Fusion rules for permutation extensions of modular tensor categories
FFusion rules for permutation extensions of modular tensorcategories
Colleen DelaneySeptember 9, 2019
Abstract
We give a construction and algorithmic description of the fusion ring of permutation extensions of anarbitrary modular tensor category using a combinatorial approach inspired by the physics of anyons andsymmetry defects in bosonic topological phases of matter. The definition is illustrated with examples,namely bilayer symmetry defects and S -extensions of small modular tensor categories like the Ising andFibonacci theories. An implementation of the fusion algorithm is provided in the form of a Mathematicapackage. We introduce the notions of confinement and deconfinement of anyons and defects, respectively,which develop the tools to generalize our approach to more general fusion rings of G -crossed extensions. Modular tensor categories and their module categories provide an algebraic framework to describe anyons,boundaries, and defects in (2+1)D bosonic topological phases of matter. A special case is when a modu-lar tensor category (MTC) with a G -action admits a G -crossed graded extension by a family of invertiblebimodule categories.When G acts not just as a group on the decategorified part of an MTC, namely its fusion ring, but actson C by braided monoidal autoequivalences, then the simple objects in the invertible bimodule categories C g describe point-like “twist defects” at the end of invertible domain walls that can be manifest by adding termsto the Hamiltonian of the anyon theory. Depending on whether the C g form a fusion category extending C ,the extension is interpreted as symmetry-enriched topological (SET) order or anomalous SET order. In thelatter case, there exists a G -crossed braided fusion category which can be interpreted as the algebraic theoryof anyons and symmetry defects in (2+1)D bosonic SET order.The obstruction theory determining the existence and subsequent classification of G xBFCs was givenin by Etingof, Nikshych, and Ostrik in [15]: the obstructions to lifting a group action on a braided fusioncategory C to a categorical group action and the obstructions to the pentagon axioms needed for the C g toform a G -crossed braided fusion category (GxBFC) are measured by cohomology classes in H ( G ; A ) and H ( G ; U (1)) , respectively. The inequivalent G xBFCs then form a H ( G ; A ) × H ( G ; U (1) -torsor.While [15] provides a classification of SET order, it is still worthwhile to give an explicit constructionof G xBFCs in terms of MTCs and suitable categorical group actions by G both from the point of view ofcomputational physics and abstract quantum algebra.A simple but important example is the case of permutation symmetry of multilayer topological order,corresponding to S n -crossed braided extensions of Deligne product MTCs of the form C (cid:2) n . While itscounterpart in conformal field theory (permutation orbifolding) is well understood, and algebraic data for1 a r X i v : . [ m a t h . QA ] S e p ermutation extensions has been described in the language of TQFT, for practitioners of condensed mattertheory it is desirable to have a construction purely in terms of MTCs.In this paper we develop an elementary approach to modeling the fusion rings of GxBFCs in order tounderstand the fusion rules satisfied by topological charges and symmetry defects in SET phases of matterand apply it to construct the fusion rings for permutation extensions (cid:0) C (cid:2) n (cid:1) × S n of MTCs.We define an H ( S n ; A (cid:2) n ) -torsor of S n -crossed ring extensions of the fusion ring of a Deligne productcategory C (cid:2) n for arbitrary modular tensor categories C . The main theorem is an explicit construction of thepossible fusion rules for ( C (cid:2) n ) × S n in terms of the fusion rules for C and a choice of 2-cocycle valued in A (cid:2) n ,the group of abelian anyon types in C (cid:2) n . Theorem (Fusion rings of permutation extensions of MTCs) . The fusion rings of S n -crossed braided exten-sions of C (cid:2) n are given by the permutation defect fusion rings ( C × S n , ⊗ ω ) , which can be computed from thedata ( C, n, ω ) . Our construction gives a short algorithm to compute the fusion rules.
Algorithm (Permutation defect fusion algorithm) . The fusion product of two permutation defects X σ(cid:126)a and X τ(cid:126)b can be computed as follows.1. σ - and τ -deconfinement:Strip the topological charges from the defects and twist with the abelian anyon ω ( σ, τ ) .2. Transposition defect annihilation:Compute the fusion product of the bare σ - and τ -defects, for every pair of indices ( ij ) permuted byboth σ and τ , pulling out a factor of (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · . στ -confinement:Confine the product of the objects from Step 1 and Step 2 with the bare στ -defect. An implementation in the form of a Mathematica package
PermutationDefectFusion.m is provided atthe author’s website, see Appendix A for a code sample showing the main function.We cast the statements and proofs of our results purely in terms of classical abstract algebra - rings,modules, and group actions - to demonstrate the principle that classical things can be understood throughclassical means. On the other hand, higher data requires higher data. But the benefit of the topologicalphase-inspired approach is not just that it is straightforward to construct the fusion rings, it becomes trans-parent how to categorify the fusion rings, which we discuss in upcoming work [10] and provides alternatecategorical proofs of the results herein.
The fusion rules for permutation defects were first understood in terms of modular functors in [5]. Theapproach taken here was independent and uses the data ( C, n, ω ) consisting of a fusion ring C of an MTC C , a choice of n , and a choice of -cocycle. 2n contemporaneous work [6], Bischoff and Jones give a more general algorithm for fusion rules forspherical G -crossed braided extensions of arbitrary fusion categories C using different methods. They ap-ply it to derive formulas for the fusion rules of maximal cyclic subextensions (cid:0) C (cid:2) n (cid:1) × Z /n Z of permutations (cid:0) C (cid:2) n (cid:1) × S n , giving an alternate description to the one that follows from our Section 4.The author thanks Christoph Schweigert for bringing his and his collaborators’ work to their attentionand to Corey Jones and Marcel Bischoff for coordinating the publication of related results. Thanks as wellto Eric Samperton for helpful advice with an earlier version of this manuscript. The perspective developedhere was heavily influenced by the author’s time at Microsoft Station Q and thanks Zhenghan Wang, MikeFreedman, and Parsa Bonderson. We begin in Section 2 with a brief review of the algebraic theory of anyons and symmetry defects andestablish the notions needed to work with them at the level of their fusion rings. In Section 3 we introducethe terminology and tools like g -confinement and g -deconfinement that are then applied in Section 4 toconstruct the permutation defect fusion ring, i.e. the fusion ring of the GxBFCs (cid:0) C (cid:2) n (cid:1) × S n . Examples when n = 2 and n = 3 are given in Section 5 to demonstrate the defect fusion algorithm and illustrate the mainfeatures of the general theory from Section 3.We conclude our discussion in Section 6 with a brief comment on generalizations of our approach forgeneral G xBFCs and applications. Although our goal is to understand permutation extensions of MTCs as categories, the approach used hereto compute the S n -crossed fusion ring doesn’t require any higher data. The fusion ring and its physicalinterpretation can be understood in terms of elementary abstract algebra, requiring only knowledge of thesymmetric group S n , Z + -based rings (in the sense of [14]), and second group cohomology. For this reasonwe present our results in a “decategorified” way and refer the reader to other sources for detailed definitionsof MTCs and GxBFCs and their interpretation as algebraic theories of anyons and symmetry defects in(2+1)D topological phases of matter see [1, 9].That being said, we will still use the notation ⊗ and ⊕ for multiplication and addition in the fusion ring.Especially in Section 4 we abuse notation and write objects to mean their isomorphism class, conflatinganyons and defects with their types.These choices have the effect of making the proofs of our results elementary, if a bit inelegant. However,the techniques we use here to construct G -crossed braided fusion rings readily suggest the form of theircategorification, which we construct in an upcoming sequel [10] and which provides a categorical proof ofthe following results. Although strictly speaking the interpretation of MTCs as topological order is only for unitary MTCs and ourresults are stated for more general braided fusion categories, we will freely use the physics terminology forUMTCs throughout.
Definition.
An anyon is a simple object in a unitary modular tensor category C . An anyon type, or topolog-ical charge, is the isomorphism class of an anyon. Z + -ring, i.e. a (braided) fusion ring. We will consistently use calligraphic fonts for fusion categories C and standard font C to denote fusion rings. Passage from the calligraphic font to standard font ( C → C ) indicates the appropriate decategorification. For groups like G and Aut( C ) their categorical groups areindicated by G and Aut( C ) . Definition (Unital based Z + -ring [14].) . Let C be a ring which is free as a Z -module. A Z + -basis of a C isa set of elements B = { b i } i ∈ I such that b i b j = (cid:80) k ∈ I N kij b k , where N kij ∈ Z + .A unital based Z + -ring is a ring with a fixed Z + -basis B such that1. ∈ B
2. there exists an involution i (cid:55)→ i ∗ of I such that the induced map a (cid:55)→ a ∗ is an anti-involution of C and whenever b i b j = (cid:80) N kij b k , N ij = (cid:40) i = j ∗ i (cid:54) = j ∗ . Given a basis of topological charges (or defect types, see Definition 2.2) equipped with the involutionthat sends every object to its dual, the data of the fusion ring is encoded by the fusion coefficients N abc . Theabelian topological charges – those a ∈ L such that for all b ∈ L there exists a unique c ∈ L with N abc (cid:54) = 0 – form an abelian group under fusion, denoted by A . Definition (Group action on fusion ring) . Let G be a group and C a braided fusion ring with fixed basis. A G -action on C is a homomorphism G −→ Aut( C ) where Aut( C ) is the group of involution-preserving ring isomorphisms of C . A G -action on a fusion ring induces the symmetry N abc = N g · a,g · bg · c of the fusion coefficients. Theobstruction that measures whether a family of G -graded C -bimodules coming from a group action on C under the isomorphism BrPic( C ) ∼ = Aut( C ) form a fusion ring is given by a cohomology class in H ( G ; A ) [15]. G -crossed braided fusion rings When the H ( G ; A ) obstruction vanishes, the extension fusion ring has the structure of a G -crossed braided( G -crossed commutative) fusion ring. Definition (G-crossed braided fusion ring) . Let C be a fusion ring of an MTC with a G -action. A G -crossedbraided fusion ring C × G is a fusion ring which admits a G -grading by C-bimodules C g where C id = C ,together with a G -action on C × G such that X g ⊗ Y = g · Y ⊗ X g (1) for all X g ∈ C g and Y ∈ C × G . An additional H ( G ; U (1)) obstruction to the existence of G xBFCs measures the failure for the invert-ible C -bimodule categories C g to satisfy the pentagon axioms and form a fusion category [15]. In otherwords, it determines whether a G -crossed braided ring extension of C lifts to a G -crossed braided extensionof C . 4 efinition 1. Given a categorical group action of G on an MTC C , a (group) symmetry defect is a simpleobject in an invertible C -bimodule category C g under the equivalence BrPic( C ) (cid:39) Aut br ⊗ ( C ) . A symmetrydefect type is its isomorphism class. When the H obstruction vanishes but the H doesn’t, the G -graded extension of C by bimodule cat-egories is interpreted as anomalous G -symmetry enriched topological order, in the sense that the fusioncannot be realized by point-like objects in a strictly (2+1)D system, but can potentially be realized as a2-dimensional slice of some ( N + 1) D system.In this case the decategorified part of the anomalous G -extension is still a fusion ring - just a fusion ringwhich cannot be lifted to a G -crossed braided fusion category. While in the case of permutation extensionsthe H ( G ; U (1)) obstruction does vanish [16], the tools we develop should be applicable to the case ofanomalous SET order and thus we keep our discussion slightly more general. Let P be the monoidal functor that acts strictly on C (cid:2) n by permutations P : S n −→ Aut br ⊗ ( C (cid:2) n ) id (cid:55)→ id : C (cid:2) n → C (cid:2) n σ (cid:55)→ T σ : C (cid:2) n → C (cid:2) n (cid:2) i X i (cid:55)→ (cid:2) i X σ ( i ) (cid:2) i f i (cid:55)→ (cid:2) i f σ ( i ) so that the tensorators U σ : T σ ( X ⊗ Y ) → T σ ( X ) ⊗ T σ ( Y ) and compositors η X : ( T ρ ◦ T σ )( X ) → T ρσ ( X ) are the identity isomorphisms for all ρ, σ ∈ S n and X, Y ∈ Obj( C (cid:2) n ) .For these (untwisted) permutation actions on MTCs it is known that the H ( G ; U (1)) obstruction van-ishes: Theorem ([16]) . The H ( S n ; U (1)) obstruction to S n -extensions of C (cid:2) n vanish for the categorical permu-tation group action P : S n → Aut br ⊗ ( C (cid:2) n ) and the equivalence classes of S n -extensions form a torsor over H ( G ; U (1)) . The permutation symmetry models a global unitary on-site symmetry of multi-layer topological order given by n layers of topological order C [1]. CC ... C S n (cid:121) (cid:0) C (cid:2) n (cid:1) × S n = (cid:76) σ ∈ S n C σ While we been interpreting the factors in the Deligne product as spatially separated layers, the permu-tation symmetry is technically on-site because C (cid:2) n can also be interpreted a monolayer topological order.This is what allows us to study the spatial symmetry using the techniques developed for on-site symmetriesin [1]. 5rior to the development of G xBFCs as the algebraic theory of symmetry defects physicists studied per-mutation defects in (2+1)D TPM under the guise of genons [2], so called because of the way they effectivelycouple layers of topological phases to create nontrivial topology [1, 2].The ability of genons to entangle the layers comes from the S n -crossed braiding, whereby exchange withdefects transports (monolayer) anyons between layers. (cid:2) i a i (cid:2) i a σ ( i ) X σ X σ g -defects Our model of symmetry defect fusion is based on a parametrization of defect types by fixed points.
Irr( C g ) = { X gf | f ∈ Irr( C ) with g · f ∼ = f } . (2)One benefit is that when σ = id one recovers the anyons in C = C id as the fixed points under the action bythe identity element of G . Thus we can think of anyons as trivial symmetry defects, although in what followswe reserve the term “symmetry defect” to indicate g (cid:54) = id . It is also worth mentioning that the defects withineach sector inherit an ordering from an ordering of C . Definition 2 (Defect charge) . Let X gf ∈ C g be a symmetry defect. Then f is called the topological chargeof the defect X gf . Since the isomorphism class of the monoidal unit is always fixed by a G -action, each sector has adistinguished object with vacuum charge. Following [1], we make the following definition. Definition 3 (Bare defect) . A symmetry defect with vacuum charge X g is called a bare defect. Anyons, being intrinsic quasiparticle excitations corresponding to the ground state of a Hamiltonian whichcan be moved by local operators without additional energy, are said to be deconfined .On the other hand, point-like symmetry defects are not finite-energy excitations. They are extrinsic inthe sense that a Hamiltonian realization of a topological phase enriched with symmetry needs additionalterms added in order for it to have excitations which correspond to the symmetry defects [1]. Moreover, theenergy needed to spatially separate defects grows differently than it does for anyons and they are said to be confined as opposed to deconfined.We make the following definitions of g -confinement and g -deconfinement, borrowing the ideas fromcondensed matter theory. These are at odds with the sense in which deconfinement is used in the context of anyon condensation, which is why we have madethe dependence on the group element explicit. .2 g -confinements and g -deconfinement of anyons and g -defects Definition 4 ( g -deconfinement) . Let X gf be a g -defect with charge f , and let d ( f ) ∈ Obj( C ) be any object,not necessarily simple, with the property that d ( f ) ⊗ X g ∼ = X gf . (3) Then we say that d ( f ) is a deconfinement, or write d g ( f ) is a g -deconfinement of the defect X gf . In general a defect has multiple deconfinements.
Definition 5 ( g -confinement) . Let a ∈ Irr( C ) , X g ∈ Irr( C g ) a bare defect, and define c ( a ) = ⊗ mk =1 g k · a (4) where | g | = m . Then the defect X gc ( a ) is called the confinement of a . Observe that g · c ( a ) ∼ = c ( a ) , so that confinement is well-defined. Now we apply the ideas introduced in the previous section to construct G -crossed braided fusion ringsdirectly in the case of G = S n acting on Deligne product MTCs C (cid:2) n . The definition of the permutationdefect fusion ring ( C (cid:2) n ) × S n proceeds by specifying a free Z -module on a basis of permutation defect typesand a binary operation ⊗ that gives it the structure of a “ G -crossed commutative” unital Z + based ring. We will consistently use ( C (cid:2) n , ⊗ ) to mean the fusion ring of C (cid:2) n with respect to a basis Irr( C (cid:2) n ) = { (cid:2) i a i | a i ∈ Irr( C ) , ≤ i ≤ n } . Since the rank of each σ -sector in an S n -extension is given by the number of fixed anyons under theaction of σ , see for example [4], one can index the isomorphism classes of simple objects in C σ by Irr( C σ ) = { X σ (cid:2) i a i | (cid:2) a i ∈ Irr( C (cid:2) n ) with (cid:2) i a σ ( i ) ∼ = (cid:2) i a i } . (5)Occasionally we use (cid:126)a := (cid:2) i a i or supress the Deligne product and write a i to simplify notation.It will soon become apparent that this parametrization of symmetry defects is the key to constructing thefusion ring in a group-theoretical way, allowing one to construct the S n -extension of C (cid:2) n from S -extensionsof C (cid:2) C in exactly the same way that S n is generated by its transpositions. Definition 6.
Permutation defect type basis Let ( C (cid:2) n ) × S n be the free Z -module on the set Irr( C (cid:2) n ) ∪ σ Irr( C σ ) which we will also write as (cid:83) σ ∈ S n { X σ(cid:126)a , σ · (cid:126)a = (cid:126)a } . The following sections endow C × S n with the structure of a G -crossed braided fusion ring, see Definition2.2. 7 .1.1 Overview of construction In Section 4.2 we define C (cid:2) n -bimodules C σ with respect to the bases { X σ(cid:126)a | σ · (cid:126)a = (cid:126)a, (cid:126)a ∈ Irr( C (cid:2) n } andin Section 4.3 define an S n -action on the Z -module C × S n which is free on the basis (cid:91) σ ∈ S n { X σ(cid:126)a , σ · (cid:126)a = (cid:126)a } . Section 4.4 defines a multiplication ⊗ on C × S n and show that this gives it the structure of an S n -graded, S n -crossed braided fusion ring C × S n = (cid:76) σ ∈ S n C σ extending C (cid:2) n . We call ( C × S n , ⊗ ) the permutation defectfusion ring.In Section 4.5 we show that this fusion ring can be realized as a basepoint for an H ( S n , A ) -torsor’sworth of fusion rings C × S n ,ω . In the final Section 4.5 we conclude that the fusion rules for permutationextensions of MTCs can be constructed from the data ( N ab,c , n, ω ) . We give an algorithm to compute thefusion product of permutation defects and an implementation (see Appendix A). We use standard permutation notation for S n . For an arbitrary permutation σ ∈ S n we write its disjointcycle decomposition as σ = σ σ · · · σ s , where each σ i is a cycle ( i i · · · i m ) .By abuse of notation we conflate permutations with the subsets of { , , . . . n } that they act nontriviallyon, and for example write σ i ∩ σ j = ∅ to mean that the two permutations are disjoint and i ∈ σ ( i / ∈ σ ) toindicate that a given i ∈ { , , . . . n } is (or isn’t) permuted nontrivially by the action of σ .Throughout we will use the physics terminology, whose corresponding mathematical meaning was givenin the next table. (2+1)D TPM UMTC multi-layer topological order Deligne product of UMTCs C (cid:2) n multi-layer SET order unitary S n xBFC (cid:0) C (cid:2) n (cid:1) × S n = (cid:76) σ ∈ S n C σ (multilayer) anyon (cid:126)a := (cid:2) i a i , where a i ∈ Irr( C ) vacuum iso. class of tensor unit (cid:2) n monolayer anyon (cid:2) · · · (cid:2) (cid:2) a (cid:2) (cid:2) · · · g -sector invertible C -bimodule category C g transposition defect simple object X ( ij ) (cid:2) i f i ∈ C ( ij ) m -cycle defect simple object X ( i i ··· i m ) (cid:2) i f i ∈ C ( i i ··· i m ) permutation defect simple object X σ (cid:2) i f i ∈ C σ bare defect X σ (cid:2) n Table 1: Mathematical definitions of the physics terminology we will use when discussing permutation-enriched topological order. 8 .2 Fusion of multilayer anyons and permutation defects
The multiplication ⊗ on C × S n that we are about to define restricts to a commutative, associative binaryoperation on C id = C (cid:2) n and for this reason in any expression involving only products in C (cid:2) n we willfreely commute elements and omit parenthesization without comment.First we restate Definitions 4 and 9 in the case of permutation symmetry of C (cid:2) n . Definition 7 ( σ -confinement) . Let (cid:126)a ∈ Irr( C (cid:2) n ) and write a disjoint cycle decomposition of σ as σ = (cid:81) j σ j . Then the σ -confinement map c σ : C (cid:2) n → C (cid:2) n is defined on basis elements by c σ ( (cid:126)a ) := (cid:2) i c i where c i = (cid:40) a i i / ∈ σ (cid:78) k ∈ σ j a k i ∈ σ j (6) and extended linearly to C (cid:2) n . Proposition 1.
The confinement map has the following properties.1. c σ ( (cid:126)a ⊗ (cid:126)b ) = c σ ( (cid:126)a ) ⊗ c σ ( (cid:126)b ) c σ ( σ k · (cid:126)a ) = c σ ( (cid:126)a ) for all (cid:126)a,(cid:126)b ∈ Irr( C (cid:2) n ) , ≤ k ≤ | σ | .Proof. Easy consequences of Definition 7.These will be applied often in the proofs that follow.
Definition 8 (Permutation sectors) . Let C σ be the free Z -module on the basis of σ -defect types Irr( C σ ) . The confinement map is the main ingredient in extending the fusion between anyons to an action on σ -defects. Definition 9 (Anyon-defect fusion) . Let (cid:126)a ∈ Irr( C ) and X σ(cid:126)b ∈ Irr( C σ ) . Then we define a binary operation ⊗ : C (cid:2) n × C σ −→ C σ on basis elements by (cid:126)a ⊗ X σ(cid:126)b := X σc σ ( (cid:126)a ) ⊗ (cid:126)b . (7) and identically for C σ × C (cid:2) n −→ C σ . Definition 10 ( σ -deconfinement) . Let X (cid:126)b ∈ Irr( C σ ) , (cid:126)a ∈ Irr( C (cid:2) n ) , and suppose they satisfy (cid:126)a ⊗ X σ(cid:126) = X σ(cid:126)b . (8) Then we say that (cid:126)a is a (left) deconfinement of X b , and define right deconfinements in the analogous wayusing the right action. X (cid:126)a = d ( (cid:126)a ) ⊗ X (cid:126) . (9)We will see that the notion of deconfinement is central to our construction, as it allows us to intuit theway that the defect theory is built from the anyon theory. Lemma 1.
Anyon-defect fusion is independent of choice of deconfinements. In other words, we can write (cid:126)a ⊗ X σ(cid:126)b = ( (cid:126)a ⊗ d σ ( (cid:126)b )) ⊗ X σ (10) for any choice of deconfinement d σ ( (cid:126)b ) .Proof. We have (cid:126)a ⊗ X σ(cid:126)b = X σc σ ( (cid:126)a ) ⊗ (cid:126)b (11) = X σc σ ( (cid:126)a ) ⊗ c σ ( d σ ( (cid:126)b )) (12)by the definitions of confinement/deconfinement, where d σ ( (cid:126)b ) is any σ -deconfinement. Then by Proposition1, X σc σ ( (cid:126)a ) ⊗ c σ ( d σ ( (cid:126)b )) = X σc σ ( (cid:126)a ⊗ d σ ( (cid:126)b )) (13) = ( (cid:126)a ⊗ d σ ( (cid:126)b )) ⊗ X σ (14) Example 1 (Fusion of monolayer anyons and bare defects) . The fusion between monolayer anyons (cid:2) i − (cid:2) a (cid:2) (cid:2) n − i and bare defects X σ(cid:126) is given by (cid:2) i − (cid:2) a (cid:2) (cid:2) n − i ⊗ X (cid:126) = X (cid:2) b j b j = (cid:40) σ ( j ) = ja σ ( j ) (cid:54) = j . (15) For example, when n = 2 , (cid:2) a ⊗ X = a (cid:2) ⊗ X = X aa (16) for all a ∈ Irr( C ) .In words, fusing an anyon in the i th layer with the bare defect results is the defect with the topologicalcharge label that has a in every layer which is in the σ -orbit of i . Proposition 2. C σ is a ( C (cid:2) n , C (cid:2) n ) -bimodule with respect to anyon-defect fusion.Proof. Distributivity was built in to the definition of the confinement map and fusion, and it is immediatethat (cid:126) ⊗ X σ(cid:126)a = X σ(cid:126)a ⊗ (cid:126) for all (cid:126)a = σ · (cid:126)a . The only axioms of a bimodule that need to be checked are10eft, right, and middle associativity. These are any easy consequence of Proposition 1 and commutativity ofassociativity of ⊗ in C (cid:2) n . For example, ( (cid:126)a ⊗ (cid:126)b ) ⊗ X σ(cid:126)c = X c σ ( (cid:126)a ⊗ (cid:126)b ) ⊗ (cid:126)c (17) = X c σ ( (cid:126)a ) ⊗ c σ ( (cid:126)b ) ⊗ (cid:126)c (18) = (cid:126)a ⊗ X c σ ( (cid:126)b ) ⊗ (cid:126)c (19) = (cid:126)a ⊗ ( (cid:126)b ⊗ X σ(cid:126)c ) (20)The other cases are similar. S n -action on defects So far we have a free Z module C × S n which is an S n -graded extension of C × by bimodules: C × S n = (cid:77) σ ∈ S n C σ . Definition 11.
Let ρ ∈ S n . Then define a map S n × C × S n −→ C × S n on basis elements by ρ · X σ(cid:126)a := X ρσρ − ρ · (cid:126)a . (21)One can check this is a well-defined action of S n on C × S n extending the action of S n on C (cid:2) n due toassociativity in S n . Proposition 3 ( S n -crossed commutativity of topological charge and defect type fusion) . X σ(cid:126)a ⊗ (cid:126)b = σ · (cid:126)b ⊗ X σ(cid:126)a Proof.
We have X σ(cid:126)a ⊗ (cid:126)b = ( (cid:126)b ⊗ d σ ( (cid:126)a )) ⊗ X σ(cid:126) (22) = X σc σ ( (cid:126)b ⊗ d σ ( a )) (23) = X σc σ ( (cid:126)b ) ⊗ c σ ( d σ ( a )) (24) = X σc σ ( σ · (cid:126)b ) ⊗ (cid:126)a ) (25) = σ · (cid:126)b ⊗ X σ(cid:126)a (26)By Lemma 1, Definition 9, and parts (1) and (2) of Proposition 1.Later we will show that the multiplication on all of C × S n is S n -crossed commutative but Lemma 3 willbe helpful for showing said multiplication is associative.11 .4 Fusion of permutation defects Finally we are ready to define a product on all of C × S n . Definition 12 (Defect fusion) . Let X ρ(cid:126)a and X σ(cid:126)b be two symmetry defects. Then their (untwisted) product isgiven by X ρ(cid:126)a ⊗ X σ(cid:126)b := d ρ ( (cid:126)a ) ⊗ d σ ( (cid:126)b ) ⊗ (cid:79) ( i k i l ) ∈ ρσ (cid:77) c ∈ Irr( C ) (cid:2) i k − (cid:2) c (cid:2) i l − i k − (cid:2) c ∗ (cid:2) n − i l ⊗ X ρσ(cid:126) . (27)It will become clear that the choice of ordering of factors in the product and left justification is arbitrary,but we will present our calculations this way consistently.The fusion product that results from the annihilation of transposition defects is an object with niceproperties that will come in handy later. The next proposition says it can teleport anyons from layer tolayer and is invariant under the layer-exchange symmetry: Let τ = ( ij ) be a transposition and write X ( ij ) (cid:126) ⊗ X ( ij ) (cid:126) = (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · as shorthand for the fusion product of bare transposition defects. Proposition 4 (Properties of the transposition fusion product) . The following equations hold.1. (cid:77) c ∈ Irr( C ) · · · a ⊗ c (cid:124) (cid:123)(cid:122) (cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · = (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ ⊗ a (cid:124) (cid:123)(cid:122) (cid:125) j · · · for all a ∈ Irr( C ) .2. ( ij ) · (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · = (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · Proof.
For (1), one can write (cid:77) c · · · a ⊗ c · · · c ∗ · · · = (cid:77) b,c · · · N acb b · · · c ∗ · · · = (cid:77) b,c · · · N b ∗ ac ∗ b · · · c ∗ · · · = (cid:77) b,c · · · b · · · N b ∗ ac ∗ c ∗ · · · = (cid:77) b · · · b · · · b ∗ ⊗ a · · · (28)Where b and c range over all of Irr( C ) and we have used symmetries of the fusion coefficients that comefrom duality-induced isomorphisms of trivalent Hom spaces in C , see for example [17, 14]. Relabelinggives equation (1). Equation (2) follows immediately from the fact that the sum is over all of Irr( C ) andrelabeling. 12he following lemma shows that it suffices to check associativity for bare transposition defects. Lemma 2.
Associativity of bare transposition defects implies associativity of all nontrivial permutationdefects.Proof.
Suppose ( X τ (cid:126) ⊗ X τ (cid:126) ) ⊗ X τ (cid:126) = X τ (cid:126) ⊗ ( X τ (cid:126) ⊗ X τ (cid:126) ) . (29)Then ( X τ (cid:126)a ⊗ X τ (cid:126)b ) ⊗ X τ (cid:126)c = (cid:16) d ( (cid:126)a ) ⊗ d ( (cid:126)b ) ⊗ (cid:16) X τ (cid:126) ⊗ X τ (cid:126) (cid:17)(cid:17) ⊗ (cid:16) d ( (cid:126)c ) ⊗ X τ (cid:126) (cid:17) = (cid:16) d ( (cid:126)a ) ⊗ d ( (cid:126)b ) ⊗ d ( (cid:126)c ) (cid:17) ⊗ (cid:16)(cid:16) X τ (cid:126) ⊗ X τ (cid:126) (cid:17) ⊗ X τ (cid:126) (cid:17) = (cid:16) d ( (cid:126)a ) ⊗ d ( (cid:126)b ) ⊗ d ( (cid:126)c ) (cid:17) ⊗ (cid:16) X τ (cid:126) ⊗ (cid:16) X τ (cid:126) ⊗ X τ (cid:126) (cid:17)(cid:17) = X τ (cid:126)a ⊗ ( X τ (cid:126)b ⊗ X τ (cid:126)c ) . (30)Since transpositions generate S n associativity for arbitrary permutation defects follows immediately.Thus it is enough to check associativity for bare transposition defects. Lemma 3 (Bare transposition defect associativity) . ( X τ (cid:126) ⊗ X τ (cid:126) ) ⊗ X τ (cid:126) = X τ (cid:126) ⊗ ( X τ (cid:126) ⊗ X τ (cid:126) ) . (31) Proof.
There are several cases to check. We start with the border cases.(a) If τ , τ , τ are pairwise disjoint, then both sides of Equation 31 simplify to X τ τ τ (cid:126) by part (1) ofProposition 5.(b) If τ = τ = τ = τ , then both sides of Equation 31 yield X τc τ ( ⊕ a ∈ Irr( C ) ··· a ··· a ∗ ··· ) by Definition 12 .(c) If | τ ∩ τ ∩ τ | = 1 , Definition 12 gives X τ τ τ (cid:126) .The remaining cases are similar and the details not particularly illuminating. Lemma 4. S n -defect type fusion is associative, making ( C × S n , ⊗ ) into a ring.Proof. Of course products involving three anyons are associative and associativity of triple-defect productsfollows from combining Lemmas 2 and 3. By Propositions 2 and 3, products involving two anyons andone defect are associative. It remains only to check associativity for products involving one anyon and twodefects.We show ( (cid:126)a ⊗ X ρ(cid:126)b ) ⊗ X σ(cid:126)c = (cid:126)a ⊗ ( X ρ(cid:126)b ⊗ X σ(cid:126)c ) (32)and claim that the other cases are similar. Note that it suffices to check the case where ρ = τ and σ = τ aretranspositions, since together with Equation 32 Lemma 3 generates equality between such parenthesizationswith arbitrary permutations. We index the confinements and deconfinements using the transposition indices , to simplify notation.On the one hand, we have ( (cid:126)a ⊗ X τ (cid:126)b ) ⊗ X τ (cid:126)c = X τ c ( (cid:126)a ) ⊗ (cid:126)b ⊗ X τ (cid:126)c =( d ( c ( (cid:126)a ) ⊗ (cid:126)b ) ⊗ d ( (cid:126)c )) ⊗ ( X τ (cid:126) ⊗ X τ (cid:126) )=( d ( c ( (cid:126)a ) ⊗ (cid:126)b ) ⊗ d ( (cid:126)c )) ⊗ X τ τ (cid:126) τ (cid:54) = τ (cid:76) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · τ = τ = ( ij ) (33)13n the other hand, (cid:126)a ⊗ ( X τ (cid:126)b ⊗ X τ (cid:126)c ) = (cid:126)a ⊗ (cid:16)(cid:16) d ( (cid:126)b ) ⊗ d ( (cid:126)c ) (cid:17) ⊗ ( X τ ⊗ X τ ) (cid:17) = (cid:16) (cid:126)a ⊗ d ( (cid:126)b ) ⊗ d ( (cid:126)c ) (cid:17) ⊗ X τ τ (cid:126) τ (cid:54) = τ (cid:76) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · τ = τ = ( ij ) (34)Now it is not necessarily the case that d ( c ( (cid:126)a ) ⊗ (cid:126)b ) ⊗ d ( (cid:126)c )) = (cid:126)a ⊗ d ( (cid:126)b ) ⊗ d ( (cid:126)c ) . However, in the case τ (cid:54) = τ , it suffices to show that their confinements with respect to τ τ are equal.If τ ∩ τ = ∅ , The confinement of d ( c ( (cid:126)a ) ⊗ (cid:126)b ) ⊗ d ( (cid:126)c )) is the object (cid:2) i X i X i = a i ⊗ b i ⊗ c i i / ∈ τ , τ a i ⊗ a j ⊗ b i ⊗ c i ⊗ c j i ∈ τ a j ⊗ a j ⊗ b i ⊗ b j ⊗ c i i ∈ τ , which one can check this is the same as the confinement of (cid:126)a ⊗ d ( (cid:126)b ) ⊗ d ( (cid:126)c ) . The case where τ and τ multiply to a 3-cycle is similar.Now let τ = τ , and write τ = ( ij ) . Write d τ ( (cid:126)b ) i,j = d i,j (35) d τ ( (cid:126)c ) i,j = e i,j (36) d τ ( c τ ( (cid:126)a ) ⊗ (cid:126)b ) i,j = f i,j (37)where the equations d i ⊗ d j = b i = b j (38) e i ⊗ e j = c i = c j (39) f i ⊗ f j = a i ⊗ a j ⊗ b i = a i ⊗ a j = b j (40)are satisfied by Definition 4.Comparising the products layer-wise indexing by k one has ( d τ ( c τ ( (cid:126)a ) ⊗ (cid:126)b ) ⊗ d τ ( (cid:126)c )) ⊗ (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · and (cid:16) (cid:126)a ⊗ d τ ( (cid:126)b ) ⊗ d τ ( (cid:126)c ) (cid:17) ⊗ (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · . Clearly when k (cid:54) = i, j the k th entry of both products are equal and hence it suffices to consider the i thand j th entries, which become (cid:77) c ∈ Irr( C ) · · · f i ⊗ e i ⊗ c (cid:124) (cid:123)(cid:122) (cid:125) i · · · f j ⊗ e j ⊗ c ∗ (cid:124) (cid:123)(cid:122) (cid:125) j · · · and 14 c ∈ Irr( C ) · · · a i ⊗ d i ⊗ e i ⊗ c (cid:124) (cid:123)(cid:122) (cid:125) i · · · a j ⊗ d j ⊗ e j ⊗ c ∗ (cid:124) (cid:123)(cid:122) (cid:125) j · By Proposition 4 and commutativity of multiplication in C id , checking whether these two objects areequal can be reduced to checking whether f i ⊗ f j ⊗ e i ⊗ e j = a i ⊗ a j ⊗ d i ⊗ d j ⊗ e i ⊗ e j Both sides simplify to a i ⊗ a j ⊗ b i ⊗ c i by Equations 38 - 40.Together with the confinement map that determines anyon-defect fusion, the following proposition showsthat bare transposition defects generate permutation defects in the same way that transpositions generate allof S n . Proposition 5.
1. Products of disjoint bare defects commute and are themselves bare defects. X ρ(cid:126) ⊗ X σ(cid:126) = X ρσ(cid:126) when ρ ∩ σ = ∅ (41)
2. Every bare defect X σ(cid:126) can be written as a product of bare transposition defects.Proof. The first is an immediate consequence of Definition 12. The second follows from Definition 12 andProposition 3 that one can write X σ(cid:126) = (cid:79) i X σ i (cid:126) = (cid:79) i,j X τ ij (cid:126) (42)where σ i = τ i τ i · · · τ im i is any transposition decomposition of the i th disjoint cycle in the decomposition of σ . S n -crossed braidingLemma 5. The multiplication is S n -crossed X ρ(cid:126)a ⊗ X σ(cid:126)b = ρ · X σ(cid:126)b ⊗ X ρ(cid:126)a . Proof.
First observe that if d σ ( (cid:126)b ) is a σ -deconfinement of X σ(cid:126)b , then ρ · d σ ( (cid:126)b ) is a ρσρ − -deconfinement of X ρσρ − ρ · (cid:126)b . Since the fusion products are independent of choice of deconfinements, we have ρ · X σ(cid:126)b ⊗ X ρ(cid:126)a = X ρσρ − ρ · (cid:126)b ⊗ X ρ(cid:126)a = d ρσρ − ( ρ · (cid:126)b ) ⊗ X ρσρ(cid:126) ⊗ X ρ(cid:126)a = ρ · d σ ( (cid:126)b ) ⊗ X ρσρ − (cid:126) ⊗ X ρ(cid:126)a = ρ · d σ ( (cid:126)b ) ⊗ X ρ(cid:126)a ⊗ X σ(cid:126) = X ρ(cid:126)a ⊗ d σ ( (cid:126)b ) ⊗ X σ(cid:126) = X ρ(cid:126)a ⊗ X σ(cid:126)b (43)15 heorem 1. The permutation defect fusion ring ( C × S n , ⊗ ) is a unital based Z + -ring which is S n -crossedcommutative.Proof. Let A be the free Z -module generated by the set (cid:8) Irr (cid:0) C (cid:2) n (cid:1) (cid:83) σ Irr ( C σ ) (cid:9) . Combining the multipli-cation coming from the fusion ring of C (cid:2) n , the confinement map in Definition 5.2, and the definition of baredefect fusion products in Definitions 5.4 and 5.5 defines an associative binary operation A × A −→ A thatgives A the structure of a ring.As a consequence of our definitions the products are all given by non-negative integer linear combina-tions of basis elements { Irr (cid:0) C (cid:2) n (cid:1) (cid:83) σ Irr ( C σ ) } , and the unit ∈ Irr( C (cid:2) n ) is simple. Therefore we have aunital Z + ring.Now since a ∼ = a ∗∗ for all simple objects in an MTC by pivotality, and ( g − ) − = g for all g ∈ G , thefollowing map defines an involution on basis elements, (cid:126)a (cid:55)→ (cid:126)a ∗ (cid:126)a ∈ Irr( C (cid:2) n ) (44) X σ(cid:126)a (cid:55)→ X σ − (cid:126)a ∗ X σ(cid:126)a ∈ Irr( C σ ) (45)which we expand linearly to an involution on all of A . Since that N ab = 1 if and only if a ∼ = b ∗ , it remainsonly to check that this involution corresponds to duality of defects.Fix σ ∈ S n , (cid:126)a ∈ Irr( C (cid:2) n ) and let X σ(cid:126)a ∈ C σ , X τ(cid:126)b ∈ C τ . Clearly N X σ(cid:126)a X τ(cid:126)b = 0 unless τ = σ − .Given a transposition decomposition of σ , σ = τ m · · · t with τ j = ( j j ) , j < j , we have X σ(cid:126)a ⊗ X σ − (cid:126)b = (cid:16) d ( (cid:126)a ) ⊗ d ( (cid:126)b ) (cid:17) ⊗ ( X σ(cid:126) ⊗ X σ − (cid:126) )= (cid:16) d ( (cid:126)a ) ⊗ d ( (cid:126)b ) (cid:17) ⊗ (cid:79) ≤ j ≤ m (cid:77) c ∈ Irr( C ) (cid:2) j − (cid:2) c (cid:2) j − j (cid:2) c ∗ (cid:2) n − j (46)In the fusion product over j there is one summand of (cid:126) , hence N X σ(cid:126)a X σ − (cid:126)b = 1 only if N d ( (cid:126)a ) ,d ( (cid:126)b )1 = 1 . Sowe must have d ( (cid:126)b ) ∼ = d ( (cid:126)a ) ∗ . In other words, the deconfinements must be dual.Now reconfinement with an arbitrary bare defect gives X ρ(cid:126) = d ( (cid:126)a ) ⊗ d ( (cid:126)b ) ⊗ X ρ(cid:126) = X ρ(cid:126)a ⊗ (cid:126)b (47)and hence (cid:126)b ∼ = (cid:126)a ∗ . Finally, Lemma 5 gives S n -crossed commutativity. The fusion ring we have defined forms a basepoint for an H ( S n , A (cid:2) n ) -torsor. Definition 13 (Twisted defect fusion) . Let ω : S n × S n → A (cid:2) n be a 2-cocycle for the permutation actionof S n on A (cid:2) n . Define X ρ(cid:126)a ⊗ ω X σ(cid:126)b := ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b ) . (48) Lemma 6.
The twisted fusion product ⊗ ω gives an S n -crossed fusion ring structure on the defects for every2-cocycle ω : S n × S n → A (cid:2) n . roof. The twisted fusion product ⊗ ω is associative:On the one hand, ( X ρ(cid:126)a ⊗ ω X σ(cid:126)b ) ⊗ ω X τ(cid:126)c = ( ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b )) ⊗ ω X τ(cid:126)c (49) = ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b )) ⊗ ω X τ(cid:126)c (50) = ( ω ( ρ, σ ) ⊗ ω ( ρ, σ )) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b ) ⊗ X τ(cid:126)c . (51)On the other hand, X ρ(cid:126)a ⊗ ω ( X σ(cid:126)b ⊗ ω X τ(cid:126)c ) = X ρ(cid:126)a ⊗ ω (cid:16) ω ( σ, τ ) ⊗ ( X σ(cid:126)b ⊗ X τ(cid:126)c ) (cid:17) (52) = ρ · ω ( σ, τ ) ⊗ (cid:16) X ρ(cid:126)a ⊗ ω ( X σ(cid:126)b ⊗ X τ(cid:126)c ) (cid:17) (53) = ρ · ω ( σ, τ ) ⊗ ω ( ρ, στ ) ⊗ (cid:16) X ρ(cid:126)a ⊗ ( X σ(cid:126)b ⊗ X τ(cid:126)c ) (cid:17) (54)where we have used the S n -crossed braiding and associativity of anyon-defect fusion. Since ω is a 2-cocyclefor the S n -action on A (cid:2) n , it satisfies ω ( ρ, σ ) ⊗ ω ( ρ, σ ) = ρ · ω ( σ, τ ) ⊗ ω ( ρ, στ ) (55)Whenever two 2-cocycles ω and ˜ ω differ by a 2-coboundary, their corresponding twisted fusion products ⊗ ω and ⊗ ˜ ω give rise to isomorphic S n -crossed fusion rings. Lemma 7. ( C × S n , ⊗ ω ) and ( C × S n , ⊗ ˜ ω ) are isomorphic as S n -crossed fusion rings if and only if ω and ˜ ω differ by a 2-coboundary.Proof. there exists a map φ : S n → A such that where φ satisfies ω ( ρ, σ )˜ ω ( ρ ) ∗ = ρ · φ ( σ ) ⊗ φ ( ρσ ) ∗ ⊗ φ ( ρ ) . (56)We check that φ defines a ring isomorphism Φ between the rings ( C × S n , ⊗ ω ) and ( C × S n , ⊗ ˜ ω ) if and onlyif ω and ˜ ω differ by a 2-coboundary. Φ( X ρ(cid:126)a ) = φ ( ρ ) ⊗ X ρ(cid:126)a (57)On the one hand, Φ( X ρ(cid:126)a ⊗ ω X σ(cid:126)b ) = φ ( ρσ ) ⊗ ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b ) . (58)On the other hand, Φ( X ρ(cid:126)a ) ⊗ ˜ ω Φ( X σ(cid:126)b ) =( φ ( ρ ) ⊗ X ρ(cid:126)a ) ⊗ ˜ ω ( φ ( σ ) ⊗ X σ(cid:126)b )=( φ ( ρ ) ⊗ ρ · φ ( σ )) ⊗ ( X ρ(cid:126)a ⊗ ˜ ω X σ(cid:126)b )=( φ ( ρ ) ⊗ ρ · φ ( σ ) ⊗ ˜ ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b )= φ ( ρσ ) ⊗ ω ( ρ, σ ) ⊗ ( X ρ(cid:126)a ⊗ X σ(cid:126)b ) (59)17 .6 The permutation defect fusion ring and algorithm We have constructed a torsor of S n -crossed fusion rings over H ( S n , A (cid:2) n ) , which classify such fusionrings extending those C (cid:2) n . Since the H ( S n , U (1)) obstruction vanishes by the work of [16], every such S n -crossed extension ring lifts to an S n -crossed braided extension [15]. Therefore we can conclude that thetwisted S n -defect fusion construction realizes the fusion rules for permutation extensions of MTCs C (cid:2) n . Theorem 2.
Given an MTC C and a 2-cocycle ω representing [ ω ] ∈ H ( S n , A (cid:2) n ) , the equivalence classesof fusion rings of S n -crossed braided extensions of C (cid:2) n are given by the isomorphism classes of the fusionrings ( C × S n , ⊗ ω ) . In particular, the following Algorithm 1 produces the correct fusion rules.
Algorithm 1 (Permutation defect fusion algorithm) . The fusion product of two permutation defects X σ(cid:126)a and X τ(cid:126)b can be computed as follows.1. σ - and τ -deconfinement:Strip the topological charges from the defects and twist with the abelian anyon ω ( σ, τ ) .2. Transposition defect annihilation:Compute the fusion product of the bare σ - and τ -defects, for every pair of indices ( ij ) permutedby both σ and τ , pulling out a factor of (cid:77) c ∈ Irr( C ) · · · c (cid:124)(cid:123)(cid:122)(cid:125) i · · · c ∗ (cid:124)(cid:123)(cid:122)(cid:125) j · · · . στ -confinement:Confine the product of the objects from Step 1 and Step 2 with the bare στ -defect. An implementation of the algorithm is given in a Mathematica
PermutationDefectFusion.m package.In Appendix A we show the main function implementing Algorithm 1; the full code is available at theauthor’s website math.ucsb.edu/˜cdelaney/research . The salient features of the S n -crossed fusion ring construction can be illustrated through a few small ex-amples. First we recall the bilayer symmetry defect fusion rules established in [1, 13, 5] from which thegeneral permutation defect fusion rules are built. The fusion rules for the trilayer Fibonacci defects of therank 24 fusion category (cid:16) Fib (cid:2) (cid:17) × S in Section 5.2.1 exhibit the overall pattern of S n -crossed fusion rings,in particular how they are determined by the fusion ring of C and bare S -defects. In Section 5.2.2 we repeatthe example for the rank 3 Ising
MTCs with fusion rules (cid:40) σ ⊗ σ = 1 ⊕ ψσ ⊗ ψ = ψ ⊗ σ = 1 , H ( S , A (cid:2) ) -torsor of possible fusion rulesof (cid:16) Ising (cid:2) (cid:17) × S with A (cid:2) ∼ = ( Z / Z ) = (cid:104) ψ, ψ , ψ (cid:105) . Z / Z -symmetry CC S (cid:121) (cid:0) C (cid:2) (cid:1) × S = C (cid:2) ⊕ C (12) Fusion rules for S -extensions of C (cid:2) C were first proven [5] in the language of modular functors. Morerecently they appeared in the physics literature in [1] and in the case of (Fib (cid:2) ) × Z / Z were computed directlyby computer assisted de-equivariantization of the MTC SU (2) [7]. Fusion rules for ( C (cid:2) C ) × S directly interms of C were then proven in [13].Recasting these results yet again in our model, the form of the general fusion rules for ( C (cid:2) C ) × S = C id ⊕ C (12) can be understood as follows.Writing Irr( C (12) ) = { X (12) a (cid:2) a | a ∈ Irr( C ) } , for the fusion rules corresponding to the trivial cocycle ω ≡ we find that every defect is fixed by the S action and related to the bare defect by fusion with a monolayeranyon a (cid:2) (also (cid:2) a ). In particular the fusion rules can be expressed as (cid:40) a (cid:2) b ⊗ X (12) c (cid:2) c = X (12) a ⊗ b ⊗ c (cid:2) a ⊗ b ⊗ c = (cid:76) d N abcd X d (cid:2) d X (12) a (cid:2) a ⊗ X (12) b (cid:2) b = (cid:76) c ∈ Irr( C ) a ⊗ c (cid:2) c ∗ ⊗ b where N abcd are generalized fusion coefficients in C .The quantum dimensions satisfy d X = (cid:115) D C (cid:80) c ∈ Irr( C ) d c (60) d X aa = d a · (cid:115) D C (cid:80) c ∈ Irr( C ) d c (61) S -symmetry FibFib S (cid:16) Fib (cid:2) (cid:17) × S = Fib (cid:2) ⊕C (12) The Fibonacci MTC is rank 2 and has nontrivial fusion rule τ ⊗ τ = 1 ⊕ τ . Writing the bilayer anyons a (cid:2) b =: ab and labeling defects by fixed points, we put Irr (cid:18)(cid:16)
Fib (cid:2) (cid:17) × S (cid:19) = Irr (cid:16) Fib (cid:2) (cid:17) ∪ Irr (cid:0) C (12) (cid:1) = { , τ, τ , τ τ, X , X ττ } . The fusion rules constructed in the previous section in the case C = Fib and n = 2 are listed in the tablebelow. 19
11 1 τ τ τ τ X X ττ
11 11 1 τ τ τ τ X X ττ τ τ ⊕ τ τ τ τ ⊕ τ τ X ττ X ⊕ X ττ τ τ τ τ ⊕ τ τ ⊕ τ τ X ττ X ⊕ X ττ τ τ τ τ τ ⊕ τ τ τ ⊕ τ τ ⊕ τ ⊕ τ ⊕ τ τ X ⊕ X ττ X ⊕ X ττ X X X ττ X ττ X ⊕ X ττ ⊕ τ τ τ ⊕ τ ⊕ τ τX ττ X ττ X ⊕ X ττ X ⊕ X ττ X ⊕ X ττ τ ⊕ τ ⊕ τ τ ⊕ τ ⊕ τ ⊕ τ τ Table 2: Fusion table for bilayer Fibonacci anyons and defects. S -symmetry The general pattern of fusion in S n -extensions can be seen from the n = 3 case. Ranks of permutationextensions are large even for small n , with rank (cid:16) (Fib (cid:2) ) × S (cid:17) = 24 and rank (cid:16) (Ising (cid:2) ) × S (cid:17) = 60 . So ratherthan providing the full fusion table for these examples we give a more compact description in the form ofthe fusion table for bare permutation defects and show how to use them to compute arbitrary products. Forgeneral n , the fusion rules can be derived from the n ( n − × n ( n − table of its bare transposition defects,but even then all transpositions behave identically and can be just as well understood through the n = 2 caseand the fusion algorithm. S -symmetry FibFibFib S (cid:16) Fib (cid:2) (cid:17) × S We write the
3! = 6 graded components as (cid:16)
Fib (cid:2) (cid:17) × S = Fib (cid:2) ⊕C (12) ⊕ C (23) ⊕ C (13) ⊕ C (123) ⊕ C (132) . Suppressing the (cid:2) in the label set { , τ } (cid:2) we label the defects in each sector C σ by the anyons fixed underthe permutation of σ . We count a rank 24 fusion category where each C σ has global quantum dimension D C σ = D Fib (cid:2) = D = (2 + φ ) / = (cid:112)
15 + 20 φ. (62)The Fibonacci MTC has no nontrivial abelian anyons, so A = 1 and H ( S , A ) is trivial. By theexistence result of [16] and the classification of G -crossed braided extensions of BFCs [15], there is aunique fusion ring shared among S -crossed extensions of Fib (cid:2) .We list the fusion between anyons and bare defects in Table 5.2.1.By Section 4, the full fusion table for the S -extension is determined by rows 1-3 and 8-10.To find the fusion rule between two arbitrary defects from the table, make any choice of deconfinements d ρ ( (cid:126)a ) and d σ ( (cid:126)b ) , and confine their product d ρ ( (cid:126)a ) ⊗ d σ ( (cid:126)b ) with the entry of the table corresponding to thebare product X ρ(cid:126) ⊗ X σ(cid:126) . 20 -sector σ -defects quantum dim. C id { , τ, τ , ττ, τ , τ τ, ττ , τττ } { , φ, φ, φ , φ, φ , φ , φ }C (12) { X , X τ , X ττ , X τττ } {√ φ, √ φ, √ φ, √ φ }C (23) { Y , Y τ , Y ττ , Y τττ } {√ φ, √ φ, √ φ, √ φ }C (13) { Z , Z τ , Z τ τ , Z τττ } {√ φ, √ φ, √ φ, √ φ }C (123) { U , U τττ } {√ φ, √
10 + 15 φ }C (132) { V , V τττ } {√ φ, √
10 + 15 φ } Table 3: Quantum dimensions of simple objects in the invertible
Fib (cid:2) -bimodule categories C σ . ⊗ X Y Z U V τ X τ Y ττ Z τ τ U τττ V τττ τ X ττ Y ττ Z τ U τττ V τττ τ X ττ Y τ Z τ τ U τττ V τττ τ τ X τττ Y ⊕ Y ττ Z τττ U ⊕ U τττ V ⊕ V τττ τ τ X τττ Y τττ Z ⊕ Z τ τ U ⊕ U τττ V ⊕ V τττ τ τ X ⊕ X ττ Y τττ Z τττ U ⊕ U τττ V ⊕ V τττ τ τ τ X τ ⊕ X τττ Y τ ⊕ Y τττ Z τ ⊕ Z τττ U ⊕ U τττ V ⊕ V τττ X ⊕ τ τ U V Y ⊕ Y τττ Z ⊕ Z τττ Y V ⊕ τ τ U Z ⊕ Z τττ X ⊕ X τττ Z U V ⊕ τ τ X ⊕ X τττ Y ⊕ Y τττ U Z ⊕ Z τττ X ⊕ X τττ Y ⊕ Y τττ V ⊕ V τττ ⊕ τ τ ⊕ τ τ ⊕ τ τ ⊕ τ τ τV Y ⊕ Y τττ Z ⊕ Z τττ X ⊕ X τττ ⊕ τ τ ⊕ τ τ ⊕ τ τ ⊕ τ τ τ U ⊕ U τττ Table 4: Fusion rules for bare defects in (cid:16)
Fib (cid:2) (cid:17) × S . Our convention is that X ρ(cid:126) ⊗ X σ(cid:126) ∈ C ρσ correspondsto the entry in the ρ -row and σ -column.For example, to compute X (12) ττ ⊗ X (132) τττ choose deconfinements τ ⊗ X (12) (cid:126) = X (12) ττ (63) τ ⊗ X (132) (cid:126) = X (132) τττ (64)and write X (12) ττ ⊗ X (132) τττ = ( τ ⊗ τ ) ⊗ ( X (12) (cid:126) ⊗ X (132) (cid:126) ) (65) = τ τ ⊗ ( X (13) (cid:126) ⊕ X (13) τ τ ) (66) = X (13) τ ⊗ τ, ,τ ⊗ τ ⊕ X (13) τ ⊗ τ ⊗ τ, ,τ ⊗ τ ⊗ τ (67) = ( X (13)111 ⊕ X (13) τ τ ) ⊕ ( X (13)111 ⊕ X (13) τ τ ) (68) = 2 X (13)111 ⊕ X (13) τ τ . (69)The fusion product between any two defect types can be computed in this manner.21 .2.2 Trilayer Ising with S -symmetry IsingIsingIsing S (cid:121) (cid:16) Ising (cid:2) (cid:17) × S We calculate the fusion rules for (Ising (cid:2) ) × S corresponding to the trivial cohomology class in H ( S , A (cid:2) ) .Table 5.2.2 contains the fusion products of the bare permutation defects. For compactness, we have usedthe shorthand (cid:126) , (cid:126)σ = σσσ, (cid:126)ψ = ψψψ in defect charge labels and X σ(cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ := X σ(cid:126) ⊕ X σ(cid:126)σ ⊕ X σ(cid:126)ψ . ⊗ X (12) (cid:126) X (23) (cid:126) X (13) (cid:126) X (123) (cid:126) X (132) (cid:126) X (12) (cid:126) ⊕ σσ ⊕ ψψ X (132) (cid:126) X (123) (cid:126) X (23) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (13) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (13) (cid:126) X (123) (cid:126) X (132) (cid:126) ⊕ σ ⊕ ψ X (12) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (23) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (23) (cid:126) X (132) (cid:126) ⊕ σσ ⊕ ψψ X (123) (cid:126) X (13) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (12) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (123) (cid:126) X (13) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (12) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (23) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (132) (cid:126) ⊕ X (132) (cid:126)ψ ⊕ σσ ⊕ σ σ ⊕ σσ ⊕ ψψ ⊕ ψψ ⊕ ψ ψ ⊕ σσψ ⊕ σψσ ⊕ ψσσX (132) (cid:126) X (23) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (13) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ X (12) (cid:126) ⊕ (cid:126)σ ⊕ (cid:126)ψ
111 3 X (123) (cid:126) ⊕ X (123) (cid:126)ψ ⊕ σσ ⊕ σ σ ⊕ σσ ⊕ ψψ ⊕ ψψ ⊕ ψ ψ ⊕⊕ σσψ ⊕ σψσ ⊕ ψσσ Table 5: Fusion rules for bare defects in (cid:16)
Ising (cid:2) (cid:17) × S . The discussion of g -confinement and g -deconfinement in Section 3 and the idea of the algorithm in Section4 can be applied to compute other fusion rings.We conjecture that fusion rules between inverse defects in G xBFCs conform to the rule X g ⊗ X g − = (cid:77) c c ⊗ g · c ∗ where c sums over a minimal subset of Irr( C ) which includes the identity, is closed under duality, and whoseorbit under the tensor product and g -action generates all of Irr( C ) .The rough idea of how to generalize our algorithm given G a finite group with presentation G = (cid:10) g , g , . . . , g k (cid:12)(cid:12) r , r , . . . , r l (cid:11) on generators is as follows. By writing arbitrary g, h ∈ G as words in gener-ators g = g g · · · g α , h = h h · · · h β , fusion of arbitrary defects proceeds by computing the product of gg − α g α h and rewriting the group labels on the resulting bare defects using the group relations. Of coursethe last step involves some art to guarantee the algorithm terminates and thus isn’t efficient in general, butfor small groups it suffices to quickly determine the fusion ring of the G -extension.22n particular all small examples of Z / Z -extensions of MTCs we checked could be described in thismanner. We include two brief examples of untwisted fusion rules, the Z / Z -toric code with electromagneticduality symmetry and Vec Z / Z anyons with charge-conjugation symmetry, see [1] or [12] for more details. Example 2 (Toric code with e ↔ m symmetry) . e ⊗ X = X f m ⊗ X = X f f ⊗ X = X e ⊗ X f = X m ⊗ X f = X f ⊗ X f = X f X ⊗ X = 1 ⊕ fX ⊗ X f = e ⊕ mX f ⊗ X f = 1 ⊕ f Example 3 ( Z -anyons with charge-conjugation symmetry) . ω ⊗ X = X ω ∗ ⊗ X = X X ⊗ X = 1 ⊕ ω ⊕ ω ∗ We conclude with a few short comments on some related directions.
While our main result shows that it is still possible to learn new things about MTCs through elementary con-siderations, already there has been work towards a more general extension theory of MTCs by mathematicalobjects with richer structure than groups, for example the Hopf monads of [8] or hypergroups of [3].Thesuccess of a classical approach here suggests it may too be possible to deduce fusion rules for more generalsymmetry-enriched categories using only the decategorified part of the symmetry.
Determining the possible quantum logical operations that can result from exchanging and measuring anyonsand defects in an SET phase requires algebraic data beyond the fusion rules. However, the fusion rulesdictate which units of quantum information e.g. qubits, qutrits, qudits, etc. can be encoded in the multi-fusion channels of objects.For example, it follows from Theorem 1 that a collection of 4 bare τ = ( ij ) defects with total vacuumcharge in (cid:0) C (cid:2) n (cid:1) × S n are always qudits with d = rank ( C ) .In upcoming work with Eric Samperton we leverage the approach to constructing fusion rings given hereto construct their categorifications and then apply it to derive algebraic data for SET phases, from which wecan derive insights into the interplay of symmetry, topological order, and quantum information, see also [9].23 Permutation defect fusion code sample PermDefectFusion[defsector1_, defcharge1_, defsector2_, defcharge2_] := Module[{i, j, k, l, m, deconfinements, righthandsector, numdiscycles1, numdiscycles2, cycle1, cycle2, permlist2, transpdecomp1, transp, newsector, transplist, index1pos,index2pos, productanyons, finaldefsector, finalanyonproduct, finalfusionproduct}, If[defsector1 == {} && defsector2 == {}, Return[AnyonFusion[defcharge1, defcharge2]]]; If[defsector1 == {} && defsector2 != {}, Return[ObjListToDefLabel[ AnyonDefectFusion[defcharge1, defsector2, defcharge2], defsector2]]]; If[defsector1 != {} && defsector2 == {}, Return[AnyonDefectFusion[defcharge2, defsector1, defcharge1]]]; deconfinements = MultilayerLabelToObj /@ {MinimalDeconfinement[defsector1, defcharge1], MinimalDeconfinement[defsector2,defcharge2]}; numdiscycles1 = Length[defsector1]; numdiscycles2 = Length[defsector2]; righthandsector = defsector2; For[i = 1, i <= numdiscycles1, i++, cycle1 = {defsector1[[i]]}; permlist2 = defsector2 // Flatten; transpdecomp1 = TranspositionDecomposition[cycle1]; For[k = 1, k <= Length[transpdecomp1], k++, transp = Cycles[{Reverse[transpdecomp1][[k]]}]; transplist = transp[[1, 1]]; newsector = PermutationProduct[transp, Cycles[righthandsector]][[1]]; righthandsector = newsector; Which[Length[Intersection[transplist, permlist2]] <= 1, permlist2 = righthandsector // Flatten; , Length[Intersection[transplist, permlist2]] == 2, index1pos = Position[defsector2, transplist[[1]] ][[1, 1]]; index2pos = Position[defsector2, transplist[[2]] ][[1, 1]]; If[index1pos != index2pos, permlist2 = righthandsector // Flatten, productanyons = BareTranspositionDefectFusion[transp[[1]]]; AppendTo[deconfinements, productanyons]; permlist2 = righthandsector // Flatten ]; ]; ]; ]; finaldefsector = righthandsector; finalanyonproduct = Fold[ObjTensor][deconfinements]; finalfusionproduct = Confinement[finalanyonproduct, finaldefsector, Table[1, numlayers]]; Return[finalfusionproduct];]; eferences [1] M. Barkeshli, P. Bonderson, M. Cheng, Z. Wang, Symmetry, defects, and gauging of topological phases .arXiv:1410.4540. (2014).[2] M. Barkeshli, C. M. Jian, X. L. Qi,
Genons, twist defects, and projective non-Abelian braiding statistics.
Phys. Rev. B, Vol. 87, No. 045130. (2013).[3] M. Bischoff.
Generalized Orbifold Construction for Conformal Nets.
Reviews in Mathematical PhysicsVol. 29, No. 1. (2017).[4] M. Bischoff.
The rank of G -crossed braided extensions of modular tensor categories. To appear in AMSContemporary Mathematics Series.[5] T. Barmeier, C. Schweigert.
A geometric construction for permutation equivariant categories from mod-ular functors.
Transformation Groups. Vol. 16, No. 2, Pgs. 287-337. (2011).[6] M. Bischoff, C. Jones.
Computing fusion rules for spherical G -extensions of fusion categories. arXivpre-print. (2019).[7] S. X. Cui, C. Galindo, J. Y. Plavnik, Z. Wang, On gauging symmetry of topological phases.
Comms. inMath. Phys. Vol. 348, No. 3. (2016).[8] S. X. Cui, M. Shokrian Zini, Z. Wang.
On generalized symmetries and structure of modular categories.
Science China Mathematics. Vol. 62, No. 3, Pgs. 417-446. (2019).[9] C. Delaney.
A categorical perspective on symmetry, topological order, and quantum information.
UCSBPhD Dissertation. (2019)[10] C. Delaney, E. Samperton. (In preparation.)[11] C. Delaney, E. Samperton.
Algebraic theory of bilayer symmetry defects . (In preparation.)[12] C. Delaney, Z. Wang.
Symmetry defects and their application to topological quantum computing.
Ac-cepted to AMS Contemporary Mathematics Series.[13] C. Edie-Michell, C. Jones, J. Plavnik.
Fusion rules for Z / Z -permutation gauging. arXiv:1804.01657.(2018).[14] P. Etinghof, S. Gelaki, D. Nikshych, V. Ostrik. Tensor categories . AMS. Mathematical surveys andmonographs. Vol. 205. (2015).[15] P. Etingof, D. Nikshych, V. Ostrik. Quantum Topology. Vol. 1, No. 209. (2010).[16] T. Gannon, C. Jones.
Vanishing of categorical obstructions for permutation orbifolds.
Comm. Math.Phys. Vol. 369, No. 1. Pgs 245-259. (2019).[17] Z. Wang.