a\times b=c in 2+1D TQFT
aa r X i v : . [ h e p - t h ] F e b QMUL-PH-20-37 a × b = c in 2+1D TQFT Matthew Buican ♦ , Linfeng Li ♣ , and Rajath Radhakrishnan ♥ CRST and School of Physics and AstronomyQueen Mary University of London, London E1 4NS, UK
We study the implications of the anyon fusion equation a × b = c on global properties of2 + 1D topological quantum field theories (TQFTs). Here a and b are anyons that fusetogether to give a unique anyon, c . As is well known, when at least one of a and b isabelian, such equations describe aspects of the one-form symmetry of the theory. When a and b are non-abelian, the most obvious way such fusions arise is when a TQFT can beresolved into a product of TQFTs with trivial mutual braiding, and a and b lie in separatefactors. More generally, we argue that the appearance of such fusions for non-abelian a and b can also be an indication of zero-form symmetries in a TQFT, of what we term“quasi-zero-form symmetries” (as in the case of discrete gauge theories based on the largestMathieu group, M ), or of the existence of non-modular fusion subcategories. We studythese ideas in a variety of TQFT settings from (twisted and untwisted) discrete gaugetheories to Chern-Simons theories based on continuous gauge groups and related cosets.Along the way, we prove various useful theorems.December 2020 ♦ [email protected], ♣ [email protected], ♥ [email protected] ontents1. Introduction 12. Discrete gauge theories and a × b = c : from groups to TQFT 8 a × b = c . . . . . . . . . . . . . . 122.3. Subgroups, subcategories, and primality . . . . . . . . . . . . . 242.4. Zero-form symmetries . . . . . . . . . . . . . . . . . . . . 282.5. Quasi-zero-form symmetries . . . . . . . . . . . . . . . . . . 302.6. Beyond Wilson lines . . . . . . . . . . . . . . . . . . . . . 33 G k , cosets, and a × b = c G k CS theory . . . . . . . . . . . . . . . . . . . . . . . 433.2. Virasoro minimal models and some cosets without fixed points . . . . . 453.3. Beyond Virasoro: cosets with fixed points . . . . . . . . . . . . . 49
4. Conclusions 52Appendix A. Wilson line a × b = c in gauge theories with order forty-eightdiscrete gauge group 54Appendix B. Genuine zero-form symmetries and quasi-zero-form symme-tries in A discrete gauge theory 55Appendix C. GAP code 561. Introduction Topological quantum field theories (TQFTs) in 2 + 1 dimensions and their anyonic excita-tions lie at the heart of important physical [1], mathematical [2], and computational [3]systems and constructions. In principle, these TQFTs can be fully characterized by solvinga set of polynomial consistency conditions [4], although proceeding in this way is oftenquite difficult as a practical matter (however, see [5, 6] for examples of some results; seealso [7] for a potentially very different approach). More generally, it is interesting to un-derstand aspects of the global structure of a TQFT and its symmetries without the needto fully solve the theory (e.g., see [8]). 1roceeding in this way, we will study anyonic fusions a × b that have a unique productanyon, c a × b = c , a, b, c ∈ T , (1.1)in a general 2 + 1 dimensional TQFT, T . Fusion rules of this type contain crucial infor-mation about the anyons a , b , and c . As a simple example, we have the relation d c = d a d b .Here d x is the so-called “quantum dimension” of the anyon x and is related to the ampli-tude for creating x and its conjugate, ¯ x , from the vacuum and re-annihilating them. Givena fusion rule of the type (1.1), a little more work shows that in a unitary TQFT d c ∈ Z if and only if d a , d b ∈ Z . This apparently simple statement has practical implications foruniversal topological quantum computation (UTQC). Indeed, anyons with d x ∈ Z are notuseful for UTQC via braiding. On the other hand, if d x Z , x can be used for UTQCby braiding alone [10]. Therefore, fusion rules of the form (1.1) give constraints on howsuitable a TQFT is for quantum computing.This simple example shows that fusion rules with unique outcomes give us importantinformation about the TQFT. Our main question is then: what does (1.1) tell us aboutthe global structure of T and its symmetries?For invertible a and b (i.e., a and b are abelian anyons), fusion rules of the form (1.1)describe the abelian 1-form symmetry group of the theory [11] (the closely related modular S matrix characterizes its ’t Hooft anomalies [12]). In the case in which, say, a is abelianand b is non-abelian, the equation (1.1) gives the fixed points of the fusion of anyons inthe theory with the one-form generator, a . Such equations have important consequencesfor anyon condensation / one-form symmetry gauging in TQFT [12, 13] as well as fororbifolding and coset constructions in closely related 2D rational conformal field theories(RCFTs) (e.g., see [14, 15]).Although these cases will play a role below, we will be more interested in the situation Throughout what follows, we only consider non-spin TQFTs. These are theories that do not require aspin structure in order to be well-defined. To understand this statement, consider a unitary TQFT with a fusion rule a × b = c . We then have c × c = ( a × a ) × ( b × b ). The trivial anyon 1 belongs to the fusion b × b . Therefore, c × c = a × a + · · · ,where the ellipses include other anyons. Let Y , X , and X be non-negative integral combinations of anyonsin the theory satisfying Y = X + X . If d Y is an integer, then d X and d X are guaranteed to be integers [9].If d c ∈ Z , then d c × c = d c is an integer. Therefore, c × c = a × a + . . . implies that d a × a = d a ∈ Z . Hence, d b ∈ Z . On the other hand, if d c Z , it trivially follows that d a Z or d b Z . In this case, b is non-invertible, and the fusion b × ¯ b = 1 + · · · necessarily contains at least one moreanyon in the ellipses.
2n which both a and b are non-abelian a × b = c , d a , d b > . (1.2)Here d a,b denote the quantum dimensions of a and b (given they are larger than one, neither a nor b are invertible). Since both a and b are non-abelian, one typically finds that theright-hand side of (1.2) has multiple fusion products. For example, fusions as in (1.2) donot occur in SU (2) k Chern-Simons (CS) theory for any value of k ∈ N . As we will see,when fusions of non-abelian a and b do have a unique outcome, there are consequences forthe global structure of T .The most trivial case in which a fusion of the type (1.2) occurs is when T factorizes(not necessarily uniquely) as T = T ⊠ T , (1.3)with T and T two separate TQFTs that have trivial mutual braiding, a ∈ T , and b ∈T . Here “ ⊠ ” denotes a categorical generalization of the direct product called a “Deligneproduct” that respects some of the additional structure present in TQFT.As we will discuss in section 3, precisely such a situation arises in the modular tensorcategories (MTCs) related to unitary A -type Virasoro minimal models with c > / MTCs are mathematical descriptions of TQFTs, and, for the theories in question, theyencapsulate the topological properties of the Virasoro primary fields. One may think ofthe, say, left-movers in these RCFTs as arising at a 1+1 dimensional interface between2+1 dimensional CS theories with gauge groups SU (2) × SU (2) k and SU (2) k +1 . In theminimal models, we have ϕ ( r, × ϕ (1 ,s ) = ϕ ( r,s ) , (1.4)where 2 ≤ r < p − ≤ s < p − r, s ) ∼ ( p − − r, p − s ), and p > Thinking in terms of cosets, we will see that (1.4)arises because the Virasoro MTC factorizes as in (1.3). In section 3, we will discuss the situation for more general G k CS theories. Note that T , may factorize further. Moreover, a may contain an abelian component in T , and b maycontain an abelian component in T . Note that in the case of the Ising model ( c = 1 / a × b = c isabelian (and the corresponding MTC does not factorize). We thank I. Runkel for drawing our attention tothe a × b = c fusion rules for non-abelian fields in Virasoro minimal models. The abelian field ϕ ( p − , ∼ ϕ (1 ,p − satisfies the fusion rule ϕ (1 ,p − × ϕ (1 ,p − = ϕ (1 , = 1. Note that this factorization does not extend to one of the RCFT. a bb a ba bc Fig. 1:
The fusions a × b and ¯ a × ¯ b have unique outcomes c and ¯ c respectively. In theleft diagram, we connect the corresponding fusion vertices. To get to the diagram on theright, we perform an F ¯ aab ¯ b transformation. Just as the left diagram has a unique internalline, so too does the diagram on the right (in this latter case, the internal line must be theidentity).To gain further insight into more general situations in which (1.2) occurs, it is usefulto imagine connecting a fusion vertex involving the a , b , c ayons with a fusion vertexinvolving the ¯ a , ¯ b , and ¯ c anyons via a c internal line as in the left diagram of figure 1.Using associativity of fusion (via a so-called F ¯ aab ¯ b symbol) we arrive at the right diagram offigure 1. The relation between these two diagrams can be thought of as a change of basison the space of internal states. Since, by construction, the left diagram in figure 1 canonly involve a c internal line, the right diagram in figure 1 can also only involve a singleinternal line. On general grounds, this line must be the identity. Therefore, we learn thata fusion rule of the form (1.2) is equivalent to the following a × ¯ a = 1 + X a i =1 N a i a ¯ a a i , b × ¯ b = 1 + X b j =1 N b j b ¯ b b j , b j ∈ b × ¯ b ⇒ b j a × ¯ a ,a i ∈ a × ¯ a ⇒ a i b × ¯ b ∀ i, j . (1.5)In other words, the fusion of a × b has a unique outcome if and only if the only fusionproduct that a × ¯ a and b × ¯ b have in common is the identity.Reformulating the problem as in (1.5) immediately suggests scenarios in which fusionsof the form (1.2) occur beyond beyond cases in which T factorizes into prime TQFTs. Forexample, if a ∈ C ⊂ T and b ∈ C ⊂ T lie in non-modular fusion subcategories of T , C , , with trivial intersection (i.e., C ∩ C = 1 only contains the trivial anyon), then wehave (1.2) and T need not factorize. More generally, when a ∈ C ⊂ T is a member of By rotating the ¯ a , ¯ b , and ¯ c vertex, we see that a × b = c is equivalent to requiring a × ¯ b = d and ¯ a × b = ¯ d (see figure 2). This logic also explains why, for non-abelian a , it is impossible to have a × a = c even if a = ¯ a . In other words, fusion of anyons in C i is closed. Moreover, the C i inherit associativity and braiding from b bada bb ac Fig. 2:
By rotating the bottom vertex in the left diagram of figure 1, we arrive at theabove diagram on the left. Again, we have a single internal line labeled by c . We get tothe diagram on the right by performing an F ¯ bab ¯ a transformation. Just as the left diagramhas a unique internal line, so too does the diagram on the right.a non-modular subcategory that does not include b (i.e., b
6∈ C ), we expect it to be morelikely to find fusions of the form (1.5) and (1.2) since a × ¯ a ∈ C , but b × ¯ b will generallyinclude elements outside C . In fact, we will see that we can often say more when the fusionof a non-abelian Wilson line carrying charge in an unfaithful representation of a discretegauge group is involved.Another scenario in which we can imagine (1.5)—and therefore (1.2)—arising is onein which zero-form symmetries act non-trivially on a (i.e., g ( a ) = a for some zero-formgenerator g ∈ G , where G is the zero-form group) and the a i = 1 but not on b . In thiscase, combinations of a i that do not form full orbits under G are forbidden from appearingin b × ¯ b . Given a particular G , this argument may suffice to show that, for all i , a i b × ¯ b .More generally, symmetries constrain what can appear as fusion products of a × ¯ a and b × ¯ b .The more powerful these symmetries, the more likely to find fusion rules of the type (1.5).Interestingly, there is a close connection between the existence of symmetries and theexistence of subcategories in TQFT. For example, as we will discuss further in section 2.4,for TQFTs corresponding to discrete gauge theories [16, 17], certain “quantum symmetries”or electric-magnetic self-dualities arise when we have particular non-modular subcategories C i ⊂ T (see [18] for a general theory of such symmetries and [19] for the case of S discretegauge theory). T , but the Hopf link evaluated on anyons in these subcategories is degenerate (as a matrix). By modularity,the C i will have non-trivial braiding with some anyons x A
6∈ C , (where A is an index running over suchanyons). On the other hand, if the Hopf links for the C i are non-degenerate, M¨uger’s theorem [8] guaranteesthat they will in fact be separate TQFTs and so we are back in the situation of (1.3). By definition, the symmetry also acts non-trivially on ¯ a so that g ( a ) = g (¯ a ) = ¯ a . On the other hand,note that one-form symmetry will act trivially on the product a × ¯ a .
5e will also find various other, more subtle, connections between symmetries and fusionrules of the form (1.5) and (1.2). Moreover, we will see that symmetry is ubiquitous: inall the theories with fusion rules of the form (1.5) and (1.2) we analyze, either there is azero-form symmetry present or else there is, at the very least, a symmetry of the modulardata that exchanges anyons (in cases where this action does not lift to the full TQFT, wecall these symmetries “quasi zero-form symmetries”).We will study fusions of the above type in two typically very different classes of 2 + 1DTQFTs: discrete gauge theories and cosets built out of CS theories with continuous gaugegroups (we will refer to these latter theories simply as “cosets”). Discrete gauge theories arealways non-chiral, whereas Chern-Simons theories and their associated cosets are typicallychiral. In the context of discrete gauge theories, whenever we have a (full) zero-form symmetrypresent, we will see that fusion rules of the type (1.5) and (1.2) have simple interpretationsin certain parent theories gotten by gauging the zero-form symmetry, G . We go from theparent theories back to the original theories by gauging a “dual” one-form symmetry, G ,that is isomorphic (as a group) to G (see [20] for a more general review of this procedure).In this reverse process, we produce the a × b = c fusion rules of the corresponding discretegauge theories via certain fusion fixed points of the one-form symmetry generators in theparent theories.Similarly, in the context of our coset theories, we will see that fusion rules of the form a × b = c arise due to certain fixed points in the coset construction (though these fixedpoints do not generally involve a , b , and c ). Cosets corresponding to the Virasoro minimalmodels lack such fixed points and so, as discussed above, they factorize. On the otherhand, more complicated cosets do sometimes have such fixed points, and we will constructan explicit example of such a prime TQFT that has fusion rules of the form (1.5) and(1.2).To summarize, this discussion leads us to the following questions we will answer insubsequent sections:1. Does (1.2) imply a factorization of TQFTs T = T ⊠ T , (1.6)with a ∈ T and b ∈ T ? As has been hinted at above, we will see in sections 2 and3 that the answer is generally no. Note that there are sometimes dualities between theories in these two classes. By a chiral TQFT, we mean one in which the topological central charge satisfies c top = 0 (mod 8).
6. Does (1.2) imply that a belongs to one fusion subcategory and b to another and thatthe intersection of these subcategories is trivial? In other words, do we have a ∈ C ⊂ T , b ∈ C ⊂ T , C ∩ C = 1 ? (1.7)As we will see in section 2, the answer is generally no, even if we relax the requirementof trivial intersection. However, we will explicitly construct such examples (with non-modular C , ⊂ T , where T is prime) in the case of discrete gauge theories.3. Does (1.2) imply that a is in some subcategory C ⊂ T that b is not a member of?In other words, do we have a ∈ C ⊂ T , b / ∈ C ? (1.8)As we will see in section 2, the answer is generally no. However, we will argue thatsuch constructions are quite easy to engineer in the context of discrete gauge theories,and we will explain when they arise. We will see that these constructions often haveinteresting interactions with symmetries.4. Given a and b as in (1.2), do they have trivial mutual braiding? In other words, dowe have S ab S b = d a , (1.9)where S is the modular S -matrix? This is true in the context of discrete gaugetheories with a simple gauge group [21]. However, non-trivial braiding does arisenaturally in the context of the fusion of non-abelian electrically charged lines withnon-abelian magnetically charged lines.5. Given a and b as in (1.2), does T have a non-trivial zero-form symmetry acting oneither a or b ? Does the TQFT have a zero-form symmetry that acts more generally?We will see in section 2 the answer to both these questions is no. However, in casesin which this is true, it seems to always be related to the existence of a certain fusionfixed point of one-form symmetry generators in a parent TQFT. Of the infinitelymany examples of untwisted discrete gauge theories we study, only gauge theoriesbased on the Mathieu groups M and M fail to have zero-form symmetries.6. Given a and b as in (1.2), does T have a non-trivial symmetry of the modular data?As we will see in sections 2 and 3, the answer seems to be yes. Clearly, it would beinteresting to see if it is possible to define parents of such theories that generalize the7elationship in (5). Note that the Mathieu gauge theories discussed in the previouspoint do have symmetries of their modular data (however, these symmetries do notlift to symmetries of the full TQFTs).As we will see, many of these questions have simpler answers when studying discretegauge theories. The reason is that powerful statements in these TQFTs can often bededuced from simple reasoning in the underlying theory of discrete groups. On the otherhand, intuition one gains from taking products of representations in various continuousgroups, like SU ( N ), turns out to be somewhat misleading for our questions above.The plan of this paper is as follows. In the next section, we start with discrete gaugetheories and explain how intuition in the theory of finite groups leads us to various answersto the above questions. Along the way, we prove various theorems about discrete gaugetheories and fusion rules of the form (1.2) and (1.5) generalizing our work in [21]. Moreover,we discuss the role that subcategories and symmetries of discrete gauge theories play in suchfusion rules. In the final part of the paper, we go to continuous groups and discuss cosettheories. We tie the existence of fusion rules of type (1.2) and (1.5) to certain fixed pointsin the coset construction. We then finish with some conclusions and future directions.
2. Discrete gauge theories and a × b = c : from groups to TQFT In this section we test the ideas presented in the introduction on discrete gauge theories[16, 17]. These TQFTs are characterized by a choice of discrete gauge group, G , and aDijkgraaf-Witten twist, ω ∈ H ( G, U (1)). The basic degrees of freedom are anyonic lineoperators of the following three types1.
Wilson lines , W π , carrying electric charge labeled by a linear representation, π , of G and trivial magnetic charge. This set of operators exists no matter the value of ω .2. Magnetic flux lines , µ [ g ] , carrying magnetic charge labeled by a conjugacy class, [ g ], ofa representative element, g ∈ G , but having trivial electric charge. In general, theirexistence depends on the choice of ω .3. Dyonic lines , L ([ g ] ,π ωg ) , carrying both magnetic flux and electric charge. In general,they carry a projective representation of G . There are redundancies / dualities in this description: see [22]. As we will see in the subsequent subsections, the physics of the various operators listedabove is qualitatively different. In order to take the shortest route to answering some ofthe questions posed in the introduction and in order to establish the existence of fusionrules of the form (1.2) in prime TQFTs, we will start with an analysis of Wilson lines.These objects form a closed fusion subcategory that is particular easy to analyze. Aswe explain, these are the most “group theoretical” and least anyonic objects in a discretegauge theory (in addition, as we see from the discussion of the above list of operators, theyare the most robust). As a result, we can borrow various useful results from the study offinite groups.In order to study the physics of other sectors of discrete gauge theories, we will find itconvenient to introduce some additional machinery for discussing subcategories (in section2.3) and symmetries (in section 2.4). We also discuss quasi zero-form symmetries and theirappearance in various discrete gauge theories of interest based on large Mathieu groups.Finally, we move beyond Wilson lines in section 2.6 and discuss fusions of the form (1.2)involving non-abelian fluxes, magnetic fluxes, and dyons.
In this section, we briefly review how to construct a discrete gauge theory given a finitegauge group, G (for a more succinct version of the review below, see [21]). Although muchof what we say is a consequence of [16], we will follow the perspective in [20]. The mainreason for this choice is that this latter perspective lends itself to generalizations to casesin which we want to gauge global symmetries of theories that already posses topologicalorder (i.e., theories that already have non-trivial anyons).To construct the G discrete gauge theory we take a set of surface defects labeled byelements g ∈ G (these are elements of a zero-form symmetry and therefore may be non-abelian). These objects comprise a G -symmetry protected topological phase ( G -SPT). Thefusion of the defects satisfies the corresponding group multiplication law, g × h = gh .To complete the definition of the fusion category, we need to choose how to implement Discrete gauge theories are, however, necessarily non-chiral. We will consider chiral coset theories insection 3. For other degrees of freedom, the story is more complicated. For example, in section 2.3, we will seethat in non-abelian discrete gauge theories, full sets of magnetic fluxes do not form fusion subcategories. ω ( g, h, k ) ∈ H ( G, U (1)).We are now ready to gauge the symmetry. To do this we pair conjugacy classes, [ g i ],with irreducible representations, π ωg i , of the corresponding centralizers, N g i (for differentchoices of the conjugacy class representatives, the centralizers are isomorphic). These twoobjects combine to give anyons in the gauged theory([ g i ] , π ωg i ) ∈ Z (Vec ωG ) , (2.1)where [ g i ] and π ωg i label the electric and magnetic charges of the discrete gauge theory,which we denote as Z (Vec ωG ). If ω is non-trivial in cohomology, we have a “twisted” discrete gauge theory with Di-jkgraaf Witten 3-cocycle ω . This twisting typically leads to the π ωg i being projective (asopposed to linear) representations of the centralizers. More precisely, to determine theprojectivity of these representations, we should compute η g ( h, k ) := ω ( g, h, k ) ω ( h, k, g ) ω ( h, g, k ) ∈ H ( N g , U (1)) , h, k ∈ N g , (2.2)as this is the phase that appears in π ωg ( h ) π ωg ( k ) = η g ( h, k ) π ωg ( hk ). If η g is non-trivial incohomology, then the representation π ωg is projective. Note that whenever g = 1, therepresentations are linear. This means that, regardless of the twisting, the sector of Wilsonlines is unchanged. More generally, even if g = 1 and ω is non-trivial, we may stillhave linear representations. As an example, we may consider G = P SL (2 ,
4) and the Z centralizer of the length twenty conjugacy class. In this case, we have H ( Z , U (1)) = Z ,so the resulting η g (with g in the length twenty conjugacy class) is cohomologically trivialno matter the choice of ω ∈ H ( P SL (2 , , U (1)) = Z × Z .The most important things for us to focus on in what follows are the fusion coefficientsappearing in ([ g ] , π ωg ) × ([ h ] , π ωh ) = X k,π ωk N ([ k ] ,π ωk )([ g ] ,π ωg ) , ([ h ] ,π ωh ) ([ k ] , π ωk ) . (2.3) Our choice of notation Z (Vec ωG ) for the discrete gauge theory reflects an alternative way to think aboutthe theory: as the Drinfeld center of the fusion category of G graded vector spaces with associator ω ∈ H ( G, C × ). The Drinfeld center of a spherical fusion category is an MTC. As a result, all of the statements that we arrive at for Wilson lines apply for both twisted and untwisteddiscrete gauge theories. More precisely, if η g is a non-trivial 2-coboundary, we will obtain projective representations that arein one-to-one correspondence with linear representations. We can remove these projective factors via asymmetry gauge transformation of the type described in [20]. Note that while linear representations canbe one-dimensional (e.g., if the centralizer is an abelian group), projective representations resulting from η g cohomologically non-trivial are necessarily higher dimensional.
10o arrive a description of such a process we must combine conjugacy classes and repre-sentations. In particular, we need to multiply elements in [ g ] and [ h ] and determine thecorresponding conjugacy classes. At the same time, we must decompose the product ofirreducible representations of the corresponding centralizers into irreducible representationsof centralizers of G . A simple prescription for doing this is given in [20] N ([ k ] ,π ωk )([ g ] ,π ωg ) , ([ h ] ,π ωh ) = X ( t,s ) ∈ N g \ G/N h m ( π ωk | N tg ∩ N sh ∩ N k , t π ωg | N tg ∩ N sh ∩ N k ⊗ s π ωh | N tg ∩ N sh ∩ N k ⊗ π ω ( t g, s h,k ) ) , (2.4)where the sum is over the double coset, we define t g := t − gt , and t π ωg | N tg ∩ N sh ∩ N k ⊗ s π ωh | N tg ∩ N sh ∩ N k ⊗ π ω ( t g, s h,k ) and π ωk | N tg ∩ N sh ∩ N k are restrictions of irreducible representations of N t g , N s h , and N k to the triple intersections of these normalizers. These restrictions are gen-erally (though crucially for us below not always) reducible representations of N t g ∩ N s h ∩ N k .The m ( a, b ) function computes inner products of the representations a and b (we will fillin further details of this function as needed later in this section). Crucially, a and b mustbe the same type of representation (i.e., they should both be linear or else transform withthe same set of projective weights) in order to be meaningfully compared.We can determine the projectivity of the t π ωg , s π ωh , and π ωk representations by a computa-tion in the relevant cohomology as in (2.2). The representation π ω ( t g, s h,k ) is one dimensional(it is a representation of the action of symmetries on the one-dimensional V k t g s h fusion spacein the G -SPT) and ensures that the arguments entering m ( a, b ) involve the same type ofrepresentations. Therefore, π ω ( t g, s h,k ) satisfies π ω ( t g, s h,k ) ( ℓ ) π ω ( t g, s h,k ) ( m ) = η k ( ℓ, m ) η t g ( ℓ, m ) η s h ( ℓ, m ) · π ω ( t g, s h,k ) ( ℓm ) . (2.5)A more basic quantity of interest to us in what follows is the modular data of thediscrete gauge theory. It is given by [23] S ([ g ] ,π ωg ) , ([ h ] ,π ωh ) = 1 | G | X k ∈ [ g ] , ℓ ∈ [ h ] ,kℓ = ℓk χ kπ ωg ( ℓ ) ∗ χ ℓπ ωh ( k ) ∗ ,θ ([ g ] ,π ωg ) = χ π ωg ( g ) χ π ωg ( e ) , (2.6)where we define χ hπ ωg ( ℓ ) as follows χ xgx − π ωg ( xhx − ) := η g ( x − , xhx − ) η g ( h, x − ) χ π ωg ( h ) . (2.7)11ere, θ is the topological spin, and S is the modular S matrix. It follows from thesedefinitions that quantum dimensions are given by d ([ g ] ,π ωg ) = S ([ g ] ,π ωg )([1] , S ([1] , , = | [ g ] | · | π ωg | , (2.8)where | [ g ] | is the size of the conjugacy class, and | π ωg | is the dimension of the representation.Non-abelian anyons have d ([ g ] ,π ω ) >
1. As a consequence, they must satisfy([ g ] , π ωg ) × ([ g − ] , ( π ωg ) ∗ ) = ([1] ,
1) + · · · , (2.9)where the ellipses necessarily contain additional terms (otherwise we would have d ([ g ] ,π ω ) =1), 1 is the trivial representation of G , and (([ g − ] , ( π ωg ) ∗ ) is the anyon conjugate to ([ g ] , π ωg ).We may write a dictionary between the non-abelian Wilson lines, flux lines, and dyonsdiscussed in the previous sections and the objects discussed in this section as follows W π ↔ ([1] , π ) , | π | > ,µ [ g ] ↔ ([ g ] , ǫg ) , | [ g ] | > , L ([ h ] ,π ωh ) ↔ ([ h ] , π ωh ) , | [ h ] | · | π ωh | > . (2.10)We have dropped the ω superscript from π in order to emphasize, as discussed above,that Wilson lines always transform under linear representations of G . We include an ǫ superscript on the trivial representation of the flux line because these objects only existwhen the relevant η g in (2.2) is trivial in cohomology. This triviality means that η g ( h, k )can be expressed in terms of a one co-chain as follows: η g ( h, k ) = ǫ g ( h ) ǫ g ( k ) ǫ g ( h · k ) . a × b = c We would like to recast the problem of constructing discrete gauge theories with fusionrules (1.2) and (1.5) in terms of the closely related problem of finding irreducible productsof irreducible finite group representations. To make this connection as direct as possible,it is useful to focus on Wilson lines of the discrete gauge theories we are studying. Indeed,by specializing (2.4) to Wilson lines, we find N (1 ,π ′′ )(1 ,π ) , (1 ,π ′ ) = m ( π ′′ , π ⊗ π ′ ) = 1 | G | X g ∈ G χ π ′′ ( g ) χ ∗ π ( g ) χ ∗ π ′ ( g ) = h χ π ′′ , χ π χ π ′ i , (2.11)where h· , ·i is the standard inner product on characters. Therefore, the Wilson lines forma closed fusion subcategory of the discrete gauge theory, C W . Moreover, the fusion rules Note that 1 ǫg is the irreducible projective representation of N g whose character is proportional to thetrivial representation of N g .
12f the Wilson lines are those of the representation semiring of the gauge group. Notethat C W is, in some sense, the “least anyonic” part of the theory: it is easy to check from(2.6) that the Wilson lines are bosonic, so θ W i = 1, and that the braiding of Wilson linesamongst themselves is trivial, so S W W = d W d W / D (here D = qP Ni =1 d i , and the sumis over all the anyons). To summarize, we see that if we can find representations of somegroup, G , satisfying χ π · χ π ′ = χ π ′′ , | π | , | π ′ | , | π ′′ | > , (2.12)where π , π ′ , and π ′′ are irreducible, then, in the corresponding G discrete gauge theory, wewill have non-abelian Wilson lines satisfying W π × W π ′ = W π ′′ . (2.13)Since, by Cayley’s theorem, every finite group is isomorphic to a subgroup of the sym-metric group, S N , (for some N ) it is natural to start our discussion with S N . In particular,to check whether π ′′ is irreducible, we want to perform the group theory analog of the F transformation discussed in the introduction (see figure 1) h χ π · χ π ′ , χ π · χ π ′ i = h χ π , χ π ′ i , (2.14)where we have used the fact that S N is ambivalent ( g and g − are in the same conjugacyclass for all g ∈ S N ) so that the characters are real. A theorem of Zisser [25] shows that χ [ N − , ∈ χ α , where [ N − ,
2] is a partition of N labeling the corresponding representationof S N , and α is any irreducible representation of dimension larger than one, | α | > S N is ambivalent, this means that χ [ N ] ∈ χ α , where χ [ N ] is the trivialrepresentation of S N . As a result, we see that the analog of (1.5) yields χ π · χ π = χ [ N ] + χ [ N − , + · · · , χ π ′ · χ π ′ = χ [ N ] + χ [ N − , + · · · ⇒ h χ π · χ π ′ , χ π · χ π ′ i > , (2.15)and so products of non-abelian representations of S N are never irreducible. Therefore, wecannot have (2.13) in S N discrete gauge theory. In fact, we have C W ≃ Rep( G ), where Rep( G ) is the category of finite dimensional representations of G over C . The Wilson lines braid non-trivially with other anyons in the theory (more formally: the Wilson linesubcategory is Lagrangian and so the M¨uger center of C W is C W itself). In fact, [24] guarantees that any such subcategory is equivalent to Rep( H ) for some group H . .2.1. Discrete gauge theories of finite simple groups Since we have A N ⊳ S N (i.e., the alternating group, A N , is a normal subgroup of S N ),it is natural to consider A N discrete gauge theories as the next possibility for realizing(2.12) [25] and hence (2.13). Moreover, since A N is simple, only pure Wilson lines can beinvolved in fusions of the form (1.2) [21], and the A N discrete gauge theories are guaranteedto be prime [26] (we will return to the question of primality in greater generality in section2.3). Therefore, finding an example of (2.13) in A N discrete gauge theories is sufficient toanswer question (1) from the introduction in the negative.To understand if going to A N is a fruitful direction, we note that there are two typesof characters that arise in going from S N to A N :(A) Characters that are restrictions of S N characters satisfying χ λ = χ [1 N ] · χ λ , where χ [1 N ] corresponds to the sign representation of S N . Let us call these “type A” characters:˜ χ λ := χ λ | A N .(B) Characters that descend from S N characters satisfying χ ρ = χ [1 N ] · χ ρ . As representa-tions of A N , they split into two representations of the same dimension, λ ± . Let uscall these “type B” characters: χ ( B ) ρ = χ ρ + + χ ρ − = χ ρ | A N .In going from S N to A N , we perform a group-theoretical version of gauging the “one-form symmetry” generated by χ [1 N ] : we identify characters related by multiplication with χ [1 N ] , and we split characters that are invariant under multiplication with χ [1 N ] . Clearly,products of type A characters cannot be irreducible since they will always contain χ ( A )[ N ] and χ ( A )[ N − , after performing the F-transformation and computing (2.15). A little more work in [25] shows that we can obtain (2.12) for A N if and only if N = k ≥ χ [ N − , · χ [ k k ] ± = ˜ χ [ k k − ,k − , . (2.16)Moreover, the Z outer automorphism of A N acts on the type B characters as g (cid:0) χ [ k k ] ± (cid:1) = χ [ k k ] ∓ , = g ∈ Out( A N ) ≃ Z . (2.17)Therefore, at the level of the non-abelian Wilson lines in the corresponding A N discretegauge theory, we learn that W [ N − , × W [ k k ] ± = W [ k k − ,k − , . (2.18) In this discussion, we have implicitly assumed that N = 4 (although, for N = 3, we should take[ N − , → [2 ,
1] to conform to usual conventions). For N = 4, we have χ ( A )[ N − , → χ ( B )[ N − , = χ [ N − , , + + χ [ N − , , − . A N ) lifts to a full zero-form symmetry of the discrete gauge theory [18], since,according to corollary 7.8 of [18]Aut br ( Z (Vec A N )) ≃ H ( A N , U (1)) ⋊ Out( A N ) ≃ Z × Z , (2.19)where the group on the left hand side is the group of braided tensor auto-equivalences ofthe MTC underlying the discrete gauge theory, Z (Vec A N ). As a result, we learn that thesymmetries of the discrete gauge theory exchange the W [ k k ] ± lines g (cid:0) W [ k k ] ± (cid:1) = W [ k k ] ∓ , = g ∈ Out( A N ) ⊳ Aut br ( Z (Vec A N ) . (2.20)In other words, we have found that, in an infinite number of prime theories, fusion rulesof the type (1.2) are generated in pairs related by symmetries of the discrete gauge theory.This discussion shows that TQFTs with fusions of the form (1.2) need not factorize andso the answer to question (1) in the introduction is “no.”Let us now drive home the importance of symmetries in arriving at (2.18) and, at thesame time, gain insight that will be useful later. To that end, let us consider gauging the Z outer automorphism symmetry of the A N discrete gauge theory. Note that this gaugingis allowed since the “defectification” obstruction described physically in [20] is trivial here: H ( Z , U (1)) = Z . Moreover, since A N is simple, the discrete gauge theory has no non-trivial abelian anyons (i.e., A = W [ N ] ) and so H ( Z , A ) = Z . Therefore, (2.19) is agenuine zero-form symmetry group (as opposed to being a 2-group).More abstractly, let us consider a generalization of the fusion rules in (2.4) to the caseof gauging a zero form group, H , of a more general G-crossed braided theory, T G × (asworked out in [20]) N ([ c ] ,π c )([ a ] ,π a ) , ([ b ] ,π b ) = X ( t,s ) ∈ N a \ H/N b m ( π c | N ta ∩ N sb ∩ N c , t π a | N ta ∩ N sb ∩ N c ⊗ s π b | N ta ∩ N sb ∩ N c ⊗ π ω ( t a, s b,c ) ) , (2.21)where a, b, c ∈ T G × , [ a ] := { h ( a ) , ∀ h ∈ H } , N a := { h ∈ H | h ( a ) = a } , and π a is a represen-tation of N a .In our case at hand, T G × = Z (Vec A N ) H × is the A N discrete gauge theory extendedby surface defects implementing the H = Z global symmetry. Moreover, a = W [ N − , , b = W [ k k ] ± , N a = Z , and N b = Z . As a result, t = s = 1, the summation in (2.21) istrivial, the various representations are all restricted to the trivial subgroup, and π ωa,b,c = 1(this latter statement follows from the fact that the action of H on the V cab fusion spacevia U ( a, b, c ) is trivial). In particular, we have N ([ W [ kk − ,k − , ] , ± )([ W [ N − , ] , ± ) , ([ W [ kk ] ± ] , +) = m ( ±| Z , ±| Z ⊗ + | Z ) = m (1 ,
1) = 1 , (2.22)15here ± denote the two representations of Z . Therefore, we learn that when we gaugethe outer automorphism group of A N , we have([ W [ N − , ] , ± ) × ([ W [ k k ] ± ] , +) = ([ W [ k k − ,k − , ] , +) + ([ W [ k k − ,k − , ] , − ) , (2.23)which is the TQFT version of the lift of (2.16) to S N . This is what we expect, since wecan always fix our choice of parameters so that gauging Z yields [27] Z (Vec A N ) Z × gauge −→ Z (Vec A N ⋊Z ) = Z (Vec S N ) , (2.24)where we have used the fact that S N ≃ A N ⋊ Z .Finally, from the general rules above, it is not hard to check that the trivial Wilson linein the A N theory lifts to a Z one-form symmetry in the S N gauge theory. The resultingnon-trivial one-form symmetry generator acts as([ W [ N ] ] , − ) × ([ W [ N − , ] , ± ) = ([ W [ N − , ] , ∓ ) , ([ W [ N ] ] , − ) × ([ W [ k k ] ± ] , +) = ([ W [ k k ] ± ] , +) , ([ W [ N ] ] , − ) × ([ W [ k k − ,k − , ] , ± ) = ([ W [ k k − ,k − , ] , ∓ ) , (2.25)where ([ W [ N ] ] , − ) = W [1 N ] .To summarize, we learn that, in order to generate the fusion rule (2.18), we can gaugea Z one-form symmetry in the S N (with N = k ≥
9) discrete gauge theory with fusionrules (2.23) and (2.25). Crucially, we need a fixed point of the one-form symmetry (as inthe second line in (2.25)) in order to generate the fusion rule of the form (2.18) in the A N discrete gauge theory. We will return to the existence of fixed points of various kindsrepeatedly throughout this paper.One may wonder if zero-form gaugings always resolve fusion rules of the form a × b = c into fusion rules with multiple outcomes. Taking G = O (5 , Indeed, in this theory, one can check that we have the following analogs of (2.18) W i × W = W i , i = 1 , , (2.26)where 5 i are the two five-dimensional representations of O (5 , i are the two complex thirty-dimensional representations (there is alsoa third, real, thirty-dimensional representation that does not appear in (2.26)). As in the This group has order 25920 and is the smallest simple group whose discrete gauge theory has a fusionof non-abelian Wilson lines with a unique outcome. O (5 , Z and it acts non-trivially on the Wilson lines involved inthe fusion above. In particular, we have W ↔ W and W ↔ W (2.27)under the action of the non-trivial element in Out( O (5 , Z (Vec O (5 , ) Z × gauge −→ Z (Vec O (5 , ⋊Z ) . (2.28)We may again apply (2.21) to find N ([ W i ] , +)([ W i ] , +) , ( W , ± ) = m (+ | Z , + | Z ⊗ ±| Z ) = m (1 ,
1) = 1 , (2.29)and conclude ([ W i ] , +) × ( W , ± ) = ([ W i ] , +) . (2.30)Such a situation arises whenever N c = Z = N a ∩ N b . This equality is special since, moregenerally, we have N a ∩ N b ⊆ N c .Before moving on to discuss other phenomena, let us note that the above discretegauge theories based on simple groups also provide answers to questions (2) and (3) fromthe introduction. Indeed, as we will see in greater detail in section 2.3, a discrete gaugetheory with a simple gauge group has no non-trivial proper fusion subcategories except thesubcategory of Wilson lines. Therefore, our above examples are enough to answer questions(2) and (3) generally in the negative (although we will see interesting examples of some ofthese ideas below). Let us now consider discrete gauge theories with unfaithful higher-dimensional (i.e., non-abelian) representations. The corresponding gauge groups are necessarily non-simple be-cause the kernel of a non-trivial unfaithful representation is a non-trivial proper normalsubgroup. As we will explain at a more pedestrian level below (and in a somewhat moresophisticated way in section 2.3), these examples illustrate the appearance of non-trivialfusion subcategories in the Wilson line sector. As a result, they demonstrate some of theideas—described in the introduction—behind constraints from subcategory structure lead-ing to fusion rules of the type (1.2). In particular, these theories provide examples whereideas in questions (2) and (3) of the introduction are realized.17o that end, let us consider some unfaithful higher-dimensional irreducible representa-tion of the gauge group, π ∈ Irrep( G ). Since π is unfaithful, it has a non-trivial kernel,Ker( π ) ⊳ G . Let us also define the set of characters whose kernel includes Ker( π ) as follows K π = (cid:8) χ ρ : χ ρ | Ker( π ) = deg χ ρ (cid:9) , (2.31)where deg χ ρ = | ρ | is the degree of the character. Now, consider χ λ , χ λ ′ ∈ K π . We claim χ λ · χ λ ′ ∈ K π . To see this, let us study χ λ | Ker( π ) · χ λ ′ | Ker( π ) = deg χ λ · deg χ λ ′ = X λ ′′ χ λ ′′ | Ker( π ) ≤ X λ ′′ (cid:12)(cid:12) χ λ ′′ | Ker( π ) (cid:12)(cid:12) . (2.32)Evaluating this expression on the identity element shows that deg χ λ · deg χ λ ′ = P λ ′′ deg χ λ ′′ .Therefore, we have χ λ ′′ | Ker( π ) = deg χ λ ′′ , and λ ′′ ∈ K π . In particular, we see that χ λ · χ λ ′ = X λ ′′ ∈ K π χ λ ′′ . (2.33)As a result, the Wilson lines with charges in K π form a closed fusion subcategory W λ × W λ ′ = X λ ′′ ∈ K π W λ ′′ ∈ C K π ≃ Rep( G/ Ker( π )) . (2.34)If we now consider the fusion of W π ∈ C K π with a non-abelian Wilson line W γ
6∈ C K π , wesee that the subcategory structure makes it more likely to find a unique outcome. Indeed, W π × W ¯ π ∈ C K π whereas W γ × W ¯ γ will typically include lines not in C K π .In fact, we can go further if we take γ | Ker( π ) to be an irreducible representation of Ker( π ).Since we are assuming that γ is a higher-dimensional representation, irreducibility of γ | Ker( π ) implies that Ker( π ) is a non-abelian group. Invoking Gallagher’s theorem (e.g., see corollary6.17 of [29]), we see that, for γ, π ∈ Irrep( G ), γ ⊗ π is an irreducible representation if therestriction γ | Ker( π ) is irreducible. Then, we are guaranteed to have the following fusion ruleof non-abelian Wilson lines W π × W γ = W πγ . (2.35)To understand this statement, let us first prove that γ K π . Suppose this were not thecase: then we arrive at a contradiction since | γ | > γ | Ker( π ) is reducible.As a result, W γ
6∈ C K π . Let us now consider the product χ γ · χ γ = χ + X i χ α i , (2.36) Such Wilson lines recently played an interesting role in [28]. Indeed, when one adds non-topologicalmatter charged under these representations, the corresponding Wilson lines can end on a point. Magneticflux lines or dyons with flux supported in Ker( π ) remain topological while lines carrying other fluxes do not. α i are irreps of G . Then we have( χ γ · χ γ ) | Ker( π ) = χ | Ker( π ) + X i χ α i | Ker( π ) . (2.37)Here, χ | ker( π ) corresponds to the trivial irreducible representation of Ker( π ), χ α i | Ker( π ) corre-sponds to an, in general, reducible representation of Ker( π ). Suppose that α i | Ker( π ) containsthe trivial irreducible representation of Ker( π ) for some i , then we will have at least twocopies of the trivial character of Ker( π ) on the right hand side of (2.37). However, weknow that ( γ ⊗ γ ) | Ker( π ) = γ | Ker( π ) ⊗ γ | Ker( π ) . Therefore, we cannot have more than one copyof the trivial character in the decomposition (2.37). Hence, α i | Ker( π ) cannot contain thetrivial representation for any i . It follows that α i | Ker( π ) ( h ) is non-trivial for at least some h ∈ Ker( π ). Therefore, it is clear that Ker( π ) cannot be in the kernel of the representations α i for any i . This shows that W α i ∈ W γ × W ¯ γ ⇒ W α i
6∈ C K π . (2.38)As a result, the subcategory structure guarantees (2.35).To better understand the above general discussion (as well as the continuing role ofsymmetries), let us consider some examples. Note that these results give explicit realiza-tions of the idea in question (3) in the introduction. The simplest discrete gauge theoriesrealizing the above discussion are based on gauge groups of order forty-eight. Interestingly,the existence of subcategory structure in the Wilson line sector, C W ≃ Rep( G ), explainsthe large ratio of orders, ∆ gap , between these groups and the smallest simple group, O (5 , gap = 2592048 = 540 ≫ . (2.39)In this section, we will discuss the examples of the binary octahedral group ( BOG ) andthe very closely related GL (2 , BOG . In this case, we have that 2 is an unfaithful (real) two-dimensional representation and that the restrictions of the other (real and faithful) two-dimensional irreducible representations to Ker(2 ) = Q ⊳ BOG , 2 , | Ker(2 ) , are irreducible.As expected from the general discussion above we have the following Wilson line fusions W × W = W × W = W . (2.40)Similarly to the simple discrete gauge theories discussed in the previous subsection, BOG ’s Z outer automorphism again lifts to a non-trivial symmetry of the TQFT, and the non-trivial element g = 1 acts as follows: g ( W ) = W .19et us note that in this case, the role of symmetries is even more pronounced. Indeed,one can check that W × W = W + W + W ∈ C K ≃ Rep(
BOG/Q ) ≃ Rep( S ) , W × W = W × W = W + W , (2.41)where 1 is a non-trivial one-dimensional irreducible representation, and 3 is a real three-dimensional irreducible representation. This latter representation satisfies χ · χ = χ (and similarly χ · χ = χ ). Therefore, we see that W generates a non-trivial one-formsymmetry in the BOG discrete gauge theory and that W , and W , form doublets underfusion with this generator while W is fixed W × W = W , W × W = W , W × W = W . (2.42)This non-trivial orbit structure then implies that W
6∈ W × W on symmetry groundsalone. Hence, in this example, both the subcategory structure and the symmetries guaran-tee the fusion rules (2.40).Before finishing this example, we should check that Z (Vec BOG ) is indeed prime. Afterwe discuss more formal aspects of subcategory structure in section 2.3, we will have moretools to use when answering this type of question. For now, let us prove that the Wilsonlines must all lie in the same TQFT factor. To that end, write down the Wilson lines ofthe
BOG discrete gauge theory W , W , W , W , W = W × W , W , W = W × W , W = W × W = W × W . (2.43)We can consider two cases: (1) W is in the same TQFT factor as W (call this factor T ) or (2) W is not in the same TQFT factor as W .Let us consider case (1) first. From the fusion equation involving W , we immediatelysee that W is also in T . Note that W cannot be written as the fusion product of twoother Wilson lines. Since there is no Wilson line of quantum dimension six, we also have W ∈ T . Now, we must clearly have that either W , ∈ T or W ,
6∈ T . However, inthe latter case we will again have a Wilson line of quantum dimension six. Therefore, we Note that since 2 , are faithful representations, a result of Burnside [30] generalized to Wilson linesshows that there exist n , ∈ N such that W × n , ⊃ W and W × n , ⊃ W . Our discussion implies n , > The same pedestrian arguments used below can be extended to the full set of lines in the theory to provethat Z (Vec BOG ) is prime. W , ∈ T . Therefore, by the W fusion rule in (2.43), all Wilson lines are inthe same TQFT factor.Let us now consider case (2). Let W ∈ T and W ∈ T with Z (Vec BOG ) = T ⊠ T .As in case (1), W cannot be written as the fusion product of two other Wilson lines, and,since there is no Wilson line of quantum dimension six, we have W ∈ T . However, thisleads to a contradiction because then W × W ′ = W . As a result, we conclude that allWilson lines must lie in the same factor of Z (Vec BOG ).Let us conclude with a brief discussion of the GL (2 ,
3) discrete gauge theory. Thisgauge group is quite similar to
BOG . For the purposes of the above discussion, the onlydifference is that 2 , become complex conjugate two-dimensional irreducible representations(otherwise, the remaining representations and remaining parts of the character tables arethe same). Therefore, (2.40) and (2.42) apply to Z (Vec GL (2 , ) as well (by identifying theseWilson lines with their relatives in Z (Vec GL (2 , ). The only change is that in the secondline of (2.41), we should take W , × W , → W × W . In particular, the roles ofsubcategory structure (again Rep( S ) ⊂ Rep( GL (2 , BOG and the GL (2 ,
3) discrete gaugetheories.Note that Gallagher’s theorem does not exhaust all cases where representations withnon-trivial kernel have irreducible products. Another interesting case is given by Gajen-dragadkar’s theorem [31]. If we have a group G which is both π -separable as well asΣ-separable, for two disjoint set of primes π and Σ, then this theorem guarantees that theproduct of a π -special character with a Σ-special character is irreducible. A character χ isknown as π -special if χ (1) is a product of powers of primes in π (a π number) and if, forevery subnormal subgroup N of G , any irreducible constituent θ of χ | N is such that o ( θ ) is a π -number. Hence, the fusion of Wilson lines corresponding to such characters have aunique outcome. Note that, in this case, one of the characters involved in the fusion is notrequired to be irreducible in the kernel of the other (unlike in Gallagher’s theorem). Let us conclude this section with a recapitulation of some of the main points above as wellas some general theorems that amplify our discussion: • In all of the infinitely many examples we studied so far, symmetries played an im-portant role. For example, zero-form symmetries had a non-trivial action on Wilson o ( θ ) is the order of the determinental character det( χ ) in the group of linear characters. A N (with N = k ≥
9) and O (5 , BOG , GL (2 , • We also saw that we could use Z one-form symmetry gauging in the S N (with N = k ≥
9) gauge theory to generate fusion rules involving non-abelian Wilson lineswith unique outcomes in the A N discrete gauge theories. We can constrain when sucha situation arises with the following theorem: Theorem 1 (one-form fixed points):
Consider a TQFT, T , with no fusion rules ofthe form (1.2). Suppose we can gauge a non-trivial one-form symmetry of this TQFT, H . After performing this gauging, we have fusion rules of the form (1.2) only if thereare a ∈ T such that fusion with at least one of the one-form generators, α ∈ Rep( H ),yields α × a = a . Proof:
Suppose this were not the case. Then, all anyons are organized into fulllength orbits under fusion with the one-form symmetry generators. When we gaugethe one-form symmetry, we identify these orbits as single elements (if the braidingwith one-form symmetry generators is trivial, these orbits become genuine lines ofthe gauged theory; if the braiding is non-trivial, these orbits become lines boundingsymmetry-generating surface operators in the gauged theory). Note that all anyonsappearing on the right hand side of fusion rules have the same braiding with theone-form symmetry generators. Therefore, the claim follows. (cid:3)
As we will see, this theorem will have echoes in the coset theories we describe in thesecond half of this paper. • In the case of O (5 ,
3) discrete gauge theories, we saw that we could gauge the outerautomorphisms and have fusion rules of form (1.2) in this gauged theory as well. Thisdiscussion inspires the following theorem:
Theorem 2 (zero-form fixed points):
Consider a TQFT, T , and suppose we cangauge a non-trivial zero-form symmetry of this TQFT, H . After performing thisgauging, we have fusion rules of the form (1.2) only if there are non-trivial a i ∈ T such that at least one of the non-trivial elements of the zero-form group fixes a i . Proof:
Suppose that all non-trivial elements of the discrete gauge theory leave allthe non-trivial anyons unfixed. Now consider anyons a, b, c ∈ T such that c ∈ a × b .22rom the general discussion around (2.21), we see that N t a ∩ N s b ∩ N c = Z and N a \ H/N b = H . Moreover, since the stabilizers are trivial, π a = π b = π c = 1 are thetrivial representations. We then have N ([ c ] , a ] , , ([ b ] , = | H | · m (1 ,
1) = | H | > . (2.44)Therefore, we cannot produce fusion rules of the desired type. (cid:3) Our discussion of the O (5 ,
3) theory also suggests the following theorem
Theorem 2A:
Consider a TQFT, T , with a fusion rule of the form a × b = c anda zero-form symmetry, H . If at least one of { a, b, c } is unfixed by H , then the onlyway for a × b = c to map to a fusion rule with unique outcome in the gauged theoryis for c to be unfixed by H . Proof: If c is unfixed by H , then N c = N a ∩ N b = Z . If either a or b are unfixedthen N a ∩ N b = Z as well (although we need not have N c = Z ). In any case, (2.21)becomes N ([ c ] ,π c )([ a ] ,π a ) , ([ b ] ,π b ) = X ( t,s ) ∈ N a \ H/N b m ( π c | Z , t π a | Z ⊗ s π b | Z ⊗ π ω ( t a, s b,c ) ) . (2.45)We have two cases: (1) N a \ H/N b = Z or (2) N a \ H/N b = Z . Consider case (1) first. In this case, all resulting fusion rules will have multiplicity | N a \ H/N b | > (2) . If c is fixed by some element of H , then we have at leasttwo possible π c (one is the trivial representation). This results in a fusion rules withnon-unique outcomes. (cid:3) • In the case of the BOG and GL (2 ,
3) discrete gauge theories we saw that both one-form symmetries and subcategory structure offered an explanation of the existence ofthe fusion rules (2.40). The following theorem further explains and generalizes thisconnection between symmetries and subcategories of the Wilson line sector:
Theorem 3 (subcategories and symmetries):
Consider a finite group, G , with anunfaithful higher-dimensional irreducible representation, π . Moreover, suppose thereare one-dimensional representations, π i , with Ker( π i ) D Ker( π ). Then, in the cor-responding (twisted or untwisted) discrete gauge theory, Wilson lines charged underrepresentations, γ , that have γ | Ker( π ) irreducible transform non-trivially under fusionwith the abelian Wilson lines, W π i . 23 roof: We have that W π i ∈ C K π , where C K π was defined around (2.34) as the sub-category of Wilson lines charged under representations whose kernels contain Ker( π )(see (2.31)). Therefore, we see that the abelian Wilson lines W π i ∈ C K π .By the discussion around (2.38), we also see that all non-identity lines W α i ∈ W γ ×W ¯ γ are not elements of C K π . As a result, W π i
6∈ W γ × W ¯ γ . On the other hand, the trivialline is clearly in W γ × W ¯ γ . This logic implies W π i × W γ × W ¯ γ = W γ × W ¯ γ , (2.46)from which the claim in the theorem trivially follows. (cid:3) This result tells us that the W γ must transform under fusion with the one-formsymmetry generators while W π need not. In the case of the BOG and GL (2 , W π × W γ = W πγ fusion rule in (2.40). Here we see it is somewhat more general. • Note that the results of this section answer questions (1)-(3) of the introductionnegatively in general. Still, we saw that in the BOG and GL (2 ,
3) discrete gaugetheories, the ideas in (3) and (1.8) do apply in some cases. We will return to aproposal for construct a theory satisfying (1.7) in question (2) in section 2.3.
In sections 2.2.2 and 2.2.3, we saw the important role subcategories play in generatingfusion rules involving non-abelian Wilson lines with unique outcomes (e.g., they explainedthe hierarchy in (2.39)). Moreover, understanding the subcategory structure is crucial toresolving the question of whether a particular discrete gauge theory is prime or not. Inthe case of theories with simple gauge groups (see section 2.2.1), we used results from [26].In the case of the examples of discrete gauge theories with non-simple groups we studied,we used an argument that does not easily generalize. Therefore, in this section, we reviewsome of the more general results of [26] on subcategories of discrete gauge theories. Wethen apply these results to generate some useful theorems that will serve us in subsequentsections.The main power of the results in [26] is that they rephrase questions about subcategoriesin discrete gauge theories in terms of data of the underlying gauge group. In particular,we have:
Theorem 4 [26]:
Fusion subcategories of discrete gauge theories with finite group G arein bijective correspondence with triples, ( K, H, B ). Here
K, H E G are normal subgroups24hat centralize each other (i.e., they commute element-by-element), and B : K × H → C × is a G -invariant bicharacter. If we have a non-trivial twist, ω , then the same conditionshold except that we demand that B is a G -invariant ω -bicharacter. Proof:
See proofs of Theorems 1.1 and 1.2 (though they are phrased using different, butequivalent, terminology) of [26]. (cid:3)
Since B is a bicharacter, it satisfies B ( k k , h ) = B ( k , h ) · B ( k , h ) , B ( k, h h ) = B ( k, h ) · B ( k, h ) . (2.47)Here G invariance means that B ( g − kg, g − hg ) = B ( k, h ) for all k ∈ K , h ∈ H , and g ∈ G .In fact, [26] also give a way to construct the subcategory, S ( K, H, B ), in question giventhe above data: S ( K, H, B ) := gen (( a, χ ) | { a ∈ K ∩ R , χ ∈ Irr( N a ) s . t . χ ( h ) = B ( a, h ) deg χ , ∀ h ∈ H } ) , (2.48)where R is a set of representatives of conjugacy classes, Irr( N a ) is the set of characters ofirreducible representations of the centralizer N a , and “gen( · · · )” means that the categoryis generated by the simple objects inside the parenthesis. A normal subgroup is a unionof conjugacy classes. Hence, K specifies all the conjugacy classes labelling the anyons inthe subcategory S ( K, H, B ). Also, all the Wilson lines in S ( K, H, B ) are such that thecorresponding representations have kernels which contain H .If we have non-trivial twist, then (2.47) and G -invariance become [26] B ( k k , h ) = η h ( k , k ) · B ( k , h ) · B ( k , h ) , B ( k, h h ) = η − k ( h , h ) · B ( k, h ) · B ( k, h ) ,B ( g − kg, h ) = η k ( g, h ) η k ( gh, g − ) η k ( g, g − ) B ( k, ghg − ) , (2.49)where η g ( h, k ) := ω ( g, h, k ) · ω ( h, k, k − h − ghk ) ω ( h, h − gh, k ) , (2.50)is a generalization of (2.2). For non-trivial twist, we also have that (2.48) becomes S ( K, H, B ) := gen (( a, χ ) | { a ∈ K ∩ R , χ ∈ Irr ω ( N a ) s . t . χ ( h ) = B ( a, h ) deg χ , ∀ h ∈ H } ) , (2.51)where the ω in Irr ω ( N a ) is a reminder that we should consider characters with projectivityphase given by (2.2) or (2.50).We can now immediately see how the subcategories we studied in previous sectionsarose: S ( G, Z , ≃ Z (Vec ωG ) is the full discrete gauge theory, S ( Z , G,
1) is the trivial25ubcategory, and S ( Z , Z , ≃ Rep( G ) ≃ C W is the full subcategory of Wilson lines. Inthe case of simple discrete gauge theories, we see that, as claimed in section 2.2.1, these arethe only subcategories. However, in the case of the Z (Vec ωBOG ), Z (Vec ωGL (2 , ), and othergauge theories based on gauge groups with unfaithful irreducible representations, π , we findadditional subcategories: S ( Z , Ker( π ) , ≃ Rep( G/ Ker( π )) and S (Ker( π ) , Z , S (Ker( π ) , Z ,
1) is the M¨uger center of S ( Z , Ker( π ) , Theorem 5:
The set of magnetic flux lines, M , of a discrete gauge theory (both untwistedand twisted) with non-abelian gauge group, G , do not form a fusion subcategory. Inparticular, M 6≃
Rep( G ). Proof:
Suppose the full set of flux lines form a subcategory. Then, we need K to includeat least one element of each conjugacy class in order to include all of M in S . However,since K normal subgroup, it must consist of full conjugacy classes. Therefore, K = G .Using theorem 4, we can label this putative subcategory as S ( G, H, B ). Since H has tocommute with all elements in G , it has to be a subgroup of the center of the group Z ( G ).Suppose the group has trivial center. This forces B = 1, and S ( G, Z ,
1) is the full discretegauge theory, which means we also include objects with charge. This is a contradiction.Suppose H is a non-trivial subgroup of Z ( G ). We know that the function B , beinga bicharacter, satisfies B ( e, h ) = 1 ∀ h ∈ H . So the Wilson line ([ e ] , π ) ∈ S ( G, H, B ) if π has H in its kernel. Recall that the irreducible representations of G/H are in one-to-onecorrespondence with irreducible representations of G with H in its kernel. Since G is non-abelian, Z ( G ) = G . Hence, G/H is a non-trivial group. It follows that there is at leastone non-trivial irreducible representation π ′ of G such that H is in its kernel. Hence, theWilson line ([ e ] , π ′ ) belongs to the subcategory S ( G, H, B ) for any B . A contradiction. (cid:3) The fact that
M 6≃
Rep( G ) has consequences in section 2.4. In particular, it explains whyelectric-magnetic self-dualities are non-trivial to engineer in theories with non-abelian gaugegroups and trivial centers. If such a duality exists and involves magnetic flux lines, thenthey will necessarily be in a Rep( G )-like subcategory with objects carrying electric charge(e.g., see the S discrete gauge theory self-duality [19], where the dimension-two flux lineis in a Rep( S ) subcategory with both dimension one Wilson lines). In any untwisted abelian gauge theory, this is not an issue as
M ≃
Rep( G ) and there is a canonicalelectric/magnetic duality. Theorem 6 [26]:
A discrete gauge theory with gauge group, G , is a prime TQFT if andonly if there is no triple ( K, H, B ) with
K, H ⊳ G normal subgroups centralizing each other, HK = G , ( G, Z ) = ( K, H ) = ( Z , G ), and B is a G -invariant bicharacter on K × H such that BB op | ( K ∩ H ) × ( K ∩ H ) is non-degenerate. In the case of non-trivial twisting, ω , theprevious conditions still hold, but B is also a G -invariant ω -bicharacter. Proof:
See proof of theorem 1.3 (though it is phrased using different, but equivalent,terminology) in [26]. (cid:3)
Note that in the statement of theorem 6, B op ( h, k ) := B ( k, h ) for all k ∈ K and h ∈ H .Given this theorem, we may prove the following result that will be useful to us insection 2.6: Theorem 7: If G is a non-direct product group with trivial center, then the corresponding(twisted or untwisted) gauge theory is a prime TQFT. Proof:
We have a non-direct product group G with trivial center. Let us assume thatRep( D ( G )) has a modular subcategory. Then, there exists two normal subgroups, K and H , commuting with each other and satisfying KH = G . So, every element of G is aproduct of an element of K with an element of H . Hence, any element in K ∩ H has tocommute with all elements of G . Since the center of G is trivial by choice, K ∩ H = Z .It follows that G has to be a direct product of K and H . A contradiction. Hence, fornon-direct product groups G with trivial center, Rep( D ( G )) is prime. (cid:3) A simple set of examples subject to this theorem include the S N discrete gauge theoriesanalyzed above and the Z ⋊ Z discrete gauge theory we will analyze further in section2.6.Finally, we conclude with a proposal for engineering an example of a theory of the typeenvisioned in question (2) in the introduction. In particular, consider a G × G discretegauge theory, Z (Vec ωG × G ). Clearly, for trivial twisting this is a non-prime theory since Z (Vec G × G ) = Z (Vec G ) ⊠ Z (Vec G ). Indeed, by theorem 6, we can take K = G × Z , H = Z × G , and B = 1. However, if we turn on a twist, ω ∈ H ( G × G, U (1)), we mightbe able to generate a prime theory. In particular, if we can find G such that ω is non-trivialand does not factorize, then we would have an example of a prime theory with Wilson linesin Rep( G × G ) = Rep( G ) ⊠ Rep( G ). Choosing one Wilson line in each Rep( G ) factor andfusing would give a unique fusion outcome. It would be interesting to see if this proposal We thank D. Aasen for suggesting the basis for this idea. G -invariant ω -bicharacter (all other requirements of theorem 6 can besatisfied). A concrete example of a theory of the type discussed in question (2) is studiedin section 2.6.1. In sections 2.2.2 and 2.2.3 we saw that zero-form symmetries played an important role ingenerating fusions rules of the form (1.2). In this section we review some relevant resultsof [18] and prove a theorem that will be useful to us in section 2.6.In three spacetime dimensions, zero-form symmetries are implemented by dimensiontwo topological defects (recall that one-form symmetries are generated by abelian lines).These defects act on lines that pierce them as in figure 3. We will say the correspondingΣ g a g ( a ) Fig. 3:
The symmetry defect Σ g , labelled by a zero-form symmetry group element g , actson an anyon a .symmetry group, H , is non-trivial iff it has a generator, h ∈ H , such that there is an anyon a ∈ T satisfying h ( a ) = a .Note that the automorphisms of the gauge group G , Aut( G ), are a natural source ofsymmetries. Indeed, in the context of the G -SPT that we gauge to generate the discretegauge theory, these automorphisms permute the symmetry defects. Therefore, we expectthey will play a role in the discrete gauge theory. To be more precise, recall that we candistinguish between the inner automorphisms Inn( G ) E Aut( G ), generated by conjugationsof the form gxg − for x, g ∈ G , and outer automorphisms, Out( G ) := Aut( G ) / Inn( G ). Sincethe discrete gauge theory involves magnetic charges labeled by conjugacy classes and electric28harges labeled by representations of centralizers, it is clear that inner automorphisms willact trivially on the discrete gauge theory (conjugacy classes are invariant under Inn( G ) andthe normalizers of different elements in a conjugacy class are isomorphic). Therefore, wecan at best expect Out( G ) to lift to a symmetry of the TQFT. Indeed, this is preciselywhat happens.More formally, we have that, in a discrete gauge theory Out( G ) lifts to a part of thegroup of braided autoequivalences of the discrete gauge theory, Aut br ( Z (Vec G )): Theorem 8 [18]:
The subgroup of braided autoequivalences that fix the Wilson linesStab(Rep( G )) ≤ Aut br ( Z (Vec G )) takes the formStab(Rep( G )) ≃ H ( G, U (1)) ⋊ Out( G ) . (2.52) Proof:
See the proof of Corollary 6.9 (though it is phrased using different, but equivalent,terminology) in [18]. (cid:3)
Note that Out(G) generally acts non-trivially on the conjugacy classes. Therefore, it willalso generally act non-trivially on the Wilson lines. However, in certain more exotic cases,all of Out(G) preserves conjugacy classes. In such cases, the Wilson lines are fixed. Notethat elements ζ ∈ H ( G, U (1)) always leave the Wilson lines invariant since they act asfollows [18] ζ (([ a ] , π a )) = ([ a ] , π g ρ g ) , ρ g ( x ) := ζ ( x, g ) ζ ( g, x ) , (2.53)where g ∈ [ a ] (in particular, g = 1 for Wilson lines). Note that ρ g ( x ) depends only on thecohomology class of ζ (it is invariant under shifts by a 2-coboundary).A second set of symmetries involves the exchange of electric and magnetic degrees offreedom. These are electric/manetic self-dualities and are inherently quantum mechanical innature. These symmetries are closely related to the existence of Lagrangian subcategories.As we briefly mentioned at the beginning of section 2.2, a Lagrangian subcategory, L , isa collection of bosons with trivial mutual braiding that is equal to its M¨uger center (e.g.,like the subcategory of Wilson lines, C W ≃ Rep( G )) . This latter condition simply meansthat the only objects that braid non-trivially with every element of L are elements of thatsubcategory.To find the set of these symmetries, it turns out to be useful to construct the categoricalLagrangian Grassmannian, L ( G ). This is the collection of all Lagrangian subcategories. The smallest group that has this feature has order 2 [32]. See [33] for an application of groups thathave at least some class-preserving outer automorphisms to quantum doubles. L ( N,µ ) ≃ Rep( G ( N,µ ) ) with | G ( N,µ ) | = | G | , is labeled by a normalabelian subgroup, N ⊳ G , and a G -invariant µ ∈ H ( N, U (1)) (the Wilson line subcategoryis L , ). For the purposes of understanding these symmetries, the important subcategoryis [18] L ⊇ L := {L ∈ L ( G ) |L ≃ Rep( G ) } . (2.54)In particular, we have Theorem 9 [18]:
The action of Aut br ( Z (Vec G )) on L ( G ) is transitive. Moreover, | Aut br ( Z (Vec G )) | = | H ( G, U (1)) | · |
Out( G ) | · | L ( G ) | . (2.55) Proof:
See proposition 7.6 and corollary 7.7 of [18]. (cid:3)
Examples of such dualities appear in the S discrete gauge theory [19] and beyond [34].Let us now apply this theorem to prove a result that will be useful for us below Theorem 10: If G ≃ N ⋊ K , where N is an abelian group, then the correspondinguntwisted discrete gauge theory has an electric-magnetic self-duality. Proof:
By theorem 9, in order to find a self-duality, we need to find a normal abeliansubgroup
N ⊳ G and a G -invariant 2-cocycle, µ ∈ H ( N, U (1)). Moreover, we need to finda corresponding G ( N,µ ) ≃ G . In particular, from remark 7.3 of [18], when µ is trivial, wehave that G ( N, ≃ b N ⋊ G/N , where b N is the character group of N . For an abelian group, b N ≃ N . Therefore, we have that G ( ˜ N, ≃ N ⋊ K = G as desired. (cid:3) This theorem will be useful in our symmetry searches in section 2.6. Note that oneimmediate consequence of the above discussion is that none of the examples discussedabove have self-dualities. Indeed, theories with simple gauge groups have no non-trivialnormal abelian subgroups. On the other hand, theories like
BOG and GL (2 ,
3) have H ( BOG, U (1)) ≃ H ( GL (2 , , U (1)) ≃ Z (and similarly for all normal abelian sub-groups). Since these groups are not semi-direct products, we conclude they lack self-dualities. In the previous subsections, we have seen that zero-form symmetries play an important rolein generating fusion rules for non-abelian anyons with unique outcomes. However, sinceour interest is simply in the existence of such fusion rules, it is natural that we shouldgeneralize our notion of symmetry to include symmetries of the modular data (and hence,30y Verlinde’s formula, automorphisms of the fusion rules) that don’t necessarily lift tosymmetries of the TQFT. The basic reason such “quasi zero-form symmetries” as we willcall them exist is that the modular data does not define a TQFT (see [36] for a consequenceof this fact). In particular, the underlying F and R symbols may not be invariant (up toan allowed gauge transformation) under a quasi zero-form symmetry even if S and T are.In fact, such “quasi-zero-form symmetries” are common, with charge conjugation beinga particular example [37]. Indeed, even in the A N (with N = k ≥
9) theories we discussedin section 2.2.1, such quasi-charge conjugation symmetries exist. These symmetries are inaddition to the genuine zero-form symmetries we described when analyzing these examples.In appendix B, we study the particular case of A discrete gauge theory in more detail andexplicitly disentangle the quasi-symmetries from the genuine symmetries.More generally, there are theories that have no genuine symmetries. One set of examplesinclude discrete gauge theories based on the Mathieu groups. These are simple groups withtrivial Out(G) and H ( G, U (1)). Moreover, since these groups have no non-trivial normalabelian subgroups, L ( G ) = L ( G ) ≃ Rep( G ), and so there are no non-trivial self-dualities.The largest Mathieu groups, M and M are of particular interest to us since theirdiscrete gauge theories have non-abelian Wilson lines that fuse together to produce a uniqueoutcome. Moreover, of the theories with fusions of type (1.2), these are the only untwisteddiscrete gauge theories that have no modular symmetries that lift to symmetries of the fullTQFTs.For M it is not hard to check that W × W = W , W × W = W , (2.56)where 22 is the real twenty-two dimensional representation, 45 , are two forty five dimen-sional complex representations, and 990 , are two nine hundred and ninety dimensionalrepresentations. Under charge conjugation W ↔ W , W ↔ W . (2.57)For M , we have a particularly rich set of fusions W × W = W , W × W = W , W × W = W In fact, most generally, we might expect automorphisms of the fusion rules that are not even symmetriesof the modular data (e.g., as studied recently in [35]). By the results of [21], these theories cannot have such fusions involving lines that carry magnetic flux. It would be interesting to know if our results here have any connection with moonshine phenomenaobserved involving M as in [38]. × W = W , W × W = W , W × W = W , W × W = W , W × W = W . (2.58)where 23 is a real twenty-three dimensional representation, 45 , are complex forty-fivedimensional representations, 231 , are two-hundred and thirty-one dimensional complexrepresentations, and 1035 , are complex one-thousand and thirty-five dimensional represen-tations, 5313 is a real five-thousand three-hundred and thirteen dimensional representation,and 10395 is a real ten-thousand three-hundred and ninety-five dimensional representation.Under charge conjugation, we have W ↔ W , W ↔ W , W ↔ W . (2.59)While we have seen similar actions in previous sections, but here the novelty is that chargeconjugation is a quasi-symmetry.More generally, as we will discuss in greater detail below, all other examples of TQFTsthat we have found with fusion rules involving non-abelian anyons with unique outcomehave at least quasi zero-form symmetries.Finally, let us conclude this section by discussing how twisting affects the quasi-zero-form symmetries. When the quasi-symmetry is charge conjugation and the group hascomplex representations, the quasi-symmetry lifts to an action on Wilson lines (see ap-pendix B for a discussion in a concrete example). In this case, the quasi-symmetry persistsregardless of the twisting.As a more complicated example, let us consider the case of BOG first discussed insection 2.2.2. This theory only has real conjugacy classes and representations. However,there is still a non-trivial charge conjugation acting on certain dyons since elements in
BOG have centralizer groups Z , Z , and Z . These latter groups admit complex representations.However, unlike the spectrum of Wilson lines, the spectrum of dyons generally changesas we change the twist. Therefore, we might imagine that the charge conjugation quasisymmetry can be twisted away.In fact, this is not the case. The main point is that any twisting ω ∈ H ( BOG, U (1)) ≃ Z of the BOG discrete gauge theory is “cohomologically trivial” in the following sense:the η g ( h, k ) ∈ H ( N g , U (1)) phases defined in (2.2) are all trivial. Indeed, this statementfollows from the fact that H ( N g , U (1)) = Z for all g ∈ BOG . Therefore, none of theanyons are lifted by the twisting, and the characters of
BOG change as follows χ π ωg ( h ) → ǫ g ( h ) · χ π ωg ( h ) (2.60)where ǫ g is a 1-cochain that gives the 2-coboundary, η g . It is not too hard to check that allchoices of the twisting leave us with complex characters. Therefore, the charge conjugation32uasi-symmetry persists (here it would be more accurate to term it a “modular symmetry”since it is apriori possible—though we have not checked—that charge conjugation becomesa symmetry of the theory for certain choices of ω ). So far, we have only constructed fusion rules of the form (1.2) using Wilson lines. In thecase of gauge theories with simple groups, this is all we can do [21]. However, when wehave non-simple gauge groups, the existence of self-dualities discussed in section 2.4 aswell as the possibility of electric-magnetic dualities between theories with different gaugegroups and Dijkgraaf-Witten twists [22, 34] suggests that we should also be able to involvenon-abelian anyons carrying flux. Indeed, we will see this is the case.To that end, let us study a fusion of the form L ([ g ] ,π ωg ) × L ([ h ] ,π ωh ) = L ([ k ] ,π ωk ) , g, h = 1 , (2.61)Carefully applying the machinery in section 2.1 reveals the following contraints
1. [ g ] · [ h ] = [ k ] = [ h ] · [ g ]2. ∃ ! π ωk such that m ( π ωk | N g ∩ N h ∩ N k , π ωg | N g ∩ N h ∩ N k ⊗ π ωh | N g ∩ N h ∩ N k ⊗ π ω ( g,h,k ) ) = 1We will apply these constraints in what follows.For an untwisted discrete gauge theory based on a group G with a non-trivial center Z ( G ), the constraints above implies that if we have a fusion of Wilson lines giving a uniqueoutcome W π × W γ = W πγ , (2.62)then we have a fusion of dyons of the form L ([ g ] ,π ) × L ([ h ] ,γ ) = L ([ gh ] ,πγ ) , (2.63) One may also wonder about the fate of the genuine Out(
BOG ) ≃ Z zero-form symmetry under twisting.First, consider ω corresponding to the order 2 element in Z . Since Out( BOG ) acts on H ( BOG, U (1))through Aut( H ( BOG, U (1))), ω should be fixed under it. Hence, it seems plausible that the twisted discretegauge theory corresponding to this choice of ω has Out( BOG ) as a subgroup of its symmetries (while theorem8 has nothing to say on this point since it assumes untwisted theories, we view the existence of a symmetry inthis case as a plausible assumption). In fact, more generally, if the action of Out( G ) leaves ω ∈ H ( G, U (1))invariant up to a 3-coboundary, then it can be shown that this is a symmetry of the modular data of thetwisted theory. It would be interesting to understand what happens for other twists as well. We refer the interested reader to the derivation in section III of [21] for further details. g, h ∈ Z ( G ). Hence, we can dress the Wilson lines with fluxes from the centerof the group to obtain fusion rules involving dyons with unique outcomes. For example,we have already seen that the discrete gauge theories corresponding to BOG and GL (2 , Z ), the above discussion immediately implies the existenceof dyonic fusions where the dyons are labelled by the non-trivial element of the centre. Infact, these two types of fusions exhaust all a × b = c type fusions in both Z (Vec BOG ) and Z (Vec GL (2 , ).In the case of the fusion of non-abelian Wilson lines with a unique outcome, we sawthat we were not guaranteed to find fusion subcategories beyond the three universal sub-categories present in any discrete gauge theory (the theory itself, the trivial TQFT, andthe Wilson line sector, C W ≃ Rep( G )). On the other hand, when we have fusions of non-abelian anyons carrying flux with a unique outcome, we are guaranteed to have fusionsubcategories. When the gauge group has a non-trivial center, Z ( G ), this statement istrivial. The following theorems guarantee this fact more generally:
Theorem 11:
Let G be a non-simple finite non-abelian group. If we have a fusion ruleinvolving two dyons or fluxes giving a unique outcome in the (twisted or untwisted) G gaugetheory, then S ( M g , Z ,
1) and S ( M h , Z ,
1) (along with S ( Z , M g ,
1) and S ( Z , M h , g and h are elements labelling the non-trivial conjugacy classes (of length >
1) involved in the fusion. M g is the normal subgroupgenerated by the elements in [ g ]. Proof:
We have an a × b = c type fusion rule involving the non-trivial conjugacy classes[ g ] and [ h ]. Let M g be the normal subgroup generated by [ g ]. In fact, it has to be aproper normal subgroup. To see this, suppose M g = G . From Lemma 3.4 of [26], weknow that [ g ] and [ h ] commute element-wise. Hence, [ h ] commutes with all elements in M g = G . It follows that [ h ] should be a subset of the elements in Z ( G ). However, elementsof Z ( G ) form single element conjugacy classes. A contradiction. Hence, M g has to be aproper normal subgroup of G . Since g = e , it is clear that M g is not the trivial subgroupeither. We can use the same argument to show that M h is also a proper non-trivial normalsubgroup of G . Therefore, by theorem 4, we have fusion subcategories corresponding tothe choices S ( M g , Z ,
1) and S ( M h , Z ,
1) (and similarly S ( Z , M g ,
1) and S ( Z , M h , (cid:3) Note that we have, L ([ g ] ,π ωg ) ∈ S ( M g , Z ,
1) and L ([ h ] ,π ωh ) ∈ S ( M h , Z , The discussion in section 2.3 guarantees that S ( Z ( G ) , Z ,
1) and S ( Z , Z ( G ) ,
1) are non-trivial subcate-gories. L ([ g ] ,π ωg )
6∈ S ( M h , Z ,
1) and L ([ h ] ,π ωh )
6∈ S ( M g , Z , h ] has atleast one element h ′ ∈ [ h ] such that [ h ′ , h ] = 1, then L ([ g ] ,π ωg ) and L ([ h ] ,π ωh ) lie in differentsubcategories Corollary 12:
Given the conditions in theorem 11, if there exists h ′ ∈ [ h ] such that[ h ′ , h ] = 1, µ [ g ] ∈ S ( M g , Z , L ([ g ] ,π ωg )
6∈ S ( M h , Z , h ↔ g .For a ∈ M g the fusion subcategory S ( M g , Z ,
1) contains anyons ([ a ] , π a ) where π a isany irrep of the centralizer N a . In an untwisted discrete gauge theory, for a fusion of fluxeslabelled by conjugacy classes [ g ] and [ h ], we can define fusion subcategories S ( M g , M h , S ( M h M g ,
1) which have a more restricted set of elements. For a ∈ M g , the anyon([ a ] , π a ) is an element of S ( M g , M h ,
1) if and only if M h ⊆ Ker( π a ). Clearly, ([ g ] , g ) ∈S ( M g , M h ,
1) and ([ h ] , h ) ∈ S ( M h , M g , g ] , g ) ( M h , M g ,
1) and ([ h ] , h )
6∈ S ( M g , M h , Theorem 13:
Let G be a non-simple group. If we have a fusion of a Wilson line anda dyon giving a unique outcome, then S (Ker( χ π ) , Z ,
1) and S ( Z , Ker( χ π ) ,
1) are properfusion subcategories of the (twisted or untwisted) discrete gauge theory. Here, π is an irrepof G labelling the Wilson line. Proof:
Suppose [ b ] is the non-trivial conjugacy labelling the flux line. Let χ π be thecharacter of an irreducible representation, π , of G labelling the Wilson line. From note 3.5of [26] we know that χ should be trivial on a subset of elements given by [ G, b ]. Since b is not in the center, [ G, b ] is guaranteed to have a non-trivial element. Hence, χ π is nota faithful representation. Ker( χ π ) is a non-trivial normal subgroup of G . Since χ π is notthe trivial representation, Ker( χ π ) = G is a non-trivial proper normal subgroup. Hence, bytheorem 4, we have a fusion subcategory given by S (Ker( χ π ) , Z ,
1) and S ( Z , Ker( χ π ) , (cid:3) Note that in this case the Wilson line is an element of S ( Z , Ker( χ π ) ,
1) while the magneticflux is not. In this sense, such fusions are “natural.” To illustrate the ideas above, let usconsider the following examples. 35 .6.1. Z ( Vec Z ⋊ Q )Let us consider the Z ⋊ Q discrete gauge theory. Even though this group has many non-trivial proper normal subgroups, we have Z ⋊ Q = HK for any proper normal subgroups H, K . Hence, using theorem 6, we have that Z (Vec Z ⋊ Q ) is a prime theory.This group has a length 2 conjugacy class [ f ] (here we are using the notation of GAP[39], where this group is entry (48 ,
18) in GAP’s small group library) and a 2-dimensionalrepresentation 2 (the third 2-dimensional representation in the character table of Z ⋊ Q on GAP). We have the following fusion of a Wilson line and a flux line giving a uniqueoutcome. W × µ [ f ] = L ([ f ] , | Nf ) , (2.64)where the restricted representation 2 | N f is irreducible.Since we have a prime theory, the existence of this fusion rule is not due to a Deligneproduct. However, it can be explained using the subcategory structure of Z (Vec Z ⋊ Q ).To that end, consider the fusion subcategory S ( Z , Ker(2 )) , W π belongs to this subcategory only if Ker(2 )is in Ker( π ). From the character table of Z ⋊ Q , we find three representations satisfyingthis constraint: 1, 1 and 2 . Here 1 is the trivial representation and 1 is the third 1-dimensional representation in the character table. Hence, the anyons contained in the fusionsubcategory S ( Z , Ker(2 ) ,
1) are the Wilson lines W , W as well as W . Moreover, wecan check the following1 × = 1; 1 × = 2 ; 2 × = 1 + 1 + 2 . (2.65)Now let us consider a fusion subcategory corresponding to the triple S ( M f , Ker(1 ) , M f is the normal subgroup generated by the elements of the conjugacy class [ f ]and 1 is the second 1 dimensional representation in the character table of Z ⋊ Q . Wehave M f = { e, f , f , f · f } . A Wilson line W π belongs to the set of generators of thissubcategory only if Ker(1 ) is in Ker( π ). Using the character table we can check that thereare only two representations which satisfy this constraint: 1 and 1 . Moreover, we have1 × = 1. Hence, the Wilson lines in S ( M f , Ker(1 ) ,
1) are W and W . Note that theflux line µ [ f ] belongs to this subcategory.Hence, we have two fusion subcategories S ( Z , Ker(2 )) ,
1) and ( M f , Ker(1 ) ,
1) withthe following structure W ∈ S ( Z , Ker(2 )) , µ [ f ] ∈ ( M f , Ker(1 ) , S ( Z , Ker(2 )) , ∩ S ( M f , Ker(1 ) ,
1) = {W } (2.66)36herefore, the fusions W × W and µ [ f ] × µ [ f ] have only W in common. This trivialintersection explains the fusion (2.64) and gives an example of the idea behind question (2)in the introduction. Z ( Vec Z ⋊Z )Let us consider the Z ⋊ Z discrete gauge theory. Since the center of the gauge groupis trivial and the group involves a semi-direct product, we know from theorem 7 that thisgauge theory is prime.This group has a length 5 conjugacy class labelled by the element f and a length2 conjugacy class labelled by the element f (here we are using the notation of GAP,where this group is entry (60 ,
7) in GAP’s small group library). We also have a length 10conjugacy class labeled by f f . It is therefore clear that we have a fusion of flux linesgiving a unique outcome corresponding to these conjugacy classes µ [ f ] × µ [ f ] = µ [ f f ] . (2.67)Based on our discussion above, let us consider the groups M f and M f generated bythe elements in the corresponding conjugacy class. It is not too hard to show that M f = [ e ] ∪ [ f ] (2.68) M f = [ e ] ∪ [ f ] ∪ [ f ] (2.69)Hence, the fusion subcategories S ( M f , M f ,
1) and S ( M f , M f ,
1) can only have Wilsonlines as common elements. The trivial Wilson line W is of course a common element.As we saw in section 2.3, a Wilson line, W π , is a member of the fusion subcategory, S ( M f , M f , χ π ( h ) := B ( e, h ) deg χ π = deg χ π , ∀ h ∈ M f , (2.70)is satisfied. Hence, M f should be in the kernel of χ π . Similarly, a Wilson line W π ′ , isa member of ( M f , M f ,
1) only if M f is in the kernel of χ π ′ . Therefore, the commonelements of the two fusion subcategories are given by the Wilson lines W ˜ π for which M f and M f are in the kernel of χ ˜ π . Using the character table of Z ⋊ Z , we find that thereis only one representation π , which satisfies this constraint.Consider the fusions µ [ f ] × µ [ f − ] = W + · · · , (2.71) µ [ f ] × µ [ f − ] = W + · · · . (2.72)37e know µ [ f ] and µ [ f ] belong to the fusion subcategories ( M f , M f ,
1) and ( M f , M f , W and W . We would liketo know whether the Wilson line, W , appears on the right hand side of these fusions. Tothat end, consider the fusion W × µ [ f ] = L ([ f ] , | Nf ) . (2.73)It turns out that 1 | N f is the trivial representation of N f . Hence, µ [ f ] is fixed underfusion with the one-form symmetry generator, W . So it is clear that W should appearin the fusion µ [ f ] × µ [ f − ] . Similarly, consider the fusion W × µ [ f ] = L ([ f ] , | Nf ) . (2.74)It is easy to check that 1 | N f is a non-trivial representation of N f . Hence, µ [ f ] is notfixed under the fusion with W . Since W is an order two anyon, it cannot appear in thefusion µ [ f ] × µ [ f − ] (because if W ⊂ µ [ f ] × µ [ f − ] , then multiplying both sides on the leftwith W implies that L ([ f ] , ) is the inverse of µ [ f ] which is clearly false).We have that the fusions µ [ f ] × µ [ f − ] and µ [ f ] × µ [ f − ] only have the trivial anyon incommon. Hence, the combination of subcategory structure and one-form symmetry explainsthe fusion rule µ [ f ] × µ [ f ] = µ [ f f ] . (2.75)It is interesting to note that this discussion parallels the one for Wilson lines in section2.2.2.This example is additionally illuminating because this theory also has a fusion involv-ing a Wilson and a flux line with unique outcome. Indeed, we have two 2-dimensionalrepresentations 2 and 2 of Z ⋊ Z whose restriction to the centralizer N f = Z ⋊ Z areirreducible. Hence, we have the fusion rules W × µ [ f ] = L ([ f ] , | Nf ) , W × µ [ f ] = L ([ f ] , | Nf ) . (2.76)Do we have trivial braiding between the anyons involved in this fusion? This question isequivalent to whether the dyons are bosons are not. For L ([ f ] , i | Nf ) to be a boson, wewant f to be in the kernel of 2 i | f , which is equivalent to the condition that f be in thekernel of 2 i . Using this condition, we can easily check to see that the anyons W and µ [ f ] braid non-trivially with each other, while W and µ [ f ] braid trivially with each other.Moreover, this theory has several fusions involving dyons which give a unique output.For example, consider the dyons L ([ f ] , ˜1 f ) and L ([ f ] , ˜1 f ) , where ˜1 f and ˜1 f are the unique38on-trivial real 1-dimensional representations of N f = Z ⋊ Z and N f = Z × D , respec-tively. We have the fusion L ([ f ] , ˜1 f ) × L ([ f ] , ˜1 f ) = L ([ f f ] , ˜1 f f ) (2.77)where ˜1 f f is the unique non-trivial 1-dimensional representation of N f f = Z .Let us also explore the zero-form symmetry of this theory. We have Out( Z ⋊ Z ) = Z and H ( Z ⋊ Z ) = Z . From theorem 10, we know that this theory features non-trivialself-duality. In fact, the group Z ⋊ Z has three non-trivial normal abelian subgroups Z , Z , Z all of which have trivial 2 nd cohomology group. So we have the Lagrangiansubcategories {L ( Z , , L ( Z , , L ( Z , , L ( Z , } (2.78)Using remark 7.3 in [18], we have L ( N, ≃ Rep(( Z ⋊ Z ) ( N, ) ≃ ˆ N ⋊ ( Z ⋊ Z ) / ˆ N (2.79)where ˆ N is the group of representations of N and N = Z , Z , Z . Also, we have theisomorphisms Z ⋊ Z ≃ Z ⋊ ( Z ⋊ Z ) ≃ Z ⋊ ( Z ⋊ Z ) (2.80)Hence, all Lagrangian subcategories above are isomorphic to Rep( Z ⋊Z ). Hence, | L ( Z ⋊Z ) | = 4. From theorem 9, we know that Aut br ( Z (Vec Z ⋊Z )) should act transitivelyon | L ( Z ⋊ Z ) | . In fact, we can use proposition 7.11 of [18] to show that H ( Z ⋊Z , U (1)) ⋊ Out( Z ⋊ Z ) ≃ Z acts trivially on | L ( Z ⋊ Z ) | . Using theorem 9, we have | Aut br ( Z (Vec Z ⋊Z )) | = 8.Finally, since Z ⋊ Z has complex characters, Z (Vec ω Z ⋊Z ) has a non-trivial quasi-zero-form symmetry given by charge conjugation. We have used the software GAP to search for groups for which the corresponding untwisteddiscrete gauge theories have fusions rules with unique outcomes. We present our resultsbelow. The relevant GAP code is given in Appendix C.
Fusion of Wilson lines
Irreducible representations of a direct product of groups is the product of representationsof the individual groups. Hence, it is natural that the first example with two Wilson linesfusing to give a unique Wilson line is the quantum double of S × S (however, this fusion39rises because the discrete gauge theory factorizes; this follows from theorem 6). Moreinteresting (non-direct-product) groups with this property only appear at order 48 (seeAppendix A). For groups of order less than or equal to 639 (except orders 384, 512, 576) we have verified that whenever the corresponding untwisted discrete gauge theory has afusion Wilson lines giving a unique outcome, Aut br Z (Vec G ) is non-trivial. In this set ofgroups, there are two which have a trivial automorphism group. They are S × ( Z ⋊ Z )and ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z . However, H ( S × ( Z ⋊ Z ) , U (1)) = Z leading tonon-trivial Aut br ( Z (Vec S × ( Z ⋊Z ) )). The group ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z has trivial H ( G, U (1)). So the theory Z (Vec ((( Z × Z ) ⋊ Q ) ⋊Z ) ⋊Z ) doesn’t have classical symmetries.((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z has only one abelian normal subgroup N = Z × Z . Moreover,we have ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z ≃ N ⋊ K where K = GL (2 , S × ( Z ⋊ Z ) and ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z have complex characters, hence thecorresponding discrete gauge theories have quasi-zero-form symmetries. Fusion of flux lines
The simplest example of an untwisted discrete gauge theory with a fusion of two flux linesgiving a single outcome is Z (Vec S × S ). The conjugacy classes of a direct product is aproduct of conjugacy classes of the individual groups. Hence, it follows that quantumdoubles of direct products naturally have such fusions. As mentioned above, it followsfrom theorem 6 that discrete gauge theories based on direct product groups are non-prime.Therefore, the fusion rules with unique outcome in this case are a consequence of theDeligne product. Since Out( S × S ) = Z , Z (Vec S × S ) has non-trivial zero-form symmetry.After S × S , we have several groups of order 48 with flux fusions giving unique outcome.The examples discussed in Appendix A (except BOG and GL (2 , br ( Z (Vec G )) is non-trivial. In fact, the only group with a trivial automorphismgroup in this set is S × ( Z ⋊ Z ). We already discussed above that this theory has non-trivial zero-form symmetries as well as non-trivial quasi-zero-form symmetries. We have not checked order 384, 512, 576 due to the huge number of groups (up to isomorphism) withthese orders. usion of a Wilson line with a flux line The simplest example with a fusion of a Wilson line and a flux line giving a single outcomeis Z (Vec S × S ). Then we have more examples in order 48. The examples discussed inAppendix A (except BOG and GL (2 , br Z (Vec G ) is non-trivial. In this set of groups, there arethree which have a trivial automorphism group. They are S × ( Z ⋊ Z ), ( Z × Z ) ⋊ QD (where QD is the semi-dihedral group of order 16) and ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z .We discussed the groups S × ( Z ⋊ Z ) and ((( Z × Z ) ⋊ Q ) ⋊ Z ) ⋊ Z above. Thegroup ( Z × Z ) ⋊ QD has trivial H ( G, U (1)). So the theory Z (Vec ( Z × Z ) ⋊ QD ) doesn’thave classical symmetries. However, ( Z × Z ) ⋊ QD has one abelian normal subgroup N = Z × Z . Moreover, we have ( Z × Z ) ⋊ QD ≃ N ⋊ K where K = QD . Therefore,using theorem 10, we know that the corresponding untwisted discrete gauge theory hasnon-trivial electric-magnetic self-duality.The group ( Z × Z ) ⋊ QD has complex characters, hence the corresponding discretegauge theory has quasi-zero-form symmetries. Fusion of general dyons
Being a Deligne product, Z (Vec S × S ) also has fusions involving dyons, and this is thesmallest rank theory with such fusions. The next example is in order 48. The examplesdiscussed in Appendix A exhausts all such groups of order 48. For groups of order less thanor equal to 100 we have verified that whenever the corresponding untwisted discrete gaugetheory has a fusion of two dyons giving a unique outcome, Aut br Z (Vec G ) is non-trivial.In fact, every group in this set has non-trivial automorphism group. Hence, they all havenon-trivial classical 0-form symmetries. G k , cosets, and a × b = c In this section, we turn our attention to a (generally) very different set of theories: TQFTsbased on G k Chern-Simons (CS) theories and cosets thereof (here G is a compact simpleLie group). Unlike the theories discussed in section 2, the theories we discuss here aretypically chiral (i.e., c top = 0 (mod 8)). 41n order to gain a sense of what such theories allow us to do in constructing TQFTswith fusion rules of the form (1.2) and (1.5), it is useful to recall the basic representationtheory of SU (2). Somewhat surprisingly, this intuition will be quite useful for more general SU ( N ) k CS theories. To that end, consider the textbook matter of the fusion of SU (2)spin j and j representations j ⊗ j = j + j X j = | j − j | j . (3.1)As in the case of the finite groups in the previous section, we would like to understand ifwe can have j ⊗ j = j for j , j > j spin. Clearly this is impossible, sincewe would have j + j > | j − j | and the sum (3.1) will have at least two contributions.While this result is rather trivial, it is useful to recast it using the group theory analogof the F -transformation described in the introduction (as well as in section 2 for the caseof discrete groups). To that end, we wish to consider j ⊗ j = j X j =0 j , j ⊗ j = j X k =0 k , | j , | > , (3.2)where | j , | are the dimensions of the representations. In particular, we see that (since j , >
0) both products in (3.2) must always contain the trivial representation and theadjoint representation. This observation also implies that j ⊗ j = j for fixed j spin.The discussion around (3.2) easily generalizes to arbitrary compact simple Lie group, G . In particular, let us consider α ⊗ ¯ α = 1 + X γ =1 N γα ¯ α γ , β ⊗ ¯ β = 1 + X δ N δβ ¯ β δ , | α | , | β | > , (3.3)where α, β and ¯ α, ¯ β are conjugate higher-dimensional irreducible representations of G ,Irr( G ). The number of times the adjoint appears in the product α ⊗ ¯ α is [40]: N adj α ¯ α = (cid:12)(cid:12)(cid:12)n λ ( α ) j = 0 o(cid:12)(cid:12)(cid:12) ≥ , (3.4)where λ ( α ) j are the Dynkin labels of α . Therefore, we learn that for all higher-dimensionalrepresentations of G α ⊗ β = γ , ∀ | α | , | β | > , α, β, γ ∈ Irr( G ) , (3.5)Of course, our interest is in the fusion algebra of G k . From this perspective, the abovediscussion is in the limit k → ∞ . As we will prove in the next section, taking G k = SU ( N ) k and imposing finite level does not lead to fusions of the form (1.2) or (1.5).42 .1. G k CS theory
Let us now consider the finite-level deformation of the fusion rules discussed in the previoussection. These are the fusion rules of Wilson lines in G k CS theory. We first consider SU (2) k as it is rather illustrative. We will then generalize to SU ( N ) k and comment onmore general G k .In the case of SU (2) k , (3.1) becomes [41, 42] j ⊗ j = min( j + j ,k − j − j ) X j = | j − j | j . (3.6)In addition to truncating the spectrum to the spins { , / , , · · · , k/ } , the above deforma-tion abelianizes the spin k/ k/ ⊗ k/ j ⊗ j = j for j non-abelian irreducible j , , . Indeed, consider j ⊗ j = min(2 j ,k − j ) X j =0 j , j ⊗ j = min(2 j ,k − j ) X j =0 j , j , = 0 , k . (3.7)The conditions j , = 0 , k are to ensure that the representation is non-abelian. In particular,we again see that the adjoint representation appears in (3.7).While the fusion rules discussed in [41,42] apply to more general groups, they are ratherdifficult to implement. Instead, using proposals suggested in [43] and finally proven in [44],the authors of [45] show that for α an irreducible representation of G k (with G a compactsimple group), we have ( k ) N adj α ¯ α = (cid:12)(cid:12)(cid:12)n ˆ λ ( α ) j = 0 o(cid:12)(cid:12)(cid:12) − , (3.8)where ˆ λ αj are the associated affine Dynkin labels.In particular, for SU ( N ) k , if | α | >
1, then ( k ) N adj α ¯ α ≥ Indeed, the abelian repre-sentations, γ i , satisfy a Z N fusion algebra and are characterized by ˆ λ ( γ i ) j = kδ ij , where i ∈ { , , ..., N − } . On the other hand, all non-abelian representations have at least twonon-zero Dynkin labels. As a result, we learn that α ⊗ β = γ , ∀ α, β, γ ∈ Irr( SU ( N ) k ) , | α | , | β | > . (3.9)Therefore, we see that we have the following fusions for non-abelian Wilson lines in SU ( N ) k CS TQFT W α × W β = W γ + · · · , | α | , | β | > , (3.10) Here we define | α | to be the quantum dimension. G k CS theory (with G compact and simple) for which the lines in questioncorrespond to affine representations with at least two non-zero Dynkin labels. We can apply the above arguments to learn about global properties of G k CS theory. Forexample, we can ask if G k CS theory is prime or not. The answer is no in general. Indeed,consider the case G = SU (2). For k ∈ N even , SU (2) k is prime. However, for k ∈ N odd ,the abelian anyon generating the Z one-form symmetry forms a modular subcategory. ByM¨uger’s theorem [8] (see also [14] for a discussion at the level of RCFT), it then decouplesand the theory resolves into a product of two prime theories SU (2) k ≃ SU (2) ⊠ SU (2) int k , if k = 1 (mod4) SU (2) ⊠ SU (2) int k , if k = 3 (mod4) . (3.11)where SU (2) int k is a TQFT built out of the integer spin SU (2) k representations. Here SU (2) is the TQFT conjugate to SU (2) (these TQFTs are sometimes called the anti-semion andsemion theories in the condensed matter literature).While G k CS theory is not prime in general, our arguments above readily prove thefollowing:
Claim 14:
Non-abelian Wilson lines in SU ( N ) k CS theory must all lie in the same primeTQFT factor. For more general G k CS theory (with G compact and simple), all Wilsonlines corresponding to affine representations with at least two non-zero Dynkin labels mustbe part of the same prime TQFT factor. Proof:
Suppose this were not the case. Then, we would find fusion rules of the form (3.10)with no Wilson lines in the ellipses. (cid:3)
Clearly, to produce fusion rules of the form (1.2) for non-abelian Wilson lines in the sameprime TQFT, we will need to go beyond SU ( N ) k CS theory. One way to proceed is toconsider coset theories and use some intuition from section 2. Indeed, since cosets can havefixed points (which we will describe below), it is natural to think they can lead to fusionrules of the form (1.2). For general G k (with G compact and simple), all fusions of non-abelian Wilson lines we are aware ofinvolve multiple fusion outcomes. It would be interesting to either find an example of a fusion of non-abelianlines with a unique outcome in such a theory or else prove a theorem forbidding it. .2. Virasoro minimal models and some cosets without fixed points We begin with a discussion of the Virasoro minimal models, as these are simple examplesof theories that are related to cosets. While these cosets do not have fixed points, theyturn out to produce factorized TQFTs that are nonetheless illustrative. In the next section,we will focus on cosets that have fixed points, and we will see how to engineer fusion rulesof the form (1.2).One way to construct the Virasoro minimal models is to take a three-dimensional space-time R × Σ and place SU (2) k − × SU (2) CS theory on I × Σ, where I is an interval in R .We can place SU (2) k CS theory outside this region. At the two 1 + 1 dimensional interfacesbetween the CS theories (which form two copies of Σ, call them Σ , ), we obtain the leftand right movers of the RCFT. Here the chiral (anti-chiral) primaries lie where endpointsof Wilson lines from the SU (2) k and SU (2) k − × SU (2) theories meet on Σ (Σ ).Another way to think about the Wilson lines related to the Virasoro minimal modelsis to start with SU (2) k − × SU (2) CS theory and change variables to make an SU (2) k subsector manifest [46]. Integrating this sector out leaves an effective coset TQFT.The end result is that the TQFT we are interested in is T p = SU (2) p − ⊠ SU (2) SU (2) p − , p ≥ . (3.13)Here, the natural number p ≥ p = 3 for the Ising model). We may construct the MTC data underlying theRCFT and the coset TQFT by taking products (e.g., see [48]) F T p = F SU (2) p − · F SU (2) · ¯ F SU (2) p − , R T p = R SU (2) p − · R SU (2) · ¯ R SU (2) p − . (3.14)In order to make (3.14) precise, we need to explain how the states in T p are related tothose in the individual SU (2) k theories that make up the coset. Let us denote the SU (2) p − , SU (2) , and SU (2) p − weights as λ , µ , and ν . Then, to build the coset we should identifyWilson lines as follows W { λ,µ,ν } := W λ × W µ × W ν ≃ ( W p − × W λ ) × ( W × W µ ) × ( W p − × W ν ) , (3.15) This is the TQFT analog of the classic result [47] for the corresponding affine algebras:Vir p ≃ b su (2) p − × b su (2) b su (2) p − . (3.12) In writing (3.13), we have used the Deligne product to emphasize the fact that the SU (2) p − × SU (2) CS theory is a product TQFT. W p − , W , and W p − are abelian Wilson lines transforming in the weight p − p − / / p − p − /
2) representations of thedifferent TQFT factors. Moreover, in order to be a valid Wilson line in T p , we shoulddemand that our Wilson lines satisfy W { λ,µ,ν } ∈ T p ⇔ λ + µ − ν ∈ Q ⇔ λ + µ + ν = 0 (mod 2) , (3.17)where Q is the SU (2) root lattice. This relation guarantees that all lines that remainhave trivial braiding with W { p − , ,p − } (which is a boson that is in turn identified with thevacuum). It is in terms of these degrees of freedom that (3.14) should be understood.Before proceeding, let us stop and note that the fusion in (3.15) has no fixed points.Indeed, this statement readily follows from the fact that SU (2) is an abelian TQFT, andabelian theories cannot have fixed points since their fusion rules are those of a finite abeliangroup (in this case Z ).Given this groundwork, we claim that T p factorizes as follows T p ≃ ( SU (2) p − ⊠ SU (2) ) int ⊠ SU (2) int p − , if p = 0 (mod 2) SU (2) int p − ⊠ SU (2) conj p − , if p = 1 (mod 2) . (3.18)The various TQFTs appearing in (3.18) are( SU (2) p − ⊠ SU (2) ) int := gen (cid:0)(cid:8) W { λ,µ } ∈ SU (2) p − ⊠ SU (2) | λ + µ = 0 (mod 2) (cid:9)(cid:1) ,SU (2) int p − := gen ( {W ν ∈ SU (2) p − | ν = 0 (mod 2) } ) ,SU (2) conj p − := gen (cid:0)(cid:8) W { λ,µ,ν } | λ + µ + ν = 0 (mod 2) , W λ , W µ abelian (cid:9)(cid:1) ,SU (2) int p − := gen ( {W λ ∈ SU (2) p − | λ = 0 (mod 2) } ) , (3.19)where “gen( · · · )” means that the TQFT is generated by the Wilson lines enclosed. Noticethat in the case that p is even, p − SU (2) int p − is precisely the decoupled TQFTfactor required by M¨uger’s theorem in (3.11) containing integer spins (even Dynkin labels).Similar logic applies to SU (2) int p − in the case that p is odd. The TQFT SU (2) conj p − has thesame fusion rules as SU (2) p − , but it is a different TQFT. Finally, for the case that p = 3(i.e., the Ising model), we see that T does not factorize. At the level of the corresponding affine algebras, this is the statement that [42] n ˆ λ, ˆ µ, ˆ ν o ≃ n a ˆ λ, a ˆ µ, a ˆ ν o , (3.16)where the hat denotes affine weights, and a is the generator of the (diagonal) O ( b su (2)) outer automorphism. Note also that Ising shares the same fusion rules as SU (2) , though they are not the same TQFTs. Forexample, the σ fields have different twists. SU (2).To that end, let us first take the case of p ≥ S matrix also takes a product form S { λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } = S ( p − λλ ′ · S (1) µµ ′ · S ( p − νν ′ , θ { λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } = θ ( p − λλ ′ · θ (1) µµ ′ · ¯ θ ( p − νν ′ , (3.20)where the superscripts on the righthand sides of the above equations refer to the corre-sponding factors in the coset (3.13). From the S matrix, Verlinde’s formula yields (see alsothe discussion in [42]) N { λ ′′ ,µ ′′ ,ν ′′ }{ λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } = N ( p − λ ′′ λλ ′ · N (1) µ ′′ µµ ′ · N ( p − ν ′′ νν ′ , (3.21)where, again, the superscripts on the righthand side denote the different coset factors in(3.13). The factor SU (2) int p − in the second line of (3.18) is clearly closed under fusion. Sotoo is SU (2) conj p − . To have factorization of the TQFT, we need only show that all Wilsonlines can be written in this way and, by M¨uger’s theorem, that one of these factors ismodular. The second part is trivial: we have already seen that SU (2) int p − is modular in thediscussion surrounding (3.11). We can confirm this statement by looking at the modular S -matrix for SU (2) p − S ( p − λλ ′ = r p sin (cid:18) ( λ + 1)( λ ′ + 1) πp (cid:19) . (3.22)and taking the submatrix involving the integer spins (even weights).Therefore, we need only check that all states in the coset (3.13) can be expressed inthis way. To that end, we can see that | SU (2) int p − | = p − , | SU (2) conj p − | = p , (3.23)where the norm denotes the number of simple elements within. Therefore, we see that wehave |T p | = p ( p − /
2, which is precisely the number of states in the coset (3.13) (notethat in these computations we have used (3.15) and (3.17)) and the corresponding A -typeVirasoro minimal model.To make contact with the fusion rules in (1.4), we need to explain precisely how cosetlines map onto the Virasoro primaries. The results above allow us to realize the, say, Vira-soro left-movers as states on the boundary of the bulk TQFT, T p ≃ SU (2) int p − ⊠ SU (2) conj p − p odd. Now, we need to see how we can map boundary endpoints of lines inthis theory to Virasoro primaries, ϕ ( r,s ) . To that end, by comparing the S -matrix for T p ≃ SU (2) int p − ⊠ SU (2) conj p − with the corresponding expressions for those of the Virasorominimal models, we have that the labels of the Virasoro primary, ϕ ( r,s ) map as follows (seealso [42]) r = λ + 1 , s = ν + 1 . (3.24)In particular, we see that the ϕ ( r, primaries are endpoints of lines in SU (2) int p − while the ϕ (1 ,s ) are endpoints of lines in SU (2) conj p − . This reasoning explains the fact that non-abelianVirasoro primaries of these types have unique fusion outcomes ϕ ( r, × ϕ (1 ,s ) = ϕ ( r,s ) , (3.25)discussed in the introduction (at least for p odd). As an example, we have T ≃ Ising (i.e.,the TQFT is the Ising MTC), which does not factorize. On the other hand, for p = 5, wehave T = ( G ) ⊠ SU (2) conj4 , (3.26)where ( G ) is the so-called “Fibnonacci” TQFT, and SU (2) conj4 is a TQFT with the samefusion rules and S -matrix as SU (2) .Let us now consider p ≥ SU (2) int p − and ( SU (2) p − ⊠ SU (2) ) int areseparately closed under fusion. Moreover, just as before, we can use the discussion around(3.11) and M¨uger’s theorem to conclude that SU (2) int p − is indeed a decoupled TQFT asclaimed in (3.18).We should again check that all states in (3.13) can be reproduced. To that end, wehave | SU (2) int p − | = p , | ( SU (2) p − ⊠ SU (2) ) int | = p − . (3.27)As a result, we have |T p | = p ( p − /
2, which is the correct number of states in the coset(3.13) and the corresponding A -type Virasoro minimal model.Our mapping is again as in (3.24), but now ϕ ( r, primaries are endpoints of lines in( SU (2) p − ⊠ SU (2) ) int , and ϕ ,s are endpoints of lines in SU (2) int p − . This again explainsthe fusion outcomes in (3.25) for the case of p even as well. As an example, note that T = Ising ′ ⊠ ( F ) , (3.28) Though, again, we stress that this factorization is not a factorization of RCFT correlators. SU (2) ), and the second factor is the time reversal of the Fibonacci theory in (3.26).As a result, we conclude that, although the TQFTs discussed in this section do havenon-abelian anyons fusing to give a unique outcome, this is due to the fact that the corre-sponding TQFTs factorize. In section 2 we saw that fixed points of various kinds gave rise to fusion rules of the form(1.5) (in particular, see theorem 1 of section 2.2.3). In the context of cosets, we can alsonaturally engineer fixed points under the action of fusion with abelian anyons generatingidentifications of fields. In the case of Virasoro, this didn’t happen (see (3.15)). Indeed,this statement followed from the fact that we had an abelian factor in the coset (3.13).The simplest way to get around this obstacle and generate fixed points is to considerinstead b T p = SU (2) p − ⊠ SU (2) SU (2) p , (3.29)where p ≥ p ≥ N = 1 super-Virasorominimal models [47, 49]. Note that the case of p = 3 corresponds to the T case discussedpreviously (i.e., to the TQFT related to the tri-critical Ising model).For the theories in (3.29), we find the following generalization of the identificationcondition in (3.15) W { λ,µ,ν } := W λ × W µ × W ν ≃ ( W p − × W λ ) × ( W × W µ ) × ( W p × W ν )= W p − − λ × W − µ × W p − ν , (3.30)In particular, if λ = ( p − / µ = 1, and ν = p/
2, we can have a fixed point. Of course, if p is odd, we don’t have a fixed point. In this case, we can again run logic similar to thatused in the Virasoro case to argue that the TQFT factorizes.However, if p is even, then we need to properly define the coset. In particular, we shouldresolve the fixed point Wilson line as follows (see [15, 50] for the dual RCFT discussion) W { ( p − / , ,p/ } → W (1) { ( p − / , ,p/ } + W (2) { ( p − / , ,p/ } . (3.31) We also require that λ + µ + ν = 0 (mod 2) so that the lines in the coset theory have trivial braidingwith the bosonic line W { p − , ,p } . This line is in turn identified with the vacuum. p = 6 b T = SU (2) ⊠ SU (2) SU (2) . (3.32)The fixed point resolution in (3.31) becomes W { , , } → W (1) { , , } + W (2) { , , } . As in the casesof one-form gauging with fixed points discussed in section 2, it is natural that there shouldbe a zero-form symmetry exchanging W (1) { , , } ↔ W (2) { , , } .As a first step to better understand the theory after resolving the fixed point, note that b T has the following number of lines | b T | = 28 . (3.33)Of these fields, twenty-six come from identifying full length-two orbits in (3.30) while twocome from resolving the fixed point. In what follows, { λ, µ, ν } will denote fields in fullorbits, while labels of the form { , , } ( i ) (with i = 1 ,
2) will denote the fixed point lines.To understand the fusion rules and the question of primality after fixed point resolution,we can compute the S matrix using the algorithm discussed in [50] (let us denote the resultby ˜ S ). It takes the form˜ S { λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } = 2 S { λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } , ˜ S { , , } ( i ) { λ ′ ,µ ′ ,ν ′ } = S { , , }{ λ ′ ,µ ′ ,ν ′ } , ˜ S { , , } ( i ) { , , } ( j ) = 12 − − ! , (3.34)where S { λ,µ,ν }{ λ ′ ,µ ′ ,ν ′ } = S ( p − λλ ′ · S (2) µµ ′ · S ( p ) νν ′ , (3.35)is the naive generalization of (3.20) to the cosets at hand. Note that the fusion rules weobtain from ˜ S for fields not involving { , , } ( i ) are the naive ones we get from S via therestrictions and identifications described above.The above discussion is sufficient to prove that b T is prime. Indeed, we see from (3.34)that the fields that come from identifying length-two orbits have the quantum dimensionsthey inherit from S . The fixed point resolution fields, on the other hand, have half thequantum dimension of the fixed point field. We therefore have the following four abeliananyons generating a Z × Z fusion algebra W { , , } ≃ W { , , } , W { , , } ≃ W { , , } , W { , , } ≃ W { , , } , W { , , } ≃ W { , , } . (3.36)By (3.34), we see that the braiding amongst abelian anyons is not affected by taking S → ˜ S .As a result, we see that the four abelian anyons all braid trivially. Therefore, they cannotform a decoupled TQFT. 50ilson lines Quantum dimensions W { , , } , W { , , } , W { , , } , W { , , } W { , , } , W { , , } , W { , , } , W { , , } cot (cid:0) π (cid:1) W { , , } , W { , , } , W { , , } , W { , , } q csc (cid:0) π (cid:1) W { , , } , W { , , } (1) , W { , , } (2) √ (cid:0) π (cid:1) W { , , } , W { , , } √ (cid:0) π (cid:1) W { , , } , W { , , } W { , , } , W { , , } csc (cid:0) π (cid:1) W { , , } , W { , , } √ W { , , } , W { , , } (cid:0) π (cid:1) W { , , } , W { , , } √ (cid:0) π (cid:1) W { , , } (cid:0) π (cid:1) Table 1:
The twenty-eight Wilson lines and associated quantum dimensions in the b T TQFT.Given this discussion, what could a putative factorized theory look like? Since b T hasorder 28 = 7 · , we see that the only way to have a non-trivial factorization is to have afactorization of the form ˜ T ⊠ ˜ T into prime TQFTs with rank fourteen and rank two, or˜ T ⊠ ˜ T with prime TQFTs of rank seven and four, or ˜ T ⊠ ˜ T ⊠ ˜ T ′ with prime TQFTs ofrank seven, two, and two.Let us consider the first factorization first. Since the abelian anyons (and any subsetthereof) cannot form a separate TQFT factor (this factor would be non-modular), theclassification in [5] implies that we have either ˜ T ≃ ( G ) or ˜ T ≃ ( F ) . In any case, thenon-trivial anyon in ˜ T has quantum dimension d τ = (1 + √ /
2. It is easy to check thatno such quantum dimension can be produced from products of quantum dimensions in thedifferent coset factors (and so restrictions cannot produce them either). Moreover, one cancheck that the resolved fixed point fields cannot have this quantum dimension either. Thissame logic applies to the ˜ T ⊠ ˜ T ⊠ ˜ T ′ factorization as well.Therefore, it only remains to consider ˜ T ⊠ ˜ T . The other factor, ˜ T , has four anyons.By [5], this theory is either ( G ) or its time reversal. In either case, we cannot producethe requisite d α = 2 cos( π/
9) quantum dimension from our coset. Therefore, we concludethat b T is indeed a prime TQFT.Moreover, we find the following fusion rules of non-abelian Wilson lines with unique51utcome W { , , } × W { , , } = W { , , } , W { , , } × W { , , } = W { , , } , W { , , } × W { , , } = W { , , } , W { , , } × W { , , } = W { , , } , W { , , } × W { , , } = W { , , } . (3.37)We can obtain additional such fusion rules by taking a product with some of the abelianlines in (3.36).Just as in the case of discrete gauge theories with fusion rules of the above type, ourtheory also has a non-trivial symmetry of the modular data. Indeed, from (3.34), it is clearthat the ˜ S -matrix has a Z symmetry under the interchange g (cid:16) W { , , } (1) (cid:17) = W { , , } (2) , = g ∈ Z . (3.38)Note that this symmetry is not charge conjugation since ˜ S is manifestly real. Moreover,since we don’t change the twists, this action lifts to a symmetry of the modular data(additionally, it should lift to a symmetry of the full TQFT).If we wish to make contact with the N = 1 minimal model, then we should note that thefermionic W { , , } line corresponds to the supercurrent of the SCFT. We can then organizethe Neveu-Schwarz (NS) sector into supermultiplets under fusion with this operator. Doingso (and paying careful attention to the fields in the resolution of the fixed point), we findnine NS sector fields and nine Ramond sector fields as required.There are many ways to generalize the example we have given here. Indeed, when thereare fixed points in the coset construction we expect to often be able to generate fusion rulesof the form (1.2). A deeper understanding of these theories and some more general methodsto characterize whether the cosets are prime (along the lines of the general criteria we havein the case of discrete gauge theories) would be useful. In any case, we see that, as in thecase of discrete gauge theories, symmetry fixed points and zero-form (quasi) symmetriesare deeply connected with fusion rules of the form (1.2).
4. Conclusions
In this paper, we have seen that the existence of fusions of non-abelian anyons havinga unique outcome is intimately connected with the global structure of the correspondingTQFT. In the case of discrete gauge theories, we saw how symmetries and subcategorystructure interact to give rise to such fusion rules. In the case of theories with a continuousgauge group, we saw that we could deduce non-factorization of the non-abelian Wilson lines52n a large class of theories from the absence of such fusion rules. Moreover, we saw thatthere is a universal connection between the existence of fixed points of various types andfusion rules of the form (1.2) in all the classes of theories we considered.The above discussion leads to various natural questions: • In the discussion around (2.39) we explained the large hierarchy between the size ofsimple and non-simple groups whose corresponding discrete gauge theories have non-abelian Wilson lines satisfying (1.2) by using symmetries and subcategory structure.It would be interesting to explore whether other related hierarchies can be explainedin a similar way. • We saw that in almost all the prime untwisted discrete gauge theories we studied, ifthere was a fusion rule of the form (1.2), then the theory had non-trivial zero-formsymmetries. The only exceptions where discrete gauge theories based on the M and M Mathieu groups discussed in section 2.5. Here we argued that there werezero-form symmetries of the modular data that did not lift to symmetries of the fulltheory. It would be interesting to understand if gauge theories based on certain finitesimple sporadic groups are the only prime theories with fusion rules of the form (1.2)that exhibit this phenomenon. • In section 3.1, we proved that the non-abelian lines of SU ( N ) k CS theory don’t havefusion rules of the form (1.2). More generally, we do not know of an example of sucha fusion in G k CS theories with G compact and simple. It would be interesting toeither find an example of such a fusion or prove a more general theorem forbidding one.Given such fusions are common for discrete gauge theories, it would be interesting tounderstand how these two statements interact with each other. • As we saw in section 3.3, it would be useful to develop new tools to understandprimality in theories built on cosets. One promising direction is to study the role ofGalois actions in such theories.
Acknowledgments
We thank D. Aasen, M. Barkeshli, S. Ramgoolam, and I. Runkel for useful discussionsand correspondence. M. B. is funded by the Royal Society grant, “New Constraints andPhenomena in Quantum Field Theory.” M. B. and R. R. are funded by the Royal Societygrant, “New Aspects of Conformal and Topological Field Theories Across Dimensions.” TheSTFC also partially supported our work under the grant, “String Theory, Gauge Theoryand Duality.” 53 ppendix A. Wilson line a × b = c in gauge theories with order forty-eight discretegauge group Let us study groups of order 48 for which the corresponding discrete gauge theories haveWilson line a × b = c type fusions .(48 ,
15) (( Z × D ) ⋊ Z ); W × W = W , W × W = W , W × W = W , W × W = W W × W = W , W × W = W , W × W = W , W × W = W . (A.1)We have Out(( Z × D ) ⋊ Z ) = Z × Z . Let r and r be the generators of this group.They act on the Wilson lines involved in the fusion above as follows r : W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; (A.2) r : W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; (A.3)Since this group has complex characters we also have a non-trivial quasi-zero-form sym-metry given by complex conjugation. Z (Vec ( Z × D ) ⋊Z ) also has all other a × b = c typefusions (involving fluxes and dyons) discussed in this paper.(48 ,
16) (( Z : Q ) ⋊ Z ); This has fusions identical to (A.1). The only difference is thatnow W and W are conjugates. The outer automorphism group and symmetry actionis identical to Z (Vec ( Z × D ) ⋊Z ). Since this group has complex characters we also have anon-trivial quasi-zero-form symmetry given by complex conjugation. We additionally haveall other a × b = c type fusions (involving fluxes and dyons) discussed in this paper.(48 ,
17) (( Z × Q ) ⋊ Z ); This has identical character table to (48 , ,
18) ( Z ⋊ Q ); Identical characters to (48 , ,
39) (( Z × S ) ⋊ Z ); We won’t discuss the direct product groups S × S , D × S and Q × S which also have such fusions(the corresponding discrete gauge theories factorize). Since we have already discussed the case of BOG and GL (2 , × W = W , W × W = W , W × W = W , W × W = W W × W = W , W × W = W , W × W = W , W × W = W . (A.4)We have Out(( Z × S ) ⋊ Z ) = Z × Z . Let r and r be the generators of this group.They act on the Wilson lines involved in the fusion above as follows r : W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; (A.5) r : W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; W ↔ W ; (A.6)Since this group has complex characters we also have a non-trivial quasi-zero-form sym-metry given by complex conjugation. Z (Vec ( Z × S ) ⋊Z ) also have all other a × b = c typefusions (involving fluxes and dyons) discussed in this paper.(48 , Z × S ) ⋊ Z )Fusion of Wilson lines giving unique output is same as (A.4). We have Out(( Z × S ) ⋊Z ) = D .Since this group has complex characters we also have a non-trivial quasi-zero-formsymmetry given by complex conjugation. Z (Vec ( Z × S ) ⋊Z ) also have all other a × b = c type fusions (involving fluxes and dyons) discussed in this paper. Appendix B. Genuine zero-form symmetries and quasi-zero-form symmetries in A discrete gauge theory Recall from section 2.2.1 that A is the simplest example of an A N (with N = k ≥ br ( Z (Vec A )) ≃ H ( A , U (1)) ⋊ Out( A ) ≃ Z × Z , (B.1)from a charge conjugation quasi zero-form symmetry [37].Let us first discuss the outer automorphisms. To that end, recall that A has an outerautomorphism corresponding to conjugation by odd elements of S ⊲ A . Acting with theouter automorphism generated by (89) ∈ S , we see that the following lines are exchanged L ([(123456789)] ,π p ) ↔ L ([(123456798)] ,π p ) , L ([(12345)(678)] ,π n ) ↔ L ([(12345)(679)] ,π n ) , (B.2)55here the relevant conjugacy classes are listed in table 2, and 0 ≤ p ≤
8, 0 ≤ n ≤
14 labelrepresentations of the corresponding Z and Z centralizers (they are also listed in table2). In fact, as described in the main text, the symmetry in (B.2) generates an action onsome of the Wilson lines involved in (2.18) W [3 ] + ↔ W [3 ] − . (B.3)This action can be read off from the character table of A or, equivalently, from the braiding S W [33]+ L ([(12345)(678)] ,πn ) S W L ([(12345)(678)] ,πn ) = χ [3 ] + ([(12345)(678)]) ∗ = −
12 (1 − i √ ,S W [33] − L ([(12345)(678)] ,πn ) S W L ([(12345)(678)] ,πn ) = χ [3 ] − ([(12345)(678)]) ∗ = −
12 (1 + i √ ,S W [33]+ L ([(12345)(679)] ,πn ) S W L ([(12345)(679)] ,πn ) = χ [3 ] + ([(12345)(679)]) ∗ = −
12 (1 + i √ ,S W [33] − L ([(12345)(679)] ,πn ) S W L ([(12345)(679)] ,πn ) = χ [3 ] − ([(12345)(679)]) ∗ = −
12 (1 − i √ . (B.4)Note that, since the [(12345)(678)] and [12345)(679)] conjugacy classes are complex, wealso have a non-trivial Z charge conjugation that acts on the modular data and swaps W [3 ] + ↔ W [3 ] − and L ([(123456789)] ,π p ) ↔ L ([(123456798)] ,π p ) . Recall from the discussion in (2.53)that elements of H ( A , U (1)) ≃ Z act trivially on the Wilson lines. Hence, we learn thatcharge conjugation cannot be a genuine symmetry of the TQFT (this statement is alsoconfirmed by the analysis in [37]).However, this is not a contradiction with what we have written, because Out( A ) alsointerchanges the real conjugacy classes [(123456789)] and [(123456798)] along with thecorresponding lines in (B.2). Since charge conjugation leaves these degrees of freedomuntouched, it is a distinct operation.Note that in the A discrete gauge theory we can also turn on a large variety of twists ω ∈ H ( A , U (1)) ≃ Z × Z × Z ≃ Z × Z . (B.5)Since the charge conjugation quasi-symmetry is a property of the Wilson line fusion rules,it remains regardless of the twist. Appendix C. GAP code
The following GAP code defines the function checkdyon() which takes in a group as anargument. It checks for a × b = c type fusions for non-abelian anyons a, b, c ∈ Z (Vec G )56onjugacy class Length Centralizer1 1 A [(12)(34)] 378 SmallGroup(480 , , , , , ( D × Z )[(123)(456)] 3360 SmallGroup(54 , , , (( Z × Z × Z ) ⋊ Z )[(1234)(56)] 7560 SmallGroup(24 , , ( S × Z )[(1234)(567)(89)] 15120 SmallGroup(12 , , ( Z )[(1234)(5678)] 11340 SmallGroup(16 , , (central product D , Z )[(12345)] 3024 SmallGroup(60 , , , ( Z × Z )[(12345)(678)] 12096 SmallGroup(15 , , ( Z )[(12345)(679)] 12096 SmallGroup(15 , , ( Z )[(123456)(78)] 30240 SmallGroup(6 , , ( Z )[(1234567)] 25920 SmallGroup(7 , , ( Z )[(123456789)] 20160 SmallGroup(9 , , ( Z )[(123456798)] 20160 SmallGroup(9 , , ( Z ) Table 2:
The eighteen conjugacy classes of A , their order, and their centralizers (recallthat the centralizers of elements in the same conjugacy class are isomorphic). The central-izer is labeled by its GAP ID (for sufficiently small groups) as “SmallGroup( a, b )” alongwith a more common name in certain cases.57nd ouputs all such fusions. Moreover, if such fusions exist, it outputs Out( G ) as well as H ( G, U (1)). Note that it requires the package HAP to function.In order to define checkdyon() we need to first define the functions comconj() andconjprof(). > conjcom:=function(a,b) > local com,i,j; > com:=[]; > for i in [1..Size(AsList(a))] do > for j in [i..Size(AsList(b))] do > Append(com, [AsList(a)[i]*AsList(b)[j]*Inverse(AsList(b)[j]*AsList(a)[i])]); > od; od; > return DuplicateFreeList(com)=[AsList(a)[1]*Inverse(AsList(a)[1])]; end;This function takes two conjugacy classes of a group G as inputs and outputs true if theycommute element-wise and false otherwise. Now, let us define the function conjprod() > conjprod:=function(a,b,c) > local prod,i,j,k; > prod:=[]; > for i in [1..Size(AsList(a))] do > for j in [i..Size(AsList(b))] do > for k in [1..Size(c)] do > if AsList(a)[i]*AsList(b)[j] in AsList(c[k]) then > Append(prod, [k]); break; fi; od; od; od; > if Size(DuplicateFreeList(prod))=1 then > return DuplicateFreeList(prod)[1]; else return 0; fi; end;This function takes three arguments. The first two arguments a, b are two conjugacy classesof a group G and the third argument c is the set of all conjugacy classes of G . The functionoutputs an integer k > k in the list of conjugacy classes c ). The function outputs 0 otherwise.58sing these two functions, we finally define the checkdyon() function.checkdyon:=function(G) > local cn,i,j,k,a,l,cen1,cen2,cen3,cenint,irrcenint,irrcen1,irrcen2,irrcen3,cen1res,cen2res,cen3res,x,y,z,w,a1,a2,A,I,F,R; > cn:=ConjugacyClasses(G); > a:=0; > for i in [1..Size(cn)] do > for j in [i..Size(cn)] do > if conjcom(cn[i],cn[j]) then > k:=conjprod(cn[i],cn[j],cn); > if k <> > cen1:=Centralizer(G,AsList(cn[i])[1]); > cen2:=Centralizer(G,AsList(cn[j])[1]); > cen3:=Centralizer(G,AsList(cn[k])[1]); > cenint:=Intersection(cen1,cen2,cen3); > irrcen1:=Irr(cen1); > irrcen2:=Irr(cen2); > irrcen3:=Irr(cen3); > cen1res:=RestrictedClassFunctions(irrcen1,cenint); > cen2res:=RestrictedClassFunctions(irrcen2,cenint); > cen3res:=RestrictedClassFunctions(irrcen3,cenint); > irrcenint:=Irr(cenint); > for x in [1..Size(cen1res)] do > for y in [1..Size(cen2res)] do > if Size(AsList(cn[i]))*DegreeOfCharacter(cen1res[x]) > > for z in [1..Size(cen3res)] do > a1:=[ ]; a2:=[ ]; > for w in [1..Size(irrcenint)] do > Append(a1,[ScalarProduct(irrcenint[w],cen1res[x]*cen2res[y])]); > Append(a2,[ScalarProduct(irrcenint[w],cen3res[z])]); > od; > if a1*a2=1 andSize(AsList(cn[i]))*DegreeOfCharacter(cen1res[x])*Size(AsList(cn[j]))*DegreeOfCharacter(cen2res[y])=Size(AsList(cn[k]))*DegreeOfCharacter(cen3res[z]) then > a:=1; > Print(IdSmallGroup(G), “ ”, StructureDescription(G), “ \ n”); > Print(“Anyon a: ”, cn[i], “ , ”, irrcen1[x], “ \ n”); > Print(“Anyon b: ”, cn[j], “ , ”, irrcen2[y], “ \ n”); > Print(“Anyon c: ”, cn[k], “ , ”, irrcen3[z], “ \ n”,” \ n”); > fi; od; fi; od;od; fi; fi; od; od; > if a=1 then > A:=AutomorphismGroup(G); > I:=InnerAutomorphismsAutomorphismGroup(A); > F:=FactorGroup(A,I); > Print(“Out(G): ”,StructureDescription(F), “ \ n”); > R:=ResolutionFiniteGroup(G,3); > Print(“H2(G,U(1)): ”,Homology(TensorWithIntegers(R),2),“ \ n”); > Print(“ \ n”,“ \ n”); fi; > end; 60 eferences [1] G. 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