Topological invariants for gauge theories and symmetry-protected topological phases
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Topological invariants for gauge theories and symmetry-protected topological phases
Chenjie Wang and Michael Levin
James Franck Institute and Department of Physics,University of Chicago, Chicago, Illinois 60637, USA (Dated: September 12, 2018)We study the braiding statistics of particle-like and loop-like excitations in 2D and 3D gaugetheories with finite, Abelian gauge group. The gauge theories that we consider are obtained bygauging the symmetry of gapped, short-range entangled, lattice boson models. We define a setof quantities — called topological invariants — that summarize some of the most important partsof the braiding statistics data for these systems. Conveniently, these invariants are always Abelianphases, even if the gauge theory supports excitations with non-Abelian statistics. We compute theseinvariants for gauge theories obtained from the exactly soluble group cohomology models of Chen,Gu, Liu and Wen, and we derive two results. First, we find that the invariants take different valuesfor every group cohomology model with finite, Abelian symmetry group. Second, we find that thesemodels exhaust all possible values for the invariants in the 2D case, and we give some evidence forthis in the 3D case. The first result implies that every one of these models belongs to a distinctSPT phase, while the second result suggests that these models may realize all SPT phases. Theseresults support the group cohomology classification conjecture for SPT phases in the case where thesymmetry group is finite, Abelian, and unitary.
I. INTRODUCTION
Topological insulators are a special case of sym-metry protected topological (SPT) phases . Thesephases can occur in quantum many-body systems ofarbitrary dimension and arbitrary symmetry. By def-inition, a gapped quantum many-body system belongsto a (nontrivial) SPT phase if it satisfies three prop-erties. First, the Hamiltonian is invariant under someset of internal symmetries, none of which are bro-ken spontaneously. Second, the ground state is short-range entangled: that is, it can be transformed into aproduct state or atomic insulator using a local unitarytransformation . Third, the ground state cannot becontinuously connected with a product state, by varyingsome parameter in the Hamiltonian, without breakingone of the symmetries or closing the energy gap. Inaddition, nontrivial SPT phases typically have robustboundary modes analogous to that of topologi-cal insulators, but this property is not part of the formaldefinition.Chen, Gu, Liu, and Wen have proposed a generalclassification scheme for SPT phases built out of bosons.Their classification scheme is based on their constructionof a collection of exactly soluble lattice boson models ofarbitrary symmetry and spatial dimension. The authorsconjecture that these models — called group cohomologymodels — have two basic properties: (i) every group co-homology model belongs to a distinct SPT phase and (ii)every SPT phase can be realized by a group cohomologymodel. If both properties hold, then it follows logicallythat there is a one-to-one correspondence between thegroup cohomology models and SPT phases. In Ref. 10,the authors assumed this to be the case, and therebyderived a classification scheme for SPT phases based ongroup cohomology.While the results and arguments of Ref. 10 represent a major advance in our understanding of SPT phases,they leave several questions unanswered. First, it is notobvious that properties (i)-(ii) hold in general (in fact,property (ii) is known to fail for SPT phases with anti-unitary symmetries ). Second, even if these prop-erties do hold at some level, the resulting classificationscheme is not completely satisfying since it doesn’t tellus how to determine to which SPT phase a microscopicHamiltonian belongs.Motivated by these problems, several proposals havebeen made for how to physically characterize and dis-tinguish SPT phases . Here, we will focus onthe suggestion of Refs.15,24 which applies to 2D and 3DSPT models with unitary symmetries. Ref. 15 showed,via a simple example, that one can probe 2D SPT modelsby gauging their symmetries and studying the braidingstatistics of the excitations in the resulting gauge the-ory. This braiding statistics data is useful because itis invariant under arbitrary symmetry preserving defor-mations of the Hamiltonian, as long as the energy gapremains open. Therefore, if two SPT models give rise todifferent braiding statistics, then they must belong to dis-tinct SPT phases. The braiding statistics approach canalso be applied to 3D SPT phases.
In that case, aftergauging the symmetry, one studies the braiding statisticsof the vortex loop excitations in the resulting gauge the-ory. More specifically, different SPT phases can be distin-guished by examining their three-loop braiding statistics— the statistics associated with braiding a loop α aroundanother loop β , while they are both linked to a third loop γ . When considered together, the braiding statistics ap-proach and the group cohomology construction raise sev-eral questions:1. Does every group cohomology model lead to dis-tinct braiding statistics?
2. Do the group cohomology models exhaust all pos-sible types of braiding statistics that can occur inSPT systems?3. If two SPT models give rise to the same braidingstatistics, do they always belong to the same phase?The answers to these questions have powerful implica-tions, especially if we can answer them affirmatively. Forexample, if we can answer the first question in the affir-mative, we can immediately conclude that every groupcohomology model belongs to a distinct phase. Likewise,if we can answer the second and third questions in theaffirmative, then we can conclude that the group coho-mology models realize all possible SPT phases. If we cananswer all three questions affirmatively, then it followsthat (1) the group cohomology classification is correctand (2) the braiding statistics data provides a universalprobe for characterizing and distinguishing different SPTphases with unitary symmetries.In this work, we consider the first and second questionsfor the case of 2D and 3D SPT phases with finite Abelianunitary symmetry group G = Q Ki =1 Z N i . We answer thefirst question in the affirmative and we find evidence thatthe same is true for the second question, as we explainbelow.We obtain our results by focusing on a subset of thebraiding statistics data that summarizes some of its mostimportant features (in fact for systems with Abelianstatistics, this subset is equivalent to the full set of braid-ing data, see section VIII). In the 2D case, this subsetconsists of 3 tensors, { Θ i , Θ ij , Θ ijk } that take values in[0 , π ], where 1 ≤ i, j, k ≤ K . In the 3D case, it consistsof 3 tensors { Θ i,l , Θ ij,l , Θ ijk,l } with 1 ≤ i, j, k, l ≤ K .These tensors — which we call topological invariants —are defined by considering the Berry phase associatedwith certain composite braiding processes of vortices orvortex loops. Conveniently, these Berry phases are al-ways Abelian phases regardless of whether the full set ofbraiding statistics is Abelian or non-Abelian.We report two main results. First, we show that thetopological invariants take different values in every groupcohomology model. Second, we show that the group co-homology models exhaust all possible values for the topo-logical invariants in the 2D case and we give some evi-dence for this in the 3D case. Our first result impliesthat the group cohomology models all belong to distinctphases. Our second result can be interpreted as evidencethat the group cohomology models realize all possibleSPT phases with finite Abelian unitary symmetry group.Some of our results have appeared previously in theliterature, though in a slightly different form. In particu-lar, in the 2D case, Ref. 32 introduced invariants similarto ours and showed that the invariants can distinguishall the 2D group cohomology models. Also, much of ouranalysis of 3D gauge theories is similar to that of ourprevious work, Ref. 24. However, this paper goes furtherthan Ref. 24 in three key ways. First, we study bothAbelian and non-Abelian loop braiding statistics, while Ref. 24 only studied Abelian statistics. Second, we con-sider a general finite Abelian symmetry group Q Ki =1 Z N i while Ref. 24 only considered groups of the form ( Z N ) K .Finally, we make a systematic comparison between thetopological invariants and the group cohomology classifi-cation, while Ref. 24 only made this comparison in a fewexamples.A note on our terminology: throughout the paper, wewill refer to gauged SPT models as simply gauge theories .Also, we will refer to the gauged group cohomology mod-els as Dijkgraaf-Witten models . The Dijkgraaf-Wittenmodels were studied long before the discovery of SPTphases, however it can be shown that they are equivalentto the gauged group cohomology models (the equivalenceis discussed in Appendix C).The rest of the paper is organized as follows. In Sec. II,we introduce the models that we will study, both the gen-eral gauged SPT models and the more specific Dijkgraaf-Witten models. In Sec. III, we discuss the general struc-ture of braiding statistics in gauged SPT models andwe define the topological invariants. Next, we computethe topological invariants in 2D and 3D Dijkgraaf-Wittenmodels in Sec. IV. In Sec. V, we show that the topologicalinvariants take different values in every Dijkgraaf-Wittenmodel. In Sec. VI, we derive general constraints that thetopological invariants must satisfy in any gauged SPTmodel. In Sec. VII, we discuss whether the Dijkgraaf-Witten models exhaust all possible values for the invari-ants. The relation between the topological invariants andthe full set of braiding statistics in the case of Abelianstatistics is discussed in Sec. VIII. Finally, in Sec. IX, weconclude and discuss the implications of our results forSPT phases. The Appendices contain several technicaldetails. II. MODELSA. Gauge theories
The main systems we will study in this paper are 2Dand 3D lattice gauge theories with finite Abelian gaugegroup, G = Q Ki =1 Z N i . More specifically, we will study aparticular class of gauge theories that are obtained from atwo step construction. The first step of the constructionis to pick a 2D or 3D lattice boson or spin model witha global Q Ki =1 Z N i symmetry. This boson model can bequite general, with the only restrictions being that (1)it has local interactions, (2) the symmetry is an internal(on-site) symmetry rather than a spatial symmetry, and(3) its ground state is gapped and short-range entangled— that is, the ground state can be transformed into aproduct state by a local unitary transformation. Here,by a local unitary transformation, we mean a unitarytranformation U of the form U = exp( iHs ), where H isa local Hermitian operator and s is a finite constant thatdoes not scale with the system size. (Note that thetransformation U need not commute with the symmetry).The second step of the construction is to gauge theglobal symmetry of the lattice boson model and couple itto a dynamical lattice gauge field with group G . In ap-pendix A, we give a precise prescription for how to imple-ment this gauging procedure. This prescription mostlyfollows the usual minimal coupling scheme . However,there is one nonstandard element that is worth mention-ing: our procedure is defined so that the gauge couplingconstant is exactly zero . More precisely, what we mean bythis is that the Hamiltonian for the gauged model com-mutes with the flux operators that measure the gaugeflux through each plaquette in the lattice. This propertyis convenient because it makes the low energy physics ofour models well-controlled. In particular, using this prop-erty it can be shown that the gauge theories constructedvia our gauging procedure are guaranteed to be gappedand deconfined as long as the original boson models aregapped and don’t break the symmetry spontaneously (seeappendix A).The above two step construction defines the class ofmodels that we will study in this paper. From now on,when we use the term gauge theory we will be referringexclusively to models of this type, unless we state other-wise.Before concluding this section, we would like to men-tion that although we find it convenient to use the par-ticular gauging prescription in appendix A, we don’t ex-pect that our results actually depend on the details ofthe gauging procedure, or on the fact that the resultinggauge theories have zero gauge coupling. Indeed, our re-sults are guaranteed to hold for any model that can becontinuously connected one of the above gauge theorieswithout closing the energy gap. We expect that the lattercategory includes models obtained from generic gaugingprocedures, as long as the gauge coupling constant is suf-ficiently small. B. Dijkgraaf-Witten models
In part of this work we will study a particular set of ex-actly soluble gauge theories, known as Dijkgraaf-Wittenmodels which are obtained by gauging the group co-homology models of Ref. 10. We now briefly reviewthe properties of the group cohomology models and thecorresponding Dijkgraaf-Witten models. For the explicitdefinition of these models, see Appendix C.The group cohomology models are exactly soluble lat-tice boson models that can be defined in any spatial di-mension d . The basic input needed to construct a d -dimensional group cohomology model is a (finite) group G and a ( d + 1) cocycle ω . Here, an n -cocycle ω is afunction ω : G n → U (1) that satisfies certain condi-tions. One may define an equivalence relation on co-cycles and the equivalence classes are labeled by the ele-ments of the cohomology group H n [ G, U (1)] (a brief in-troduction to group cohomology is given in AppendixB). It can be shown that the models constructed from equivalent cocycles are identical so we will say that the d -dimensional group cohomology models are labeled byelements of H d +1 [ G, U (1)].Like the group cohomology models, the basic inputneeded to construct a Dijkgraaf-Witten model in spa-tial dimension d is a group G and a ( d + 1) − cocycle ω .Also, like the group cohomology models, the Dijkgraaf-Witten models constructed from equivalent cocycles arethe same, so we will say that they are labeled by differ-ent elements of H d +1 [ G, U (1)]. Here we will focus on thecase G = Q Ki =1 Z N i and d = 2 , C. Braiding statistics and phases of SPT modelsand gauge theories
Before proceeding further, we briefly review some re-sults on the relationship between SPT models, gaugetheories, and braiding statistics. We begin by defining phases of SPT models and phases of gauge theories. Theformer definition is relatively simple: we say that twolattice boson models with the same symmetry group be-long to the same SPT phase if they can be continuouslyconnected to one another by varying some parameter inthe (symmetry-preserving) Hamiltonian, without closingthe energy gap.Defining phases of gauge theories is more subtle. Infact, there are two inequivalent ways to define this con-cept, both of which have their merits. In the first defini-tion, two gauge theories belong to the same phase if theycan be continuously connected by varying some parame-ter in the (gauge invariant) Hamiltonian without closingthe energy gap. In the second definition, not only dowe require the existence of an interpolating Hamiltonianwith an energy gap, but we also demand that the interpo-lating Hamiltonian has vanishing gauge coupling — thatis, the Hamiltonian must commute with the flux opera-tors that measure the gauge flux through each plaquettein the lattice. While the first definition is very natural ifone is interested in gauge theories for their own sake, thesecond definition is more relevant to the study of SPTphases. In this paper, our primary interest is in SPTphases so we will use the second definition.In parallel to the two ways of defining phases of gaugetheories, there are also two ways to define what is meansfor two gauge theories to have the “same” braiding statis-tics data. In the first definition, two gauge theories havethe same braiding statistics data if one can map the ex-citations of one gauge theory onto the excitations of theother gauge theory such that the corresponding excita-tions have identical braiding statistics. In the second def-inition, the corresponding excitations are required bothto have the same braiding statistics and the same gaugeflux. In this paper, we will use the second definition,since it fits more naturally with our definition of phasesof gauge theories.With these definitions in mind, we can now discusssome results. An important observation is that if two lat-tice boson models belong to the same SPT phase, thenthe corresponding gauged models must also belong tothe same phase. To see this, note that our gauging pre-scription (Appendix A) maps gapped lattice boson mod-els onto gapped zero-coupling gauge theories; hence, anycontinuous interpolation between two SPT models canbe gauged to give an interpolation between the two cor-responding gauge theories.Another important observation is that if two gaugetheories belong to the same phase, then they must havethe same braiding statistics. One way to see this isto note that braiding statistics data can only take ondiscrete values and cannot change continuously. (Thisdiscreteness property is known as
Ocneanu rigidity ).Combining the above two observations, we derive a use-ful corollary: if two lattice boson models belong to thesame SPT phase then they must give rise to the samebraiding statistics after gauging their symmetries. Theconverse of this statement may also be true, but it is notobvious. III. DEFINING THE TOPOLOGICALINVARIANTS
In this section, we construct a set of topological invari-ants for gauge theories with gauge group G = Q Ki =1 Z N i .(Here, when we say “gauge theory”, we mean a gaugetheory of the type discussed in section II). These invari-ants are defined in terms of the braiding statistics ofthe excitations of the gauge theory. They are denoted byΘ i , Θ ij , Θ ijk in the 2D case and Θ i,l , Θ ij,l , Θ ijk,l in the3D case, where the indices i, j, k, l range over 1 , . . . , K .For pedagogical purposes, we first define the invariantsin the case where the braiding statistics are Abelian, andthen discuss the general case (where the statistics maybe Abelian or non-Abelian). A. 2D Abelian case
We start with the simplest case: we consider 2D gaugetheories with group G = Q Ki =1 Z N i and with Abelianbraiding statistics.
1. Excitations and braiding statistics
We first discuss the excitation spectrum of these gaugetheories. In general, every excitation α in a discretegauge theory can be labeled by the amount of gaugeflux φ α that it carries. In our case, the gauge flux φ α can be described by a K -component vector φ α =( φ α , φ α , . . . , φ Kα ) where each component φ iα is a multi-ple of πN i , and is defined modulo 2 π . Excitations can bedivided into two groups: charge excitations that carryvanishing gauge flux and vortex excitations that carrynonzero gauge flux. As far as their topological properties go, charge exci-tations are uniquely characterized by their gauge charge q = ( q , q , . . . , q K ) where each component q i is de-fined modulo N i . In contrast, vortex excitations are not uniquely characterized by the amount of gauge flux thatthey carry: in fact, there are | G | = Q Ki =1 N i differenttypes of vortices carrying the same flux φ . All of thesevortices can be obtained by attaching charge excitationsto a fixed reference vortex with flux φ . Throughout thispaper, we will use Greek letters α, β, γ, . . . to denote vor-tices as well as general excitations, and we will use theletter q to denote both a charge excitation and its gaugecharge.Before proceeding further, we would like to point outa possible source of confusion: given what we have saidabout the different types of vortices, it is tempting to tryto label vortex excitations by both their gauge flux andtheir gauge charge. The problem with this approach isthat we do not know any physically meaningful way todefine the absolute charge carried by a vortex excitationin a discrete gauge theory. Therefore we will avoid usingthis notion in this paper. Instead, we will only use theconcept of relative charge: we will say that two vortices α, α ′ differ by charge q if α ′ can be obtained by attachinga charge excitation q to α .Let us now consider the braiding statistics of the dif-ferent excitations. There are three different braiding pro-cesses to consider: braiding of two charges, braiding ofa charge around a vortex, and braiding of two vortices.The first process is easy to analyze: indeed, it is clearthat the charges correspond to local excitations in theungauged short-range-entangled bosonic state. There-fore the charges are all bosons and have trivial (bosonic)mutual statistics. The braiding between a charge anda vortex is also easy to understand, as it follows fromthe Aharanov-Bohm law. More specifically, the statis-tical Berry phase θ associated with braiding a charge q around a vortex with flux φ is given by θ = q · φ, (1)where “ · ” is the vector inner product. Note that attach-ing a charge to the vortex does not change the Aharanov-Bohm law since the charges have trivial mutual statisticswith respect to one another.From the above arguments, we see that the charge-charge and charge-vortex statistics are completely fixedby the gauge group, leaving no room for variation. There-fore, the only braiding process that has potential for dis-tinguishing gauge theories with the same gauge group isvortex-vortex braiding. Motivated by this observation,we will define the topological invariants Θ i , Θ ij in termsof the vortex-vortex braiding statistics.
2. The topological invariants
Let α be a vortex carrying a unit flux πN i e i , where e i = (0 , . . . , , . . . ,
0) with a 1 is the i th entry and 0everywhere else. Let β be a vortex carrying a unit flux πN j e j . Here, i and j can take any value in 1 , . . . , K . Wedefine Θ ij = N ij θ αβ , Θ i = N i θ α , (2)where θ αβ is the mutual statistics between α and β , θ α isthe exchange statistics of α , and N ij is the least commonmultiple of N i and N j (More generally, throughout thepaper, we use N ij...k to denote the least common multipleof N i , N j , . . . , N k and use N ij...k to denote the greatestcommon divisor of N i , N j , . . . , N k ).For the quantities Θ ij and Θ i to be well-defined, weneed to check that N ij θ αβ and N i θ α only depend on i and j , and not on the choice of the vortices α, β . Tosee that this is the case, imagine that we replace α, β with some other vortices α ′ , β ′ carrying flux πN i e i and πN j e j . Then clearly the vortices α and α ′ differ only bythe attachment of charge, as do β and β ′ . Therefore,according to the Aharonov-Bohm law, the change in Θ ij that occurs when we replace α → α ′ , β → β ′ isΘ ij → Θ ij + 2 πN ij (cid:18) xN i + yN j (cid:19) (3)where x, y are integers that describe the amount of type- i and type- j charge that is attached to β and α , respec-tively. But N ij is divisible by both N i and N j so wesee that this replacement does not change Θ ij modulo2 π . Similarly, the Aharonov-Bohm law tells us that thechange in Θ i that occurs when we replace α → α ′ isΘ i → Θ i + 2 πN i zN i , (4)where z is the type- i charge that is attached to α . ThusΘ i is also unchanged modulo 2 π . We conclude that thequantities Θ ij and Θ i are both well-defined.In addition to being well-defined, it is possible to showthat Θ ij and Θ i have another nice property: they containthe same information as the full set of braiding statistics.We will derive this result in Sec. VIII. B. 2D general case
In this subsection, we move on to general 2D gaugetheories with gauge group G = Q Ki =1 Z N i . Unlike theprevious section, we do not assume that the braidingstatistics of the excitations is Abelian. This additionalgenerality is important because, contrary to naive expec-tations, gauge theories with Abelian gauge groups cansometimes have excitations with non-Abelian statistics.For example, this phenomenon occurs in 2D Q Ki =1 Z N i Dijkgraaf-Witten models when K ≥ i , Θ ij , Θ ijk . The first two Θ i , Θ ij reduce tothose defined in (2) when restricted to Abelian statistics.The third invariant Θ ijk is new to the non-Abelian case,and vanishes in the Abelian case.
1. General aspects: excitations, fusion rules, and braidingstatistics
Many features of the Abelian case carry over to thegeneral case without change. First, we can still la-bel every excitation α by the amount of gauge flux φ α = ( φ α , . . . , φ Kα ) that it carries, where φ iα is a mul-tiple of πN i and is defined modulo 2 π . Also, we canstill divide excitations into two groups: charges , thatcarry vanishing flux, and vortices that carry nonzeroflux. Charge excitations are still characterized uniquelyby their gauge charge q = ( q , . . . , q K ) with q i definedmodulo N i , while vortices are still characterized non-uniquely by their gauge flux. Finally, charges are stillAbelian particles with trivial charge-charge statistics,and with charge-vortex statistics given by the Aharonov-Bohm law: θ = q · φ where φ is the gauge flux carried bythe vortex. The main new element in the general case isthat vortices can be non-Abelian, i.e., they can have non-Abelian fusion rules and non-Abelian braiding statisticswith one another .While the possibility of non-Abelian vortices compli-cates our analysis, we can still make some general state-ments about the fusion rules and braiding statistics inthese systems. In what follows, we focus on the fusionrules, and we list some properties which will be useful inour later arguments (see Appendix D for proofs and de-tails). To begin, imagine we fuse together two excitations α and β . In general, there may be a number of possiblefusion outcomes corresponding to other excitations γ : α × β = X γ N γαβ γ, (5)where N γαβ is the dimension of the fusion space V γαβ . Oneproperty of these fusion rules is that φ γ = φ α + φ β (6)for any fusion product γ . In particular, if φ α + φ β = 0,then all the γ ’s that appear on the right hand side of (5)are pure charges.A second property is that the fusion of a charge q andan excitation α always results in a single excitation α × q = α ′ (7)where α ′ is not necessarily distinct from α and φ α ′ = φ α .A third property is that if two excitations α, α ′ have thesame flux, φ α ′ = φ α , then there exists at least one charge q with α ′ = α × q .To describe the final property, let α and β be two ex-citations, and let γ be one of their fusion channels. Let α ′ and β ′ be two other excitations with φ α ′ = φ α and φ β ′ = φ β , and let γ ′ be one of their fusion channels. Thefinal property states that there exist charges q and q such that α ′ = α × q , β ′ = β × q and γ ′ = γ × q × q . α β (a) α βγ (b) FIG. 1: Space-time trajectories of the vortices in the braidingprocesses associated with Θ ij [panel (a); N ij = 3] and Θ ijk [panel (b)]. The arrow of time is upward.
2. The topological invariants
Similarly to the Abelian case, we define the topo-logical invariants Θ i , Θ ij , Θ ijk in terms of the braidingstatistics of vortices. Let α, β, γ be three vorticescarrying unit fluxes πN i e i , πN j e j , πN k e k respectively. Thetopological invariants Θ i , Θ ij , Θ ijk are defined as follows. Definitions: • Θ i = 2 πN i s α , where s α is the topological spin of α ; • Θ ij is the Berry phase associated with braiding α around β for N ij times; • Θ ijk is the Berry phase associated with the follow-ing braiding process: α is first braided around β ,then around γ , then around β in the opposite direc-tion, and finally around γ in the opposite direction.Fig. 1 shows the space-time trajectories of the vorticesin the braiding processes of Θ ij and Θ ijk . We note thatthe definitions of Θ i and Θ ij reduce to our previous def-initions (2) in the Abelian case, since 2 πs α = θ α forAbelian quasiparticles. We can also see that Θ ijk = 0 inthe Abelian case.Before we show that these quantities are well defined,we comment on the definition of Θ i . In defining Θ i , wehave used the notion of topological spin. The topologicalspin s α , 0 ≤ s α <
1, of an anyon α is defined to be e i πs α = 1 d α X γ d γ tr( R γαα ) , (8)where d α and d γ are the quantum dimensions of α and γ respectively, R γαα is the braiding matrix associated witha half braiding (exchange) of two α ’s, and the summationis over the γ ’s appearing in the fusion product of α × α .We see that Θ i is rather abstract since s α does not havea direct physical interpretation. In contrast, Θ ij , Θ ijk aredefined in terms of concrete physical braiding processes. One might wonder if there is a more concrete definitionof Θ i . Indeed, when N i is even, we find an alternativedefinition of Θ i : • Θ i is the phase associated with exchanging two α ’sfor N i times, where α is any vortex carrying unitflux πN i e i .This alternative definition provides a direct way to “mea-sure” Θ i when N i is even. The equivalence between thisalternative definition and the original topological spindefinition of Θ i follows from two facts: First, exchangingtwo identical α vortices N times is equivalent to braid-ing one around the other N times. Second, braiding twoidentical α vortices around one another gives a pure phase e i πs α . (The latter claim, which is less obvious, is provedin Appendix E).One problem with the above definition is that it doesnot make sense when N i is odd, since in this case the uni-tary matrix associated with the exchange process is notnecessarily a pure phase. Fortunately, we will see laterthat when N i is odd, Θ i is uniquely determined by Θ ii ,and the latter can be directly “measured” using a con-crete braiding process. This point can be obtained fromthe constraints on the invariants, which we will study inSec. VI B.
3. Proving the invariants are well-defined
For the invariants to be well defined, we need to provetwo points: (i) We need to show that the unitary trans-formations associated with the above braiding processesare always Abelian phases regardless of the fact that thevortices may be non-Abelian; (ii) We need to show thatthese Abelian phases are functions of i, j, k only and donot depend on the choice of vortices α, β, γ as long as theycarry fluxes πN i e i , πN j e j , πN k e k respectively. Only point (ii)is needed for showing Θ i is well-defined.Let us start with proving point (ii) for Θ i . We firstreview some key properties of topological spin (a detaileddiscussion of topological spin can be found in Ref. 38.) If α is an Abelian anyon, 2 πs α is just the exchange statisticsof α . In general, s α = s ¯ α , where ¯ α is the anti-particle of α . An important property of the topological spin is R γβα R γαβ = e i π ( s γ − s α − s β ) id V γαβ . (9)Here V γαβ is the fusion space of α, β in the fusion channel γ and R γαβ is the braiding matrix associated with a halfbraiding of α and β in the fusion channel γ . The notationid V γαβ denotes the identity matrix in the fusion space V γαβ .With these properties in mind, we now show that Θ i is well defined, i.e., we show that2 πN i s α ′ = 2 πN i s α (10)for any two vortices α, α ′ carrying unit flux πN i e i . Inthe first step, we note that we can assume without lossof generality that α ′ = q × α for some charge q sinceaccording to the properties discussed in Sec. III B 1, anyvortex with unit flux πN i e i can be constructed from a fixedvortex by fusing charges with it. To prove the result forthis case, we substitute β = q and γ = α ′ = q × α intoEq. (9), obtaining R α ′ qα R α ′ αq = e i π ( s α ′ − s α − s q ) = e i π ( s α ′ − s α ) (11)where in the second equality we used the fact that q is aboson so s q = 0. At the same time, we know R α ′ qα R α ′ αq = e πNi × integer (12)since the braiding of a charge around a vortex can becomputed from the usual Aharonov-Bohm law. Com-bining these two relations, we see that 2 πN i ( s α ′ − s α )vanishes modulo 2 π , proving (10).Next, we prove points (i) and (ii) in the case of Θ ij .Let α and β be two vortices carrying unit flux πN i e i and πN j e j respectively. Imagine we perform a full braiding of α around β when they are in some fusion channel δ . Fromthe general theory of non-Abelian anyons , we knowthat the unitary matrix associated with a full braidingof α around β in a fixed fusion channel δ , is a pure phasefactor (this result is a corollary of Eq. (9)). Denoting thisphase factor by e iθ δαβ , the quantity Θ ij can be computedas Θ ij = N ij θ δαβ (13)In order to establish properties (i) and (ii) above, it suf-fices to show that N ij θ δαβ = N ij θ δ ′ α ′ β ′ (14)for any other vortices α ′ , β ′ carrying unit flux πN i e i , πN j e j ,and for any other fusion channel δ ′ . Indeed, the inde-pendence of N ij θ δαβ with respect to the fusion channel δ implies point (i), while the independence of N ij θ δαβ withrespect to α, β implies point (ii).In fact, it is enough to prove (14) for the case where α ′ = α × q , β ′ = β × q , and δ ′ = δ × q × q for somecharges q , q since according to the general propertiesdiscussed in Sec. III B 1, any α ′ , β ′ , δ ′ can be obtained inthis way. But it is easy to prove (14) in this case. Indeed,from the Aharonov-Bohm law we can deduce the relation N ij ( θ δ ′ α ′ β ′ − θ δαβ ) = 2 πN ij (cid:18) q i N i + q j N j (cid:19) (15)where q i and q j are integers that describe the amountof type- i and type- j charge carried by q and q , respec-tively. We then observe that the expression on the righthand side vanishes modulo 2 π since N ij is divisible byboth N i and N j . This establishes (14) and proves prop-erties (i) and (ii) for Θ ij .The proof of points (i) and (ii) for Θ ijk is more tech-nical and is given in Appendix F. C. 3D Abelian case
Having warmed up with the 2D gauge theories, we nowconsider 3D gauge theories with gauge group G = Q i Z N i and with Abelian loop statistics. The discussion thatfollows is a generalization of the case G = ( Z N ) K , studiedin Ref. 24.
1. Excitations and three-loop braiding statistics
Discrete gauge theories in three dimensions supporttwo types of excitations: charges and vortices. Chargeexcitations are particle-like and are characterized by theamount of gauge charge q that they carry, where q =( q , . . . , q K ) with q i defined modulo N i . Vortex excita-tions are string-like and are characterized by the amountof gauge flux φ that they carry where φ = ( φ , . . . , φ K )with the component φ i being a multiple of πN i , and de-fined modulo 2 π . We will refer to vortex excitations as vortex loops or simply loops , since we will generally as-sume that the system is defined on a closed manifold withno boundary so that vortex excitations necessarily formclosed loops . We will use Greek letters α, β, γ to de-note vortex loop excitations, and will use φ α to denotethe gauge flux carried by the loop excitation α .As in the 2D case, it is important to keep in mindthat while charge excitations are uniquely characterizedby their gauge charge, vortex loop excitations are not uniquely characterized by their gauge flux: in fact, thereare | G | = Q Ki =1 N i different types of vortex loop excita-tions carrying the same gauge flux φ . All of these exci-tations can be obtained by attaching charges to a fixedreference loop with flux φ .Also, just as in 2D, there is some subtlety in definingthe absolute charge carried by a vortex loop excitation.Therefore, throughout this paper we will only use theconcept of relative charge: we will say that two vortexloops α and α ′ differ by charge q if α ′ can be obtained byattaching a charge excitation q to α . The only exceptionto this rule involves unlinked vortex loops: when a vortexloop is not linked to any other loops, then there is anatural way to define how much charge it carries: we willsay that such a vortex loop is neutral if it can be shrunkto a point and annihilated by local operators. Similarly, (a) (b) (c) α βγ β Ω α γ × × γα β FIG. 2: Three-loop braiding process. (a) The gray curvesshow the paths of two points on the moving loop α . (b)Cross-section of the braiding process in the plane that γ liesin. (c) A torus Ω α is swept out by α during the braiding,which encloses the loop β (dashed circle). we will say that an unlinked vortex loop carries charge q if it can be obtained by attaching charge q to a neutralloop.Let us now consider the braiding statistics of these ex-citations. There are several types of processes we canconsider: braiding of two charges, braiding of a chargearound a vortex loop, and braiding involving several vor-tex loops. As in the 2D case, it is easy to see that thecharge-charge statistics are all bosonic and the statisticsbetween a charge q and a vortex loop carrying a flux φ follow the Aharonov-Bohm law θ = q · φ. (16)We note that the above Aharonov-Bohm law holds quitegenerally: it does not depend on the amount of chargeattached to the loop, nor on whether the loop is unlinkedor linked with other loops.What is left are braiding processes involving severalvortex loops. In general, there are many kinds of loopbraiding processes we can consider — including processesinvolving two loops , three loops , or evenmore complicated configurations. Here, we will followRef. 24 and focus on the three-loop braiding process de-picted in Fig. 2, in which a loop α is braided around aloop β while both are linked to a third “base” loop γ .Ref. 24 argued that this three-loop braiding process isa useful probe for characterizing and distinguishing 3Dtopological phases. As in Ref. 24, we denote the sta-tistical phase associated with this braiding process by θ αβ,c , where c is an integer vector that characterizes theamount of flux carried by γ . More specifically, c is de-fined by φ γ = ( πN c , . . . , πN K c K ). We use the notation θ αβ,c rather than θ αβ,γ , because the statistical phase isinsensitive to the amount of charge attached to γ anddepends only on its flux φ γ which is parameterized by c .We will also consider an exchange or half-braiding pro-cess, in which two identical loops α , both linked to thebase loop γ , exchange their positions. We denote theassociated three-loop exchange statistics by θ α,c .
2. The topological invariants
We define our topological invariants, Θ i,l and Θ ij,l , interms of the the three-loop braiding statistics of vortexloops. Let α and β be two vortex loops carrying unit flux πN i e i and πN j e j , where e i = (0 , . . . , , . . . ,
0) with the i thentry being 1 and all others being 0. Suppose both α and β are linked to a third vortex loop γ carrying unit flux πN l e l . We defineΘ ij,l = N ij θ αβ,e l , Θ i,l = N i θ α,e l (17)Using arguments similar to those for Θ i , Θ ij fromSec. III A 2, one can show that the quantities Θ i,l , Θ ij,l depend only on i, j, l and not on the choice of vortices α, β, γ . Thus, Θ ij,l and Θ i,l are well-defined quantities. β β γ β × β γ FIG. 3: Fusion of two loops β and β , both linked to γ .We denote this type of fusion by β × β . (This is differentnotation from Ref. 24, where this type of fusion was denotedby β + β ). In addition to being well-defined, it is possible to showthat Θ ij,l and Θ i,l contain all the information about thethree-loop braiding statistics in the gauge theory. Wewill derive this result in Sec. VIII.
D. 3D general case
In this section, we move on to general 3D gauge the-ories with gauge group G = Q Ki =1 Z N i . Unlike the lastsection, we do not assume that the three-loop braidingstatistics is Abelian. We will define three topologicalinvariants Θ i,l , Θ ij,l and Θ ijk,l . The first two, Θ i,l , Θ ij,l ,reduce to those defined in (17) when restricted to Abelianstatistics. The third invariant Θ ijk,l is new to the non-Abelian case, and vanishes in the Abelian case.
1. General aspects of non-Abelian loop braiding
In order to analyze the general case, it is important torecognize the analogy between 3D loop braiding and 2Dparticle braiding. This analogy can be seen most easilyby examining a 2D cross-section of a loop braiding pro-cess, as shown in Fig. 2(b). Here we see that a braidingprocess involving two loops α, β that are linked to a baseloop γ , can be mapped onto a braiding process involvingtwo point-like particles in two dimensions. More gener-ally, any braiding process involving loops α , ..., α N thatare linked to a base loop γ can be mapped onto a braidingprocess involving N point-like particles in two dimen-sions. It can be shown that this mapping between 3Dloop braiding and 2D particle braiding is one-to-one, sothat the 3D braid group for loops (when linked to a baseloop) is identical to the 2D braid group for particles. In addition to braiding, there is also a close analogy be-tween fusion processes in two and three dimensions. Justas two particles can be fused together to form anotherparticle, two loops α, β that are linked to the same loop γ can be fused to form a new loop that is also linked to γ (Fig. 3).This correspondence between the 2D and 3D cases im-plies that the algebraic structure of fusion and braiding in2D anyon theories can be carried over without changeto the theory of 3D loop excitations. In particular, forany loops α and β that are both linked with γ , we candefine an associated fusion space V δαβ,c , where δ denotestheir fusion channel. Also, we can define an F -symbol F δαβµ,c that describes a unitary mapping between twodifferent ways of parameterizing the fusion of three loops F δαβµ,c : M ξ V ξαβ,c ⊗ V δξµ,c → M η V δαη,c ⊗ V ηβµ,c , (18)and that satisfies the pentagon equation . Likewise, wecan define an R -symbol R δαβ,c which is a unitary trans-formation R δαβ,c : V δαβ,c → V δβα,c (19)and that satisfies the hexagon equation. As in the 2Dcase, the R -symbol describes a half-braiding of loops: afull braiding of two loops α, β that are in a fusion channel δ is given by R δβα,c R δαβ,c . Finally, we can define quantumdimensions and topological spins of loop excitations. Thetopological spin of a loop α that is linked to γ is givenby e i πs α,c = 1 d α,c X δ d δ,c tr( R δαα,c ) (20)where d α,c and d δ,c are quantum dimensions.For all of the above quantities, V δαβ,c , F δαβµ,c , etc., thedependence on the base loop γ enters through the in-dex ‘ c ’ where c is an integer vector defined by φ γ =( πN c , . . . , πN K c K ). The reason we use the notation V δαβ,c , etc. rather than V δαβ,γ , etc. is because it is clearthat these quantities depend only on the flux carried by γ which is parameterized by c .
2. The topological invariants
Having established the analogy between 3D loop braid-ing and 2D particle braiding, we now define the 3D topo-logical invariants using the same approach as in the 2Dcase. Let α, β, γ be three vortex loops that are linkedwith another loop σ . Suppose that α, β, γ, σ carry unitflux πN i e i , πN j e j , πN k e k , πN l e l , respectively. The topologicalinvariants Θ i,l , Θ ij,l , Θ ijk,l are defined as follows. Definitions: • Θ i,l = 2 πN i s α,e l , where s α,e l is the topological spinof α when it is linked to σ ; • Θ ij,l is the Berry phase associated with braidingthe loop α around β for N ij times, while both arelinked to σ ; • Θ ijk,l is the phase associated with the followingbraiding process: α is first braided around β , thenaround γ , then around β in a opposite direction,and finally around γ in a opposite direction. Here α, β, γ are all linked with σ . Similarly to the 2D case, there is an alternative and moreconcrete definition of Θ i,l when N i is even: Θ i,l can bedefined as the phase associated with exchanging two α loops for N i times.We need to prove two points to show these quanti-ties are well-defined: (i) We need to show that the uni-tary transformations associated with the above braid-ing processes are always Abelian phases regardless ofthe fact that the vortex loops may be non-Abelian; (ii)We need to show that these Abelian phases are func-tions of i, j, k, l only and do not depend on the choiceof vortex loops α, β, γ, σ as long as they carry fluxes πN i e i , πN j e j , πN k e k , πN l e l . These two properties can be es-tablished using similar arguments to those given in the2D case in Sec. III B. E. Examples
To see some examples of these invariants, we considerthe gauged group cohomology models of Ref. 10 or equiv-alently, the Dijkgraaf-Witten models of Ref. 34. We willcompute the invariants for these models in the next twosections. All the results listed below follow from two for-mulas which we will derive later, namely (42a-42c) and(49a-49c).The simplest nontrivial example is given by the 2 D Dijkgraaf-Witten models with symmetry group G = Z .In this case, H ( Z , U (1)) = Z so we can construct twoDijkgraaf-Witten models . The only independent in-variant in this case is Θ , which describes the phase as-sociated with exchanging two identical π vortices twice .The values of Θ in the two Dijkgraaf-Witten models areTrivial model: Θ = 0Non-trivial model : Θ = π Importantly, we can see that Θ takes different values inthe two models, which proves that they belong to distinctphases.More generally, for G = Z N , we have H ( Z N , U (1)) = Z N , so we can construct N Dijkgraaf-Witten modelsin this case. Similarly to the Z case, these modelscan be distinguished from one another by the topo-logical invariant Θ , which takes a different value in0 , πN , . . . , πN ( N −
1) for each of the N models.Another interesting example is given by the 2DDijkgraaf-Witten models with symmetry group G = Z N × Z N × Z N . In this case, H ( Z N × Z N × Z N , U (1)) = Z N so we can construct N models. Interestingly, thereare also seven independent topological invariants in thiscase: Θ , Θ , Θ , Θ , Θ , Θ , Θ . The invariant Θ is the topological spin of a vortex thatcarries πN (1 , ,
0) flux, multipled by N . The invariantΘ is the phase associated with braiding a vortex car-rying πN (1 , ,
0) flux around a vortex carrying πN (0 , , N times. The invariant Θ is the phase associ-ated with braiding πN (1 , ,
0) flux around πN (0 , ,
0) fluxin the counterclockwise direction, then around πN (0 , , πN (0 , ,
0) flux and πN (0 , ,
1) flux in the clockwisedirection. The meanings of the other invariants are sim-ilar. The invariant Θ is an indicator of non-Abelianstatistics: if Θ = 0, the corresponding statistics isAbelian; otherwise, the statistics is non-Abelian. We findthat all seven invariants take values in0 , πN , πN . . . , π ( N − N . and that these values distinguish all of the N Z N × Z N × Z N models. Again, this result proves that the N models each belong to a different phase according to thedefinition given in Sec. II C.Finally, we consider 3D Dijkgraaf-Witten models withsymmetry group G = Z × Z . In this case, H ( Z × Z , U (1)) = Z so we can construct 4 models . We findthere are two independent topological invariants in thiscase, namely, Θ , and Θ , . While there exist otherinvariants such as Θ , , Θ , , they are not independent,as we show in Sec. VI. The invariant Θ , is the phaseassociated with exchanging two identical loops that carrya ( π,
0) flux while both loops are linked to a third loopthat carries a (0 , π ) flux. The meaning of Θ , is similar.We find that Θ , = 0 or π, Θ , = 0 or π. Each of the four combinations of the values occurs in adifferent model. Hence, once again, the invariants dis-tinguish all of the 3D Z × Z Dijkgraaf-Witten modelsand prove that they belong to distinct phases accordingto the definition given in Sec. II C.
IV. THE INVARIANTS INDIJKGRAAF-WITTEN MODELS
In this section, we compute the topological invariantsfor all 2D and 3D Dijkgraaf-Witten models with Abeliangauge group G = Q Ki =1 Z N i . We obtain explicit expres-sions of the invariants in terms of the cocycle ω that isused to define the Dijkgraaf-Witten model. A. 2D topological invariants
1. Review of braiding statistics in 2D Dijkgraaf-Wittenmodels
In this section, we summarize some previously knownresults on the braiding statistics in 2D Dijkgraaf-Wittenmodels. Although these results do not provide an explicit formula for the braiding statistics in Dijkgraaf-Witten models, they do the next best thing: they give a well de-fined mathematical procedure for how to compute thesestatistics in terms of the 3-cocycle ω that defines themodel. This procedure involves a mathematical struc-ture known as the twisted quantum double algebra . Thetwisted quantum double formalism is quite general andcan be applied to any finite group G , including non-Abelian groups. However, in the following discussion, wewill specialize to the case of Abelian G , and we will onlygive a minimal review of the ingredients that are neces-sary for the computation of the invariants Θ i , Θ ij , Θ ijk .For more details, readers may consult Ref. 37,47.The first component of the twisted quantum doubleformalism is a scheme for labeling quasiparticle excita-tions. Let us consider a Dijkgraaf-Witten model cor-responding to an Abelian group G = Q Ki =1 Z N i and 3-cocycle ω . According to the formalism, each excitationin this model can be uniquely labeled by a doublet α = ( a, ρ ) (21)where a = ( a , ..., a K ) is a group element of G with0 ≤ a i ≤ N i −
1, and ρ is an irreducible projective repre-sentation of G satisfying ρ ( b ) ρ ( c ) = χ a ( b, c ) ρ ( b + c ) , (22)for all b, c ∈ G . Here, χ a is a phase factor defined by χ a ( b, c ) = ω ( a, b, c ) ω ( b, c, a ) ω ( b, a, c ) , (23)and is called the slant product of ω . The two labels ( a, ρ )have a simple physical meaning: the first component a describes the amount of flux φ α = ( πN a , ..., πN k a k ) car-ried by the excitation α , while the second component ρ is related to the amount of charge attached to α . For amore precise correspondence between the mathematicallabels ( a, ρ ) and the physical notions of gauge flux andgauge charge, we refer the reader to Appendix G.The second component of the formalism is a formulafor the fusion rules of the excitations. Specifically,( a, ρ ) × ( b, µ ) = X σ N σρµ ( a + b, σ ) (24)where the fusion multiplicities N σρµ are computed as fol-lows. First, we define a projective representation ρ ∗ µ by ( ρ ∗ µ )( g ) = ρ ( g ) ⊗ µ ( g ) · χ g ( a, b ) (25)for any g ∈ G . The N σρµ are then defined in terms of thedecomposition of ρ ∗ µ into irreducible projective repre-sentations, σ : ρ ∗ µ = M σ N σρµ σ (26)In addition to fusion rules, this formalism provides aconvenient way to parameterize the degenerate ground1states associated with a collection of (non-Abelian) exci-tations. Consider a system of n excitations α i = ( a i , ρ i ), i = 1 , ..., n , and suppose that these excitations fuse tothe vacuum. The ground state manifold associated withthese excitations can be obtained in two steps. First, weconstruct the tensor product V = V ⊗ · · · ⊗ V n where V i is the vector space on which ρ i is defined. Then, weproject onto the subspace of V that corresponds to thevacuum fusion channel. This projection is implementedby the operator P = 1 | G | X g ∈ G ρ ( g ) ∗ · · · ∗ ρ n ( g ) (27)The degenerate ground states associated with α , ..., α N can be parameterized by vectors that lie in the image of P : V → V .We are now ready to present a formula for the braid-ing statistics of the excitations. Suppose that the abovesystem of n excitations are arranged in a line in the order α , ..., α n . Then, the unitary transformation associatedwith braiding α i around its neighbor α i +1 , is given by B α i α i +1 = P · id V ⊗ · · · ⊗ id V i − ⊗ ρ i ( a i +1 ) ⊗ ρ i +1 ( a i ) ⊗ id V i +2 ⊗ · · · ⊗ id V n · P, (28)where id V i denotes the identity matrix on the vectorspace V i of ρ i . (The exchange statistics for the exci-tations can be computed in a similar fashion, but we willnot discuss them here as they are not necessary for ourpurposes).The final result we will need is an expression for thetopological spin. According to the twisted quantumdouble formalism, the topological spin of an excitation α = ( a, ρ ) is given by e i πs α = 1dim( ρ ) tr ρ ( a ) , (29)where dim( ρ ) is the dimension of the representation ρ .This formula can equivalently be written as e i πs α id V = ρ ( a ) (30)where id V is the identity matrix in the vector space V of ρ . The reason that (29) is equivalent to (30) is that ρ ( a ) is always a pure phase. Indeed, this property followsfrom Schur’s lemma and the observation that χ a ( a, b ) = χ a ( b, a ) so that ρ ( a ) commutes with any other matrix ρ ( b ). (As an aside, we note that the fact that ρ ( a ) is apure phase is consistent with the results in Appendix E.)
2. Explicit formulas for the invariants
We now compute the invariants Θ i , Θ ij and Θ ijk fora 2D Dijkgraaf-Witten model with group G = Q Ki =1 Z N i and 3-cocycle ω . Let α, β, γ be three vortices carryingunit flux 2 πe i /N i , 2 πe j /N j and 2 πe k /N k respectively. Using the notation from the previous section, we canlabel these vortices as α = ( e i , ρ ), β = ( e j , µ ) and γ = ( e k , ν ) for some projective representations ρ, µ, ν .We will denote the vector spaces associated with ρ, µ, ν by V, W, X . To compute Θ i , Θ ij and Θ ijk , we need tofind the topological spin of these vortices and to analyzevarious braiding processes involving them.We begin with Θ i . From (30) we derive e i Θ i id V = e i πN i s α id V = ρ ( e i ) N i (31)We then rewrite the right hand side as ρ ( e i ) N i = N i − Y n =0 ( ρ (( n + 1) e i ) − ρ ( e i ) ρ ( ne i ))= N i − Y n =0 χ e i ( e i , ne i ) id V (32)where the second line follows from equation (22). Weconclude that the invariant Θ i is given byexp( i Θ i ) = N i − Y n =0 χ e i ( e i , ne i ) (33)Similarly, we can obtain expressions for Θ ij , Θ ijk using(28), e i Θ ij id V ⊗ id W = ρ ( e j ) N ij ⊗ µ ( e i ) N ij ,e i Θ ijk id V = ρ ( e k ) − ρ ( e j ) − ρ ( e k ) ρ ( e j ) , which can then be related to χ using (22):exp( i Θ ij ) = N ij Y n =1 χ e i ( e j , ne j ) χ e j ( e i , ne i )exp( i Θ ijk ) = χ e i ( e k , e j ) χ e i ( e j , e k ) (34)Equations (33, 34) are the formulas we seek, where χ isdefined by (23).The properties of Θ i , Θ ij , Θ ijk that were derived inSec. III B from more general considerations are manifestin the above expressions. We can see that Θ i , Θ ij , Θ ijk only depend on i, j, k and not on the choice of vortices α, β, γ , since the representations ρ, µ, ν do not appear inthe final expressions (33,34). In addition, it is easy toverify that these formulas are invariant under the change ω → ω · ν if ν is a coboundary (see Appendix B for thedefinition of a coboundary). This is to be expected, sincetwo cocycles that differ by a coboundary are known todefine the same Dijkgraaf-Witten model, and thereforemust give the same values for the invariants. B. 3D topological invariants
1. Dimensional reduction
Our approach for computing the 3D invariants is basedon dimensional reduction: we derive a relationship be-2tween vortex loop statistics in 3D Dijkgraaf-Witten mod-els and vortex statistics in 2D Dijkgraaf-Witten models,and then we analyze the latter using previously known 2Dresults. A similar dimensional reduction approach wasused in Ref. 24. The derivation we discuss here is moregeneral than that of Ref. 24 in some ways and less generalin other ways. It is more general because it applies evenif the vortex loop statistics are non-Abelian, but it is alsoless general because it is only valid for Dijkgraaf-Wittenmodels, while Ref. 24 derived a dimensional reductionformula without restricting to these exactly soluble sys-tems.To begin, consider a Dijkgraaf-Witten model associ-ated with a group G = Q Ki =1 Z N i and a 4-cocycle ω . Letus define this model on a thickened 2D torus, i.e., themanifold T × [0 ,
1] (Fig. 4). Consider a state consistingof two loop excitations α, β which wind around the innerhole of the torus, and suppose that there is a flux φ = ( 2 πN a , ..., πN k a k ) (35)threading the inner hole. This geometry is equivalentto a standard three-loop setup in which α, β are linkedwith another loop carrying flux φ . Our task is to findthe unitary transformation associated with braiding α around β in the presence of the flux φ .To this end, we redraw the thickened torus T × [0 , α, β can be drawn aslines connecting the top and bottom faces of the cube.To proceed further we make use of a special propertyof Dijkgraaf-Witten models: these models can be definedfor any triangulation of space-time, and their propertiesdo not depend on the choice of triangulation. There-fore, we are free to triangulate our cube however we likeand it won’t affect the braiding statistics between α and β . Making use of this freedom, we consider a triangula-tion with translational symmetry in the x , y and z di-rections. More specifically, we consider triangulation inwhich there is only one unit cell in the z direction, butmany unit cells in the x and y direction. With this choice (a) (b) (c) α βφ α β FIG. 4: (a) A thickened torus T × [0 ,
1] with a flux φ threadingthe inner hole. (b) The thickened torus drawn as a cube withthe top and bottom faces as well as the front and back facesidentified. of triangulation, the cube can be viewed as a 2D system.Furthermore, the braiding process involving the vortexloops α and β can be viewed as a process involving twovortices in this effective 2D system.By the same argument as in the supplementary mate-rial of Ref. 24, one can show that this effective 2D systemis identical to a 2D Dijkgraaf-Witten model with group G and a 3-cocycle χ a ( b, c, d ) = ω ( b, a, c, d ) ω ( b, c, d, a ) ω ( a, b, c, d ) ω ( b, c, a, d ) (36)where a = ( a , ..., a K ) is defined in terms of φ as in equa-tion (35). Here, χ a is known as the slant product of ω .One can check that χ a is indeed a 3-cocycle for any choiceof a .Putting this all together, we conclude that the vortexloop statistics for a 3D Dijkgraaf-Witten model with 4-cocycle ω and with a base loop with flux φ , are identicalto the vortex statistics in a 2D Dijkgraaf-Witten modelwith 3-cocycle χ a .
2. Explicit formulas for the invariants
We are now ready to compute the invariantsΘ i,l , Θ ij,l , Θ ijk,l for a 3D Dijkgraaf-Witten model withgroup G = Q Ki =1 Z N i and 4-cocycle ω .Let α, β, γ be three vortex loops carrying unit flux2 πe i /N i , 2 πe j /N j and 2 πe k /N k respectively, and sup-pose that all three are linked with another loop σ car-rying unit flux 2 πe l /N l . To compute Θ i,l , Θ ij,l , Θ ijk,l ,we need to analyze various braiding processes involving α, β, γ . We can accomplish this task with the help ofthe dimensional reduction results of the previous section:according to those results, the braiding statistics of thevortex loops α, β, γ in a 3D Dijkgraaf-Witten model withcocycle ω are identical to the braiding statistics of vor-tices in a 2D Dijkgraaf-Witten model with 3-cocycle χ e l (36). Therefore the 3D invariants Θ i,l , Θ ij,l , Θ ijk,l can beobtained from the corresponding 2D invariants (33, 34)simply by substituting χ e l ( a, b, c ) for ω ( a, b, c ). In thisway, we obtainexp( i Θ i,l ) = N i Y n =1 χ e l ,e i ( e i , ne i )exp( i Θ ij,l ) = N ij Y n =1 χ e l ,e i ( e j , ne j ) χ e l ,e j ( e i , ne i )exp( i Θ ijk,l ) = χ e l ,e i ( e k , e j ) χ e l ,e i ( e j , e k ) (37)where χ e l ,e i is defined as χ e l ,e i ( b, c ) = χ e l ( e i , b, c ) χ e l ( b, c, e i ) χ e l ( b, e i , c ) (38)and where χ e l is defined in (36).3 V. SHOWING THE INVARIANTSDISTINGUISH ALL DIJKGRAAF-WITTENMODELS
In this section, we show that the invariants take differ-ent values for each of the Dijkgraaf-Witten models withgroup G . This result has two implications: (i) each ofthe Dijkgraaf-Witten models belongs to a distinct phaseand (ii) the invariants can distinguish these phases. A. 2D case
We now show that the invariants Θ i , Θ ij , Θ ijk can dis-tinguish all the 2D Dijkgraaf-Witten models correspond-ing to the group G = Q Ki =1 Z N i (A similar result wasobtained previously in Ref. 32). Our proof is based ona counting argument: let N DW2 D be the number of 2DDijkgraaf-Witten models with group G , let N phase2 D be thenumber of phases of 2D Dijkgraaf-Witten models withgroup G , and let N Θ2 D be the number of distinct val-ues that the invariants take over all 2D Dijkgraaf-Wittenmodels with group G . Clearly we must have N Θ2 D ≤ N phase2 D ≤ N DW2 D . (39)What we will prove is the opposite inequality: N Θ2 D ≥ N DW2 D (40)It will then follow that N Θ2 D = N phase2 D = N DW2 D , whichshows that each of the Dijkgraaf-Witten models belongsto a distinct phase and the invariants can distinguish allthe phases (according to the definition of phases given inSec. II C).To prove (40) we consider the set of 3-cocycles of theform ω ( a, b, c ) = exp { i π X ij P ij N i N j a i ( b j + c j − [ b j + c j ]) }× exp { i π X ijk Q ijk N ijk a i b j c k } , (41)where P ij is an integer matrix and Q ijk is an integer ten-sor with Q ijk = 0 if i, j, k are not all distinct. (The latterrestriction on Q ijk is not essential, and we include it onlyto simplify some of the formulas that follow). Here, thesymbol [ b j + c j ] is defined as the residue of b j + c j mod-ulo N j with values taken in the range 0 , ..., N j −
1. (Onecan verify that ω obeys the 3-cocycle condition (B2) withstraightforward algebra). Inserting the cocycle ω into theexpressions in (33,34), one immediately obtainsΘ i = 2 πN i P ii (42a)Θ ij = 2 πN ij ( P ij + P ji ) (42b)Θ ijk = − πN ijk ( Q ijk + Q jki + Q kij − Q jik − Q ikj − Q kji )(42c) From the above formulas, we see that the invariants Θ ij can take on N ij different values as P ij ranges over allinteger matrices. Similarly, Θ i can take on N i differentvalues while Θ ijk can take on N ijk different values. Fur-thermore, the values of Θ i , and Θ ij with i < j , and Θ ijk with i < j < k , can be varied independently from oneanother. Therefore, we have the lower bound N Θ2 D ≥ Y i N i Y i We now show that the invariants Θ i,l , Θ ij,l , Θ ijk,l candistinguish all the 3D Dijkgraaf-Witten models corre-sponding to the group G = Q Ki =1 Z N i . The argumentclosely follows the 2D case: let N DW3 D be the number of3D Dijkgraaf-Witten models with group G , let N phase3 D be the number of phases of 3D Dijkgraaf-Witten modelswith group G , and let N Θ3 D be the number of distinct val-ues that the invariants take over all 3D Dijkgraaf-Wittenmodels with group G . Clearly, we have N Θ3 D ≤ N phase3 D ≤ N DW3 D (46)What we will show is that N Θ3 D ≥ N DW3 D . (47)It will then follow that N Θ3 D = N phase3 D = N DW3 D , whichshows that each of the Dijkgraaf-Witten models belongsa distinct phase and the invariants distinguish all thephases (according to the definition of phases given inSec. II C).To prove (47), we consider the set of 4-cocycles of theform ω ( a, b, c, d ) = exp { i π X ijk M ijk N ij N k a i b j ( c k + d k − [ c k + d k ]) }× exp { i π X ijkl L ijkl N ijkl a i b j c k d l } . (48)where M ijk is an arbitrary integer tensor and L ijkl is aninteger tensor with L ijkl = 0 if i, j, k are not all distinct.4(As in the 2D case, it is simple to verify that ω obeys the4-cocycle condition (B3)). Inserting the 4-cocycle ω intothe expressions in (37), we obtainΘ i,l = 2 πN il ( M ili − M lii ) (49a)Θ ij,l = 2 πN ij N il N j ( M ilj − M lij ) + 2 πN ij N jl N i ( M jli − M lji )(49b)Θ ijk,l = − πN ijkl X ˆ p sgn(ˆ p ) L ˆ p ( i )ˆ p ( j )ˆ p ( k )ˆ p ( l ) (49c)where ˆ p is a permutation of i, j, k, l and sgn(ˆ p ) = ± p . From the above formulas, we seethat different choices of M and L give different valuesof Θ i,l , Θ ij,l and Θ ijk,l . More precisely, it can be shownthat as M and L range over the set of allowed integer ten-sors, the invariants Θ i,l , Θ ij,l and Θ ijk,l take on at least Q i In this section, we derive general constraints on theinvariants that hold for any gauge theory with group G = Q Ki =1 Z N i . In the next section, we will discuss whetherthese constraints are complete, i.e. whether any solutionto these constraints can be realized by an appropriategauge theory. A. 2D Abelian case Let us start with the case of 2D gauge theories withgauge group G = Q Ki =1 Z N i and Abelian braiding statis-tics. According to Sec. III A, there are two invariants Θ i and Θ ij in this case. We will now argue that Θ i and Θ ij must satisfy the following general constraints:Θ ii = 2Θ i , (53a)Θ ij = Θ ji , (53b) N ij Θ ij = 0 , (53c) N i Θ i = 0 , (53d)where all equations are defined modulo 2 π . To prove thisstatement, we first recall the following general propertiesof Abelian braiding statistics: θ αβ = θ βα , (54a) θ αα = 2 θ α (54b) θ α ( β × β ) = θ αβ + θ αβ , (54c) θ ( α × β ) = θ α + θ β + θ αβ . (54d)Here θ αβ denotes the mutual statistics of α, β , while θ α denotes the exchange statistics of α and α × β denotes theexcitation created by fusing together α and β . Each ofthese identities follow from simple physical arguments.The symmetry relation (54a) comes from the fact thatbraiding α around β is topologically equivalent to braid-ing β around α . The relation (54b) follows immediatelyfrom the definition of exchange statistics. The linearityrelation (54c) comes from the fact that fusing β and β must commute with braiding α around them. The otherlinearity relation (54d) has a similar flavor.We can see that equations (53a) and (53b) follow im-mediately from these general constraints. To prove (53c),consider braiding a vortex α that carries a unit flux πN i e i around N j identical vortices β , with each β carrying aunit flux πN j e j . Clearly the associated statistical phaseis N j θ αβ . At the same time, according to the linearityrelation (54c), this statistical phase is equal to the phaseassociated with braiding α around the fusion product ofall the β vortices. Since the total flux of N j β vorticesis zero, they fuse to a charge. It then follows from theAharanov-Bohm law that the latter quantity is a multipleof πN i . We conclude that N j θ αβ = 2 πN i × integer . Equation (53c) follows immediately using N ij N ij = N i N j together with the definition (2) of Θ ij .To prove (53d), imagine exchanging a set of N i α ’s withanother set of N i α ’s. According to the linearity relation(54d) and the relation (54b), the associated exchange sta-tistical phase is N i θ α = N i Θ i . However, this phase mustbe a multiple of 2 π since N i α ’s fuse to a charge which isa boson. Thus, equation (53d) must hold. B. 2D general case We now consider the case of general 2D gauge theo-ries with gauge group G = Q Ki =1 Z N i . There are three5 γα β FIG. 5: A Borromean ring obtained by closing up the trajec-tories in Fig. 1(b). invariants, Θ i , Θ ij , Θ ijk in this case. We will now arguethat Θ i , Θ ij again satisfy the constraints (53a)-(53d), asin the Abelian case. In addition, the third invariant Θ ijk satisfies Θ ijk = sgn(ˆ p )Θ ˆ p ( i )ˆ p ( j )ˆ p ( k ) , (55a)Θ iij = 0 , (55b) N ijk Θ ijk = 0 , (55c)where ˆ p is a permutation of the indices i, j, k and sgn(ˆ p ) = ± π . We note that the constraint (55a) tells usthat Θ ijk is fully antisymmetric modulo 2 π , while (55b) isa stronger constraint on Θ iij than its antisymmetry, sincethe antisymmetry only requires that 2Θ iij = 0 (mod 2 π ).We first prove the above constraints for Θ ijk ; after-wards we will prove (53a)-(53d). To prove that Θ ijk isfully antisymmetric, i.e. (55a), we consider the space-time trajectories (Fig. 1b) of the vortices in the braidingprocess associated with Θ ijk . Since the unitary transfor-mation associated with the braiding process is Abelian,we can close up the space-time trajectories in Fig.1b, sothat they form the Borromean ring in Fig.5. The closed-up trajectories are associated with the following process:we first create three particle-antiparticle pairs α, ¯ α , β, ¯ β and γ, ¯ γ out of the vacuum, where α, β, γ carry unit flux πN i e i , πN j e j , πN k e k respectively, then braid α, β, γ in theway that leads to the phase Θ ijk , and finally annihilatethe three pairs to return to the vacuum. The fact thatwe can annihilate the particles at the end of the processis guaranteed by the fact that the braiding results in apure phase, otherwise the particle-antiparticle need notbe in the vacuum fusion channel after the braiding.With this picture in mind, we can see that Θ ijk is equalto the phase associated with the Borromean ring space-time trajectories. Since the Borromean ring is cyclicallysymmetric, we deduce that Θ ijk = Θ jki = Θ kij . It isalso not hard to see that reversing the braiding processassociated with Θ ijk gives rise to the phase Θ ikj . So,Θ ijk = − Θ ikj . Putting these relations together, we seeΘ ijk is fully antisymmetric.To prove (55b), we make use of the result in AppendixE which shows that braiding a vortex α around another ( i ) α βγ γ γ ( ii ) α βγ γ γ FIG. 6: Thought experiment to prove the constraint (55c)( N k = 3 is taken for illustration). For clarity, we split thecomposite braiding into two steps ( i ) and ( ii ). In the thoughtexperiment, ( ii ) follows immediately after ( i ). α gives only a pure phase. Consider three vortices α, α, β with φ α = πN i e i and φ β = πN j e j and imagine the braidingprocess associated with Θ iij : α is braided around theother α , then around β , then around the other α in theopposite direction, and finally around β in the oppositedirections. In this four-step process, we can switch theorder of the second and third step since the third stepgives only a pure phase. Thus, it is obvious that thisfour-step process neither changes the state of the system,nor leads to any Abelian phase. Hence, equation (55b)holds.To prove (55c), we consider a collection of excitations α, β, γ , . . . , γ N k , where φ α = πN i e i , φ β = πN j e j and φ γ t = πN k e k for any t = 1 , . . . , N k . We consider a com-posite braiding process shown in Fig. 6: first we braid α around β in the counterclockwise direction, then se-quentially round γ , . . . , γ N k in the counterclockwise di-rection, then around β in the clockwise direction, and se-quentially around γ N k , . . . , γ in the clockwise direction.This braiding process can be described by a product ofoperators B ≡ B − αγ . . . B − αγ Nk B − αβ B αγ Nk . . . B αγ B αβ , where B αβ , B αγ , . . . , B αγ Nk are the operators associ-ated with braiding α around β, γ , . . . , γ N k respectively.Now according to the definition of Θ ijk , we have B − αγ t B − αβ B αγ t B αβ = e i Θ ijk ˆ I for any t = 1 , . . . , N k . Itthen follows that B = e iN k Θ ijk ˆ I (56)where ˆ I is the identity operator. On the other hand,braiding α around γ , . . . , γ N k in sequence is equivalentto braiding α around the fusion product of γ , . . . , γ N k .Since γ , . . . , γ N k fuse to some charge q , we derive B = B − αq B − αβ B αq B αβ , (57)where B αq denotes the operator associated with thebraiding of α around q . Now, by the Aharonov-Bohmformula, we know that B αq is a pure phase, which impliesthat B is just the identity operator, B = ˆ I . Therefore, e iN k Θ ijk = 1, i.e., N k Θ ijk = 0. Similarly, we can show6 N i Θ ijk = N j Θ ijk = 0. Putting this together, we derivethe constraint (55c).Next, we prove the constraints (53a)-(53d) in the non-Abelian case. The constraint (53a) follows immediatelyfrom Appendix E while (53b) is obvious. To prove (53c),we consider a vortex α carrying a flux πN i e i , together with N j vortices β , . . . , β N j all carrying a flux πN j e j . Imagine α is braided around β for N ij times, then around β for N ij times, and so on. The result is a total phase N j Θ ij .This sequence of braiding processes can be described bya product of operators B ′ = B N ij αβ Nj · · · B N ij αβ B N ij αβ (58)where B αβ t represents the operator associated withbraiding α around β t once. Any two operators B αβ t and B αβ s commute, because the commutator B − αβ t B − αβ s B αβ t B αβ s = e i Θ ijj and Θ ijj = 0 according to(55b). Therefore, the operator B ′ can be rewritten as B ′ = ( B αβ Nj · · · B αβ B αβ ) N ij , (59)which means B ′ is equivalent to braiding α around β , . . . , β N j as a whole for N ij times. However, the vor-tices β , . . . , β N j fuse to a pure charge, and when α isbraided around any charge for N ij times, the result is nophase at all. Therefore, we obtain N j Θ ij = 0. Similarly,one can show that N i Θ ij = 0. Putting this together, wederive the constraint (53c).Finally, to prove the constraint (53d), we use the dia-grammatical representation of the topological spin = e i πs α α αα α (60)Let us imagine N i identical α ’s, which should fuse tosome charge q . Consider the following diagram ααα = α α α (61)where the case N i = 3 is shown for simplicity. The lefthand side equals the topological spin e i πs q = 1, whilethe right hand side equals e i πN i s α + P Ni − n =0 i πns α = e i πN i s α = e iN i Θ i , (62)where the result of Appendix E is used. We concludethat N i Θ i = 0 modulo 2 π , which proves the constraint(53d). β γ β γ β ◦ β γ γ FIG. 7: Fusion of two loops β and β that carry the sameamount of flux φ β = φ β and that are linked to differentbase loops. We denote this type of fusion by β ◦ β . This isdifferent notation from Ref. 24, where this type of fusion wasdenoted by β ⊕ β . C. 3D Abelian case We now consider the case of 3D gauge theories withgauge group G = Q Ki =1 Z N i and Abelian braiding statis-tics. In this case, there are two invariants Θ i,l and Θ ij,l .We will argue that these invariants must satisfy the fol-lowing general constraints:Θ ii,l = 2Θ i,l , (63a)Θ ij,l = Θ ji,l , (63b) N ijl Θ ij,l = 0 , (63c) N il Θ i,l = 0 , (63d) N ijl N ij Θ ij,l + N ijl N jl Θ jl,i + N ijl N li Θ li,j = 0 , (63e)Θ i,l N il N i + Θ il,i = 0 , (63f)Θ i,i = 0 . (conjectured) (63g)We note that the above constraints are a generalizationof those derived in Ref. 24.To prove these constraints, we make use of the follow-ing general properties of Abelian three-loop statistics: θ αα,c = 2 θ α,c , (64a) θ αβ,c = θ βα,c , (64b) θ α ( β × β ) ,c = θ αβ ,c + θ αβ ,c , (64c) θ ( α × β ) ,c = θ α,c + θ β,c + θ αβ,c , (64d) θ ( α ◦ α )( β ◦ β ) , ( c + c ) = θ α β ,c + θ α β ,c , (64e) θ α ◦ β,c + c = θ α,c + θ β,c , (64f)Note that these equations involve two types of fusions ofloops, i.e. the “ × ” and “ ◦ ” fusions, shown in Fig. 3 andFig. 7 respectively. The “ × ” fusion involves two loopsthat are linked to the same base loop, while the “ ◦ ” fusioninvolves two loops that carry the same amount of fluxesbut are linked to different base loops (Note that thesetypes of fusion were denoted by “+” and “ ⊕ ” in Ref. 24).One can see that the first four equations resemble Eqs.(54a)-(54d) in 2D systems, while the last two are new to73D systems. We call (64b) the symmetry relation and call(64c)-(64f) the linearity relations . Like the 2d relations,the linearity relations encode the fact that braiding andexchanging of loops commute with fusion of loops. Thelinearity relations (64c, 64d) only involve a single baseloop and are similar to the 2D linearity relations (54c,54d). The linearity relations (64e, 64f) involve the “ ◦ ”fusion with different base loops, and have no analogue in2D systems. A graphical proof of (64e) can be found inRef.24 and the proof for (64f) is similar.With the help of these general properties, we will nowprove the constraints (63a-63f). The constraints (63a)and (63b) are the simplest to prove, as they follow im-mediately from (64a) and (64b). To prove (63c), we firstimagine braiding a vortex loop α around N j identical vor-tex loops β , where both α and β are linked to a base loop γ and φ α = πN i e i , φ β = πN j e j , φ γ = πN l e l . Clearly the to-tal statistical phase is N j θ αβ,e l . On the other hand, fromthe linearity property (64c), we know that this phase isequal to that of braiding α around the fusion product ofthe β ’s. Since the N j β ’s fuse to a pure charge, it followsthat the latter quantity is a multiple of 2 π/N i . Hence,we derive N j θ αβ,e l = 2 πN i × integer . (65)Comparing with the definition of Θ ij,l , we deduce that N ij Θ ij,l = 0 , (66)modulo 2 π . Next, we imagine a collection of N l identicalthree-loop linked structures. In each structure, loop α and β are linked with γ , and α is braided around β . Thetotal phase in the N l braiding processes is N l θ αβ,e l . Wethen fuse together all the linked structures and make useof the linearity relation (64e), and obtain N l θ αβ,e l = θ AB,N l e l . (67)where A, B denote the two loops A = α ◦ · · · ◦ α and B = β ◦ · · · ◦ β . Now since the N l e l = 0, the two loops A and B are not linked to any base loop so the statisticalphase on the right hand side can be computed using theconventional Aharonov-Bohm law (see Ref. 24 for a moredetailed argument): θ AB,N l e l = 2 πN j q A · e j + 2 πN i q B · e i (68)where q A , q B denote the amount of charge carried by the(unlinked) loops A, B . We conclude that N l θ αβ,e l = 2 πN i × integer + 2 πN j × integer . (69)which implies that N l Θ ij,l = 0 (70)modulo 2 π . Combining (66) and (70), we immediatelyderive the constraint (63c). The proof of (63d) is similar — the only difference being that one needs to considerexchange statistics rather than mutual statistics, and thelinearity relations (64d, 64f) rather than (64c, 64e).The proof of the “cyclic relations” (63e) and (63f) fol-lows the same philosophy as above, and involves consider-ing certain thought experiments. These thought experi-ments are described in Ref. 24 in the case of G = ( Z N ) K .It is not hard to extend these thought experiments to G = Q i Z N i , so we do not repeat them here and insteadrefer the reader to Ref. 24.It is unfortunate that we are not able to prove thelast constraint (63g). Therefore, this relation is just aconjecture. However, from (63d) and (63f), we can provethe weaker constraint 3Θ i,i = 0. D. 3D general case To complete our discussion, we now consider the caseof general 3D gauge theories with gauge group G = Q Ki =1 Z N i . In this case, there are three invariants Θ i,l ,Θ ij,l and Θ ijk,l . We will now argue that Θ i,l , Θ ij,l sat-isfy the constraints (63a-63d), as in the Abelian case. Inaddition, the third invariant Θ ijk,l satisfiesΘ ijk,l = sgn(ˆ p )Θ ˆ p ( i )ˆ p ( j )ˆ p ( k ) ,l (71a)Θ iij,l = 0 (71b) N ijk Θ ijk,l = 0 , (71c)Unlike the other cases that we have discussed, we ex-pect that the above list of constraints is incomplete : thatis, there are likely further constraints on the invariantsbeyond the ones listed here. One reason for this beliefis that in the case of the Dijkgraaf-Witten models, theinvariants obey the cyclic constraints (63e), (63f), and(63g), and Θ ijk,l satisfies the stricter constraintΘ ijk,l = sgn(ˆ p )Θ ˆ p ( i )ˆ p ( j )ˆ p ( k ) , ˆ p ( l ) Θ iij,l = 0 N ijkl Θ ijk,l = 0 , where ˆ p is a permutation of the indices i, j, k, l andsgn(ˆ p ) = ± is its parity. We find it plausible that theseadditional constraints may apply more generally than tothe Dijkgraaf-Witten models, but we have not been ableto prove this fact.The easiest constraints to establish are (71a-71c).These constraints follow identical arguments to the 2Dresults (55a-55c). Likewise, the two constraints (63a)and (63b) follow from the same logic as the 2D results(53a) and (53b), which were proved in the general casein section VI B.To prove (63c), we first notice that N ij Θ ij,l = 0can be established in the same way as its 2D analogue(53c). Next, we consider a thought experiment with N l identical three-loop linked structures { α, β, γ } with φ α = πN i e i , φ β = πN j e j , φ γ = πN k e k . We imagine braiding8each α around the corresponding β for N ij times. Theresult of each braiding process is an Abelian phase Θ ij,l .Therefore, the result of braiding all the α ’s simultane-ously is N l Θ ij,l . Now, since all the phases are Abelian,a linearity relation like (64e) applies in this case. Morespecifically, one can argue that the phase associated withthis braiding process is equal to that of braiding the loop A = α ◦ · · · ◦ α around B = β ◦ · · · ◦ β for N ij times,while both A, B are linked to a base loop which carries aflux N l φ l . Now since N l φ l = 0 modulo 2 π , A, B are notlinked to any base loop so this phase can be computedfrom the Aharonov-Bohm law, as in equation (68). Inthis way, we deduce that N l Θ ij,l = N ij (cid:18) πN i × integer + 2 πN j × integer (cid:19) = 0 . (72)Combining this with N ij Θ ij,l = 0, we derive the con-straint (63c). The proof of (63d) is similar and involvesthe use of a linearity relation for Θ i,l analogous to (64f). VII. DO THE DIJKGRAAF-WITTEN MODELSEXHAUST ALL POSSIBLE VALUES FOR THEINVARIANTS? In this section we ask and partially answer the follow-ing question: Q: Do there exist Abelian gauge theories for which theinvariants acquire values beyond those given by theDijkgraaf-Witten models? If the answer to this question is “yes”, then it followsthat there exist gauge theories that do not belong tothe same phase as any of the Dijkgraaf-Witten mod-els. On the other hand, if the answer is “no”, we cannotmake any rigorous statements about the existence or non-existence of gauge theories beyond the Dijkgraaf-Wittenmodels, since we cannot rule out the possibility that twogauge theories may share the same invariants but still be-long to distinct phases. That being said, a negative an-swer can be interpreted as circumstantial evidence thatthe Dijkgraaf-Witten models exhaust all possible Abeliangauge theories.To address this question, we compare the general con-straints derived in the previous section with our explicitcomputation of the invariants in the Dijkgraaf-Wittenmodels. We first consider the 2D case. In that case, weknow that the invariants must obey constraints (53a)-(53d) as well as (55a)-(55c). At the same time, weknow that the Dijkgraaf-Witten models can realize any(Θ i , Θ ij , Θ ijk ) of the form given in equations (42a) -(42c). Comparing these two results, one can easily verifythat the Dijkgraaf-Witten models can realize all possi-ble values of the invariants that are consistent with thegeneral constraints. We conclude that in the 2D case,the Dijkgraaf-Witten models exhaust all possible valuesof the invariants. Next we consider the 3D Abelian case: that is, 3Dgauge theories with gauge group G and Abelian three-loop statistics. In this case, we know that the invariantsmust obey the constraints (63). We also know that theDijkgraaf-Witten models with Abelian statistics canrealize any (Θ i,l , Θ ij,l ) of the form given in equations(49a)-(49b). From these two facts, one can show thatthe Abelian Dijkgraaf-Witten models realize all possi-ble values of the invariants that are consistent with thegeneral constraints; this derivation is given in appendixH. One loophole is that the last constraint (63g) is sim-ply a conjecture . Therefore, all we can say is that theDijkgraaf-Witten models exhaust all possible values ofthe invariants, assuming this conjecture is correct.Finally, let us consider the general 3D case: that is, 3Dgauge theories with gauge group G and any type of three-loop statistics (Abelian or non-Abelian). In this case,we have only managed to prove very weak constraintson the invariants, as discussed in the previous section.As a result, the Dijkgraaf-Witten models only realize asmall subset of the invariants that are consistent withour constraints. Hence, we cannot make any statementsas to whether the Dijkgraaf-Witten models exhaust allpossible values of the invariants. VIII. RELATION BETWEEN THEINVARIANTS AND BRAIDING STATISTICS In this section, we show that the topological invariantscontain the same information as the full set of braidingstatistics data, for 2D or 3D gauge theories with gaugegroup G = Q Ki =1 Z N i and Abelian statistics. We do notknow whether a similar result holds for gauge theorieswith gauge group G and non-Abelian statistics. A. 2D case We begin with the 2D case. What we will show is thatif two 2D gauge theories with Abelian statistics have thesame values for the invariants Θ i and Θ ij , then all theirbraiding statistics must be identical. In this sense thetopological invariants contain all the information aboutthe braiding statistics in these systems.Before presenting our argument, let us recall our defini-tion for when two gauge theories have the “same” braid-ing statistics. As discussed in Sec. II C, we say that twogauge theories have the same braiding statistics if thereexists a one-to-one correspondence between the quasipar-ticle excitations in the two theories that (1) preserves allthe algebraic structure associated with braiding statis-tics, e.g. R -symbols, fusion rules, F -symbols, etc and (2)preserves the gauge flux of excitations. In other words,for each excitation in one gauge theory, there should be acorresponding excitation in the other gauge theory thathas the same braiding statistics properties and the samegauge flux.9Given the above definition, our task is as follows. Con-sider two 2D gauge theories with group G = Q Ki Z N i and Abelian statistics. Suppose that the gauge theorieshave the same values for the invariants Θ i and Θ ij . Wehave to show that there exists a one-to-one correspon-dence between the excitations in the two theories thatpreserves their exchange statistics, mutual statistics, andgauge flux.We construct this correspondence as follows. In thefirst gauge theory, for each i = 1 , . . . , K , we choose oneof the | G | types of vortices that carry unit flux πN i e i , anddenote it by ˆ v i . Similarly, in the second gauge theory,for each i = 1 , . . . , K , we choose one of the | G | typesof vortices carrying unit flux πN i e i and denote it by ˆ w i .Given that the two gauge theories have identical valuesof Θ i and Θ ij , we know that the exchange statistics andmutual statistics of { ˆ v i } and { ˆ w i } are related by θ ˆ v i = θ ˆ w i + 2 πx i N i , θ ˆ v i ˆ v j = θ ˆ w i ˆ w j + 2 πy ij N ij (73)for some integers x i , y ij , with y ii = 2 x i .In the next step, we fuse some gauge charge q i =( q i , ..., q iK ) onto each vortex ˆ w i , to obtain another unitflux vortex ˆ w ′ i . We choose the q i so that the new vorticesˆ w ′ i obey θ ˆ v i = θ ˆ w ′ i , θ ˆ v i ˆ v j = θ ˆ w ′ i ˆ w ′ j (74)To see that we can always do this, note that θ ˆ w ′ i = θ ˆ w i + 2 πq ii N i θ ˆ w ′ i ˆ w ′ j = θ ˆ w i ˆ w j + 2 πq ij N j + 2 πq ji N i (75)by the Aharonov-Bohm formula. Hence, we can arrangefor equation (74) to hold if we choose the gauge charges q i so that they satisfy q ii = x i (mod N i )1 N ij ( N i q ij + N j q ji ) = y ij (mod N ij ) (76)We are now ready to construct the desired one-to-onecorrespondence between the excitations in the two gaugetheories. We note that every excitation in the first gaugetheory can be written uniquely as a fusion product α = (ˆ v ) a × · · · × (ˆ v K ) a K × q. (77)where a i are integers with 0 ≤ a i ≤ N i − 1, and where q = ( q , . . . , q K ) is some gauge charge with 0 ≤ q i ≤ N i − 1. Similarly, every excitation in the second gaugetheory can be written uniquely as α ′ = ( ˆ w ′ ) a × · · · × ( ˆ w ′ K ) a K × q. (78)We define a one-to-one correspondence between two setsof excitations by mapping α = (ˆ v ) a × · · · × (ˆ v K ) a K × q ↔ α ′ = ( ˆ w ′ ) a × · · · × ( ˆ w ′ K ) a K × q (79) It is clear that this correspondence preserves gauge fluxsince φ α = (cid:18) πa N , . . . , πa k N k (cid:19) = φ α ′ (80)To see that this correspondence preserves the exchangestatistics and mutual statistics of the excitations, we needto check that θ α = θ α ′ , θ αβ = θ α ′ β ′ (81)for any α, β in the first gauge theory and correspond-ing α ′ , β ′ in the second gauge theory. These relationsfollow immediately from (74) together with the linearityrelations (54c) and (54d). This completes our argument:we have shown that if two gauge theories have the samevalues of the invariants Θ i , Θ ij , then all their braidingstatistics is identical. B. 3D case We now repeat the argument in the 3D case. Con-sider two 3D gauge theories with group G = Q Ki Z N i and Abelian three-loop statistics. Suppose that the gaugetheories have the same values for the invariants Θ i,l andΘ ij,l . We will show that the two gauge theories haveidentical three-loop statistics. More precisely, we willshow that for each gauge flux φ , there exists a one-to-onecorrespondence between the loop-like excitations in thetwo theories that are linked with base loops with flux φ ,such that the corresponding excitations have the samethree-loop statistics and the same gauge flux.The derivation closely follows the 2D case. To begin,we focus on the first gauge theory, and we fix a baseloop that carries unit flux πN l e l . For each i = 1 , ..., K , wechoose one of the | G | types of vortex loops that carry flux πN i e i and are linked with the base loop and we denoteit by ˆ v i . We repeat this process for the second gaugetheory. That is, we fix a base loop with flux πN l e l and wechoose vortex loops { ˆ w i } that are linked with the baseloop and carry flux πN i e i . Using the fact that the twogauge theories have identical values of Θ i,l and Θ ij,l , weknow that the three-loop statistics of the { ˆ v i } and { ˆ w i } loops are related by θ ˆ v i ,e l = θ ˆ w i ,e l + 2 πx i,l N i , θ ˆ v i ˆ v j ,e l = θ ˆ w i ˆ w j ,e l + 2 πy ij,l N ij for some integers x i,l , y ij,l , with y ii,l = 2 x i,l . Note thatthe integers x i,l and y ij,l may depend on l — that is theymay take different values for different base loops.We next fuse some gauge charge q i = ( q i , ..., q iK ) ontoeach vortex loop ˆ w i , to obtain another vortex loop ˆ w ′ i .We choose the q i so that the new vortex loops ˆ w ′ i obey θ ˆ v i ,e l = θ ˆ w ′ i ,e l , θ ˆ v i ˆ v j ,e l = θ ˆ w ′ i ˆ w j ,e l (82)The fact that we can always find such a q i follows fromthe same reasoning as in the 2D case.0We now construct a one-to-one correspondence be-tween the loop excitations in the two gauge theories, fo-cusing on the excitations that are linked with a base loopwith flux πN l e l . The construction is identical to the 2Dcase. First, we note that every loop excitation α in thefirst gauge theory can be written uniquely as a fusionproduct of the ˆ v i vortex loops together with some gaugecharge, as in equation (77). Similarly, every loop excita-tion in the second gauge theory can be written uniquelyas a product of the ˆ w ′ i vortex loops as in equation (78).We can therefore define a one-to-one correspondence us-ing the mapping in equation (79). For the same reasonsas in the 2D case, it is clear that this correspondencepreserves the statistics of the loop excitations, as well astheir gauge flux.To finish the derivation, we need to generalize theabove one-to-one correspondence to the case where thebase loop carries arbitrary flux φ . This generalizationis easy to prove since we can construct base loops witharbitrary flux by fusing together base loops with unitflux using the “ ◦ ” fusion process (see Fig. 7). Further-more, using the linearity relations (64e, 64f) we can seethat the three-loop statistics associated with these moregeneral base loops is completely determined by the three-loop statistics for the unit flux base loops. Putting thesepieces together, we can construct a similar one-to-onecorrespondence between loop excitations that are linkedwith base loops with arbitrary flux. This completes ourargument and proves that the two gauge theories haveidentical three-loop statistics. IX. CONCLUSION In this paper, we have studied the braiding statisticsof 2D and 3D gauge theories with group G = Q Ki =1 Z N i ,and we have defined topological invariants that summa-rize some of the most important aspects of this braid-ing structure. In the 2D case, these invariants consist ofthree tensors, { Θ i , Θ ij , Θ ijk } , while in the 3D case, theyconsist of three tensors, { Θ i,l , Θ ij,l , Θ ijk,l } . These ten-sors are defined in terms of certain composite braidingprocesses involving vortices and vortex loops.Using these invariants, we have obtained two results.First, we have shown that the invariants distinguish all2D and 3D Dijkgraaf-Witten models (= gauged group co-homology models) with group G . Second, we have shownthat the Dijkgraaf-Witten models with group G exhaustall possible values of the invariants in the 2D case and wehave derived similar, but weaker, results in the 3D case.So far our discussion has focused on gauge theories, ormore precisely, gauged SPT models. We now return tothe questions raised in the introduction, and discuss theimplications of our findings for ungauged SPT models.We begin with our result that the invariants take differ-ent values for each of the 2D and 3D Dijkgraaf-Wittenmodels (= gauged group cohomology models) with group G . This result has an immediate implication: the 2D and 3D group cohomology models with group G all belong todistinct phases.Next, we consider our finding that the 2D Dijkgraaf-Witten models exhaust all possible values for the invari-ants. While we cannot draw rigorous conclusions fromthis result, it is at least consistent with the possibilitythat the 2D group cohomology models realize all possibleSPT phases with symmetry group G . Our 3D results onthis topic are also consistent with this possibility, thoughthey are somewhat weaker.In short, our results support the group cohomologyclassification conjecture of Chen, Gu, Liu, and Wen ,for the case of finite, Abelian symmetry group G . In ad-dition, our results allow us to go further: they providea simple diagnostic for determining whether a specificmicroscopic Hamiltonian belongs to the same phase asa specific group cohomology model. The diagnostic in-volves gauging the Hamiltonian and then computing thetopological invariants of the associated gauge theory. Ifthe invariants for the microscopic Hamiltonian are differ-ent from that of the group cohomology model, then wemay conclude that the Hamiltonian belongs to a distinctphase. If the invariants are the same, then it may belongto the same phase, though we cannot be certain becausewe do not know if the invariants are complete in the sensethat they distinguish all possible SPT phases.We see this work as a first step towards answering thethree questions raised in the introduction. We have man-aged to make some progress on these questions for thespecial case of Abelian symmetry groups, but many issuesremain unresolved. First, although we have partially an-swered questions 1-2 from the introduction, we have saidnothing at all about question 3 — which asks whether thebraiding statistics data can distinguish all possible SPTphases. Another important direction for future work is tounderstand the braiding statistics in gauge theories with non-Abelian gauge group and see if similar topologicalinvariants could be defined in that context. Finally, itwould be interesting to study gauge theories associatedwith fermionic SPT models. Such systems may have aneven richer braiding statistics structure than the bosoniccase analyzed here. Acknowledgments We thank C.-H. Lin, Z. Wang, J. Wang, X. Chen andP. Ye for helpful discussions. This work is supported bythe Alfred P. Sloan foundation and NSF under grant No.DMR-1254741. Appendix A: Gauging prescription In this Appendix, we give a general prescription forhow to gauge a lattice boson model with an Abeliansymmetry group G = Q Ki =1 Z N i . The procedure we de-scribe follows the usual minimal coupling scheme for lat-1tice gauge theories . The only nonstandard element isthat we will set the gauge coupling constant to zero inorder to maximize our control over the models, as inRefs. 15,50. More precisely, what we mean by this is thatthe Hamiltonians for the gauged models commute withthe flux operators that measure the gauge flux througheach plaquette in the lattice. This property has a niceconsequence: the gauge theories we construct are guaran-teed to be gapped and deconfined as long as the originalboson models are gapped and don’t break the symmetryspontaneously.For concreteness, we will focus on a particular kind ofboson model with Q Ki =1 Z N i symmetry. Specifically, wewill focus on lattice boson models built out of K speciesof bosons, where the particle number of the i th species isconserved modulo N i and where the different species ofbosons live on the sites p of some 2D or 3D lattice. Wedenote the boson creation operator for the i th species onlattice site p by b † p,i . We assume that the boson model haslocal interactions, so that the Hamiltonian of the bosonscan be written as H = X A H A ( { b p,i } ) (A1)where the sum is taken over localized regions A , andwhere H A is some operator composed out of boson cre-ation and annihilation operators acting on region A .We now discuss our procedure for gauging such aHamiltonian. The first step is to introduce a Hilbertspace H pq of dimension | G | = Q Ki =1 N i for each link pq ofthe lattice. This Hilbert space is spanned by basis states {| m i} , where m = ( m , . . . , m K ) with 0 ≤ m i ≤ N i − µ pq,i for each species i = 1 , . . . , K . Theseoperators are defined by µ pq,i | m i = e ± i πNi m i | m i . (A2)with a + or − sign depending on whether pq is parallelor anti-parallel to some fixed orientation that we assignto every link of the lattice. Likewise, we define a setof unitary shift operators S pq,a for each group element a ∈ G : S pq,a | m i = | m ± a i . (A3)with a + or − sign depending on whether pq is parallel orantiparallel to the prescribed orientation. Here, we haveused a = ( a , . . . , a k ), 0 ≤ a i ≤ N i − 1, to denote thegroup elements of G = Q Ki =1 Z N i .In the second step, we replace each operator H A ( { b p,i } )by a gauged operator ˜ H A ( { b p,i , µ pq,i } ) following the min-imal coupling procedure: H A ( { b p,i } ) → ˜ H A ( { b p,i , µ pq,i } ) , (A4)For example, a nearest neighbor hopping term undergoesthe following substitution under minimal coupling b p,i b † q,i → b p,i b † q,i µ pq,i (A5) For more complicated terms involving multiple sites,the substitution contains a product of µ operators actingon a path that connects the sites. One subtlety is thatthere is an ambiguity for how to choose the path. Thisambiguity is eliminated by the third step in the gaugingprocedure. In this step, we multiply ˜ H A by a projectionoperator P A which projects onto the states that havevanishing gauge flux through each plaquette that belongto A . That is, the projector P A can be written as aproduct P A = Y h pqr i∈ A P h pqr i (A6)where P h pqr i projects onto states with vanishing fluxthrough a particular plaquette h pqr i , which we have as-sumed to be triangular for concreteness. The projector P h pqr i can be explicitly written as P h pqr i = 1 | G | K Y i =1 N i − X k =0 ( µ pq,i µ qr,i µ rp,i ) k ! . (A7)After multiplying ˜ H A by P A , one can show that all pathsenclosed by A lead to the same term so that the minimalcoupling procedure is unambiguous. It is not hard to seethat P h pqr i is a Hermitian operator.In the last step of the gauging procedure, we add aterm of the form − ∆ P h pqr i to the Hamiltonian for eachplaquette, so that it costs finite energy to create vortexexcitations. After applying these steps, the final gaugedHamiltonian is˜ H = X A ˜ H A ( { b p,i , µ pq,i } ) P A − ∆ X h pqr i P h pqr i (A8)where we assume ∆ is large and positive. This Hamil-tonian is defined on a Hilbert space consisting of gaugeinvariant states, i.e. states | Ψ i satisfying T p,a | Ψ i = | Ψ i (A9)for every p, a , where T p,a is the gauge transformationassociated with group element a ∈ G and site p : T p,a = e i P i πaiNi b † p,i b p,i Y q ∈ neigh.( p ) S qp,a (A10)This constraint is the analogue of the usual Gauss’s lawof electromagnetism, ∇ · E = ρ .The gauging prescription we have just described hasa special property: the Hamiltonian ˜ H commutes withthe flux through each plaquette of the lattice. That is,[ ˜ H, µ pq,i µ qr,i µ rp,i ] = 0 for every plaquette h pqr i and ev-ery i = 1 , ..., K . This property has an important conse-quence: the gauge theory ˜ H is guaranteed to be gappedand deconfined as long as H is gapped and doesn’t breakthe symmetry spontaneously. To see this, note that[ ˜ H, P h pqr i ] = 0, so the eigenstates of ˜ H are also eigen-states of P h pqr i . As long as ∆ is large, then all the low2energy eigenstates | Ψ i will have vanishing flux: that is, P h pqr i | Ψ i = | Ψ i . At the same time, it is easy to seethat, within the zero flux sector, the energy spectrumof ˜ H is identical to the energy spectrum of the originalboson model H . We conclude that ˜ H and H have iden-tical low energy spectra. Hence, ˜ H is guaranteed to begapped if H is gapped. Similar reasoning shows that ˜ H is deconfined as long as H doesn’t break the symmetryspontaneously. Appendix B: Group cohomology In this Appendix, we review the basic ingredients ofthe cohomology of finite groups. We focus on thecohomology group H n [ G, U (1)].Let G be a finite group. The basic objects that groupcohomology studies are n -cochains . An n -cochain is a U (1) valued function c ( g , . . . , g n ): c : G × G × · · · × G | {z } n times → U (1) . The collection of n -cochains form an Abelian group C n ,where the group operation is defined by( c · c )( g , . . . , g n ) = c ( g , . . . , g n ) · c ( g , . . . , g n ) . The coboundary operator δ is a map δ : C n → C n +1 ,defined by δc ( g , . . . , g n +1 ) = c ( g , . . . , g n +1 ) c ( g , . . . , g n ) ( − n +1 × n Y i =1 [ c ( g , . . . , g i g i +1 , . . . , g n +1 )] ( − i . (B1)It is easy to check that the coboundary operator satisfies δ ( c · c ) = δc · δc . More importantly, one can checkthat δ is nilpotent: δ = 1.With the help of the coboundary operator, we can nowdefine n -cocycles and n -coboundaries . An n -cocycle isan n -cochain ω that satisfies δω = 1. For example, 3-cocycles satisfy ω ( g , g , g ) ω ( g , g g , g ) ω ( g , g , g ) ω ( g g , g , g ) ω ( g , g , g g ) = 1 , (B2)and 4-cocycles satisfy ω ( g , g , g , g ) ω ( g , g g , g , g ) ω ( g , g , g , g g ) ω ( g g , g , g , g ) ω ( g , g , g g , g ) ω ( g , g , g , g ) = 1 . (B3)Likewise, an n -coboundary is an n -cochain ν that canbe written as ν = δc where c ∈ C n − . The nilpotenceof δ implies that a coboundary must also be a cocycle.This allows us to define an equivalence relation for thecocycles: two n -cocycles ω and ω are said to be co-homologically equivalent if and only if ω = ω · δc , forsome c ∈ C n − . The equivalence classes of the n -cocycles form an Abelian group, called the n th cohomology group ,which is denoted by H n [ G, U (1)].In this paper, we will only need the cohomology group H n [ G, U (1)] for G = Q Ki =1 Z N i and for n = 3 and 4.These cohomology groups can be computed explicitly us-ing the Kunneth formula: H [ G, U (1)] = Y i Z N i Y i In this Appendix, we briefly review the group cohomol-ogy models of Ref. 10, as well as the Dijkgraaf-Wittenmodels of Ref. 34. In addition, we show that couplingthe group cohomology models to a lattice gauge fieldgives exactly the Dijkgraaf-Witten models. For conve-nience, we describe these models as well as the gaugingprocedure using a path integral formulation in Euclideanspace-time. This is different from the gauging procedurein Sec. A which is described in a Hamiltonian formula-tion. We expect the similar results can be derived in aHamiltonian formulation (e.g., see Ref. 52,53 for a Hamil-tonian description of Dijkgraaf-Witten models). 1. Group cohomology models The basic data needed to construct a d +1-dimensionalgroup cohomology model with (finite) group G is (1) a d + 1-cocycle ω together with (2) a triangulation of d + 1-dimensional Euclidean space-time. To build the model,we label the vertices of the triangulation by an orderedsequence i, j, . . . , the links by [ ij ] , [ jk ] , . . . , and the tri-angular plaquettes by [ ijk ] , . . . , etc. We will refer to thevertices as “0-simplices”, the links as “1-simplices” thetriangular plaquettes as “2-simplices” and so on. Thebasic degrees of freedom in the model are group elements g i ∈ G that live on the vertices i of the triangulation.For every space-time configuration { g i } , we assign a localweight [ ω ( g − i g j , . . . , g − k g l )] σ ij...kl to each d + 1-simplex[ ij . . . kl ] ( i < j < · · · < k < l ) where σ ij...kl = ± ij . . . kl ]. The action cor-responding to { g i } is given by the product of the localweights e − S ( { g i } ) = Y [ ij...kl ] [ ω ( g − i g j , . . . , g − k g l )] σ ij...kl . (C1)Summing over all the configurations { g i } , we obtain thepartition function Z = 1 | G | N v X { g i } e − S ( { g i } ) , (C2)3where | G | Nv is a normalization factor, | G | is the size ofthe group, and N v is the number of vertices. One caneasily check that the action (C1) is invariant under theglobal symmetry g i → gg i , (C3)for all g ∈ G . According to the arguments in Ref. 10, theground states of these models are gapped and short-rangeentangled for any G and ω . Moreover, two cocycles thatdiffer by a coboundary define the same group cohomologymodel. Thus, the group cohomology models are labeledby equivalence classes of cocycles, i.e., by elements of thecohomology group H d +1 [ G, U (1)]. 2. Dijkgraaf-Witten models The basic data needed to construct a d +1-dimensionalDijkgraaf-Witten model with (finite) group G is the sameas that for a group cohomology model: (1) a d + 1-cocycle ω together with (2) a triangulation of d + 1-dimensional Euclidean space-time. Unlike the group co-homology SPT models, the basic degrees of freedom ina Dijkgraaf-Witten model are group elements h ij ∈ G that live on the links [ ij ] of the triangulation. For everyspace-time configuration { h ij } , the corresponding action e − S ( { h ij } ) is defined as follows. First, one needs to de-termine if the configuration is flat, i.e., h ij h jk h ki = 1for every 2-simplex [ ijk ]. If it is flat, we assign a lo-cal weight [ ω ( h ij , . . . , h kl )] σ ij...kl to each d + 1-simplex[ ij . . . kl ] ( i < j < · · · < k < l ) where σ ij...kl = ± ij . . . kl ]. The action is thengiven by e − S ( { h ij } ) = Y [ ij...kl ] [ ω ( h ij , . . . , h kl )] σ ij...kl . (C4)If the gauge configuration is not flat, then e − S ( {{ h ij } ) =0. Summing over all the configurations { h ij } , we obtainthe partition function Z = 1 | G | N v X ′{ h ij } Y [ ij...kl ] [ ω ( h ij , . . . , h kl )] σ ij...kl , (C5)where | G | Nv is again a normalization factor, and P ′ is asummation over flat gauge configurations. The partitionfunction Z describes the Dijkgraaf-Witten models. Onecan check that if two cocycles ω, ω ′ differ by a cobound-ary, then they define the same Dijkgraaf-Witten models.Thus, the Dijkgraaf-Witten models are labeled by ele-ments of H d +1 [ G, U (1)], just like the group cohomologymodels. 3. Connection between the two classes of models We now show that the Dijkgraaf-Witten model withgroup G and cocycle ω is equivalent to the group coho- mology model of the same group G and cocycle ω afterthe global symmetry of the latter is gauged.To gauge the symmetry in the group cohomologymodel, we introduce lattice gauge fields h ij ∈ G thatlive on the links [ ij ] of the triangulation. We then couplethe matter degrees of freedom { g i } and gauge degrees offreedom { h ij } by replacing each g − i g j in the action (C1)by g − i h ij g j , following the minimal coupling procedure.After this step, the action becomes e − ˜ S ( { g i } , { h ij } ) = Y [ ij...kl ] (cid:2) ω ( g − i h ij g j , . . . , g − k h kl g l ) (cid:3) σ ij...kl . (C6)The next step is to choose a value for the gauge cou-pling constant. Here, as in Appendix A, we choose thegauge coupling constant to be 0. This means that weset e − ˜ S ( { g i } , { h ij } ) = 0 if the gauge configuration is notflat. With this choice of coupling constant, the gaugedpartition function acquires the form˜ Z = 1 | G | N v X { g i } X ′{ h ij } Y [ ij...kl ] (cid:2) ω ( g − i h ij g j , . . . , g − k h kl g l ) (cid:3) σ ij...kl , (C7)where the summation P ′ is taken only over flat gaugeconfigurations, and where we have included a normaliza-tion factor | G | Nv . By construction, the partition function˜ Z has a local gauge symmetry g i → α i g i , h ij → α i h ij α − j , (C8)for all α i ∈ G and all i, j. To see the connection between the gauged group coho-mology models ˜ Z and the Dijkgraaf-Witten models Z ,we fix the gauge in ˜ Z . The gauge that we choose is g i = 1 , for all i. (C9)In this gauge, the partition function ˜ Z becomes˜ Z = 1 | G | N v X ′{ h ij } Y [ ij...kl ] [ ω ( h ij , . . . , h kl )] σ ij...kl , (C10)where the numerical factor | G | N v comes from performingthe sum over { g i } . We can see that ˜ Z is identical to Z ,proving that the gauged group cohomology model withgroup G and cocycle ω is equivalent to the Dijkgraaf-Witten model with the same G and ω . Appendix D: Properties of fusion rules in Abeliandiscrete gauge theories In this Appendix, we derive some properties of the fu-sion rules of excitations in 2D gauge theories with group G = Q i Z N i .4The first property is that any excitation γ that ap-pears in the fusion product α × β = P γ N γαβ γ must obey φ γ = φ α + φ β . This property is clear from the follow-ing thought experiment. Imagine braiding an arbitrarycharge q around α and β . One can braid q around α and β sequentially, which gives a phase q · ( φ α + φ β ). Or, onecan first fuse α and β into some γ , then braid q around γ , leading to a phase q · φ γ . Clearly the two processesshould give the same phase, so q · ( φ α + φ β ) = q · φ γ .Since q is arbitrary, we have φ α + φ β = φ γ .Another property of the fusion rules is that when anexcitation α is fused with a charge q , there is exactly onefusion outcome: q × α = α ′ , (D1)where α ′ might be the same as α . To prove this property,we imagine four excitations α, ¯ α and q, ¯ q , where ¯ α denotesthe antiparticle of α and we suppose that the overall fu-sion channel for the four excitations is the vacuum. Wenow count the degeneracy of this four-excitation space intwo different ways. First, we note that q, ¯ q are Abelianparticles, so they must fuse to the vacuum, which in turnforces α, ¯ α to fuse to the vacuum. So, the four-excitationspace is non-degenerate. On the other hand, we can alsofuse the particles in a different order: we first fuse q with α and fuse ¯ q with ¯ α , then fuse the resulting particles.If we fuse the particles in this way, we can see that thedegeneracy is given by the number of different fusion out-comes in q × α . We conclude that there is a unique fusionoutcome in q × α . Therefore, (D1) holds.The third property is that an excitation α and its anti-particle ¯ α can only fuse to charges, i.e. α × ¯ α = ∅ + q + q + . . . . (D2)where ∅ denotes the vacuum. Moreover, the coefficientof each q i appearing in the fusion rule is 1. The firststatement is easy to prove: all particles on the right sidemust be charges since φ α + φ ¯ α = 0. To prove that thecoefficient associated with each q i is 1, we use the factthat N γαβ = N ¯ βα ¯ γ from the general algebraic theory ofanyons . From this fact, we derive N q i α ¯ α = N αα ¯ q i = 0 , q i appearing in (D2)are exactly those that follow the fusion rule α × q i = α .With the above properties, we now prove two claims.The first claim states that for any two excitations α, α ′ with φ α ′ = φ α , there exists at least one charge q with α ′ = α × q . The second claim is more complicated. Toexplain it, consider two excitations α, β in a fusion chan-nel γ , and two other excitations α ′ , β ′ in a fusion channel γ ′ . The claim states if φ α ′ = φ α and φ β ′ = φ β then thereexist charges q and q such that α ′ = α × q , β ′ = β × q and γ ′ = γ × q × q .To prove the first claim, consider an arbitrary excita-tion α ′ with φ α ′ = φ α . Imagine fusing together α, ¯ α, α ′ .We can do the fusion in two different orders,( α × ¯ α ) × α ′ = ( ∅ + q + . . . ) × α ′ , (D3) and α × (¯ α × α ′ ) = α × ( q ′ + q ′ + . . . ) , (D4)where the “ . . . ” means some charges. Since the orderof fusion can’t affect the final result, we conclude that α ′ = α × q ′ i , where q ′ i is one of the charges in (D4). Hence,the claim holds.To prove the second claim, let α, β, α ′ , β ′ be any exci-tations with φ α ′ = φ α and φ β ′ = φ β . Let γ be one ofthe fusion channels of α, β and let γ ′ be one of the fu-sion channels of α ′ , β ′ . Our task is to show that, givenany state in V γαβ , we can construct at least one state in V γ ′ α ′ β ′ by fusing charges onto α and β . To show this,we note that γ ′ = γ × q for some charge q , by the firstclaim, proven above. Let us consider the fusion product( α ′ × β ′ ) × (¯ α × ¯ β ) × ¯ q . The vacuum fusion channel mustappear at least once in this fusion product since( α ′ × β ′ ) × (¯ α × ¯ β ) × ¯ q = ( γ ′ + . . . ) × (¯ γ + . . . ) × ¯ q (D5)and γ ′ × ¯ γ × ¯ q contains the vacuum. We can now reorderthe fusion product as( α ′ × ¯ α ) × ( β ′ × ¯ β ) × ¯ q (D6)This reordering shouldn’t change the result so the vac-uum must also appear in this product. We conclude q canbe written as a product q = q × q where q appears inthe fusion product of α ′ × ¯ α and q belongs to the fusionproduct of β ′ × ¯ β . Clearly q and q satisfy q × α = α ′ and q × β = β ′ . Also, we know that q × q × γ = q × γ = γ ′ .This proves the claim. Appendix E: A property of R βαα In this Appendix, we show that when an excitationis braided around another identical excitation in a 2Dgauge theory with group G = Q Ki =1 Z N i , the resultingunitary transformation is a pure phase, i.e., proportionalto the identity matrix. Note that this result only holdsfor a full braiding: the unitary transformation associatedwith an exchange of two identical excitations need not bea pure phase.Consider an arbitrary excitation α in a 2D gauge the-ory with group G = Q Ki =1 Z N i . In the first step, we showthat the unitary transformation associated with braiding α around its antiparticle ¯ α is a pure phase. To see this,note that according to Appendix D, the only fusion out-comes for α and ¯ α are charges: α × ¯ α = ∅ + q + . . . . Forany q that appears in this fusion rule, we can use theformula (9) to derive R q ¯ αα R qα ¯ α = e i π ( s q − s α − s ¯ α ) = e − i πs α , (E1)where we have used the facts dim( V qα ¯ α ) = 1, s q = 0 and s α = s ¯ α . Examining the above identity, we can see thatthe statistical phase associated with braiding α around ¯ α α βγ = u − α = X q, ˜ q u − α α βγ ¯ α ¯ αα = X q, ˜ q u − α e iq · φ γ − i ˜ q · φ β α βγ ¯ α ¯ ααq ˜ q = C α,φ β ,φ γ α βγ ¯ αα α ¯ αα ¯ α α ¯ α αq ˜ q α βγ FIG. 8: Diagrammatic proof that the unitary matrix associated with the braiding process in the definition of Θ ijk is an Abelianphase. is independent of the fusion channel q . Hence, braiding α around ¯ α gives a pure phase.Next, we show that all the charges q that appear in thefusion product α × ¯ α have vanishing braiding statisticswith α , i.e., θ αq = 0. To see this, imagine we have twoexcitations α and ¯ α in the vacuum fusion channel. If wenow fuse q to ¯ α , the excitation ¯ α will remain unchanged(i.e. q × ¯ α = ¯ α ) but after this fusion process, the twoexcitations α and ¯ α will be in the fusion channel q . Letus imagine braiding α around ¯ α before and after fusing q into ¯ α . Clearly the two processes will differ by a phasefactor e iθ αq . Hence R q ¯ αα R qα ¯ α = R ∅ ¯ αα R ∅ α ¯ α e iθ αq . (E2)Then since R q ¯ αα R qα ¯ α is independent of q , we derive θ αq =0. To complete the argument, we consider a process inwhich α is braided around a pair of α and ¯ α excitations.Independent of the fusion channel of the α and ¯ α exci-tations, the unitary transformation associated with thisprocess must be the identity since θ αq = 0. At the sametime, this braiding process can be divided into two pieces:first α is braided around ¯ α , and then around another α .Since the first piece is a pure phase e − i πs α , the secondmust also be a pure phase. This proves the claim. Inaddition, we have derived the formula R βαα R βαα = e i πs α (E3)where β is any excitation in the fusion product α × α . Appendix F: Proving Θ ijk is well defined In this Appendix, we prove that Θ ijk is a well definedquantities. More specifically, we show that (i) the uni-tary transformation associated with the braiding processdefining Θ ijk is always an Abelian phase even if the vor-tices are non-Abelian; we also show that (ii) the Abelianphase is a function of i, j, k only and does not depend onthe choice of vortices α, β, γ as long as they carry fluxes πN i e i , πN j e j , πN k e k respectively.To prove points (i) and (ii), we make use of a diagram-matic technique to compute the unitary matrix associ-ated with the braiding process in the definition of Θ ijk (for more details about this diagrammatic technique, seeRef. 38.) The technique uses space-time trajectories,where the arrow of time is drawn upward. We will notuse the technique to carry out an actual calculation butonly to show that the unitary transformation associatedwith Θ ijk is a pure phase. We will make use of two dia-grammatic relations in our proof. The first relation is (a) = u α αα ¯ α αα (F1)This relation allows us to turn downward a trajectoryof α by introducing its antiparticle ¯ α , with a compensa-tion of a complex factor u α . The factor u α is related tothe quantum dimension d α by d α = | u α | − . The secondrelation is (b) = X γ,n α βα β α βα βγ nn (F2)which means that the propagation of two particles α, β can be decomposed into a sum over their possible fusionchannels, where γ ranges over the fusion channels in α × β = P γ N γαβ γ and n ranges over 1 , . . . , N γαβ . The vertices γα βn γα βn mean the n th way of splitting and fusing α, β into γ re-spectively.Now let α, β, γ be vortices carrying unit flux πN i e i , πN j e j , πN k e k respectively. Consider the space-timetrajectories of α, β, γ associated with the braiding pro-cess in the definition of Θ ijk (Fig. 8). Making using ofthe relation (F1), we establish the first equation in Fig. 8.The second equation is established by using the relation6(F2) in the two shaded regions in the second diagram inFig. 8. The charges q, ˜ q are those appearing in the fusionrule of α × ¯ α . We have used the fact that N qα ¯ α = 1 forany q , which is proven in Appendix D. To establish thethird equation, we notice that q, ˜ q have Abelian statis-tics with the vortices β, γ . Winding q around γ in thecounterclockwise direction gives rise to a phase e iq · φ γ ,and winding ˜ q around β in the clockwise direction givesrise to a phase e − i ˜ q · φ β . In the fourth diagram in Fig. 8,we see that β and γ are decoupled from α , while α stillhas some “self-interaction”. However complicated this“self-interaction” is, it only depends on α, q, ˜ q , but noton β, γ . Therefore, we can denote everything as a com-plex number C α,φ β ,φ γ , after the summation over q, ˜ q hasbeen performed. We see that C α,φ β ,φ γ only depends onthe flux of β, γ , but not on the choice of β, γ , nor on theirfusion channel. Since the overall transformation must beunitary, C α,φ β ,φ γ is a pure phase.To complete the argument, we have to show that thefactor C α,φ β ,φ γ = e i Θ ijk only depends on the flux of α but not on the choice of α . This can be proven by fusingcharges to α . Let the outcome of the fusion be a vortex α ′ . According to the Aharanov-Bohm law, one can easilysee that C α ′ ,φ β ,φ γ = C α,φ β ,φ γ e iq · φ β + iq · φ γ − iq · φ β − iq · φ γ = C α,φ β ,φ γ . (F3)With this, we have shown that Θ ijk only depends on theflux of α, β, γ , i.e. it only depends on i, j, k . Hence, Θ ijk is well defined. Appendix G: Correspondence between labels ( a, ρ ) and physical notions of gauge flux and gauge charge In this appendix, we show how to translate betweenthe mathematical labels α = ( a, ρ ), used to denote exci-tations in 2D Dijkgraaf-Witten models, and the physicalnotions of gauge flux and gauge charge. As discussed inthe main text, the basic outline of correspondence is sim-ple: the first component a describes the amount of fluxcarried by the excitation α , while the second component ρ is related to the amount of charge attached to α . Wenow explain how this works in more detail.For each group element a = ( a , ..., a K ), the corre-sponding gauge flux is given by φ = ( φ , ..., φ K ) where φ i = πN i a i . Likewise, for each representation ρ we shoulddefine a corresponding gauge charge q = ( q , ..., q K ).This correspondence is easy to define for the case where α is a pure charge excitation: α = (0 , ρ ). Indeed in thiscase, Eq. (22) implies that ρ is a linear representationof G — provided that we choose a “gauge” such that ω ( a, b, c ) = 1 if any of a, b, c is 0. It follows that ρ canbe written in the form ρ ( h ) = exp( P k πiN k q k h k ) for someinteger vector ( q , ..., q K ). This defines the desired corre-spondence ρ ↔ q = ( q , ..., q K ). How does the correspondence work for vortex excita-tions α = ( a, ρ ) where a = 0? In this case, ρ is a projec-tive representation, so there is no natural way to trans-late ρ into an integer vector ( q , ..., q K ). This is relatedto the general point made in Section III A 1: we do notknow a physically meaningful way to define the abso-lute charge carried by a vortex excitation. On the otherhand, if we compare two vortex excitations with the sameflux, α = ( a, ρ ) and α ′ = ( a, ρ ′ ), and we find that ρ, ρ ′ are related by ρ ′ ( h ) = ρ ( h ) · exp( P k πiN k q k h k ) for some( q , ..., q k ), then we can say that α ′ can be obtained from α by attaching charge q = ( q , ..., q K ). Appendix H: Showing that all solutions to theconstraints (63) can be written in the form (49a),(49b) In this section, we show that if Θ i,l and Θ ij,l obey theconstraints (63a)-(63g), then they can be written in theform (49a)-(49b), i.e.,Θ i,l = 2 πN il ( M ili − M lii ) (H1)Θ ij,l = 2 πN ij N il N j ( M ilj − M lij ) + 2 πN ij N jl N i ( M jli − M lji )for some integer tensor M ijk . The first step is rewrite thesecond equation asΘ ij,l = 2 πN ijl N jl N ijl ( M ilj − M lij ) + 2 πN ijl N il N ijl ( M jli − M lji )(H2)This new form of the second equation can be derived fromthe relations N ij N il N j = N ijl N jl N ijl , N ij N jl N i = N ijl N il N ijl (H3)which in turn follow from the identities N ij = N i N j N ij , N ijl = N ij N il N jl N ijl N i N j N l (H4)We now construct the integer tensor M ijk . First, weset M iii = 0 for all i , and we set M lii = M iil = 0 , M ili = N il π Θ i,l , (H5)for all i = l . Next, for each i < j < l , we define M ilj = − b N ijl π Θ ijl ,M jil = N ijl π ( a Θ jli + b Θ lij ) ,M lji = − a N ijl π Θ ijl ,M lij = 0 , M jli = 0 , M ijl = 0 (H6)7where a and b are integers such that a N ijl N il − b N ijl N jl = 1 (H7)(The existence of a, b will be established below). If onesubstitutes the tensor M ijk into (H1, H2), it is straight-forward to check that the resulting expressions exactlyreproduce the invariants Θ i,l and Θ ij,l as long as theseinvariants obey the constraints (63).At this point, we have successfully constructed a tensor M ijk that satisfies equations (H1, H2). However, thereare two gaps in our derivation that need to be addressed.First, we have to show that the components of M ijk areall integers. Second, we have to show that we can alwaysfind integers a, b satisfying (H7). The fact that the com-ponents of M ijk are all integers is easy to prove, as it fol-lows immediately from the two constraints (63d), (63c).As for the second statement, this will follow if we canshow that N ijl /N il and N ijl /N jl are relatively prime.The latter property can be derived from a simple obser-vation: we note that the only prime factors appearingin N ijl /N il are those that divide into j more times thaneither i or l . Similarly, the only prime factors appearingin N ijl /N jl are those that divide into i more times than j or l . We conclude that these two numbers do not shareany prime factors so they are relatively prime. Appendix I: Counting the number of values of Θ i,l , Θ ij,l , Θ ijk,l In this section, we consider the formulas (49a)-(49c),reprinted below for convenience:Θ i,l = 2 πN il ( M ili − M lii ) (I1a)Θ ij,l = 2 πN ijl N jl N ijl ( M ilj − M lij ) + 2 πN ijl N il N ijl ( M jli − M lji )(I1b)Θ ijk,l = 2 πN ijkl X ˆ p sgn(ˆ p ) L ˆ p ( i )ˆ p ( j )ˆ p ( k )ˆ p ( l ) (I1c)What we will show is that invariants Θ i,l , Θ ij,l , Θ ijk,l takeon at least Y i Cohomology of groups , (Springer, 1982). M. de Wild Propitius, PhD dissertation (1995), availableat arXiv: hep-th/9511195. A. Kitaev, Ann. of Phys. , 2(2006). Besides simple loops such as circles or linked circles, loopscan be knotted or even branched. In this paper, we onlyconsider simple loops without knotting or branching. C. Aneziris, A. P. Balachandran, L. Kauffman, and A. M.Srivastava, Int. J. Mod. Phys. A , 2519 (1991). M. G. Alford, K.-M. Lee, J. March-Russell, and J. Preskill,Nucl. Phys. B , 251 (1992). J. C. Baez, D. K. Wise, and A. S. Crans, Adv. Theor.Math. Phys. , 707 (2007). C.-M. Jian and X.-L. Qi, Phys. Rev. X , 041043 (2014). Z. Bi, Y.-Z. You, and C. Xu, Phys. Rev. B , 081110(2014). Note that this correspondence between loop and particlebraiding only holds if the loops are linked to a base loop γ ; the braid group for unlinked loops is different from the2D case, as discussed in Ref. 43. R. Dijkgraaf, V. Pasquier and P. Roche, Nucl. Phys. B , 60 (1990). X. Chen, Y.-M. Lu, and A. Vishwanath, Nat. Commun. ,3507 (2014). It is not hard to see that for any cohomology class in H ( G, U (1), we can always find a representative cocycle ω with the property that ω ( a, b, c ) = 1 if any of a, b, c is 0. Z.-C. Gu and M. Levin, Phys. Rev. B , 201113 (2014). It can be shown that the Dijkgraaf-Witten model corre-sponding to the cocycle ω (48) is Abelian if L ijkl = 0. Y. Hu, Y. Wan, and Y.-S. Wu, Phys. Rev. B , 125114(2013). Y. Wan, J. Wang, and H. He, arXiv:1409.3216.54