Anomaly of 2 1 -Dimensional Symmetry-Enriched Topological Order from 3 1 -Dimensional Topological Quantum Field Theory Weicheng Ye1 2and Liujun Zou1

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Anomaly of (2 + 1)-Dimensional Symmetry-Enriched Topological Order

from (3 + 1)-Dimensional Topological Quantum Field Theory

Weicheng Ye1, 2 and Liujun Zou1

1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada N2L 2Y5

2Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Symmetry acting on a (2+1)Dtopological order can be anomalous in the sense that they possess

an obstruction to being realized as a purely (2+1)Don-site symmetry. In this paper, we develop

a (3+1)Dtopological quantum ﬁeld theory to calculate the anomaly indicators of a (2+1)Dtopo-

logical order with a general symmetry group G, which may be discrete or continuous, Abelian or

non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indi-

cators are partition functions of the (3+1)Dtopological quantum ﬁeld theory on a speciﬁc manifold

equipped with some G-bundle, and they are expressed using the data characterizing the topological

order and the symmetry actions. Our framework is applied to derive the anomaly indicators for

various symmetry groups, including Z2×Z2,ZT

2×ZT

2,SO(N), O(N)T,SO(N)×ZT

2, etc, where Z2

and ZT

2denote a unitary and anti-unitary order-2 group, respectively, and O(N)Tdenotes a sym-

metry group O(N) such that elements in O(N) with determinant −1 are anti-unitary. In particular,

we demonstrate that some anomaly of O(N)Tand SO(N)×ZT

2exhibit symmetry-enforced gapless-

ness, i.e., they cannot be realized by any symmetry-enriched topological order. As a byproduct, for

SO(N) symmetric topological orders, we derive their SO(N) Hall conductance.

Contents

I. Introduction 1

A. Relation to prior work 3

B. Outline and summary 4

II. Review of topological order with symmetry G4

A. Review of UMTC notation 4

B. Global symmetry 5

III. (3+1)DTQFT with ﬁnite group symmetry G7

A. Characterizing the anomaly by

bulk-boundary correspondence 8

B. General construction of TQFT 8

C. Handle decomposition 11

D. Recipe for calculating the partition

function 12

IV. Examples: ﬁnite group symmetry 15

A. No symmetry 15

B. ZT

216

C. Z2×Z217

D. ZT

2×ZT

219

1. All-fermion Z2topological order 21

V. Generalization to connected Lie group

symmetry 24

A. Example: SO(N)25

1. Anomaly indicator for N⩾526

2. SO(N) Hall conductance 27

VI. Other symmetry groups 28

A. O(N)T28

B. SO(N)×ZT

230

VII. Discussion 31

A. Derivation of Eq. (44) 34

1. Vector Spaces 34

2. Partition functions 34

3. Inner Products 35

4. Requirement from Invertibility 36

B. An explicit expression of the η-factor 37

C. Consistency check of TQFT 37

1. Independence on the handle

decomposition 38

2. Invariance under change of defects 42

3. Gauge invariance 44

4. Cobordism invariance 45

5. Invertibility 46

6. Generalization to connected Lie groups 46

D. Identifying the manifold Mfrom bordism 47

E. More information about handle decomposition

of manifolds 48

1. CP248

2. RP448

3. RP3×S149

4. RP2×RP249

References 50

I. Introduction

Topological orders are interesting gapped quantum

phases of matter beyond the conventional paradigm,

and their discovery is one of the main forces that rev-

olutionized modern quantum many-body physics [1].

Instead of being characterized by local order param-

eters associated with symmetries, in (2+1)Dthey are

arXiv:2210.02444v3 [cond-mat.str-el] 30 May 2023

characterized by anyons, quasiparticle excitations with

nontrivial statistics that may be neither bosonic nor

fermionic. The physical properties of a topological or-

der are nicely summarized using the language of ten-

sor category [2–5], and in particular in (2+1)Dbosonic

systems the data of anyons forms an elegant mathemat-

ical structure called unitary modular tensor category

(UMTC). In this paper, we focus on bosonic topologi-

cal orders in (2+1)D, and will use the terms topological

order and UMTC interchangeably.

There is rich interplay between topological order and

symmetry1. Two topological orders that have the same

set of anyon excitations but cannot be smoothly con-

nected to each other in the presence of some symmetry

are referred to as diﬀerent symmetry-enriched topolog-

ical orders (SETs). A non-trivial aspect of symmetry

actions on a topological order is symmetry fractional-

ization, in the sense that symmetry actions on anyons

may not form a representation of the symmetry group,

but a projective representation. So we sometimes say

that anyons carry “fractional” quantum numbers. Dif-

ferent symmetry actions on anyons, reﬂected in how

anyons are permuted by symmetries and symmetry

fractionalization patterns, diﬀerentiate diﬀerent SETs.

Interestingly, some SETs are anomalous, in the sense

that their symmetry fractionalization patterns cannot

be realized in a purely (2+1)Dstyle with on-site sym-

metry actions. On the contrary, it has to be realized on

the boundary of a (3+1)Dsymmetry-protected topolog-

ical phase (SPT), so that the symmetry actions can be

on-site. This is believed to be equivalent to the notion

of a ’t Hooft anomaly [6]. Given a symmetry group G,

possible anomalies are classiﬁed by group cohomology

or cobordism, and these diﬀerent classes are in one-to-

one correspondence with the SPT states in the (3+1)D

bulk that can potentially cancel the anomaly and host

this anomalous SET on its boundary [7–9].

Understanding the anomaly of SETs, or general

quantum many-body systems, is very important be-

cause the anomaly constrains the low-energy dynamics

in a powerful way. If the system has some ’t Hooft

anomaly, then its ground state cannot be trivial, i.e.,

either the symmetries are spontaneously broken, or the

ground state is gapless or topologically ordered. Going

one step further, even more powerful constraint comes

from anomaly matching. Since the anomaly can be

viewed as a property of the higher dimensional bulk,

it is an invariant under deformations of the original

system. In particular, it is an invariant under renor-

malization group that should be the same in the UV

and IR. For strongly interacting ﬁeld theories, we do

not have too many handles on their low-energy dynam-

ics so far, and understanding their ’t Hooft anomalies

and considering anomaly matching serve as a powerful

1Unless otherwise stated, all symmetries in this paper are 0-

form invertible internal symmetries.

approach [10–17]. More speciﬁcally, similar to topolog-

ical orders, in a general strongly interacting ﬁeld theory

with one-form symmetries, the action of Gon charged

line operators is speciﬁed by symmetry fractionaliza-

tion and serves as a further constraint on the IR phase

of these theories [13,16,17]. Therefore, understand-

ing the anomaly of SETs can deﬁnitely shed light into

the understanding of a general theory with one-form

symmetries.

In the context of symmetry-enriched topological

orders in condensed matter systems, it is also of

paramount importance to understand the anomaly of

SETs under similar veins. A fundamental task of con-

densed matter physics is to understand whether a cer-

tain quantum phase or phase transition can emerge

from a many-body system. For this purpose, an

emergibility hypothesis based on matching the Lieb-

Schultz-Mattis-type anomaly of a lattice system and

the anomaly of a quantum phase or transition is pro-

posed [11,18]. In particular, the Lieb-Schultz-Mattis-

type anomalies of a large class of lattice systems rele-

vant to experimental and numerical studies are worked

out in Ref. [18]. To apply the emergibility hypothesis

to an SET, we need to know its anomaly. Further-

more, although there is great progress in understand-

ing the characterization and classiﬁcation of SETs (es-

pecially in (2+1)D), such understanding mostly ap-

plies to topological orders with internal symmetries,

i.e., symmetries that do not change the spatial loca-

tions of the degrees of freedom. However, lattice sym-

metries are important in condensed matter systems,

yet the characterization and classiﬁcation of topolog-

ical orders with lattice symmetries are relatively less

understood, despite the partial progress [19–25]. In

the spirit of Refs. [11,18], understanding the anoma-

lies of a topological order with lattice symmetries (and

possibly also with internal symmetries) and applying

the emergibility hypothesis provide a route to clas-

sify such symmetry-enriched topological orders in con-

densed matter systems.

The main goals of this paper are two folds. First, for

any SET with any symmetry group G(which may be

discrete or continuous, Abelian or non-Abelian, con-

tain anti-unitary elements and/or permute anyons),

we develop a (3+1)Dtopological quantum ﬁeld theory

(TQFT) deﬁned on manifolds with a G-bundle struc-

ture, which describes the SPT state whose boundary

can host this SET. Based on this TQFT, we estab-

lish a framework to calculate the anomaly of a (2+1)D

topological order with symmetry group G, by calculat-

ing the partition function of the corresponding TQFT

on certain manifolds with some G-bundle structure.

This procedure is spelled out in great detail for ﬁnite

group symmetries and connected Lie groups. For dis-

connected Lie groups, we also have a formal construc-

tion of the partition function, although we have not

rigorously proved that it satisﬁes all consistency con-

ditions of a TQFT. Second, we apply this framework

to speciﬁc examples. In particular, we calculate the

anomaly indicators of various symmetry groups, in-

cluding ZT

2,Z2×Z2,ZT

2×ZT

2,SO(N), O(N)Tand

SO(N)×ZT

2, where Z2and ZT

2refer to a unitary and

anti-unitary order-2 symmetry group, respectively, and

O(N)Tdenotes a symmetry group O(N) such that el-

ements in O(N) with determinant −1 are anti-unitary.

Here anomaly indicators of symmetry group Grefer

to a family of quantities, expressed in terms of the

data characterizing an SET, that can completely de-

termine the anomaly of any topological order enriched

by the symmetry group G. In addition, a byproduct

of our analysis is an explicit formula for the SO(N)

Hall conductance of an SO(N) symmetric topologi-

cal order, expressed in terms of the data character-

izing this SET (up to contributions from (2+1)Din-

vertible states). Moreover, for O(N)T, N ⩾5 and

SO(N)×ZT

2, N ⩾4, we show that certain anoma-

lies cannot be realized by any SET, demonstrating the

phenomenon of “symmetry-enforced gaplessness” [26].

In the rest of this introduction, we ﬁrst comment

on the relation between our work and prior work, and

then give an outline and summary of the paper.

A. Relation to prior work

There are already multiple papers that discuss the

anomaly of a topological order from various perspec-

tives. See for example Refs. [27–46]. In particular,

based on the idea of G-crossed braided fusion categories

[27], Refs. [28,29] derived a formula to calculate the

anomaly of a general topological order with a unitary

symmetry that does not permute anyons. Ref. [31] con-

sidered anomalies of Abelian topological orders with a

ﬁnite unitary Abelian symmetry that does not permute

anyons, by explicitly studying the bulk-boundary cor-

respondence. Later, for reﬂection symmetry ZR

2and

time reversal symmetry ZT

2that may permute anyons,

Refs. [32–34] gave their anomaly indicators which ap-

ply to any topological order. The anomaly indica-

tors for U(1) ⋊ ZT

2and U(1) ×ZT

2symmetries were

later given in Ref. [35], with their lattice-symmetry-

versions discussed in Ref. [36]. Ref. [36] also gave

anomaly indicators for SO(3) ×ZT

2and SU (2) ×ZT

Refs. [30,37] derived a general formula to calculate

the relative anomaly between two diﬀerent symmetry-

enriched topological orders, i.e., the diﬀerence between

the anomalies of a given topological order with diﬀer-

ent symmetry fractionalization classes. Ref. [38] gave

a state-sum construction to calculate the anomaly of

a general bosonic symmetry-enriched topological or-

der with a general ﬁnite group symmetry, which may

contain anti-unitary elements and/or permute anyons.

This work was later generalized to fermonic symmetry-

enriched topological orders [39] (see related work in

Refs. [41–44]) and to incorporate a U(1) subgroup in

the symmetry [40].

In this work, we calculate the anomalies and

anomaly indicators via (3+1)DTQFTs, in a similar

spirit to Refs. [34,38–40]. Diﬀerent from Ref. [38–

40], where the TQFTs are based on cellulations of 4-

manifolds, we utilize handle decompositions in our con-

struction, following the idea of Ref. [34,47,48]. The

handle-decomposition-based formulation greatly sim-

pliﬁes the calculations. In this way, we explicitly derive

the anomaly indicators for Z2×Z2,ZT

2×ZT

2,SO(N),

O(N)Tand SO(N)×ZT

2symmetries (besides repro-

ducing the known anomaly indicators in the literature

[32–36]). Our framework has wide applicability, and

now the calculation of anomaly indicators for any sym-

metry group Gis equally straightforward.

There is also a vast number of works done regard-

ing constructing a (3+1)DTQFT from the data of

a UMTC (or tensor category in general), including

Refs. [34,38,47–57]. Our work builds on the construc-

tion in Refs. [34,48] to build up our TQFT, and in

particular we spell out in detail how to deal with man-

ifolds with a G-bundle structure and categories with

aG-action, for general symmetry group G. When G

is ﬁnite, our work can also be thought of as a handle

version of the state sum construction in Ref. [38]. As

mentioned before, our formulation makes the calcula-

tion much easier and explicit formulae possible. More-

over, our framework generalizes in a straightforward

manner to continuous symmetries.

We remark that symmetries considered in this pa-

per are all “exact symmetries”, which are supposed to

be present in the system microscopically. They are in

contrast to “emergent symmetries”, which do not exist

microscopically but emerge as good approximate sym-

metries at low energies and long distances, sometimes

in the form of generalized symmetries [58–60]. A possi-

ble approach to calculate the anomaly associated with

an exact symmetry is to ﬁrst ﬁgure out the full emer-

gent symmetry of a theory and its associated anomaly,

and then use some “pullback” to get the anomaly

of the exact symmetry (see, e.g., Refs. [17,45,46]).

This approach is certainly elegant. However, as more

and more emergent generalized symmetries are discov-

ered, it appears subtle to know whether we obtain the

complete set of emergent symmetries and how exactly

the anomaly of the exact symmetry is related to the

anomaly of the emergent symmetry (see Point 7 of

Discussion in Sec. VII). Speciﬁcally, one might won-

der, within such an approach, if one has to ﬁrst un-

derstand all emergent non-invertible symmetries and

their anomalies, which seems complicated. In this pa-

per, we avoid this subtlety by directly working with the

exact symmetry of a topological order, without refer-

ring to its full emergent symmetry. In particular, the

construction of the (3+1)DTQFT does not explicitly

take the full emergent symmetry as an input.

B. Outline and summary

The outline and summary of the rest of the paper

are as follows.

•In Sec. II, we brieﬂy review relevant concepts and

notations of UMTC and symmetry fractionaliza-

tion.

•In Sec. III, for a ﬁnite symmetry group G, we

present the general construction of the (3+1)D

TQFT deﬁned on 4-manifolds equipped with an

extra G-bundle structure (see Sec. III B) and an

explicit recipe to calculate its partition function

(see Sec. III D). This partition function is ex-

pressed compactly in Eq. (44).

•In Sec. IV, we apply the general framework to

calculate the anomaly indicators of various ﬁ-

nite group symmetries. First, we reproduce the

anomaly indicators of the ZT

2symmetry (see Eqs.

(46),(50)), ﬁrst proposed in Ref. [32] and later

proved in Ref. [34] (see also Ref. [33]). We then

derive the anomaly indicators of the Z2×Z2

(see Eqs. (53),(54)) and ZT

2×ZT

2symmetries (see

Eqs. (55),(56)), which are unavailable in the prior

literature as far as we know. To illustrate the us-

age of these anomaly indicators, in Sec. IV D 1 we

classify all symmetry fractionalization classes of

the all-fermion Z2topological order with ZT

2×ZT

symmetry, and calculate the anomalies for all

these classes.

•In Sec. V, we generalize the construction to con-

nected Lie group symmetries, where the expres-

sion of the partition function is given by Eq. (65).

We then apply it to derive the anomaly indica-

tors of SO(N) (see Eqs. (78)). As a byproduct,

we also derive the SO(N) Hall conductance of

an SO(N) symmetric topological order (up to

contributions from (2+1)Dinvertible states), ex-

pressed in terms of the data characterzing an

SET (see Eqs. (80),(81)).

•In Sec. VI, we explain a simple way to use the

results we have already derived to obtain the

anomaly indicators of many other groups, includ-

ing O(N)T,SO(N)×ZT

2,Zn×ZT

2,Zn⋊ ZT

Zn⋊ Z2,O(N), etc. In particular, we derive the

anomaly indicators of O(N)T(see Eqs. (95),(96))

and SO(N)×ZT

2(see Eq. (106)), and demon-

strate that certain anomaly of them cannot be re-

alized by any symmetry-enriched topological or-

der, showcasing the phenomenon of “symmetry-

enforced gaplessness” [26,61,62].

•We ﬁnish with some discussion in Sec. VII.

•The appendices contain further details of our

framework and calculations. Appendix A

presents the derivation that leads to our main

formulae Eq. (44). In Appendix B, for ﬁnite

group symmetry G, we give a more explicit ex-

pression of the “η-factor” that will enter the par-

tition function in Eq. (44). In Appendix C, we

explicitly perform various consistency checks for

the partition functions, given by Eq. (44) for a

ﬁnite group symmetry Gand Eq. (65) for a con-

nected Lie group symmetry G. In Appendix D,

we give some introduction about identifying man-

ifolds relevant to calculating the anomaly indi-

cators. In Appendix E, we present more details

on the handle decomposition of various manifolds

explicitly used in the paper.

II. Review of topological order with symmetry G

A. Review of UMTC notation

In this subsection we brieﬂy review relevant concepts

and notations that we use to describe UMTCs. For

a more comprehensive review of these concepts and

notations, see e.g., Refs. [29,63,64] for a more physics

oriented introduction, or Refs. [2,3,65,66] for a more

mathematical treatment.

A category consists of objects and morphisms be-

tween those objects. In a UMTC C, there is a ﬁnite set

of simple objects a. They are referred to as (simple)

anyons in the context of topological orders. The set of

morphisms Hom(a, b) between two objects aand bin

a UMTC Cforms a C-linear vector space. The vector

space is referred to as the topological state space in the

context of topological order. For example, Hom(a, b)

can be viewed as the Hilbert space of states on a 2-

sphere that hosts anyons aand ¯

b(see Eq. (35)).

Moreover, a UMTC Chas the structure of fusion and

braiding. Fusion means that there is a bifunctor ×such

that acting it on anyons aand bwe have

a×b∼

abc(1)

where Nc

ab is interpreted as the dimension of the topo-

logical state space of two anyons aand bfusing into a

third anyon c. There are two related vector spaces, Vc

and Vab

c, referred to as the fusion and splitting vector

spaces, respectively. The two vector spaces are dual to

each other, and depicted graphically as:

(dc/dadb)1/4

=⟨a, b;c|µ∈Vc

ab,(2)

(dc/dadb)1/4

=|a, b;c⟩µ∈Vab

c,(3)

where µ= 1, . . . , Nc

ab,dais the quantum dimension

of a, and the factors dc

dadb1/4are a normalization

convention for the diagrams.

In this paper, we will use the convention that the

splitting space is referred to as the vector space, corre-

sponding to “ket” in Dirac’s notation, while the fusion

space is the dual vector space, corresponding to “bra”

in Dirac’s notation. Diagrammatically, inner products

of the vector space are formed by stacking vertices so

the fusing/splitting lines connect

a b

c′

µ′

=δcc′δµµ′rdadb

,(4)

which can be applied inside more complicated dia-

grams.

More generally, for any integer nand mthere are

vector spaces Va1,a2,...,an

b1,b2,...,bm, which are referred to as the

fusion space of manyons into nanyons. These vector

spaces have a natural basis in terms of tensor prod-

ucts of the elementary splitting spaces Vab

cand fusion

spaces Vc

ab. For instance, we have

Vabc

d∼

Vab

e⊗Vec

d∼

Vaf

d⊗Vbc

f(5)

The two vector spaces are related to each other by a

basis transformation referred to as F-symbols, which

is diagrammatically shown as follows

a b c

f,µ,ν Fabc

d(e,α,β),(f,µ,ν)

a b c

(6)

The basis transformations are required to be unitary

transformations, i.e.

hFabc

d−1i(f,µ,ν)(e,α,β)=hFabc

d†i(f,µ,ν)(e,α,β)

=Fabc

d∗

(e,α,β)(f,µ,ν).(7)

There is also a trivial anyon denoted by 1 such that

1×a=a×1 = a. We denote aas the anyon conjugate

to a, for which N1

aa = 1, i.e.

a×a= 1 + · · · (8)

Note that ¯ais unique for a given a.

The R-symbols deﬁne the braiding properties of the

anyons, and are deﬁned via the the following diagram:

νRab

cµν

.(9)

Under a basis transformation, Γab

c:Vab

c→Vab

c, the

Fand Rsymbols change according to:

Fabc

def →˜

Fabc

d= Γab

eΓec

dFabc

def [Γbc

f]†[Γaf

d]†

Rab

c→˜

Rab

c= Γba

cRab

c[Γab

c]†.(10)

where we have suppressed splitting space indices and

dropped brackets on the F-symbol for shorthand. In

this paper, we refer to this basis transformation as a

vertex basis transformation.

On the other hand, physical quantities, like the topo-

logical twist θaand the modular S-matrix Sab, should

always be basis-independent combinations of the data.

The topological twist θais deﬁned via the diagram:

θa=θa=X

c,µ

[Raa

c]µµ =1

(11)

Finally, the modular S-matrix Sab, is deﬁned as

Sab =D−1X

θc

θaθb

dc=1

a b

,(12)

where D=pPad2

ais the total dimension of the

UMTC.

B. Global symmetry

We now consider a UMTC Cwhich is equipped with

a global symmetry group G. Mathematically speak-

ing, by deﬁnition, Gassociates a monoidal functor ρg

modulo natural isomorphism to each g∈G, which

should satisfy various consistency conditions. In this

subsection we break down the deﬁnition and review

the concepts and notations related to global symme-

try G. For a more comprehensive review, see e.g.,

Refs. [27,29,65].

First of all, as a functor, ρgacts on the anyon labels

and the topological state spaces. For an individual el-

ement g∈G,gcan permute the anyons and we use ga

to denote the (simple) anyon we get after the gaction

on the (simple) anyon labeled by a. Moreover, galso

has an action on the topological state space, which is a

C-linear or C-anti-linear operator on the fusion space,

depending on whether gis unitary or anti-unitary. We

denote this action on individual topological state space

as ρgas well:

ρg:Vab

c→Vgagb

gc.(13)

And in particular we have

Ngc

gagb=Nc

ab (14)

To account for anti-unitary symmetry, we associate a

Z2grading q(g) (and related σ(g)) as follows

q(g) = 0 if gis unitary

1 if gis anti-unitary (15)

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Anomalyof(2+1)-DimensionalSymmetry-EnrichedTopologicalOrderfrom(3+1)-DimensionalTopologicalQuantumFieldTheoryWeichengYe1,2andLiujunZou11PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,CanadaN2L2Y52DepartmentofPhysicsandAstronomy,UniversityofWaterloo,Waterloo,Ontario,CanadaN2L3G1Symmetryacti...

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Anomaly of 2 1 -Dimensional Symmetry-Enriched Topological Order from 3 1 -Dimensional Topological Quantum Field Theory Weicheng Ye1 2and Liujun Zou1

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