the U(1) Hopf fiber gauged, and thus the theory flows to an S2 NLSM. Since the CS level in (5.4.9) is even, there is no topological Hopf term [145] in the S2 NLSM, reproducing the result from the deformation of S4. This gives evidence that the

(1) ﬁxed point T− could exist.

(1) Interestingly, the description (5.4.9) of T− and the fermionic description in (5.4.8) of (1) T+ almost ﬁt in the dualities (5.0.1)-(5.0.2) but fail to be dual because their parameters

21As in [67], U(1) ⊂ U(k) is a spinc connection and we must add a transparent line to the theories in order for the duality to be valid. This transparent line does not aﬀect the critical behavior.

225 are outside the allowed region. For instance the two theories

NOT SO(2)2 with 2 φ ←→ SO(2)−1 with 2 ψ (5.4.10)

(1) fail to give a dual pair (as advocated in [3] by a diﬀerent argument). In this example T± appear in the same RG diagram, but are indeed distinct.

5.4.3 A family of RG ﬂows with O(4) global symmetry

(k) We can consider a diﬀerent deformation of the SO(5) invariant theories T0 , obtained by using a quartic operator in O14 that preserves an O(4) . We will call this operator Ob(14). It can be written in terms of a Spin(4) invariant tensor η0 as22

0jk 0il Ob(14) = (O14)ijkl η η . (5.4.11)

(k) Deforming T0 by Ob(14) breaks SO(5) → O(4). Thus, we study the theory USp(2)k with 2

2 Φ and quartic deformations O1 and Ob(14). More easily, this is SU(2)k with two scalars and

4 4 2 2 a potential V = α |Φ1| + |Φ2| + 2β|Φ1| |Φ2| with α 6= β. In order to have a theory with

2 a potential bounded from below, O1 should have positive coeﬃcient while the coeﬃcient

λb14 of Ob(14) should not be too large in absolute value. As before, we expect two diﬀerent (k) RG ﬂows for λb14 ≷ 0, separated by T0 with enhanced SO(5) symmetry.

The only other O(4) invariant relevant deformation is O1, which is an equal mass for all scalars (O5 does not contain O(4) invariants), and we can study the combined eﬀect

(k) of O1 and Ob(14) on T0 —in a way similar to what we did in ﬁgure 5.2. With positive

22To deﬁne η0 it is convenient to use a diﬀerent basis than before, namely Ω = ω 0 with ω = 0 −1 . 0 ω 1 0 C T1 0 C We embed Spin(4) into USp(4) as where T1,2 ∈ SU(2), while the charge conjugation is o Z2 0 T2 Z2 0 1 0 ω 0 C = 1 0 . Quotient by −1 gives an embedding of O(4) into USp(4)/Z2. Then η = 0 −ω is invariant C under a Spin(4) and is odd under Z2 .

226 Global symmetry Quadratic Φ2 Quartic Φ4 2 SO(5) O1 O1 2 SO(3) × O(2) O1 O1, Oe(14) 2 O(4) O1 O1, Ob(14) 2 SO(4) O1, O(5) O1, Ob(14), O1O(5) 2 C 2 U(1) o Z2 O1, O(5) O1, Oe(14), Ob(14), O1O(5)

(k) Table 5.1: Relevant deformations of T0 depending on the preserved global symmetry.

mass-squared we flow to the TQFT SU(2)k, which is not affected by Ob(14) because the latter is decoupled. With negative mass-squared we flow to deformations of the S4 NLSM.

2 2 2 2 2 In the NLSM coordinates, Ob(14) = −(ρ1 + ρ2 + ρ3 + ρ4) + 4ρ5. Therefore, λb14 > 0 leads to

3 C an S NLSM, while λb14 < 0 leads to two gapped vacua with spontaneous breaking of Z2 .

4 3 The WZ term kSWZ in the S NLSM descends to a θ-term πkQ in the S NLSM, where

3 23 Q ∈ Z is the wrapping number in π3(S ) = Z (in other words θ = kπ).

In the presence of the deformation Ob(14), with either sign of its coupling λb14, a tuning on O1 may or may not lead to a ﬁxed point. At the moment we do not have candidate dual descriptions for those ﬁxed points, and we leave the question open.

(k) There are more general deformations of T0 we can consider, depending on the amount of symmetry we want to preserve. A few examples, some of which we have already discussed, are in Table 5.1. By O(5) we mean the speciﬁc component of O5 that is a singlet under the preserved symmetry group under consideration.

For instance, if we want to preserve only SO(4) ⊂ SO(5), in terms of the two SU(2) doublets Φ1 and Φ2 we can turn on the following relevant deformations: there are two mass

2 2 4 4 2 2 terms |Φ1| and |Φ2| , and three quartic terms |Φ1| , |Φ2| , |Φ1| |Φ2| . This is a diﬀerent basis than the one in Table 1. With many operators at our disposal, the precise breaking pattern depends on the ratios between the various terms.

23Restricting the NLSM maps σ to an equatorial S3 in S4, the WZ term gives 0 on a map that does not 3 3 wrap S , and π on a map that wraps S once. By linearity, SWZ = πQ(σ).

227 5.4.4 Relation with a Gross-Neveu-Yukawa-like theory

We can compare the USp(2)k theory with two scalars with a diﬀerent model, discussed in [123], which also exhibits SO(5) global symmetry and a phase described by the S4 NLSM with WZ term.

Consider a Gross-Neveu-Yukawa-like theory (GNY) with 5 real scalars, 4k complex fermions and schematic Lagrangian [123]

2 4 a L = (∂φ) + Ψ/∂Ψ − φ + φ ΨΓaΨ . (5.4.12)

The scalars transform in the vector representation 5 of Spin(5), the fermions in k copies of the spinor representation 4, and Γa are gamma matrices of Spin(5). The Lagrangian (5.4.12) enjoys a USp(4)×USp(2k) /Z2 global symmetry, and the quartic interaction is the only one preserving that symmetry. In addition, the theory also preserves a time-reversal

T Z2 symmetry under which φ is odd.

With a tuning, the Lagrangian (5.4.12) is expected to flow to a fixed point with the full global symmetry. The tuning is on the scalar mass deformation (while the fermion mass is odd under T and is thus set to zero by imposing that symmetry). We could also think of the fixed point as the IR limit of the O(5) Wilson-Fisher fixed point with 4k complex

a decoupled fermions perturbed by the relevant operator φ ΨΓaΨ.

As discussed in [123], if we deform (5.4.12) by a negative scalar mass-squared, the scalars condense breaking spontaneously SO(5) → SO(4) and leading to an S4 NLSM. In addition, because of the Yukawa interaction the fermions become massive. Integrating them out produces a WZ interaction kSWZ [1]. Deformation by a positive mass-squared leads to 4k complex massless free fermions.24

24At intermediate energies one ﬁnds a Gross-Neveu-like model of 4k complex massless fermions with quartic interactions, which however are irrelevant and disappear in the IR.

228 (k) The GNY fixed point (5.4.12) and the fixed point T0 of the USp(2)k theory with two scalars discussed above, despite sharing the S4 NLSM phase with a WZ term, are clearly different. Even their global symmetries are different. The GNY fixed point has a (k) USp(4)×USp(2k) /Z2 symmetry and T-reversal symmetry, while the fixed point T0 has only an SO(5) global symmetry. In fact, even the S4 NLSM phases are slightly different.

The one obtained from (5.4.12) has the time-reversal symmetry T, preserved by SWZ, while the one from SU(2)k with two scalars has higher-derivative corrections that break T, because time-reversal symmetry is not present in the UV. In addition, the phase obtained by positive mass-squared is diﬀerent in these two models.

5.5 Appendix A: Derivation of the Wess-Zumino term

in the 3D S4 NLSM

4 Here we show that when SU(2)k with two scalars ﬂows to the S NLSM by mass deformation, the Chern-Simons term induces a Wess-Zumino interaction at level k at low energies [116].

Insisting on SO(5) global symmetry and turning on a negative mass-squared, the minima

P 2 7 of the potential lie along a,i |Φai| = λ (a = 1, 2, i = 1, 2) which is S (here λ is a mass scale). The SU(2) action corresponds to the Hopf fibration SU(2) → S7 → S4, thus what we gauge is the SU(2) fiber. Recall that SU(2) bundles over S4 are completely classified

7 by π3 SU(2) = Z which is the second Chern class, and S has minimal class:

1 Z 2 Tr G ∧ G = 1 , (5.5.1) 8π S4 where G = dC − iC2 is the curvature of the SU(2) bundle and C(Φ) is a function of Φ.

229 The 3D gauge theory has a CS term

Z Z k 2i 3 k 3 SCS = Tr ada − a = Tr F ∧ F, S = ∂Mf4 , (5.5.2) 3 4π S 3 4π Mf4 where S3 is the topology of spacetime and the second deﬁnition is the proper one. At low energies the scalars are constrained to P |Φ|2 = λ and we integrate out the gauge ﬁeld. Starting with the schematic Lagrangian

2 k L = DµΦ + LCS(a) , (5.5.3) 4π the equation of motion for a is

k 0 = a ΦΦ† + ΦΦ†a − iΦ∂ Φ† + i∂ ΦΦ† + F νρ . (5.5.4) µ µ µ µ 2π µνρ

This equation contains aµ as well as its first derivative, and it is non-linear. If we drop ↔ † the last term, the equation is simply Jµ ≡ −iΦDµΦ = 0 setting to zero the SU(2) gauge current. This means that aµ is identified with the connection C(Φ) of the SU(2) bundle over S4. To take into account the last term, we notice that the first four terms in (5.5.4) are of order λ (because |Φ|2 ∼ λ) while the last term is of order λ0 and it contains a derivative. We can then solve the equation as a series expansion in λ−1, and since λ−1 is dimensionful, the series is actually a derivative expansion. Thus in the IR limit we have

µ aµdx = C(Φ) + ... (5.5.5) where the dots are higher-derivative corrections.

Having identiﬁed the ﬁeld strength F in (5.5.2) over the extended spacetime manifold

Mf4 with the curvature G(Φ) of the Hopf ﬁbration (up to higher-derivative corrections), we

230 obtain Z SCS → k SWZ + ··· = 2πk ω4(Φ) + ..., (5.5.6) Mf4

4 where ω4 is the volume form on S normalized to have integral 1. Notice that (5.5.4), because of the last term, is not invariant under time reversal T. Therefore the higher- derivative corrections to (5.5.5) do not transform homogeneously under T, and they break T in (5.5.6).

Finally, consider coupling the UV theory to SO(5) background ﬁelds, namely consider the theory SU(2)k × USp(4)L /Z2 with a bifundamental scalar. As discussed in Section 2.3, for odd k the action has a sign dependence on the extension of the SO(5) background

ﬁelds to Mf4. By (5.5.6), this implies that also the WZ term coupled covariantly to SO(5) background ﬁelds [68] has the same anomalous dependence.

5.6 Appendix B: Comments on Self-Dual QED with

Two Fermions

Building on the interesting fermion/fermion duality of [112, 126, 141], the authors of [153] proposed the self-duality of a U(1) theory with two fermions. This was later generalized in [32] to the self-duality of a U(1) gauge theory with two fermions, one with charge 1 and one with charge k odd. As emphasized in [67,121,122], the coefficients in the Lagrangians in [112,126,141] are improperly quantized. This was fixed in [121] by adding more fields and more terms to the Lagrangian. Then, a proper derivation of the self-duality of the theory with k = 1 was given in [67]. That perspective was also consistent with the spin/charge relation and described the proper coupling of background gauge fields. Here we will present a similar derivation of the self-duality of the theory with generic odd k. This will lead us to a more detailed analysis of the global symmetries and ’t Hooft anomalies of the problem.

231 We start with the fermion/fermion duality of [121]:

1 2 1 1 iΨ/D Ψ ←→ iχD/ χ + adu − udu + udA − AdA − 2CS . (5.6.1) A a 2π 4π 2π 4π grav

Next, we follow the steps in [67]. We take a product of the theory in (5.6.1) and of its time-reversed version in which we substitute A → kA − 2X (X is a background U(1) gauge ﬁeld):

1 iΨ D/ Ψ1 + iΨ D/ Ψ2 ←→ iχ D/ χ1 + iχ D/ χ2 + a du − a du 1 A 2 kA−2X 1 a1 2 a2 2π 1 1 2 2 2 1 + u du − u du + a da 4π 2 2 1 1 4π 2 2 1 1 1 + u dA + u d2X − kA − AdA . (5.6.2) 2π 1 2π 2 4π

Note that for odd k this is consistent with the spin/charge relation. We add the following

1 1 1 counterterms, 2π Ad Y − kX + 4π XdX − Y dY + N 4π AdA + 2CSgrav , to the two sides of the duality. Here N = (k2 + 1)/2 and Y is a background U(1) field. The specific counterterms and the value of N were picked such that we can integrate out most of the fields on the right hand side. Then we can promote A to a dynamical field (more precisely, a spinc connection) a. On the left hand side we find

N 1 1 iΨ D/ Ψ1+iΨ D/ Ψ2+ ada+ adY −kX+ XdX−Y dY +2NCS . (5.6.3) 1 a 2 ka−2X 4π 2π 4π grav

We will call this Lagrangian L0(X,Y ). On the right hand side there are several gauge ﬁelds,

0 0 k+1 0 but we can integrate most of them out. We redeﬁne a = a + 2u1 and u2 = u2 + ku1 + 2 a , then the Lagrangian is linear in u1 and it can be integrated out to set a1 = ka2 − 2Y . Finally we can integrate out a0 to ﬁnd

N 1 1 iχ D/ χ1 +iχ D/ χ2 + ada+ adX −kY + Y dY −XdX+2NCS , (5.6.4) 1 kea−2Y 2 ea 4πe e 2πe 4π grav

232 where we relabelled a2 = ea. Note that all terms are properly quantized with ea being a spinc connection. We see that (5.6.3) and (5.6.4) are related by relabeling the dynamical ﬁelds and by exchanging X ↔ Y . This establishes the self-duality of the model, namely

25 L0(X,Y ) ←→ L0(Y,X).

As a check, for k = 1 we can substitute a → a + X in (5.6.3), ea → ea + Y in (5.6.4) 1 and subtract the counterterm 2π XdY from both sides, to ﬁnd the same duality (5.3.1) as in [67].

Let us examine the global symmetry of the problem. First, there is a U(1)X × U(1)Y . Second, there is a charge-conjugation symmetry acting as C(a) = −a, C(X) = −X,

C(Y ) = −Y (and C(ea) = −ea in the dual). We will denote the combined group for these two symmetries as SO(2)X × O(2)Y . Third, because of the duality there is the

C Z2 transformation that exchanges X ↔ Y . Fourth, there is a time-reversal transformation with T(a) = a, T(X) = X, T(Y ) = −Y (and T(ea) = −ea) that acts on the theory as

h i 2 T L (X,Y ) = L (X,Y ) + (XdX + Y dY ) − 2(k2 − 1)CS , (5.6.6) 0 0 4π grav i.e. it is a symmetry up to an anomalous shift of CS counterterms. Next we determine

i the symmetry that acts faithfully. Operators constructed out of polynomials in Ψ , Ψi and derivatives have even U(1)X charge and vanishing U(1)Y charge. Similarly, operators

i Y made out of polynomials in χ , χi and derivatives have even U(1) charge and vanishing

X X U(1) charge. We can also consider monopole operators of a or ea: they have odd U(1) 25 1 In order to compare with [32], for every fermion coupled withD / A we should add the terms − 8π AdA − CSgrav. This turns (5.6.3) and (5.6.4) into 1 1 iΨ D/ Ψ1 + iΨ D/ Ψ2 + Y da − (XdX + Y dY ) ←→ 1 a 2 ka−2X 2π 4π 1 1 iχ D/ χ1 + iχ D/ χ2 + Xda − (XdX + Y dY ) (5.6.5) 1 kea−2Y 2 ea 2π e 4π where we removed the gravitational Chern-Simons term. Up to the last counterterm (which we cannot remove because of the spin/charge relation) this agrees with the equations in [32].

233 and odd U(1)Y charge. Hence the symmetry that acts faithfully on the space of operators

X Y C is S O(2) × O(2) /Z2. Including Z2 and time reversal, we ﬁnd the symmetry group X Y C T S O(2) × O(2) /Z2 o Z2 o Z2 .

Let us consider background gauge ﬁelds for the symmetry group that does not act on

R dX spacetime. Because of the Z2 quotient, we allow background ﬁelds X,Y with 2π mod 1 = R dY 1 X Y 2π mod 1 = 2 . The restriction on the U(1) × U(1) charges of local operators should make such backgrounds consistent. However, one can check (e.g. by deﬁning ordinary U(1)

fields Z± = X ± Y ) that the two sides (5.6.3)-(5.6.4) of the duality are not well defined in the presence of such fluxes. This is an anomaly.

As in the other cases, in particular the one in Section 4, we have diﬀerent options.

1. We can leave the (2 + 1)d Lagrangian L0(X,Y ) as it is, but then we can only couple

X Y C it to S O(2) × O(2) o Z2 background ﬁelds with no fractional ﬂuxes.

2. We add to the two sides (5.6.3)-(5.6.4) of the duality the Chern-Simons counterterms

1 − 4π (XdX − Y dY ). These counterterms violate the spin/charge relation. Now we X Y C can have S O(2) × O(2) /Z2 backgrounds, but Z2 is violated.

3. We can attach the theory to a (3 + 1)d bulk, add suitable bulk terms and obtain a well-deﬁned theory on general backgrounds, but whose partition function depends on the extension of the background ﬁelds to the bulk.

X Y C Let us explore the third option. The θ-parameters of S O(2) × O(2) /Z2 o Z2 are subject to the periodicities

(θX , θY ) ∼ (θX + 8π, θY ) ∼ (θX + 4π, θY + 4π) , (5.6.7)

234 and the restrictions (θX , θY ) ∼ (−θX , −θY ) ∼ (θY , θX ) up to periodicities. We add a

1 boundary term − 4π (XdX − Y dY ), i.e. we consider the boundary theory

1 L (X,Y ) = L (X,Y ) − XdX − Y dY , (5.6.8) 1 0 4π

and also add a bulk term SB with (θX = 2π, θY = −2π). Now the boundary theory is well

X Y C deﬁned on S O(2) × O(2) /Z2 backgrounds. The Z2 transformation is anomalous, and 2 maps L1(X,Y ) → L1(X,Y ) + 4π (XdX − Y dY ) (making use of the duality), however this is precisely oﬀset by an opposite anomalous transformation of SB.

0 In order to preserve time-reversal as well, we add another boundary term SB with k2−1 R (θX = 2π, θY = 2π) and also − 192π Tr R∧R. In the presence of a boundary, the variation 0 of SB under T precisely cancels the one of L0 (while the variations of the added boundary term to get L1 and of SB cancel among themselves).

5.7 Appendix C: More ’t Hooft anomalies

We list here the ’t Hooft anomalies for other cases discussed in the main text. Consider the

theories U(N)k with Nf scalars and SU(k) Nf with Nf fermions. The global symmetry −N+ 2 is U(Nf )/Zk and charge conjugation that we will neglect. Following the same steps as in Section 2.2, one ﬁnds that for generic choices of the CS counterterms and with the same conventions as in (5.1.9) and (5.1.11), the anomaly is

Z (k) (Nf ) 2 N P(w2 ) L P(w2 ) J Fe Sanom = 2π − + 2 2 . (5.7.1) M4 k 2 Nf 2 D 8π

kNf Here d = gcd(k, Nf ), D = lcm(k, Nf ) = d , F is the ﬁeld strength of U(1) ⊂ U(Nf ) while

(Nf ) Fe = DF is the well-deﬁned and integer ﬁeld strength of the U(1)/ZD bundle, w2 is the

235 (k) second Stiefel-Whitney class of the PSU(Nf ) bundle and w2 is deﬁned by the constraint

Fe Nf (k) k (Nf ) = w + w mod D. (5.7.2) 2π d 2 d 2

With the choice J ∈ DZ, using the square of the previous relation the anomaly simpliﬁes to

Z (k) (Nf ) J + Nk P(w2 ) J − Nf L P(w2 ) J (k) (Nf ) Sanom = 2π 2 + 2 + w2 ∪ w2 . (5.7.3) M4 k 2 Nf 2 kNf

The case k = 0 is special and the formulae above do not directly apply.

So, consider the theory U(N)0 with Nf scalars. In this case the global symmetry is

PSU(Nf ) × U(1)M , as well as charge conjugation and time reversal that we neglect. The scalars are coupled to a U(Nf ) gauge ﬁeld B (where U(1) ⊂ U(Nf ) is dynamical) and a dynamical gauge ﬁeld b, with Nf Tr db = NTr dB. The coupling to the magnetic U(1)

N background field BM is described by the ill-defined expression (Tr B)dBM which needs 2πNf to be moved to the bulk. This highlights that the global symmetry suffers from an ’t Hooft anomaly. Including a CS counterterm at level L for SU(Nf ) (which could be set to zero), the anomaly is characterized by the bulk term

Z (Nf ) N (Nf ) dBM L P(w2 ) Sanom = 2π w2 ∪ − , (5.7.4) M4 Nf 2π Nf 2

1 (Nf ) where we have identiﬁed 2π Tr dB = w2 mod Nf . This expression can be regarded as a singular limit of (5.7.1).

N Similarly, the theory U(k) f with Nf fermions has global symmetry U(Nf )/ZNf , besides 2 charge conjugation that we neglect. The expression (5.1.13) for the anomaly does not directly apply (since N = 0). Following similar steps as before, we ﬁnd that the anomaly

236 is characterized by the bulk term

Z (Nf ) k (Nf ) dBM L P(w2 ) Sanom = 2π w2 ∪ − . (5.7.5) M4 Nf 2π Nf 2

The other time-reversal invariant theory is U(k)0 with Nf fermions, which requires Nf to be even. The UV symmetry is U(Nf )/ZNf /2 together with charge conjugation and time reversal. Applying (5.1.14) with N = Nf /2, the anomaly is

Z (Nf /2) (Nf ) 2 2k P(w2 ) L P(w2 ) J Fe Sanom = 2π − − + 2 2 (5.7.6) M4 Nf 2 Nf 2 Nf 8π

where Fe satisﬁes (5.1.12) with d = Nf /2, D = Nf . Besides, under time reversal there

4J−2Nf k is an anomalous shift by SU(Nf )−2L−k × U(1)−2Kf /ZNf where Kf = 2 . For the Nf special case k = 1, Nf = 2 we can choose the counterterms L = J = 0 such that there is no anomaly for the U(2) = SU(2)×U(1) /Z2 symmetry, but there is a time-reversal anomaly that shifts the theory by SU(2)−1 × U(1)2 /Z2. We elaborate more on the anomaly for

U(1)0 with two fermions in Section 4.

237 Chapter 6

Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)d

6.1 Introduction

Time-reversal symmetry is an important property of a variety of systems relevant to both high-energy and condensed matter physics. In this paper we clarify some aspects of time- reversal symmetry in gauge theories. Our methodology here is that most natural in continuum field theory. We will start with a continuum Lagrangian defining our model at short distances and try to ascertain its long-distance behavior. A first step in this analysis is a precise determination of the global symmetry of the model. This includes the ordinary unitary global symmetries, as well as spacetime symmetries such a time-reversal. As we describe below, in general these symmetries are linked in a non-trivial algebra.

We will focus here on three-dimensional systems based on some gauge group and fermions transforming in some representation. We will mostly study U(1) and SO(N) gauge theories (U(1) is a special case of SO(N) with N = 2, but with many special features) and fermions in the vector or the two index tensor of SO(N) (in U(1) these are

238 fermions of charge 1 or 2). In order to determine the IR behavior of the system we need to understand in detail its symmetries and in particular its time-reversal symmetry.

6.1.1 T

Time-reversal symmetry T is an antiunitary transformation that acts on the time coordinate as t → −t combined with some action on the ﬁelds in the theory. In Euclidean spacetime it reverses the orientation of spacetime. In general, the T symmetry of the theory is not unique. We can redeﬁne T by combining it with a global symmetry transformation. For example, many systems have a unitary symmetry that acts as an outer automorphism of the gauge group, which is called charge conjugation C. Then we can say that the basic time-reversal symmetry is T or CT .

Neither of these choices is universally natural. For instance, the standard definition of time-reversal in four-dimensional free Maxwell theory acts on the electric and magnetic fields as E → E and B → −B, while charge conjugation reverses the sign of both. However, electromagnetic duality exchanges E and B and therefore maps T to CT . In this paper we will follow [150,151] and define the symmetry T to act on a gauge field a as T (a(t)) = a(−t). In components this reads

T (a0(t)) = −a0(−t) , T (ai(t)) = ai(−t) . (6.1.1)

One advantage of the above is that it makes sense even for systems where there is no natural notion of charge conjugation. (Note that if the U(1) gauge ﬁeld is that of ordinary electromagnetism, this symmetry is usually called CT .) In the condensed matter literature

T on models with a global U(1) our convention (6.1.1) is known as U(1) × Z2 as opposed to

T U(1) o Z2 (see e.g. [47, 92, 94, 151]). Notice that as a consequence of these deﬁnitions, the

239 electric charge Q is odd under T , while the magnetic charge M is even.

T QT −1 = −Q, T MT −1 = M. (6.1.2)

In particular, this means that on a charged fermion ﬁeld ψ in an abelian gauge theory, we have:

∗ T (ψ) = γ0ψ(−t) , CT (ψ) = γ0ψ(−t) . (6.1.3)

Although time-reversal is an antiunitary symmetry, the operator T 2 is a unitary symmetry. In systems that depend on spin structure (like the models with fermions of interest here) there is also a fermion number symmetry (−1)F , and this leads to several elementary possibilities for the unitary symmetry T 2.

2 T • Non-spin theories with T = 1. We refer to this as Z2 . In Euclidean signature this symmetry algebra means that the theory may be formulated on any unorientable manifold. These systems can have an ’t Hooft anomaly for the time-reversal symmetry

valued in Z2 × Z2 [18,77,134,138].

2 F T • Spin theories with T = (−1) . We refer to this as Z4 . This is also the algebra realized on the charged fermions in (6.1.3). In Euclidean signature this symmetry algebra means that the theory may be formulated on unorientable manifolds with a

+ P in structure. Any system with this symmetry has an ’t Hooft anomaly ν ∈ Z16 characterizing its behavior on such manifolds [47,86,92,139,150].

2 T F • Spin theories with T = 1. We refer to this as Z2 × Z2 . In Euclidean signature this symmetry algebra means that the theory may be formulated on unorientable manifolds with a P in− structure. Unlike the cases above, there are no possible ’ t Hooft anomalies for this symmetry algebra [82].

240 As we describe below, the possibilities listed above are by no means exhaustive, and we give examples of time-reversal invariant gauge theories where T 2 is a more general unitary symmetry. Similar phenomena have been observed in [92,139].

One particularly interesting class of time-reversal invariant theories are certain spin topological field theories defined by Chern-Simons gauge theories at specific non-zero values of the level. These models are not classically time-reversal invariant but they enjoy level- rank duality that changes the sign of the level and hence defines a T symmetry of quantum theory satisfying T 2 = (−1)F [3,35,67]. 1 A summary of these theories and there associated value of ν is given in table 6.1.

T -invariant spin-TQFT Anomaly ν (mod 16) U(n)n,2n 2 Sp(n)n 2n SO(n)n n 1 O(n)n,n+3 n 1 O(n)n,n−1 n Table 6.1: Time-reversal invariant spin TQFTs and their associated anomaly ν. These anomalies have been computed by various methods [19, 31, 62, 129, 130, 134]. The anomaly for the O(n)1 theories is determined based on the consistency of the conjectured phase diagrams of [35, 62]. Note that in general redefining the orientation of spacetime changes ν → −ν [150]. For a given theory the sign is convention dependent, but the relative sign between theories is meaningful. For instance, in the U(n)n,2n sequence it is natural to fix the sign to be (−1)n−1 [62]. Some of the TQFTs above also admit unitary global symmetries of order two and these can be combined with T to produce other antiunitary symmetries of the model with a different value of ν. An example that will occur below is the T-Pfaffian theory vs. the CT-Pfaffian theory [47, 92, 94]. In addition, the value of ν can depend on other choices like the eigenvalue of T 2 on the anyons. Several special cases of these theories have been previously considered. The example SO(2)2 ↔ U(1)1,2 is known as the semion- fermion [47]. Meanwhile SO(3)3 was analyzed in [47], and SO(4)4 was studied in [19,22,31]. The theory O(2)2,1 is equivalent to the T-Pfaffian theory [25,29,35], and we have the duality O(2)2,5 ↔ U(2)2,4 [35].

1Certain special cases of the level also deﬁne bosonic TQFTs. However, in general the dualities below only hold when the theories are promoted to spin theories [3, 67].

241 6.1.2 What is the Global Symmetry?

As discussed above, the models of interest to us in this paper all admit an ultraviolet definition as a gauge theory with gauge group H. To analyze their global symmetry group G, it is often useful to discuss the related model defined by restricting the dynamical gauge fields to be classical.

In this theory we have a set of fields with a global symmetry K. The global symmetry action is characterized by some ’t Hooft anomaly. This means that in the presence of background K gauge fields, the system is not gauge invariant and this lack of gauge invariance cannot be fixed by adjusting any local term. Instead, the anomaly is characterized by a local term in one higher dimension.

Next, we try to gauge a subgroup H ⊂ K. This can be done only when the anomaly vanishes when restricted to H gauge ﬁelds. What is the global symmetry after this gauging? In many cases it is given by the group

G ∼= N(H,K)/H , (6.1.4) where N(H,K) is the normalizer of H in K, and we mod out by the gauged subgroup H.

As an example of this construction that will occur below, we can start with 2Nf real ∼ ∼ Majorana fermions and then K = O(2Nf ). The subgroup H = U(1) that acts on the ﬁelds as Dirac fermions of unit charge is anomaly free and can be gauged. The resulting global

C symmetry group G is then PSU(Nf ) o Z2 , where C is a charge conjugation symmetry.

There are two phenomena that can make the answer (6.1.4) wrong:

• The anomaly might mean that if the gauge ﬁelds of H are dynamical, then the some of the elements in N(H,K) are no longer a symmetry. This means that only a subgroup of G is the true global symmetry.

242 • When the gauge ﬁelds of H are dynamical, we can have a new emergent symmetry

Hb. Examples that we will see below are that in 2+1d when H ∼= U(1) we have an emergent magnetic symmetry Hb ∼= U(1). Similarly, when H ∼= SO(N) we have an ∼ emergent magnetic symmetry Hb = Z2.

These two phenomena often mix with each other. One aspect of this is that the symmetries Hb and G can form a nontrivial algebra. We refer to this possibility by saying that G is deformed by Hb (examples were studied in [22, 35, 53]). For instace, we will describe systems with a time-reversal symmetry T ∈ G where the unitary symmetry T 2 is neither of the two elementary possibilities discussed above (i.e. T 2 = (−1)F or T 2 = 1), but instead is an element of the emergent magnetic symmetry T 2 ∈ Hb. These symmetry algebras have been explored in [92, 139]. We will also see examples where the time-reversal symmetry T participates in a non-abelian algebra with elements of Hb and G.

6.1.3 Monopole Operators and Their Quantum Numbers

Many of our results follow from a careful analysis of monopole operators in abelian gauge theory and their quantum numbers under various global symmetries.

Consider a U(1) gauge theory with gauge field b coupled to Dirac fermions ψi of charge qi. We follow standard conventions and label theories by an effective level k defined for massless fermions. The effective level is partitioned into two parts. The first is the integral bare level kbare ∈ Z, which controls the level in the UV Lagrangian. The second piece is in general half-integral and encodes the contribution from the fermions. The level shifts under mass deformation as shown below.

mψ < 0 mψ = 0 mψ > 0 (6.1.5)

1 P 2 P 2 kbare k ≡ kbare + 2 i qi kbare + i qi

243 Note that when the fermions are massive, the level is always an integer.

In addition to the dynamical gauge ﬁeld b, it is also instructive to introduce a background gauge ﬁeld A, which couples to all fermions with charge one. As in [122], A can be viewed as a spinc connection, and we can include a mixed Chern-Simons term for A and b in the theory with bare level Q. Thus Lagrangian of interest is

Q k i L = bdA + bare bdb + iψ (/∂ +A / + q /b)ψi . (6.1.6) 2π 4π i

Our focus is on time-reversal invariant theories, which must have vanishing eﬀective levels and hence we adjust the counterterms to

1 X 1 X Q = − q , k = − q2 . (6.1.7) 2 i bare 2 i i i

Since Q and kbare must be integers this means that T symmetry requires that the number

Nodd of fermions with odd charge qi must be even [10,107,118].

The fact that all the elementary fermions carry charge one under the background field A and all the elementary bosons are neutral means that all of these models superficially satisfy the spin/charge relation stating that all the fermions carry odd charge under A and all the bosons have even charge. More mathematically, this is the statement that if A is a spinc connection, we can formulate the theory without a choice of a spin structure (or even on a non-spin manifold), but with a choice of a spinc structure. However, although this is true for all the perturbative states, we should also examine monopole operators. The condition that the spin/charge relation is satisfied for them is [122]

Q = kbare mod 2 . (6.1.8)

244 Now let us turn to the action of time-reversal. On operators constructed from the elementary fields, we have the standard relation T 2 = (−1)F . However on states carrying magnetic charge this relation can be modified. In [92, 139, 151] our dynamical gauge field b was interpreted as a classical background field and a mixed anomaly was found between time-reversal symmetry T and the U(1) global symmetry coupling to b. When the field b is instead dynamical we interpret this result to mean that the symmetry algebra is modified as in the discussion of section 6.1.2. Specifically we find:

  F (−1) M Nodd = 2 mod 4 , T 2 = (6.1.9)  F (−1) Nodd = 0 mod 4 , where in the above, M ≡ (−1)M (6.1.10) generates the Z2 subgroup of the U(1) magnetic global symmetry. Notice also that Nodd/2 =

2 kbare mod 2, and thus the above relation can also be expressed by saying that T contains the magnetic symmetry whenever kbare is odd. We review aspects of this result as they arise below.

6.1.4 Summary of Models

Our ﬁrst class of examples is three-dimensional quantum electrodynamics U(1)0 with Nf fermionic ﬂavors of unit charge, where as described above we choose Nf to be even to ensure time-reversal symmetry.

The global symmetries of these models are easily diagnosed. The continuous part is

(SU(Nf ) × U(1)M )/ZNf , where the U(1)M factor is the magnetic global symmetry that

245 acts on monopole operators. Additionally we have charge conjugation C and time-reversal T .

Following the discussion around (6.1.9), the unitary symmetry T 2 depends on the number of flavors. Specifically we find that

2 F 2 F Nf = 0 mod 4 =⇒ T = (−1) ,Nf = 2 mod 4 =⇒ T = (−1) M . (6.1.11)

T In the latter case time-reversal is an order four symmetry (denoted Z4 ) and is mixed with

T the magnetic symmetry as (U(1)M o Z4 )/Z2. We also observe that, although there are fermions in the ultraviolet Lagrangian, these models do not have any gauge invariant local fermionic operators.

The special case Nf = 2 is worthy of separate analysis. For this theory there is a conjectured self-duality [67,153] which implies that the IR limit has enhanced global symmetry [22,67,136,153]. Including both C and T the pattern of enhancement is

SU(2) × P in−(2) T O(4) T UV : o Z4 −→ IR : o Z4 . (6.1.12) Z2 × Z2 Z2

Some aspects of these models have been investigated in [151] and in related analysis in the condensed matter literature [92, 139]. Note that compared to these works we do not say that time-reversal symmetry is anomalous. Indeed in all of these theories, T is a global symmetry of the model and the spectrum is organized into associated representations.

However, if Nf = 2 mod 4, the symmetry T satisﬁes the non-standard algebra stated in (6.1.11). Thus in this case it is not meaningful to compute the quantity ν. Instead we must

T separately classify and compute anomalies for the correct symmetry (U(1)M o Z4 )/Z2.

Our next class of time-reversal invariant gauge theories is QED with a single fermion of even charge q. These theories have a U(1)M magnetic symmetry as well as charge

246 conjugation C and time-reversal T , which obey a familiar symmetry algebra. In particular:

T 2 = (−1)F , C2 = 1 , TCT −1 = C , T exp(iαM)T −1 = exp(−iαM) . (6.1.13) Taking as input the dualities in [94, 121, 126, 140] we derive new fermionic particle-vortex dualities which determine the long-distance behavior of these models. For instance, in the simplest case of a charge two fermion, the infrared is a free Dirac fermion together with a decoupled topological ﬁeld theory U(1)2.

U(1)0 + ψ with charge two ←→ free Dirac fermion χ + U(1)2 , (6.1.14) where the time-reversal symmetry in the UV acts on the topological sector in the IR via level-rank duality as in table 6.1.

We deform the theory (6.1.14) by adding monopole operators to the Lagrangian, which

M breaks the magnetic symmetry to Z2 generated by (−1) = M and preserves a new antiunitary time-reversal symmetry T 0. The algebra of these symmetries is non-abelian. In particular:

T 0CT 0−1 = CM , T 02 = (−1)F , (CT 0)2 = (−1)F M . (6.1.15)

Both the models described above admit extensions to higher rank gauge theories. One possible such extension is to consider U(N) gauge theory. Instead here we will examine

SO(N). In this case for N > 2 the magnetic symmetry is Z2 with generator M.

In the case of QED3 with Nf ﬂavors the natural generalization is to SO(N) Chern-

Simons theory coupled to Nf fermions in the vector representation. These theories have been recently studied in [3, 35, 88] and participate in many dualities. These models have an O(Nf ) ﬂavor symmetry with ﬂavor charge conjugation symmetry Cf as well as the Z2

247 magnetic symmetry M, and we demonstrate that their algebra with time-reversal is non- abelian

−1 TCf T = Cf M . (6.1.16)

Analogously, QED3 with a charge two fermion naturally generalizes to SO(N)0 coupled to a fermion in the symmetric tensor representation. These models have global symmetries C, M, and T and we demonstrate that the algebra is non-abelian

TCT −1 = CM . (6.1.17)

As we describe below, this algebra is intimately related to the jump across tensor transitions of certain discrete θ-parameters in these models [35]. We study the algebra (6.1.17) in the context of the phase diagram of these theories determined in [88], and compute the time- reversal anomaly ν using both the UV and IR descriptions.

We also brieﬂy discuss similar theories of SO(N)0 coupled to adjoint fermions, which also enjoy the algebra (6.1.17).

The outline of this paper is as follows. In section 6.2 we analyze QED3 with Nf fermions of unit charge and derive the algebra (6.1.9) by analyzing the monopole operators. In section

6.3, we consider QED3 with a single fermion of general even charge, and we determine its long-distance behavior both with and without monopole operator deformations. In section

6.4 we consider SO(N)0 coupled to Nf vector fermions and derive the non-abelian algebra

(6.1.16). Finally, in section 6.5 we analyze SO(N)0 theories with tensor fermions. We demonstrate the algebra (6.1.17), and elucidate its interplay with the IR phase diagram.

248 6.2 QED3 with Nf Fermions of Charge One

In this section we study U(1) gauge theories with time-reversal symmetry. We consider

i models with Nf species of fermions ψ of charge one, where i is a ﬂavor index. We will be interested in the global symmetry algebra, including the interplay of unitary symmetries and time-reversal. As reviewed in the introduction, we must have Nf ∈ 2Z to have T symmetry.

The unitary global symmetries of this model have been analyzed in detail in [22]. They form the group

SU(Nf ) × U(1)M C o Z2 , (6.2.1) ZNf where the SU(Nf ) subgroup acts on the fundamental ﬁelds, the U(1)M factor is the

C monopole symmetry, and Z2 is the charge conjugation symmetry. Finally the ZNf subgroup deﬁning the quotient is generated by:

2πi/Nf (e INf , −1) ∈ SU(Nf ) × U(1)M , (6.2.2)

where INf is the identity matrix. Note that this quotient does not lead to the group U(Nf ).

Let us review the derivation of (6.2.1). On the elementary ﬁelds the magnetic symmetry

C does not act and the PSU(Nf )oZ2 symmetry is given by the construction described around (6.1.4). Meanwhile, the precise global form of the group including the magnetic symmetry

U(1)M can be determined by a careful analysis of the monopole operators. It is instructive to first view the gauge field as classical, and to work out the spectrum including charged operators. We then restore the fact that the gauge field is dynamical, and select the gauge invariant local operator.

i In the background of a minimally charged monopole, each fermion ψ and ψi has a single zero mode with spatial wavefunction ρ(x). Considering only the zero modes, we expand the

249 ﬁelds as

i i † ∗ ψ = α ρ(x) , ψi = αi γ0(ρ(x)) , (6.2.3)

i † where α and αi are creation and annihilation operators that have equal time commutation

i † i relations {α , αj} = δj. Since the ﬁelds have charge ±1, these creation and annihilation operators have no spin.

We now quantize this spectrum of zero modes. Let |0i denote the bare monopole state

† deﬁned to be annihilated by αi for all i. It has zero spin, and electric charge kbare = −Nf /2. Quantizing the Cliﬀord algebra of zero modes leads to the state space

i i i i i i |0i , α 1 |0i , α 1 α 2 |0i , ··· α 1 α 2 ··· α Nf |0i . (6.2.4)

We then deﬁne monopole operators Mi1i2···i` via the associated state

|Mi1i2···i` i ≡ αi1 αi2 ··· αi` |0i . (6.2.5)

These operators transform in totally antisymmetric representations of SU(Nf ) with ` indices, and they are all bosonic. The gauge invariant monopole operator has ` = Nf /2.

Note that under the center of SU(Nf ), this has Nf -ality Nf /2, which leads to the quotient (6.2.2).

Having understood the monopole operators, we continue our investigation of the symmetries. Observe that gauge invariant operators constructed out of the elementary fermions or gauge ﬁeld strengths must have an even number of fermions and are therefore bosons. As our analysis of the monopole operators illustrates they are also bosons. Therefore we conclude that all gauge invariant local operators in this theory are bosons. Another way to see this is that the theory satisﬁes the spin/charge relation where the dynamical gauge

250 ﬁeld is a spinc connection, and thus gauge invariant local operators must have integral spin [122].

We now turn to our main focus which is the time-reversal symmetry T of these models. On the elementary gauge non-invariant fermion ﬁelds this symmetry acts as stated in (6.1.3). This shows that on operators built from fundamental fermions we have a standard algebra T 2 = (−1)F .

To determine the action of T in sectors with non-vanishing monopole number we observe that the sum of the operators Mi1i2···i` for general ` form a Cliﬀord algebra representation for Spin(2Nf ), and the operator T is an anti-linear involution on this spinor representation. The operator T 2 is then either +1 or −1 depending on whether the representation is real or pseudoreal. Combining this with the discussion of the spin above we deduce

  F (−1) M Nf = 2 mod 4 , T 2 = (6.2.6)  F (−1) Nf = 0 mod 4 , where M = (−1)M .

More explicitly, time-reversal symmetry organizes the spectrum of fermion zero-modes into singlets and Kramers doublets. Using the mode expansion (6.2.3) we see that the

i −1 † creation and annihilation operators are related as T α T = αi . From this it follows that the bare monopole |0i is mapped by T to the top state in (6.2.4). More generally, time-reversal acts on the monopole operators as

`(`−1) (−1) 2 i1i2···i` −1 j1j2···jN −` T M T = εi i ···i j j ···j M f . (6.2.7) 1 2 ` 1 2 Nf −` (Nf − `)!

Notice that time-reversal changes fundamental indices of SU(Nf ) into antifundamental indices. The gauge invariant monopole operator is in a representation of SU(Nf ) that is

251 isomorphic to its complex conjugate and therefore (6.2.7) maps the monopole operator to itself. Using this formula it is straightforward to reproduce (6.2.6).

It is also instructive to consider a deformation of this theory that reduces the global symmetry. We add to the Lagrangian an operator of monopole number two:

i1i2···iN /2 j1j2···jN /2 δL = δi j δi j ··· δi j M f M f + c.c. . (6.2.8) 1 1 2 2 Nf /2 Nf /2

This deformation breaks the U(1)M symmetry down to a Z2 subgroup generated by M. It also breaks the ﬂavor symmetry (6.2.1) down to the subgroup that preserves δij, which is the orthogonal group O(Nf ). In particular, it preserves a Z2 charge conjugation element

Cf that acts by reﬂection on one of the ﬂavor indices. Of course, there are other ways to contract the indices in the double monopole (6.2.8). For example, for Nf = 0 mod 4 we can replace all δij → Jij with the standard antisymmetric Jij to break SU(Nf ) → Sp(Nf /2).

We will focus on (6.2.8) with the breaking to O(Nf ) because it exists for all even Nf and it ﬁts the discussion of SO(N) in section 6.4 below.

Let us investigate the algebra formed by time-reversal T and the symmetry Cf . Clearly on local operators constructed by polynomials in ﬁelds these operators commute. However, on monopole operators we ﬁnd a more interesting result. Since T acts as (6.2.7), the Cf charge of a monopole operator (i.e. ±1) is changed by the action of T . Thus T and Cf do not commute in a sector with monopole charge, but instead they obey the algebra

−1 TCf T = Cf M . (6.2.9)

One implication of this symmetry algebra is that

2 2 (Cf T ) = T M . (6.2.10)

252 The operator Cf T is another antiunitary symmetry that reverses the orientation of time and hence gives another time-reversal symmetry of this theory. What we see from (6.2.9)

2 2 is that one or the other of T or (Cf T ) must always involve the magnetic symmetry M.

6.2.1 Nf = 2: O(4) Unitary Symmetry

2 The simplest model that exhibits T = M is the case Nf = 2. This model is special because it has been conjectured to ﬂow to an infrared ﬁxed point with unitary O(4) symmetry [22, 67, 136, 153]. Here we will discuss the interplay between the enhanced symmetry and time-reversal.

The basis for the claim that the theory has enhanced global symmetry in the IR is a conjectural self-duality [67, 153], which acts on the (SU(2) × U(1)M )/Z2 global symmetry discussed in the previous section. Speciﬁcally, the duality exchanges a U(1) subgroup of the SU(2) symmetry that acts on the fundamental fermions with the U(1)M magnetic symmetry. Since the former is part of an SU(2) the latter must be as well.

More precisely, the duality in question states the equivalence of long-distance limits of the following two Lagrangians [67]2

1 2 1 1 1 iψ D/ ψ + iψ D/ ψ − ada + adY + Y dY a+X 1 a−X 2 4π 2π 4π 1 1 1 ←→ iχ1D/ χ + iχ2D/ χ − ada + adX + XdX . (6.2.11) ea+Y 1 ea−Y 2 4πe e 2πe 4π

3 In the above a, ea are dynamical U(1) gauge fields, and ψ, χ are Dirac fermions. The fields X and Y are background U(1) gauge fields coupling to the global symmetries that are exchanged under the duality. In the first line Y couples to the magnetic symmetry and X

2Below and in the following we omit gravitational Chern-Simons terms in our description of dualities. 3More precisely they are spinc connections [122].

253 couples to the charged ﬁelds, while in the dual Lagrangian on the second line their roles are reversed.

In order to determine the enhanced IR symmetry and the properties of the time-reversal symmetry T in this model, we must ﬁrst describe the complete UV symmetries. We use the language of the ﬁrst line in (6.2.11). In addition to the (SU(2) × U(1)M )/Z2 symmetry described in general in the previous section, there is also an order two charge conjugation symmetry C that acts as

C(ψ) = ψ , C(a) = −a , C(X) = −X, C(Y ) = −Y. (6.2.12)

The theory also has time-reversal symmetry T with T (a) = a, T (Y ) = −Y. We also deﬁne   0 1 X X   be the order four element in SU(2) given by the matrix =   . These −1 0 discrete symmetries of the UV theory are summarized in table 6.2.

Symmetry a X Y ea C −1 −1 −1 −1 X +1 −1 +1 −1 CY ≡ CX −1 +1 −1 +1 T +1 +1 −1 −1

Table 6.2: Symmetries and their eigenvalues in U(1)0 with two fermions of charge one. Note that Y is charged under T , and that CY T commutes with the unitary global symmetry. Under the duality (6.2.11), X ↔ Y and a ↔ ea.

Consider in particular the element CY deﬁned above. This stabilizes the SU(2), but acts on the U(1)M magnetic symmetry as reﬂection. Moreover, using the quotient described in (6.2.2), we see that (CY )2 = (−1)M . It follows that including charge conjugation extends

− 4 U(1)M to the group P in (2)M . Thus the unitary global symmetry in the ultraviolet is [136]

SU(2) × P in−(2) . (6.2.13) Z2 4 This corrects a small misidentiﬁcation of the global symmetry group in [22].

254 Now let us consider the implication of the self-duality (6.2.11). Since the duality exchanges the Cartan subgroup of the SU(2) with the U(1)M magnetic symmetry, it is clear that (6.2.13) must be enhanced in the IR to a group where the two factors in the numerator are on equal footing. The simplest possibility is O(4)

SU(2) × P in−(2) UV : −→ IR : O(4) . (6.2.14) Z2

Note that the exchange of the two SU(2) subgroups is now implemented by the duality which acts as a global symmetry.

The group O(4) has a Z2 center subgroup with non-trivial element z. From the point of view of the ﬁrst duality frame in (6.2.11) we recognize that z = (−1)M . Using our analysis of monopole operators in the previous section we therefore have:

T 2 = (CT )2 = z . (6.2.15)

This is consistent with the duality, which acts on the discrete symmetries in table 6.2 as

C ←→ C , T ←→ CT , X ←→ CY , CX ←→ Y . (6.2.16)

6.3 QED3 with Fermions of Even Charge

In this section we consider quantum electrodynamics with a single fermion ψ of general even charge q. Since the charge is even, we can adjust the bare Chern-Simons level to achieve a

C time-reversal invariant theory U(1)0. These theories have unitary symmetry U(1)M o Z2 .

As in our analysis in section 6.2, our goal is to elucidate the properties of T . On the local operators built from polynomials in the ﬁelds we ﬁnd as usual T 2 = (−1)F . (In fact all the elementary gauge invariant local operators are bosons.) Thus we now turn to the

255 monopole operators. The analysis is similar to that of the previous section, and hence we will be brief. For a complete treatment see [151].

In the background of a monopole of unit charge, the ﬁeld ψ now has q zero modes, which form an irreducible representation under the Lorentz group of spin j = (q − 1)/2. Notice that since q is even, the modes carry half-integral spin. These zero modes act on

2 the bare monopole state, |0i, which has zero spin and electric charge kbare = −q /2. The gauge invariant monopole operator M is dressed by q/2 zero modes and hence we deduce that its statistics is correlated with the electric charge as

  fermionic q = 2 mod 4 , M is (6.3.1)  bosonic q = 0 mod 4 .

Meanwhile, we can also compute the sign produced by T 2 acting on monopole operators. In this case, the modes ﬁll out a Dirac spinor of Spin(2q), and T 2 on these states is +1 if the spinor is real, and −1 if the spinor is pseudoreal. We can compare this to the statistics of the monopole and we ﬁnd, for all charge q, the expected relation T 2 = (−1)F . Thus the algebra of symmetries involving time-reversal is simple

T 2 = (−1)F , TCT −1 = C , T exp(iαM)T −1 = exp(−iαM) . (6.3.2)

Notice also that for q = 2 mod 4 the only fermionic operators are those with odd monopole number and hence (−1)F = (−1)M , while for q = 0 mod 4 all gauge invariant local operators are bosons.

Let us also discuss the possible anomalies of the time-reversal symmetry T . We can compute the time-reversal anomaly ν valued in Z16 by counting Majorana fermions λ in the UV lagrangian. In this calculation, a given fermion can have a sign σ that appears in

256 the formula T (λ) = σγ0λ, and the contribution to ν of λ is σ. This leads to the formulas

νT = 2 , νCT = 0 . (6.3.3)

6.3.1 Infrared Behavior

The models discussed above have simple long-distance description that can be derived assuming the particle-vortex dualities studied in [94,121,126,140]. This duality states that the following two Lagrangians describe the same IR physics (note the carefully normalized coeﬃcients [121]):

1 1 2 1 iψD/ ψ − AdA ←→ iχD/ χ − adb + bdb − bdA . (6.3.4) A 4π a 2π 4π 2π

In the above, our conventions are such that lower case letters (such as a, b) indicate dynamical abelian gauge ﬁelds, while capital letters (such as A) indicate classical background ﬁelds that couple to global symmetry currents.

Let us briefly summarize several aspects of this duality. The right-hand-side above is an interacting Chern-Simons matter theory with χ a Dirac fermion of charge one. In particular it defines a non-trivial RG flow. Meanwhile, the left-hand-side is free theory of a Dirac fermion ψ, which can therefore be viewed as the long-distance limit of the interacting theory. Under the duality, the magnetic global symmetry of the interacting theory is exchanged with the flavor symmetry in the dual free Dirac description. Thus, the operator ψ is dual to the monopole operator in the interacting description.

Duality for a Charge Two Fermion

We now assume (6.3.4) and use it to derive the IR behavior of U(1)0 coupled to an even charge fermion beggining with the case q = 2.

257 2 1 We substitute A → 2U and add classical terms 4π UdU + 2π UdB on both sides of (6.3.4), where U and B are new background ﬁelds. We then set U → u and promote u to be dynamical. On the right-hand-side, if we change variables to ub = u − b then the ﬁeld b becomes a Lagrange multiplier that can be integrated out. This results in the fermion-fermion duality5

2 1 2 1 iψD/ ψ − udu + udB ←→ iχD/ χ + udu + udB . (6.3.5) 2u 4π 2π B 4π b b 2π b

The left-hand side above is QED with charge-two Dirac fermion ψ. In this duality frame, the classical field B couples to the U(1)M magnetic global symmetry. The dual description on the right-hand side is a free Dirac fermion and the TQFT U(1)2, which we can view as the IR limit of the interacting theory. In this frame B couples to the χ flavor symmetry and to the U(1)2 sector. Thus, as in the particle-vortex duality (6.3.4), the monopole operator becomes a free field at long distances.

Notice that both sides of the duality have T symmetry. On the left-hand side this is simply because the bare Chern-Simons level for the dynamical gauge ﬁeld u has been ad- justed to make the theory time-reversal invariant. On the right-hand side the time-reversal symmetry exists because level-rank duality U(1)2 ↔ U(1)−2 (see table 6.1). Although both theories are time-reversal invariant, their two antiunitary symmetries are exchanged under the duality T ←→ CT . (6.3.6)

5We can repeat this analysis with the substitution A → A + 2U and then make U dynamical, while keeping a nontrivial background spinc connection A. Then we have a duality similar to (6.3.5), which is valid on a spinc manifold. However, if we do that the time-reversal symmetry is modiﬁed to T (A) = A, T (B) = −B + 2A.

258 This can be seen by comparing the transformation properties of the various gauge ﬁelds. For instance on the left-hand side T (u) = u and hence T (B) = −B. Therefore the dual description of this symmetry must act with a minus sign on ub and hence is CT . We can also compare the time-reversal anomalies for these theories across the duality.

In the description as U(1)0 with a charge two fermion the anomalies are read oﬀ from the Lagrangian resulting in (6.3.3). This matches with the free Dirac description if we use the fact that the anomaly for the semion-fermion spin TQFT U(1)2 is

νT (U(1)2) = −2 , νCT (U(1)2) = 2 , (6.3.7) where T and CT denote two distinct ways that the time-reversal anyon permutation symmetry can couple to the theory (they are also called SF− and SF+ in the literature) [92,130,134].

One way to check the duality (6.3.5) is to deform both sides by relevant operators. Assuming that the RG flow is smooth, the resulting theories after deformation must still be dual. Across the duality (6.3.5) the fermion mass terms map to each other with a relative sign ψψ ↔ −χχ. For positive coefficient of ψψ, the two sides of the duality flow to U(1)2 coupled to a background magnetic gauge field. For negative coefficient of ψψ the duality becomes:

2 1 2 1 1 − udu + udB ←→ udu + udB + BdB . (6.3.8) 4π 2π 4π b b 2π b 4π

Again, these two theories are equivalent via level-rank duality.

It is useful to explore how this theory and its long-distance behavior are modiﬁed when we add monopole operators to the Lagrangian. These results will also enable us to anticipate many features of the higher rank SO(N)+ tensor gauge theories described in section 6.5.

259 We perturb the theory by a bosonic operator of monopole charge two [62]:6 δL =

2 iM + h.c.. This interaction breaks the U(1)M magnetic symmetry down to Z2 generated by (−1)M ≡ M. This interaction also breaks the symmetry T . However it preserves a new symmetry T 0 ≡ T eiπM/2 . (6.3.9)

It is straightforward to determine the algebra of the unbroken symmetries C, M, T 0 using their embedding in the algebra (6.3.2). We have

T 0CT 0−1 = T eiπM/2Ce−iπM/2T −1 = (T eiπM/2T −1)(TCT −1)(T e−iπM/2T −1) = CM . (6.3.10) Thus the algebra of symmetries is now non-abelian. By similar manipulations we can also determine that (T 0)2 = (−1)F , (CT 0)2 = (−1)F M . (6.3.11)

Note also that the time-reversal anomaly νT 0 is simply equal to νT since the operators only diﬀer by a magnetic symmetry in the UV. Meanwhile for the antiunitary symmetry CT 0 the anomaly ν is no longer meaningful.

We can also ﬁnd the same result using the infrared description (6.3.5). The UV monopole

2 2 operator interaction maps to a mass term (χ1) −(χ2) , where we have written the complex fermion in terms of Majorana components. This mass term breaks the ﬂavor symmetry

2 down to a Z2 subgroup. Each mass term χi is odd under T , and hence time-reversal is also broken. However, the combination of T with a flavor rotation by π/2 is preserved and is identified with T 0. 6 In general we can add the operator αM2 + h.c. for complex coefficient α. Here we consider the C- even monopole perturbation. If instead we add a C-odd deformation, then we find equivalent physics. Specifically, the perturbed theory preserves C0 = CeπiM/2 and T , and the two symmetries again do not commute: T −1C0T = C0M.

260 The eﬀect of this mass term is to split the Dirac point into two distinct Majorana points. ∼ The phase in the middle is U(1)2 = SO(2)2 as illustrated in ﬁgure 6.1.

푈(1)0 with charge 2 + double monopole

Free Majorana 휒 + 푆푂(2) Free Majorana 휒1 + 푆푂(2)2 2 −2

푚휓 ≪ 0 푚 ≫ 0 푆푂(2)2 푆푂(2)2 푆푂(2)−2 휓 ↕

푆푂(2)−2

Figure 6.1: The phase diagram of U(1) gauge theory coupled to a Dirac fermion with charge two, together with a double monopole perturbation. The monopole perturbation splits the free Dirac point into two Majorana points. These transitions separate TQFTs.

Duality for General Even Charge

We now extend our analysis to theories with a general even charge q. Starting from (6.3.5)

q/2 1 we add the classical terms − 2π BdX + 2π XdBb to both sides. We then set B → b and

X → x with x, b dynamical (on the right we also rename x as xb). On the left-hand side, the integral over b is trivial leading to the duality

q2/2 1 2 1 q/2 1 iψD/ qxψ − xdx + xdBb ←→ iχD/ bχ + udu + udb − bdx + xdB.b 4π 2π 4π b b 2π b 2π b 2π b (6.3.12)

Note that in the special case q = 2, we can also integrate out xb on the right and reproduce the duality (6.3.5).

The left-hand side of the duality (6.3.12) is QED with even charge q fermion ψ. The right-hand side is a free Dirac fermion χ together with a U(1)2. The ﬁeld xb is a Lagrange

261 multiplier, which reduces b to a Zq/2 gauge field which couples to both χ and the topological sector U(1)2. On local operators the effect of the Zq/2 gauge field is simply to quotient the spectrum. In particular, the right-hand side is effectively free and hence can be viewed as the IR limit of the interacting theory on the left-hand side.

Many of the essential features of this duality are similar to the case of charge two.

• The unit charge monopole operator M in the QED description maps across the duality

q/2 to the operator χ , which is the minimal local operator consistent with the Zq/2

quotient. Note (via integrating out xb) that both operators couple to the background ﬁeld Bb with unit charge and that the statistics of these operators agree from our general result (6.3.1).

• Both theories are time-reversal invariant. Across the duality T and CT are exchanged and the time-reversal anomalies agree using (6.3.7).

• The fermion mass terms match again up to a relative sign. Deforming the duality by

these relevant operators we ﬁnd that both theories ﬂow to the TQFT U(1)±q2/2.

As a further consistency check of the general charge q duality, we can match the one- form symmetry and its ‘t Hooft anomaly. Both theories in (6.3.12) have Zq one-form 1 symmetry [54]: on the left it is generated by x → x + q dθ for periodic scalar θ ∼ θ + 2π, while on the right it is generated by x → x + 1 dθ and u → u + 1 dθ. We can turn on b b q b b 2 background gauge ﬁeld B2 for this one-form symmetry and study its ‘t Hooft anomaly. Since the fermion mass term is invariant under the one-form symmetry, the anomaly can be computed from the resulting TQFT U(1)±q2/2 under the mass perturbation mψψ with m positive or negative. This gives the same Z2 ⊂ Zq valued anomaly on both sides of the duality

Z P(B ) π 2 , (6.3.13) Y 2 262 where Y is a closed four-manifold, P is the Pontryagin square with coeﬃcient in Zq, and

B2 is the background two-form Zq gauge ﬁeld for the one-form symmetry.

Monopole Deformation of the Charge Four Theory

Let us now specialize from general charge q and consider some aspects of the theory with q = 4. This is the same theory as Spin(2)0 coupled to a symmetric tensor fermion [35] and hence our results here anticipate the higher-rank generalizations of section 6.5.

Again we ﬁnd it useful to add a monopole operator interaction to break the magnetic

U(1)M symmetry. In the Spin(2)0 theory the unit charge monopole M of the SO(2)0 theory is absent and instead, the basic allowed monopole operator is the charge two monopole M2

2 of SO(2)0. As in the analysis of section 6.3.1, we add the perturbation δL = iM + h.c., but in this case, the U(1)M symmetry is completely broken.

This deformation breaks the time-reversal symmetry T , but preserves the antiunitary symmetry T 0 = T eiπM . Therefore after deformation the unbroken symmetries are T 0 and charge conjugation C. By using the algebra (6.3.2) we ﬁnd that after the deformation the symmetry algebra is standard

T 0CT 0−1 = C , C2 = 1 , T 02 = (−1)F (6.3.14)

Moreover, the time-reversal anomalies are unmodiﬁed from their values before the symmetry breaking perturbation, i.e. νT 0 = νT and νCT 0 = νCT .

The long-distance behavior for the theory with q = 4 and the symmetry breaking monopole perturbation can be obtained by gauging the Z2 magnetic symmetry M in the phase diagram of the theory with q = 2 presented in ﬁgure 6.1. In the IR, the monopole

2 2 deformation is a mass term (χ1) − (χ2) for the two Majorana fermions, and integrating out these massive ﬁelds generates a non-trivial Lagrangian for the new Z2 gauge theory.

263 Speciﬁcally this is the theory (Z2)1, where the subscript indicates that this is the minimal consistent level in Z2 gauge theory. (See [35] for additional discussion.)

Taking into account the new topological sector from gauging M we ﬁnd that in the presence of the monopole perturbation, the infrared theory is the TQFT

U(1)8 × (Z2)1 ←→ O(2)2,1 , (6.3.15) Z2 where the quotient on the left-hand side gauges the one-form symmetry generated by the product of a charge 4 line in U(1)8 and the Wilson line of the Z2 gauge theory. This is equivalent to the T-Pfaﬃan spin TQFT [25,29], and it is also dual to O(2)2,1 [35].

푈(1)0 with charge 4 + monopole

3 3 푂(2)2, + 퐶- odd Majorana 휒1 푂(2)−2,− + 퐶- odd Majorana 휒2 2 2

푚휓 ≪ 0 푚휓 ≫ 0

푂(2)2,1 (T-Pfaffian) 푈(1)−8 푈(1)8 ↕ ↕ ↕ 푂(2) 푂(2)2,2 −2,−1 푂(2)−2,−2

Figure 6.2: The phase diagram of U(1) gauge theory coupled to a Dirac fermion of charge four, with a minimally charged magnetic monopole perturbation. In the two dual descriptions the Majorana fermion χ couples to the Z2 gauge ﬁeld of O(2) by the transformation χ → −χ. The low energy TQFT is the T-Pfaﬃan theory.

In particular, the time-reversal anomalies νCT = 0, νT = 2 agree with those of T-

Pfaﬃan+ (the subscript indicates a particular deﬁnition of the antiunitary symmetry).

Note that the names T , CT are reversed in the literature, see [92, 130] for νCT for T-

264 Pfaﬃan, and [47, 94] for νT . The latter symmetry of T-Pfaﬃan is the diagonal subgroup of the conventional time-reversal symmetry and the magnetic symmetry of O(2). (In the duality U(1)8 ↔ O(2)2 the charge conjugation symmetry of U(1)8 maps to the magnetic symmetry of O(2)2 [35].)

The resulting phase diagram is illustrated in ﬁgure 6.2

6.4 SO(N)0 with Vector Fermions

In this section we consider SO(N)0 with Nf fermions in the vector representation, where Nf is even to avoid the standard parity anomaly. These theories have a time-reversal symmetry T , whose basic properties we discuss below. In section 6.2 we analyzed the case N = 2, and many features of those models are common to the case N > 2.

We first describe the global unitary symmetries. There is a flavor symmetry O(Nf ), which includes a flavor charge conjugation Cf that acts on one of the flavor indices as

7 reﬂection. There is also charge conjugation C and a Z2 magnetic symmetry M. Notice that in comparison with the abelian gauge theories analyzed in section 6.2 the magnetic symmetry in SO(N) for N > 2 is always Z2 (measured by the Z2-valued second Stiefel-

Whitney class). Thus the global symmetries agree with those of U(1)0 with Nf ﬂavors after the deformation by a monopole operator of charge two.

Now let us consider the time-reversal symmetry T . Clearly on local operators constructed out of the elementary fields we find the standard algebra (i.e. T 2 = (−1)F ). Meanwhile on monopole operators the analysis of time-reversal symmetry and its square is similar to that of section 6.2. The basic monopole can be described by a gauge field

7 Depending on Nf and N, there can be discrete identiﬁcations on these global symmetries when acting on gauge invariant local operators.

265 conﬁguration with unit ﬂux in SO(2) ⊂ SO(N), and each of the Nf fermions in the vector representation has one zero mode.

Therefore, we similarly ﬁnd that the time-reversal symmetry T does not commute with

Cf on operators carrying magnetic charge

−1 TCf T = Cf M . (6.4.1)

In addition we have

   F  F (−1) M Nf = 2 mod 4 , (−1) Nf = 2 mod 4 , 2  2  T = (Cf T ) = (6.4.2)  F  F (−1) Nf = 0 mod 4 , (−1) M Nf = 0 mod 4 .

We can also compute the time-reversal anomaly ν of these theories. For an antiunitary symmetry that squares to (−1)F , ν is given by the number of Majorana fermions ψ that transform as ψ(x, t) → γ0ψ(x, −t) minus the number of Majorana fermions ψ0 that transform as ψ0(x, t) → −γ0ψ0(x, −t). Therefore

νT = NNf , νCf T = NNf − 2N. (6.4.3)

Notice that it is not obvious that the answer (6.4.3) is gauge invariant since the sign in the time-reversal transformation of a fermion can be changed by a gauge symmetry.

Consider for instance combining T or Cf T with the Z2 ⊂ SO(N) gauge transformation diag(1, · · · − 1, ··· ) with 2p total minus signs. The time-reversal anomalies (6.4.3) change to

∆νT = 4Nf p , ∆νCf T = 4(Nf − 2)p . (6.4.4)

266 However as emphasized above, the anomaly ν of an antiunitary symmetry is only meaningful when that symmetry squares to fermion parity. According to (6.4.2) when this is so, the ambiguity above vanishes in Z16, and the anomaly ν is well-deﬁned.

As a ﬁnal remark on these models, let us consider gauging the magnetic symmetry M. This changes the gauge group from SO(N) to Spin(N)[35]. From (6.4.2), we conclude that

F in the Spin(N)0 gauge theory coupled to vector fermions both T and Cf T square to (−1) . In particular, the anomaly ν is meaningful for both symmetries.8 This is also compatible with the calculation (6.4.4). The group Spin(N) is a double covering of SO(N), and the gauge transformation used in (6.4.4) is an Z2 element only if p is even.

6.5 SO(N)0 with Two-Index Symmetric Tensor Fermion

For our ﬁnal class of models, we consider SO(N)0 with one fermion in the two-index symmetric tensor representation. Across the transition where the fermion becomes massless the Chern-Simons level jumps from −(N + 2)/2 to +(N + 2)/2. Therefore, N must be even to achieve a time-reversal invariant theory at zero fermion mass. Aspects of these theories were discussed in [35,62]. The special case N = 2 is U(1) gauge theory coupled to a charge two fermion and was analyzed in section 6.3.

These models have unitary global symmetry C and M which form Z2 × Z2, as well as time-reversal symmetry T . Notice that these are the symmetries present in U(1) plus a charge two fermion, after deformation by a monopole operator of magnetic charge two. Therefore many of aspects of these models are similar.

8More generally, whenever we have the algebra T 2 = (−1)F X with some X we could try to gauge the symmetry generated by X to ﬁnd a new theory with T 2 = (−1)F . However, there is a subtlety in doing it. A mixed anomaly between T and X in the original theory can mean that after gauging T is no longer a symmetry. More precisely, as we said in the introduction, the gauging of X leads to a one-form global symmetry generated by Xb and the mixed anomaly can lead to a new symmetry, a 2-group, which mixes T and Xb [128]. We will not analyze it in detail here. We simply point out that in the example above and in section 6.5.3 this phenomenon does not happen.

267 6.5.1 Time-Reversal Symmetry and its Anomaly

As in all our previous analysis, on elementary ﬁelds the time-reversal symmetry T satis- ﬁes T 2 = (−1)F and hence we turn to the sector with non-trivial monopole charge. The bare classical monopole operator transforms in the (N + 2)/2-index symmetric tensor representation of SO(N). This can be derived, for instance, by giving the fermions mass and using N + 2 k = − . (6.5.1) bare 2

If N = 0 mod 4, this representation is charged under the center of SO(N) and cannot be screened by any elementary fermion ﬁeld. Thus in this case, there is no local monopole operator. By contrast when N = 2 mod 4, the bare monopole is neutral under the center of SO(N) and hence can be dressed by fermions to form a gauge invariant local operator.

Regardless of whether the charge of the monopole can be screened by fundamental ﬁelds, we can always produce sectors with monopole charge by attaching an appropriate Wilson line to classical conﬁguration. The resulting object is gauge invariant, but nonlocal since it now contains the line.

The algebra of global symmetries in sectors with magnetic charge can again be understood by analyzing zero modes in a monopole background. To be speciﬁc, we consider a unit magnetic ﬂux in the N,N − 1 direction of the gauge group which breaks SO(N) to

(O(2) × O(N − 2)) /Z2. The symmetric tensor fermion decomposes in this background into the following ﬁelds:

• A Dirac fermion of charge 2 under O(2) which is neutral under O(N − 2). This ﬁeld

has two zero modes which form a spin 1/2 doublet. We indicate them by ψa for a = 1, 2. There are also complex conjugate ﬁelds.

268 • A Dirac fermion of charge 1 under the O(2) which transforms as a vector of O(N −2). This ﬁeld has N − 2 zero modes which are Lorentz scalars. We denote them via their embedding in the symmetric tensor as (ψ(N,N−j) +iψ(N−1,N−j)), where j = 1, ··· N −2. There are also complex conjugate ﬁelds.

• (N 2 −3N +2)/2 fermions which are neutral under the O(2). These ﬁelds have no zero modes in the monopole background and hence decouple from our analysis.

Let |0i indicate the bare monopole described above. Quantizing the zero-modes leads to a Hilbert space of states

N−1 Y (N,N−j) (N−1,N−j) |0i , ··· , ψ1ψ2 (ψ + iψ )|0i , (6.5.2) j=2 any of which can be made gauge invariant by attaching a suitable Wilson line. The action of time-reversal symmetry T exchanges the top and bottom states listed in (6.5.2). Notice that charge conjugation C acts by a sign on ﬁelds with gauge index one. Therefore, the bottom and top states above have opposite charge under C, and we see that the algebra of symmetries is non-abelian TCT −1 = CM . (6.5.3)

The non-abelian algebra above is closely related to the behavior of discrete θ-parameters discussed in [35]. Speciﬁcally, for any level k we can consider the SO(N)k+tensor fermion

C theory coupled to a background Z2 gauge ﬁeld B for the charge conjugation global symmetry. As the mass of the tensor is transitioned from negative to positive the eﬀective

R C action shifts by the coupling π X B ∪ w2 where w2 is the second Stiefel-Whitney class of the SO(N) bundle, which measures the charge M. This means that across such a tensor transition the symmetry C is exchanged with CM. In the speciﬁc case of a time-reversal invariant theory this implies (6.5.3) since the fermion mass is odd under T .

269 We can also compute the action of T 2 and (CT )2. Since the monopole is eﬀectively abelian T 2 is ﬁxed by the parity of the bare Chern-Simons level as described in section 6.1.3. Using the formula (6.5.1) we conclude that

   F  F (−1) N = 2 mod 4 , (−1) M N = 2 mod 4 , T 2 = (CT )2 = (6.5.4)  F  F (−1) M N = 0 mod 4 , (−1) N = 0 mod 4 .

It is also straightforward to compute the time-reversal anomaly ν for these theories [62]

N 2 + N − 2 N 2 − 3N + 2 ν = , ν = . (6.5.5) T 2 CT 2

As a consistency check, consider mixing T or CT with a Z2 ⊂ SO(N) gauge transformation diag(1, · · · − 1, ··· ) with 2p minus signs. The change in the anomalies is

2 2 ∆νT = 8p − 4Np , ∆νCT = 8p + 8p − 4Np . (6.5.6)

Exactly when T or CT square to fermion parity, these changes above vanish modulo sixteen as expected.

6.5.2 Time-Reversal Symmetry in the IR

The long-distance behavior of these models has been analyzed in [62] leading to the proposed phase structure summarized in ﬁgure 6.3.

In particular, for small fermion mass the theory is conjectured to ﬂow to a quantum phase described by the spin TQFT SO(n)n with n = (N +2)/2. This theory is time-reversal invariant as a spin TQFT by level-rank duality [3] (see table 6.1)

SO(n)n ←→ SO(n)−n . (6.5.7)

270

푆푂(푁)0 with 푆 symmetric

푍2 × 푍2 symmetry (퐶, 푀)

푁+2 푁+2 ̂ 푆푂 ( ) + adjoint 휆 푆푂 ( ) 3푁+2+ adjoint λ 2 3푁+2 2 − 4 4 (푀, 퐶푀) (퐶푀, 퐶)

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0 푁 + 2 푆푂 ( ) ( ) 푆푂(푁) 푁+2 푁+2 푆푂 푁 푁+2 − 2 2 2 2 ↕ (퐶, 푀) (퐶푀, 퐶) ↕ (퐶푀, 푀) ↕ 푁 + 2 푁 + 2 푁 + 2 푆푂 ( ) 푆푂 ( ) 푆푂 ( ) 2 푁 푁+2 2 −푁 2 − (푀, 퐶) 2 (퐶, 퐶푀) (푀, 퐶푀)

Figure 6.3: The phase diagram of SO(N) gauge theory coupled a symmetric tensor fermion S. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. Each of these transitions can be described by a dual theory with adjoint fermions, in which the transition can be seen at weak coupling. These dual theories cover part of the phase diagram. These ﬁgures are identical to those in [62], with the map of the Z2 × Z2 unitary global symmetry determined from [35].

Let T IR be the time-reversal symmetry of the infrared TQFT that squares to (−1)F . It is related to the UV symmetries discussed in the previous section via

  UV T N = 2 mod 4 , T IR = (6.5.8)  UV UV C T N = 0 mod 4 .

The time-reversal anomaly ν for the TQFT SO(n)n is known to be n [31]. Combining this with the map (6.5.8) of UV and IR symmetries we can check the phase diagram of ﬁgure 6.3 using anomaly matching. We have:

N 2 + N − 2 N + 2 N 2 − 4 N = 2 mod 4 =⇒ ν − ν = ν UV − n = − = , (6.5.9) UV IR T 2 2 2

271 N 2 − 3N + 2 N + 2 N 2 − 4N N = 0 mod 4 =⇒ ν − ν = ν UV UV − n = − = . UV IR C T 2 2 2

Both expressions vanish modulo sixteen as expected.

The charge conjugation and magnetic symmetries in the UV and IR are related by

MUV = CIRMIR, CUV = CIR . (6.5.10)

In particular, the algebra (6.5.3) for CUV, MUV and T UV implies in the infrared the time- reversal symmetry satisﬁes

T IRCIR = MIRT IR , (6.5.11) which is consistent with the fact that CIR, MIR are exchanged under level-rank duality [3,35].

6.5.3 Gauging the Magnetic Symmetry: Spin(N)0 + Tensor Fermion

Consider gauging the Z2 magnetic symmetry M. This changes the theory to Spin(N)0 coupled to a symmetric tensor fermion. These models are a natural generalization of the

U(1)0 with a charge four fermion discussed in section 6.3.1. From (6.5.4) we immediately see that both time-reversal symmetries T and CT are standard (see footnote8)

TCT −1 = C, T 2 = (CT )2 = (−1)F . (6.5.12)

In particular, the anomaly ν for both T and CT is well-deﬁned for all even N. This is compatible with the anomaly computation (6.5.6) since there p is required to be even for the Z2 gauge transformation to be an element of Spin(N).

272

푆푝푖푛(푁)0 with 푆 symmetric

1 1 − 푁+2 푁+2 2 푂 ( )2 + adjoint λ̂ 푂 ( ) 3푁+2 3푁 + adjoint 휆 2 3푁+2 3푁 2 , − ,− 4 4 4 4

푚푆 ≪ 0 small |푚푆| 푚푆 ≫ 0

푁 + 2 1 푆푝푖푛(푁) 푁+2 푂 ( ) 푆푝푖푛(푁)푁+2 − 푁+2 푁 2 2 , 2 2 2 ↕ ↕ ↕ 0 0 푁 + 2 푁 + 2 1 푁 + 2 푂 ( ) 푂 ( ) 푂 ( ) 2 푁,푁 푁+2 푁 2 −푁,−푁 2 − ,− 2 2

Figure 6.4: The phase diagram of a Spin(N) gauge theory coupled to a fermion S in the two-index symmetric tensor representation. It can be obtained from SO(N) gauge theory by gauging the magnetic symmetry in the UV, which corresponds to gauging the diagonal CM in the long-distance TQFT. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. Each of these transitions can be described by a dual theory with adjoint fermions, in which the transition can be seen at weak coupling. These dual theories cover part of the phase diagram. These ﬁgures are identical to those in [35].

The phase diagram of the Spin(N) gauge theory was derived in [35] and is reproduced in figure 6.4. This is related to the phase digram in figure 6.3 by gauging. In particular, for small fermion mass the theory is conjectured to flow to a quantum phase described by the

1 spin TQFT O(n)n,n−1 (the notation as in [35]) with n = (N + 2)/2, which is time-reversal invariant as a spin TQFT by level-rank duality [35] (see table 6.1)

1 1 O(n)n,n−1 ←→ O(n)−n,−n+1 . (6.5.13)

This theory is a higher rank generalization of the T-Pfaﬃan theory O(2)2,1 discussed in section 6.3.1 for N = 2. (See also appendix H of [35].)

273 From the phase diagram in ﬁgure 6.3 we can deduce that the unitary symmetries in the UV and IR are related as

CUV(Spin(N)) = MIR(O(n)) . (6.5.14)

Thus from (6.5.8), the antiunitary symmetries in the UV and IR are related as

   UV  UV UV T N = 2 mod 4 , C T N = 2 mod 4 , T IR = MIRT IR =  UV UV  UV C T N = 0 mod 4 , T N = 0 mod 4 . (6.5.15)

We proceed to match the anomaly ν between the short-distance and long-distance descriptions. The anomaly for T IR always matches the corresponding UV anomaly, since they agreed in the SO(N) theory before gauging MUV [62]. Therefore, we focus on the anomaly for the antiunitary symmetry MIRT IR.

The UV computation is as in the SO(N) theories (6.5.5). In the IR we need to compute

1 νMT (O(n)n,n−1). This can be done using the formalism in [129, 134]. The answer depends on choices we do not know how to determine, like the eigenvalue of T 2 on some anyons and on a choice of orientation (i.e. the sign of ν).

We will split the discussion depending on N mod 4. For N = 0 mod 4, n is odd and we have (see appendix D of [35]):

1 O(n)n,n−1 ←→ SO(n)n × (Z2)2(n−1) . (6.5.16)

1 Furthermore, for odd n the magnetic symmetry in O(n)n,n−1 does not permute the anyons [35], and thus both T IR and MIRT IR deﬁne the same permutation action on the anyons.

Therefore, the anomaly ν can be obtained from that of SO(n)n [31] and that of (Z2)0, (Z2)4

274 [47,92,134]

ν((Z2)0) = 0 or 8 , ν((Z2)4) = 0 or + 4 or − 4 , (6.5.17) where the values depend on the choices mentioned above. With appropriate choices here we match that anomalies with those of the UV theory!

1 For N = 2 mod 4, n is even and the magnetic symmetry of O(n)n,n−1 permutes the anyons. This makes the computation of the anomaly slightly more involved and we have not carried it out explicitly. Instead, we use the anomaly matching with the UV theory to

1 conjecture that for even n we have νMT (O(n)n,n−1) = ±5(n − 2). Note that for n = 2 this agrees with the expected answer for the T-Pfaﬃan+ theory. (This is to be contrasted with the known value νT = ±n of these theories.)

푆푝푖푛(푁)0 with adjoint 휆 for 푁 > 4

1 1 − 푁 − 2 2 푁 − 2 2 푂 ( ) 푂 ( ) 3푁−2 3푁+4 3푁−2 3푁+4 2 , 2 − ,− 4 4 4 4 + symmetric S + symmetric Ŝ

SUSY

푚휆 ≪ 0 푚휆 ≫ 0

푁 − 2 1 푂 ( ) 푆푝푖푛(푁) 푁−2 푁−2 푁 푆푝푖푛(푁)푁−2 − 2 , +2 2 2 2 2 ↕ ↕ massless Goldstino ↕ 0 푁 − 2 0 푁 − 2 1 푁 − 2 푂 ( ) 푂 ( ) 푂 ( ) 2 푁−2 푁 2 −푁,−푁 푁,푁 2 − ,− −2 2 2

Figure 6.5: The phase diagram of a Spin(N) gauge theory coupled to a fermion λ in the adjoint representation. The phase transitions are visible at weak coupling in a dual theory with symmetric tensor fermions. This can be derived from the SO(N) gauge theory by gauging the UV magnetic symmetry MUV, which corresponds to gauging the symmetry CIRMIR in the long-distance TQFT [35].

The entire discussion of this section can be repeated with a fermion in the adjoint representation. In particular, for gauge group SO(N) these theories also have the non-

275 commutative algebra of symmetries (6.5.3). Similar phase diagrams for SO(N) and Spin(N) gauge groups are conjectured in [35, 62]. Now the infrared theory consists of a massless Goldstino (take N > 4) and a time-reversal invariant TQFT. For gauge group

1 Spin(N) the TQFT is O(n)n,n+3 with n = (N − 2)/2 (see ﬁgure 6.5), and we can match both νT and νCT between the UV and the IR. Again, the matching of νT IR follows from the matching in SO(N)[62]. The matching of νMIRT IR is a new test of the conjectured phase diagram. In particular, for N = 0 mod 4, n is odd and we can use the relation

1 O(n)n,n+3 ↔ SO(n)n × (Z2)2(n+1) [35]. Then, with an appropriate choice in (6.5.17) we match the UV and the IR values.

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