Minimal modular extensions for super-Tannakian categories
aa r X i v : . [ m a t h . QA ] A ug MINIMAL MODULAR EXTENSIONS FORSUPER-TANNAKIAN CATEGORIES
C´ESAR F. VENEGAS-RAM´IREZ
Abstract.
In this paper, we continue with the ideas presented in[GVR17]. In this opportunity, we apply the fermionic action conceptto classify in cohomology terms the minimal modular extensions of asuper-Tannakian category. For this goal, we study some properties ofequivariantization and de-equivariantization processes and cohomologydata for the fermionic case. Introduction
Given a braided fusion category B , the construction of modular categoriesfrom B is studied in [M¨00] and [Bru00a] modifying the structure of B .However, it is wished to find modular categories from B without modifyingit. In other words, it is wanted to find a modular category M with a copyof B . In particular, modular categories with a copy of B satisfying someminimality condition are called minimal modular extensions in [M¨ug03].Find the minimal modular extensions of a braided fusion category is anopen problem proposed by Muger in [M¨ug03].The problem of finding the minimal modular extensions for symmetricfusion category was addressed in [LKW16a] . For a symmetric fusioncategory, its set of minimal modular extensions has a structure of abeliangroup. Moreover, for the category of representations of a finite group, calledTannakian category, was presented a complete description of all minimalmodular extensions. A description of a category of representations of asuper-group, called super-Tannakian categories, is left as an open problem.In [GVR17], it is studied the connection between the concept of fermionicaction of a super-group and minimal modular extensions for a super-Tannakian category. Here, it is proposed the obstruction to the existence ofminimal modular extensions for a braided fusion category. Moreover, it isproved that the homomorphism D : M ext (Rep( e G, z )) → M ext (SVec)defined in Equation (7) is surjective if and only if e G = G × Z / Z .In this paper, it is studied braided crossed fusion categories whose actionis a fermionic action as well as 2-homomorphism induced of these, accordingto [ENO10, Theorem 7.12]. As a result, it is obtained a fermionic version of[ENO10, Theorem 7.12] in Theorem 6.14. t is established a correspondence between minimal modular extensions ofa super-Tannakian category and braided crossed extensions with fermionicactions. This correspondence allows us to mean the minimal modularextensions as 2-homomorphisms of 2-groups; therefore, it is possible topresent a description in cohomology terms. This idea is developed inProposition 7.10, Corollary 7.11, Theorem 7.12, and Corollary 7.14.Finally, some examples are presented in Theorem 7.15 and Example 7.17where the results obtained here are applied .I thank Cesar Galindo at Universidad de los Andes for time, advice, andguidance in the development of this project .2. Mathematical background
Fusion categories.
By a fusion category , we mean a rigid monoidalcategory, C -linear, semisimple, with finite-dimensional Hom-spaces, and afinite number of isomorphism classes of simple objects that include the unitobject. We denote by Irr( C ), the set of isomorphism classes of simple objectsin a fusion category C .If K ( C ) denotes the Grothendieck ring, there exists a unique ringhomomorphism FPdim : K ( C ) → R such that FPdim( X ) > X ∈ Irr( C ), see [EGNO15, Proposition 3.3.6]. The Frobenius-Perrondimension of a fusion category C is defined asFPdim( C ) = X X ∈ Irr( C ) FPdim( X ) . Example 2.1.
Consider a finite group G and ω ∈ Z ( G, C × ) a 3-cocyclewith coefficients in C × . Vec ωG is the fusion category of finite dimensional G -graded vector spaces, the tensor product ⊗ is the tensor product of G -graded vectors spaces, the associativity constraint is given by a δ g ,δ h ,δ k = ω ( g, h, k ) id δ ghk , and the unit constraints are given by l δ g = ω ( e, e, g ) − id δ g and r δ g = ω ( g, e, e ) id δ g .Given a fusion category C , the set of isomorphism classes of invertibleobjects is denoted by Inv( C ), see [EGNO15].2.2. Braided fusion categories.
A fusion category B is called a braided fusion category if it is endowed with a family of natural isomorphisms c X,Y : X ⊗ Y → Y ⊗ X, X, Y ∈ C , satisfying the hexagon axioms, see [JS93].If B is a braided fusion category with braiding c , the reverse braided fusioncategory is defined as follows: B rev is equal to B as fusion category, but thebraiding is given by c revX,Y := c − Y,X for
X, Y ∈ B . This project is supported by Faculty of Science of Universidad de los Andes,Convocatoria 2018-2019 para la financiaci´on de proyectos de investigaci´on y presentaci´onde resultados en eventos ac´ademicos categor´ıa: estudiantes de doctorado candidatos. ext, we present a way to construct braided fusion categories from groupcohomology data. For this, we give the abelian cohomology concept.Let A be a finite abelian group, an abelian 3-cocycle is pair ( ω, c ) suchthat ω ∈ Z ( A, C × ) and c : A × A → C × satisfying the following equations: c ( g, hk ) c ( g, h ) c ( g, k ) = ω ( g, h, k ) ω ( h, k, g ) ω ( h, g, k ) c ( gh, k ) c ( g, k ) c ( h, k ) = ω ( g, k, h ) ω ( g, h, k ) ω ( k, g, h ) , for all g, h, k ∈ A. (1)We denote by Z ab ( A, C × ) the abelian group of all abelian 3-cocycles ( ω, c ),see [EML53, EML54],.An abelian 3-cocycle ( ω, c ) ∈ Z ab ( A, C × ) is called an abelian 3-coboundary if there is α : A × → C × , such that ω ( g, h, k ) = α ( g, h ) α ( gh, k ) α ( g, hk ) α ( h, k ) ,c ( g, h ) = α ( g, h ) α ( h, g ) , for all g, h, k ∈ A.B ab ( A, C × ) denotes the subgroup of Z ab ( A, C × ) of abelian 3-coboundaries.The quotient group H ab ( A, C × ) := Z ab ( A, C × ) /B ab ( A, C × ) is called the thirdgroup of abelian cohomology of A . Definition 2.2.
Given ( ω, c ) ∈ Z ab ( A, C × ), we define the braided fusioncategory Vec ( ω,c ) A as follows.Vec ( ω,c ) A = Vec ωA as fusion category.The braiding of Vec ( ω,c ) A is defined by the map c , and it will be denoted bythe same letter. c ( δ g , δ h ) = c ( g, h )id δ gh , for each g, h ∈ A. The hexagon axioms are equivalent to (1).If ( ω, c ) ∈ Z ab ( A, C × ), the map q : A → C × , q ( l ) := c ( l, l ) , l ∈ A, is a quadratic form on A ; that is, q ( l − ) = q ( l ) for all l ∈ A and thesymmetric map b q ( k, l ) := q ( kl ) q ( k ) − q ( l ) − , k, l ∈ A, is a bicharacter. Definition 2.3.
A pointed (braided) fusion C is a (braided) fusion categorywhere any simple object is invertible. The set of isormorphism classes ofsimple objects A := Inv( C ) is a (abelian) group with product induced by thetensor product. t is known that a (braided) fusion category C is (braided) equivalent toVec ( ω,c ) A for some finite (abelian) group A and (abelian) 3-cocycle.The Muger center of B is the fusion subcategory Z ( B ) := { Y ∈ B : c Y,X ◦ c X,Y = id X ⊗ Y , for all X ∈ B} . A braided fusion category B is called symmetric if Z ( B ) = B , i.e., if c Y,X ◦ c X,Y = id X ⊗ Y for each pair of objects X, Y in B .It is well known that symmetry fusion categories are equivalent to one ofthe following two examples:(a) Tannakian categories . The category Rep( G ) of finite dimensionalcomplex representation of a finite group G , with standard braiding c X,Y ( x ⊗ y ) := y ⊗ x for x ∈ X and y ∈ Y .(b) Super-Tannakian categories . A finite super-group is a pair ( e G, z ) where e G is a finite group and z is a central element of order two. An irreduciblerepresentation of e G has one degree if z acts as − id, and has zero degreeif z acts as id. We denote the degree of a simple object X ∈ Rep( e G )by | X | ∈ { , } .We define the braiding c ′ of two simple object X , Y by c ′ X,Y ( x ⊗ y ) = ( − | X || Y | y ⊗ x. The category Rep( e G ) with the braiding c ′ is called a super-Tannakian category, and it will be denoted by Rep( e G, z ).The super-Tannakian category Rep( Z / Z , [1]) is called the category of super-vector spaces and it will be denoted by SVec.Deligne establishes that every symmetry fusion category is braidedequivalent to Rep( G ) or Rep( e G, z ) for a unique finite group G or super-group ( e G, z ), see [Del02].A braided fusion category ( B , c ) is called non-degenerate if Z ( B ) ∼ =Vec. A modular category means a non-degenerate spherical braided fusioncategories. Non-degenerancy in this case is equivalente to the invertibilityof the S -matrix, see [DGNO10]. Example 2.4 (Pointed braided fusion categories of dimension four) . Non-degenerate braided pointed fusion categories of dimension four were classifiedin [DGNO10, Appendix A.3], in terms of the associated metric group. Next,we present the braided structure of such categories as categories of finitedimensional A -graded vectors spaces Vec ( ω,c ) A . We present abelian 3-cocyclesassociated to the metric groups. For this, we will identify the group of allroots of unity in C with Q / Z . The description that we show below will beof great importance later.(a) If A has a presentation given by A := { , v, f, v + f : 2 f = 0 , v = 0 } and k ∈ Q / Z such that 4 k = 0.We define ( ω k , c k ) ∈ Z ab ( A, Q / Z ) as follows: if x = x v v + x f f , y = y v v + y f f , and z = z v v + z f f then k ( x, y, z ) = ( if y v + z v < kx v if y v + z v ≥ ,c k ( x, y ) = 12 ( x v + x f ) y f + kx v y v The case k = 0 corresponds to the Drinfeld center of Vec Z / Z alsocalled the Toric Code MTC. The case k = corresponds to ( D , k = ± corresponds to twocopies of Semion MTC.(b) If A has a presentation given by A := { , v, f, v + f : 2 f = 0 , v = f } and k ∈ Q / Z such that 4 k = .We define ( ω k , c k ) ∈ Z ab ( A, Q / Z ) as follows: if x = x v v , y = y v v ,and z = z v v then ω k ( x, y, z ) = ( if y v + z v < if y v + z v ≥ ,c k ( x, y ) = kx v y v . In all cases above, the quadratic form q : A → Q / Z is given by q ( f ) = 12 , q ( v ) = q ( v + f ) = k, q (0) = 0 , and the number k = q ( v ) ∈ { s : 0 ≤ s < } is a complete invariant. See[RSW09] for more details about the classification of modular categories ofdimension four.2.3. Drinfeld center of a fusion category.
An important class of non-degenerate fusion categories arises using the Drinfeld center Z ( C ) of a fusioncategory ( C , a, ), see [DGNO10, Corolary 3.9]. The center constructionproduces a non-degenerate braided fusion category Z ( C ) from any fusioncategory C . Objects of Z ( C ) are pairs ( Z, σ − ,Z ), where Z ∈ C and σ − ,Z : − ⊗ Z → Z ⊗ − is a natural isomorphism such that the diagram Z ⊗ ( X ⊗ Y ) a Z,X,Y / / ( Z ⊗ X ) ⊗ Y ( X ⊗ Y ) ⊗ Z σ X ⊗ Y,Z ♥♥♥♥♥♥♥♥♥♥♥♥ a X,Y,Z ( ( PPPPPPPPPPPP ( X ⊗ Z ) ⊗ Y σ X,Z ⊗ id Y h h PPPPPPPPPPPP X ⊗ ( Y ⊗ Z ) id X ⊗ σ Y,Z / / X ⊗ ( Z ⊗ Y ) a − X,Z,Y ♥♥♥♥♥♥♥♥♥♥♥♥ commutes for all X, Y, Z ∈ C . The braided tensor structure is the following: the tensor product is ( Y, σ − ,Y ) ⊗ ( Z, σ − ,Z ) = ( Y ⊗ Z, σ − ,Y ⊗ Z ) where σ X,Y ⊗ Z := a Y,Z,X (id Y ⊗ σ X,Z ) a − Y,X,Z ( σ X,Y ⊗ id Z ) a X,Y,Z for X ∈ C . • the braiding is the isomorphism σ X,Y .We have that FPdim( Z ( C )) = FPdim( C ) for any fusion category C , see [EGNO15, Proposition 9.3.4]. Morevoer, fora braided fusion category B , there is braided embedding functor B → Z ( B ) , X ( X, c − X ) . Group actions on fusion categories.
Let C be a fusion category,Aut ⊗ ( C ) denotes the monoidal category where objects are tensorautoequivalences of C , arrows are monoidal natural isomorphisms, the tensorproduct is the composition of functors, and unit object Id C . Similarly,we define the monoidal category Aut br ⊗ ( B ) of braided autoequivalences ofa braided fusion category B .An action of a finite group G on a fusion category C is a monoidalfunctor ∗ : G → Aut ⊗ ( C ) where G denotes the discrete monoidal categorywith objects indexed by elements of G and tensor product given by themultiplication of G .An action ∗ : G → Aut ⊗ ( C ) of G over C has the following data: • tensor functors g ∗ : C → C , for each g ∈ G , • natural tensor isomorphisms φ ( g, h ) : ( gh ) ∗ → g ∗ ◦ h ∗ , for all g, h ∈ G ,and • a monoidal natural isomorphism ν : e ∗ → Id C ,which satisfy some conditions of coherence, see [Tam01, Section 2].An action of a finite group G on a braided fusion category B is definedsimilarly. In this case the monoidal functor ∗ : G → Aut br ⊗ ( B ) is defined overAut br ⊗ ( B ) and the data satisfy the same coherence conditions.An action on a (braided) fusion category as the one described here is alsocalled a bosonic action . Example 2.5 ([Tam01, Section 7]) . Let G and A be finite groups. Given ω ∈ Z ( A, C × ), an action of G on Vec ωA is determined by a homomorphism ∗ : G → Aut( A ) and normalized maps µ : G × A × A → C × γ : G × G × A → C × uch that ω ( a, b, c ) ω ( g ∗ ( a ) , g ∗ ( b ) , g ∗ ( c )) = µ ( g ; b, c ) µ ( g ; a, bc ) µ ( g ; ab, c ) µ ( g ; a, b ) ,µ ( g ; h ∗ ( a ) , h ∗ ( b )) µ ( h ; a, b ) µ ( gh ; a, b ) = γ ( g, h ; ab ) γ ( g, h ; a ) γ ( g, h ; b ) ,γ ( gh, k ; a ) γ ( g, h ; k ∗ ( a )) = γ ( h, k ; a ) γ ( g, hk ; a ) , for all a, b, c ∈ A , and g, h, k ∈ G .The action is defined as follows: for each g ∈ G , the associated monoidalfunctor g ∗ is given by g ∗ ( δ a ) := δ g ∗ ( a ) , constraint ψ ( g ) a,b = µ ( g ; a, b )id δ g ∗ ( ab ) and the tensor natural isomorphism is φ ( g, h ) δ a = γ ( g, h ; a )id δ ( gh ) ∗ ( a ) , for each pair g, h ∈ G, a ∈ A .We present a bosonic action of the cyclic group of order 2 on the categoriesVec ( ω,c ) A presented in Example 2.4 where A is an abelian group of order 4,and ( ω, c ) ∈ Z ab ( A, C × ).Given that H n ( G, C × ) ∼ = H n ( G, Q / Z ), we use normalized maps µ and γ with coefficients in Q / Z . Example 2.6.
Let A be an abelian group of order 4, let Vec ( ω k ,c k ) A be thecategories presented in Example 2.4, and let C = h u i be the cyclic groupof order 2 generated by u . Then, the following data defines an action of C over Vec ( ω k ,c k ) A .(a) If A has a presentation given by A := { , v, f, v + f : 2 f = 0 , v = 0 } ,then C has an action on Vec ( ω,c ) A defined by u ∗ ( f ) = f, u ∗ ( v ) = v + f, where the normalized maps µ and γ are defined by the tables: µ ( u ; − , − ) v f f + vv f f + v γ ( u, u ; − ) v f f + v (b) If A has a presentation given by A = { , v, f, v + f : 2 f = 0 , v = f } ,then C has an action on Vec ( ω,c ) A defined by u ∗ ( f ) = f, u ∗ ( v ) = v + f, where the normalized maps µ and γ are defined by the tables: µ ( u ; − , − ) v f f + vv f f + v γ ( u, u ; − ) v f f + v efinition 2.7. Let ρ : G → Aut ⊗ ( C ) be a group homomorphism where C is a fusion category and G is a finite group. A lifting of ρ is a monoidalfunctor ˜ ρ : G → Aut ⊗ ( C ) such that the isomorphism class of ˜ ρ ( g ) is ρ ( g ) foreach g ∈ G .If ρ : G → Aut ⊗ ( C ) is a group homomorphism, the finite group G actson \ K ( C ). Let us fix a representative tensor autoequivalence F g : C → C foreach g ∈ G and a tensor natural isomorphism θ g,h : F g ◦ F h → F gh for eachpair g, h ∈ G . Define O ( ρ )( g, h, l ) ∈ \ K ( C ) by the commutativity of thediagram(2) F g ◦ F h ◦ F lF g ( θ h,l ) (cid:15) (cid:15) ( θ g,h ) Fl / / F gh ◦ F lθ gh,l (cid:15) (cid:15) F ghlO ( ρ )( g,h,l ) (cid:15) (cid:15) F g ◦ F hl θ g,hl / / F ghl . Proposition 2.8 ([Gal11, Theorem 5.5]) . Let C be a fusion category and ρ : G → Aut ⊗ ( C ) a group homomorphism. The map O ( ρ ) : G × → \ K ( C ) defined by the diagram (2) is a 3-cocycle and its cohomology class dependson ρ . The map ρ lifts to an action e ρ : G → Aut ⊗ ( C ) if and only if O ( ρ )] ∈ H ρ ( G, \ K ( C )) . If [ O ( ρ )] = 0 the set of equivalence classes ofliftings of ρ is a torsor over H ρ ( G, \ K ( C )) . Proposition 2.8 says us that there exists an action of H ρ ( G, \ K ( C )) onthe liftings of ρ ; we denoted this action by ⊲ . Moreover, if e ρ is a lifting of ρ ,any other lifting can be obtained in the form µ ⊲ e ρ for µ ∈ H ρ ( G, \ K ( C )).2.4.1. Equivariantization. processes of equivariantization and de-equiva-riantization are some of the main tools that we will use throughout thismanuscript. we present a description of these processes as well as the mostrelevant results in this regard. Most of the results presented here appear in[DGNO10].Given an action ∗ : G → Aut ⊗ ( C ) of a finite group G on a fusion category C with monoidal structure given by φ . The G -equivariantization of C is thefusion category denoted by C G and defined as follows. An object in C G is apair ( V, τ ), called G -object, where V is an object of C and τ is a family ofisomorphisms τ g : g ∗ ( V ) → V , g ∈ G , such that τ gh = τ g g ∗ ( τ h ) φ ( g, h ) , for all g, h ∈ G . A morphisms σ : ( V, τ ) → ( W, τ ′ ) between G -objects is amorphism σ : V → W in C such that τ ′ g ◦ g ∗ ( σ ) = σ ◦ τ g , for all g ∈ G . The ensor product is defined by( V, τ ) ⊗ ( W, τ ′ ) := ( V ⊗ W, τ ′′ )where τ ′′ g = τ g ⊗ τ ′ g ψ ( g ) − V,W , and the unit object is ( , id ). The Frobenius-Perron dimension of C G is | G | FPdim( C ), see [DGNO10, Proposition 4.26]. Theorem 2.9 ([ENO11a, Proposition 2.10]) . Let D be a fusion categoryand let G be a finite group. If there exists a braided tensor functor Rep( G ) → Z ( D ) such that its composition with the forgetful functor is fullyfaithful, then there is a fusion category C and an action of G on C such that D ∼ = C G . De-equivariantization.
In this part, we describe the oppositeconstruction to equivariantization called de-equivariantization.
Definition 2.10 ([DGNO10]) . A central functor from a braided fusioncategory B to a fusion category C is a braided functor B → Z ( C ).If E is a symmetric fusion category, C is called a fusion category over E ifit is endowed with a braided inclusion E → Z ( C ) such that its compositionwith the forgetful fuctor is an inclusion in C . If B is braided, B is a braidedfusion category over E if it is endowed with a braided inclusion E → Z ( B ),see [DGNO10, ENO11a].Let C be a fusion category and Rep( G ) ⊂ Z ( C ) be a Tannakiansubcategory which embeds into C via the forgetful functor Z ( C ) → C .The algebra O ( G ) of functions on G is a commutative algebra in Z ( C ).The category of left O ( G )-modules in C is a fusion category called de-equivariantization of C by Rep( G ), and it is denoted by C G , see [DGNO10]for more details. It follows from [DGNO10, Lemma 3.11] thatFPdim( C G ) = FPdim( C ) | G | . There is a canonical fully faithful monoidal functorRep( G ) ∼ = Vec G → C G . This functor canonically decomposes as Rep( G ) → Z ( C G ) → C G . Thus C G is a fusion category over Rep( G ), see [DGNO10, Section 4.2.2]. Proposition 2.11 ([DGNO10, Theorem 4.18, Proposition 4.19, Proposition4.22]) . Equivariantization defines an equivalence between the 2-category of(braided) fusion categories with an action of G and the 2-category of fusioncategories over Rep( G ) . The de-equivariantization functor is inverse to theequivariantization functor. In short, equivariantization and de-equivariantization are mutually inversefunctors. .5. The picard group.
Let C be a fusion category, a left C -modulecategory M is a C -linear semisimple category with an action of C denotedby ( X, M ) X ∗ M for X ∈ C and M ∈ M . The action has an associativity and unit constraint denoted by a X,Y,M :( X ⊗ Y ) ∗ M → X ∗ ( Y ∗ M ) and l M : ∗ M → M for X, Y ∈ C and M ∈ M that satisfy certain conditions. A right C -module category is defined in asimilar way, see [ENO10]Given C and C ′ fusion categories, a ( C , C ′ )-bimodule category is a left( C ⊠ C ′ op )-module category.If M is a right C -module category and N is a left C -module category, thetensor product of M and N over C can be understood as the category ofexact left C -module functors M ⊠ C N := Fun C ,re ( M op , N ), see [ENO10].In [ENO10, Remark 3.6], it is ensured that the tensor product over D ofa ( C , D )-bimodulo by a ( D , C ′ )-bimodulo has a ( C , C ′ )-bimodule structure. Definition 2.12 ([ENO10]) . A ( C - C ′ )-bimodule category M is invertible ifthere exist bimodule equivalences such that M op ⊠ C M ∼ = C ′ , and M ⊠ C ′ M op ∼ = C . If B is a braided fusion category, any left B -module category can beendowed with a structure of ( B , B )-bimodule, so we can speak aboutinvertible left B -module. Definition 2.13.
For a braided fusion category B , the Picard 2-group isdenoted by Pic( B ) and described as follows: Objects are invertible left B -modules, 1-morphisms are module equivalences, and 2-morphisms areisomorphisms between such equivalences. Pic( B ) can be truncated to acategorical group Pic( B ) if we forget the 2-morphisms and consider 1-morphisms up to isomorphism. Similarly, Pic( B ) can be truncated to thegroup Pic( B ) called the Picard group of B .If M is an invertible module category over B and B ∗M = Fun B ( M , M ),we obtain an equivalence B ⊠ B rev ∼ = Z ( B ∗M ). The compositions α + : B = B ⊠ ⊂ B ⊠ B rev ∼ = Z ( B ∗M ) → B ∗M ,α − : B = ⊠ B rev ⊂ B ⊠ B rev ∼ = Z ( B ∗M ) → B ∗M , are called alpha-induction functors, see [Ost03, ENO10]. In particular, forevery invertible module M , the alpha-inductions are equivalences. Thus α + = α − ◦ θ M , where θ M : B → B is a braided autoequivalence. For a more specificdefinition of the alpha-induction functors, see [DN + or each M ∈
Pic( B ) the functors α ± are defined as follows: α ±M : B → B ∗M : X → X ⊗ − ;The tensor structure for α ±M is defined by: α + M ( X )( Y ⊗ M ) = X ⊗ Y ⊗ M c x,y / / Y ⊗ X ⊗ M = Y ⊗ α + M ( X )( M ) , (3) α −M ( X )( M ⊗ Y ) = X ⊗ Y ⊗ M c − y,x / / Y ⊗ X ⊗ M = α −M ( X )( M ) ⊗ Y, (4)for each X, Y ∈ B and M ∈ M . Theorem 2.14. [ENO10, Theorem 5.2.]
For a non-degenerate braidedfusion category B , the functor M → θ M is an equivalence between Pic( B ) and Aut br ⊗ ( B ) . Fermionic fusion categories
All definitions and results of this section was presented in [GVR17].
Definition 3.1 ([GVR17]) . Let C be a fusion category. An object ( f, σ − ,f ) ∈Z ( C ) is called a fermion if f ⊗ f ∼ = 1 and σ f,f = − id f ⊗ f .(a) A fermionic fusion category is a fusion category with a fermion. Afermionic fusion category C with fermion ( f, σ − ,f ) is denoted by thepair ( C , ( f, σ − ,f )).(b) A braided fusion category B with braiding c and a fermion of the form( f, c − ,f ) is called a spin-braided fusion category . This spin-braidedfusion category will be denoted by ( B , f ) because the half-braiding isdetermined in an obvious way by c − ,f . Example 3.2.
The categories Vec ( ω k ,c k ) A presented in Example 2.4 are spin-braided fusion categories with fermion f . Example 3.3 (Ising categories as spin-braided fusion categories) . By anIsing fusion category, we mean a non-pointed fusion category of Frobenius-Perron dimension 4.Ising categories have 3 classes of simple objects , f, σ . The Ising fusionrules are σ = 1 + f, f = 1 , f σ = σf = σ. The associativity constraints are given by the F -matrices F σσσσ = ǫ √ (cid:18) − (cid:19) , F σfσf = F fσfσ = − , where ǫ ∈ { , − } .The Ising fusion categories admit several braided structures, and in allcases f is a fermion. An Ising braided fusion category is always non-degenerate. In [DGNO10, Appendix B], it was proven that there are 8equivalence classes of Ising braided fusion categories. efinition 3.4 ([GVR17]) . Let ( C , ( f, σ − ,f )) and ( C ′ , ( f ′ , σ ′− ,f ′ )) befermionic fusion categories. A tensor functor ( F, τ ) :
C → C ′ is called a fermionic functor if F ( f ) ∼ = f ′ and the diagram(5) F ( V ⊗ f ) τ V,f (cid:15) (cid:15) F ( σ V,f ) / / F ( f ⊗ V ) τ f,V (cid:15) (cid:15) F ( V ) ⊗ F ( f ) id F ( V ) ⊗ φ (cid:15) (cid:15) F ( f ) ⊗ F ( V ) φ ⊗ id F ( V ) (cid:15) (cid:15) F ( V ) ⊗ f ′ σ ′ F ( V ) ,f ′ / / f ′ ⊗ F ( V )commutes for each V ∈ C , where φ is an isomorphism between F ( f ) and f ′ .Let ( C , ( f, σ − ,f )) be a fermionic fusion category. We will denote byAut ⊗ ( C , f ) the full monoidal subcategory of Aut ⊗ ( C ) whose objects arefermionic tensor autoequivalences. The group of isomorphism classes ofautoequivalences in Aut ⊗ ( C , f ) is denoted by Aut ⊗ ( C , f ). Example 3.5.
If ( B , f ) and ( B ′ , f ′ ) are spin-braided fusion categories, and F : B → B ′ is a braided functor such that F ( f ) ∼ = f ′ ; then, F is a fermionicfunctor. In fact, by definition of a braided functor, F satisfies Diagram (5),see [ENO05, Definition 8.1.7.].For spin-braided fusion categories, we will denote by Aut br ⊗ ( B , f ) thefull monoidal subcategory of Aut br ⊗ ( B ) whose objects are braided tensorautoequivalences described in Example 3.5. The group of isomorphismclasses of spin-braided autoequivalences in Aut br ⊗ ( B , f ) is denoted byAut br ⊗ ( B , f ).Next, we will present the fermionic action concept, but before we willrecall some important facts.If ( e G, z ) is a super-group, the exact sequence1 −→ h z i −→ e G −→ e G/ h z i −→ , defines and is defined by a unique element α ∈ H ( e G/ h z i , Z / Z ). Fromnow on we will identify a super-group ( e G, z ) with the associated pair (
G, α )according to the convenience of the case. We set the following notation G := e G/ h z i and α ∈ H ( G, Z / Z ). Lemma 3.6 ([GVR17]) . Let G be a finite group. There is a canonicalcorrespondence between equivalence classes of monoidal functors e ρ : G −→ Aut br ⊗ (SVec) and elements of H ( G, Z / Z ) . Given a functor e ρ : G −→ Aut br ⊗ (SVec), the corresponding element in H ( G, Z / Z ) will be denoted by θ e ρ . efinition 3.7 ([GVR17]) . Let ( C , ( f, σ − ,f )) be a fermionic fusion categoryand ( G, α ) a super-group. A fermionic action of (
G, α ) on ( C , ( f, σ − ,f )) isa monoidal functor e ρ : G −→ Aut ⊗ ( C , f ) , such that the restriction functor e ρ : G −→ Aut ⊗ ( h f i ) satisfies θ e ρ ∼ = α in H ( G, Z / Z ), see Lemma 3.6.A fermionic action of a finite super-group ( G, α ) on a spin-braided fusioncategory ( B , f ) is defined in a similar way. In this case, the monoidal functor e ρ : G → Aut br ⊗ ( B , f ) is defined over Aut br ⊗ ( B , f ) and the data satisfies thesame condition.Two of the most important results presented in [GVR17] are the following: Theorem 3.8 ([GVR17]) . Let ( e G, z ) be a finite super-group. Then theequivariantization and de-equivariantization processes define a biequivalenceof 2-categories between fermionic fusion categories with a fermion action of ( e G, z ) and fusion categories over Rep( e G, z ) . Corollary 3.9 ([GVR17]) . Let ( e G, z ) be a finite super-group. Thenequivariantization and de-equivariantization processes define a biequivalenceof 2-categories between spin-braided fusion categories with fermionic actionof ( e G, z ) compatible with the braiding, and braided fusion categories D over Rep( e G, z ) such that Rep( G ) ⊆ Z ( D ) . Given a finite super-group (
G, α ), a fermionic fusion category( C , ( f, σ − ,f )), and a group homomorphism ρ : G → Aut ⊗ ( C , f ). An α -liftingof ρ is a fermionic action e ρ : G → Aut ⊗ ( C , f ) of ( e G, α ) over ( C , ( f, σ − ,f ))such that the isomorphism class of e ρ ( g ) is ρ ( g ) for each g ∈ G .4. Obstruction to fermionic actions If ρ : G → Aut ⊗ ( C , f ) is a group homomorphism, the question that weanswer in this part is when there exists a fermionic action that realizes ρ ,i.e., we want to know when there exists a α -lifting of ρ .The fermionic obstruction was defined in [GVR17] to determine when agroup homomorphism ρ is an α -lifting.The existence of an α -lifting implies the existence of a lifting for ρ in thesense of Definition 2.7. Thus, we have that the obstruction O in Theorem2.8 vanishes.Let us recall the fermionic obstruction. We consider the G -modulehomomorphism r : \ K ( C ) → \ K ( h f i ) ∼ = Z / Z defined by restriction. When r is non-trivial, the exact sequence1 / / Ker( r ) (cid:31) (cid:127) i / / \ K ( C ) r / / / / Z / Z / / / H ( G, Ker ( r )) i ∗ / / H ( G, \ K ( C )) r ∗ / / H ( G, Z / Z ) d / / / / H ( G, Ker ( r )) i ∗ / / H ( G, \ K ( C )) r ∗ / / H ( G, Z / Z ) d / / . . . This long exact sequence is used in the following definition.
Definition 4.1.
Let ρ : G → Aut ⊗ ( C , f ) be a group homomorphism and e ρ : G −→ Aut ⊗ ( C , f ) a lifting of ρ . For each α ∈ H ( G, Z / Z ), we define O ( ρ, α ) := ( θ e ρ /α ∈ H ( G, Z / Z ) if r is trivial, d ( θ e ρ /α ) ∈ H ( G, Ker( r )) if r is non-trivial,where r : \ K ( C ) → \ K ( h f i ) is the restriction map defined above.In Theorem 4.2, we establish the independence of the obstruction onthe choice of the lifting e ρ , the existence of an α -lifting in terms ofa cohomological value, and a correspondence of liftings with a certainsubgroup in H ( G, \ K ( C )). Theorem 4.2 ([GVR17]) . Let ( G, α ) be a finite super-group and ρ : G → Aut ⊗ ( C , f ) a group homomorhism with e ρ : G → Aut ⊗ ( C , f ) a lifting of ρ .Then (a) the element O ( ρ, α ) does not depend on the lifting. (b) The homomorphism ρ has lifting to a fermionic action of ( G, α ) if andonly if O ( ρ, α ) = 0 . (c) The set of equivalence classes of α -liftings of ρ is a torsor over Ker (cid:16) r ∗ : H ( G, \ K ( C )) → H ( G, Z / Z ) (cid:17) . Fermionic actions on non-degenerate spin-braided fusioncategories of dimension four
Up to equivalence, all braided fusion categories with dimension four waspresented in Example 2.4, as pointed braided fusion categories with fusionrules given by an abelian group of order four, and in Example 3.3, as Isingcategories.Next, we present some facts about fermionic actions over spin-braidedfusion categories of dimension four. For more details can be consulted[GVR17].
Proposition 5.1.
Only trivial super-groups act fermionically on a spin-braided Ising category.
Theorem 5.2.
Let ( G, α ) be a finite super-group and Vec ( ω k ,c k ) A a pointedspin-modular category of dimension four. Then a) Aut br ⊗ (Vec ( ω k ,c k ) A , f ) ∼ = Z / Z . (b) A group homomorphism G → Z / Z ∼ = Aut br ⊗ (Vec ( ω k ,c k ) A , f ) is alwaysrealized by a bosonic action. (c) A group homomorphism G → Z / Z ∼ = Aut br ⊗ (Vec ( ω k ,c k ) A , f ) is realized bya fermionic action of a super-groups ( G, α ) associated to ρ if and onlyif d ( α ) = 0 . (d) If d ( α ) = 0 then the equivalence classes of fermionic actions of ( G, α ) associated to ρ is a torsor over Ker (cid:16) r ∗ : H ( G, A ) → H ( G, Z / Z ) (cid:17) . Here d : H ( G, Z / Z ) → H ( G, Z / Z ) is the connecting homomorphismassociated to the G -module exact sequence → Z / Z → A r → Z / Z → . Braided ( e G, z ) -crossed extensions In this section, we finish establishing some properties of fermionicactions on fermionic fusion categories that arise under the processesof equivariantization and de-equivariantization. These properties referexclusively to the classification of a particular type of extensions of fermionicfusion categories, which we call braided ( e G, z )-crossed extensions.6.1.
Braided G -crossed fusion categories.Definition 6.1. Let G be a finite group. A G -grading for a fusion category C is a decomposition C = M g ∈ G C g into a direct sum of full abelian subcategories such that the tensor productdefines a functor from C g × C h into C h for all g, h ∈ G . We assume that thegrading is faithful, i.e., C g = 0 for all g ∈ G . Definition 6.2. A G -extension of a fusion category D is a G -graded fusioncategory C such that C e is equivalent to D . Definition 6.3 ([Tur00]) . A fusion category C is called a braided G -crossedfusion category if it is equipped with the following data:(a) a grading C = L g ∈ G C g ,(b) an action G → Aut ⊗ ( C ) of G on C such that g ∗ ( C h ) ⊂ C ghg − , and(c) a G - braiding , that is, natural isomorphisms c X,Y : X ⊗ Y → g ∗ ( Y ) ⊗ X, X ∈ C g , g ∈ G, and Y ∈ C . If φ g,h : ( gh ) ∗ → g ∗ h ∗ is the monoidal structure of the functor g → g ∗ ,and µ g is the tensor structure of g ∗ , we need to hold some compatibilityconditions.Note that the trivial component C e is itself a braided fusion category withan action of G by braided autoequivalences of C e . heorem 6.4 ([DGNO10]) . The equivariantization and de-equivarian-tization constructions define a bijection betweeen equivalence classes ofbraided G -crossed fusion categories and equivalence classes of braided fusioncategories containing Rep( G ) as a symmetric fusion subcategory. Theorem 6.4 tells us that the de-equivariantizacion by G of a braidedfusion category containing Rep( G ) is a braided G-crossed fusion category.The following theorem shows how we can find the trivial componentof de-equivariantization and the commutativity relation between takingcentralizers and taking de-equivariantization. These results can be foundin [Tur10, Theorem 3.8] and [DGNO10, Proposition 4.30]. Proposition 6.5.
Let B be a braided fusion category containing Rep( G ) asa full subcategory, and D be the de-equivariantization of B by G . (a) The trivial component of the braided G -crossed fusion category D is C B (Rep( G )) G . In particular, if Z ( C B (Rep( G ))) = Rep( G ) the braidedfusion category D e is modular. (b) Equivariantization and de-equivariantization define an isomorphismbetween the lattice of fusion subcategories of B containing Rep( G ) andthe lattice of G -stable fusion subcategories of D , i.e., if e D is a G -stable subcategory of D then e D G = e B contains Rep( G ) , and if e B isa subcategory of B containing Rep( G ) then e B G = e D is a G -stablesubcategory of D . (c) Suppose B is a braided fusion category over Rep( G ) . The isomorphismof (b) commutes with taking centralizer, i.e., C B ( e D G ) = ( C D ( e D )) G and C D ( e B G ) = ( C B ( e B )) G . Theorem 6.6 ([ENO10, Theorem 1.3.]) . Graded extensions of a fusioncategory C by a finite group G are parametrized by triples ( c, M, α ) , where c : G → BrPic( C ) is a group homomorphism, M belongs to a certaintorsor T c over H ( G, Inv( Z ( C ))) (where G acts on Inv( Z ( C )) via c ), and α belongs to a certain torsor T c,M over H ( G, C × ) . Here the data c , M mustsatisfy the conditions that certain obstruction O ( c ) ∈ H ( G, Inv( Z ( C ))) and O ( c, M ) ∈ H ( G, C × ) vanish. Theorem 6.6 establishes the basis for the classification of all extensionsof fusion categories, but we are interested in particular braided G -crossedextensions that we presente below . For those extensions Theorem 6.7 ismore relevant. Theorem 6.7 ([ENO10, Theorem 7.12]) . Let B be a braided fusion category.Equivalence classes of braided G -crossed extensions of B (with faithful G -grading) are in bijection with morphisms of categorical 2-groups G → Pic( B ) . H -Obstruction. The obstruction that we will study next, called H -obstruction, measures when a bosonic action on a non-degenerate fusioncategory e ρ : G → Aut br ⊗ ( B ) can be lifted to a 2-group homomorphism e ρ : G → Pic( B ). According to Theorem 2.14, we are identifying Aut br ⊗ ( B )with Pic( B ).We want to construct a 2-homomorphism from G to Pic( B ) fora non-degenerate braided fusion category. We can characterize this2-homomorphisms in cohomology terms. Firtly, Computing the O -obstruction to know if a group homomorphism G → Aut br ⊗ ( B ) admits alifting e ρ , and secondly, computing the H -obstruction to determine if ithas a lifting to a 2-homomrphism. Since 2-homomorphisms classify braided G -crossed extensions of B , the process gives us an algorithm to find thesebraided G -crossed extensions. That is one of our goals, specially extensionswith specific properties that we will discuss later.Suppose a non-degenerate braided fusion category B , a bosonic action e ρ is determined by data e ρ := ( g ∗ , ψ g , ϕ g,h ) : G → Aut br ⊗ ( B ), and the setof equivalence classes of bosonic actions is a torsor over H ρ ( G, Inv( B )).If we suppose that e ρ admits a lifiting ee ρ : G → Pic( B ), a 2-cocycle µ ∈ Z ρ ( G, Inv( B )) has a bosonic action denoted by µ ⊲ e ρ .The H -obstruction of the pair ( e ρ, µ ) is defined as a 4-cocycle O ( e ρ, µ ) ∈ H ( G, C × ) described by the formula O ( e ρ, µ ) = c µ g ,g , ( g g ) ∗ ( µ g ,g ) (6) a ( g g ) ∗ ( µ g ,g ) ,µ g ,g ,µ g g ,g g a − g g ) ∗ ( µ g ,g ) , ( g ) ∗ ( µ g ,g g ) ,µ g ,g g g a ( g ) ∗ ( µ g ,g ) , ( g ) ∗ ( µ g g ,g ) ,µ g g g ,g a − g ) ∗ ( µ g ,g ) ,µ g ,g g ,µ g g g ,g a µ g ,g ,µ g g ,g ,µ g g g ,g a − µ g ,g , ( g g ) ∗ ( µ g ,g ) ,µ g g ,g g ϕ g ,g ( µ g ,g )( ψ g ) − (( g ) ∗ ( µ g ,g ) , µ g ,g g ) ψ g ( µ g ,g , µ g g ,g ) , where a is the associative constraint of the category B . Proposition 6.8 ([CGPW16, Proposition 9]) . If B is a non-degeneratebraided fusion category, the homomorphism of categorical groups ( µ ⊲ e ρ ) : G −→ Pic( B ) can be lifted if and only if O ( e ρ, µ ) defined by (6) is trivial. More details about the H -obstruction used in this paper can be consultedin [ENO10] and [CGPW16].6.3. Braided ( e G, z ) -crossed extensions. Of all braided G -crossedextensions of a braided fusion category B , we are interested in studyingthose extensions with a fermionic action of ( e G, z ) over the trivial component. he idea of this part is to classify this “fermionic” extensions in terms of2-homomorphisms as happens in Theorem 6.7. Definition 6.9. A braided ( e G, z ) -crossed fusion category ( D , f ) is a braided G -crossed fusion category D where ( D e , f ) is a spin-braided fusion categoryin the sense of Definition 3.1, and the action that corresponds to thestructure of braided G -crossed category is a fermionic action of ( e G, z ) on( D e , f ).In this case, a functor between braided ( e G, z )-crossed fusion categoriesis a functor between braided G -crossed fusion categories which is also afermionic functor.The next corollary is the fermionic version of Theorem 6.4. This one givesa bijection between braided fusion categories over Rep( e G, z ) and braided( e G, z )-crossed fusion categories.
Corollary 6.10.
Let ( e G, z ) be a finite super-group. Equivariantization andde-equivariantization define a bijection between braided ( e G, z ) -crossed fusioncategories ( D , f ) , up to equivalence, and braided fusion categories C over Rep( e G, z ) , up to equivalence.Proof. There is a bijection between braided G -crossed fusion categories D and braided fusion categories over Rep( G ) by Theorem 6.7.If C is a braided fusion category over Rep( e G, z ), it is a braided fusioncategory over Rep( G ), so D := C G is a braided G -crossed fusion category,and D is a fermionic fusion category by Theorem 3.8.Finally, as Rep( e G, z ) ⊆ Z (Rep( G )) then SVec ⊆ D e , and the fermion in D belongs to its trivial component.Conversely, if ( D , f ) is a braided ( e G, z )-crossed fusion category, theequivariantization C = D G is a braided fusion category such thatRep( e G, z ) = SVec G ⊆ C . (cid:3) Definition 6.11. A braided ( e G, z ) -crossed extension of a spin-braided fusioncategory ( B , f ) is a braided ( e G, z )-crossed fusion category ( D , f ) whosetrivial component ( D e , f ) is equivalent to ( B , f ). Definition 6.12.
If ( B , f ) is a spin-braided fusion category, we considerPic( B , f ) ⊆ Pic( B ) as the full subcategory with objects M ∈
Pic( B ) suchthat θ M ( f ) = f .The condition θ M ( f ) = f for an invertible module category M isequivalent to say that the module functors − ⊗ f and f ⊗ − are isomorphicautoequivalences of M . In fact, this is a consequence of the definition of thefunctors α ± in (3), (4) and θ . More details can be found in [DN + Proposition 6.13.
Consider a non-degenerate spin-braided fusion category ( B , f ) . There is an equivalence between Aut br ⊗ ( B , f ) and Pic( B , f ) . roof. This proposition is a direct consequence of Definition 6.12. In fact, θ : Pic( B ) → Aut br ⊗ ( B ) is an equivalence of categories. In particular, for each M ∈
Pic( B , f ), we have θ M ( f ) = f by definition, so θ M ∈ Aut br ⊗ ( B , f ). (cid:3) Theorem 6.14 classifies braided ( e G, z )-crossed extensions of spin-braidedfusion categories in terms of homomorphisms of 2-groups from G toPic( B , f ). This theorem is the key result that we use to find minimal modularextensions for super-Tannakian categories, as we will see later. Theorem 6.14.
Let ( B , f ) be a spin-braided fusion category. Equivalenceclasses of braided ( e G, z ) -crossed categories C having a faithful G -gradingwith trivial component B are in bijection with homomorphism of categorical -groups ee ρ : G → Pic( B , f ) , such that e ρ is a fermionic action of ( e G, z ) on ( B , f ) .Proof. According to Theorem 6.7, a 2-group homomorphism ee ρ : G → Pic( B , f ) corresponds to a braided G -crossed extension D of B . The actionof G on D e = B is given by e ρ , so D is a braided ( e G, z )-crossed extension of B . Conversely, if D is a braided ( e G, z )-crossed extension of B , it correspondsto a 2-group homomorphism ee ρ : G → Pic( B ), whose action induced isfermionic since the action on B is fermionic too. Now, the G -action for g ∈ G on B is given by the equivalence of Fun B ( D g , D g ) with the leftand right multiplication by elements of B , see [ENO10]. In particular, if D g ∈ Pic( B , f ) then g ∗ is a fermionic functor, i.e., g ∗ ∈ Aut br ⊗ ( B , f ). (cid:3) Minimal modular extensions
Muger defines minimal modular extensions of a braided fusion category B in [M¨00]. For a Tannakian category, a complete description of its minimalmodular extensions was presented in [LKW16a]. Nevertheless, for the super-Tannakian case, such description is yet an open problem. We use Corollary6.10 and Theorem 6.14 to give an approach to the solution.7.1. Minimal modular extensions.Definition 7.1.
Let B be a braided fusion category. A minimal modularextension of B is a pair ( M , i ), where M is a modular fusion category suchthat i : B → M is a braided full embedding andFPdim( M ) = FPdim( B ) FPdim( Z ( B )) . Two minimal modular extensions ( M , i ) and ( M ′ , i ′ ) are equivalent ifthere exists a braided equivalence F : M −→ M ′ such that F ◦ i ∼ = i ′ . Example 7.2 (Modular extensions of SVec) . For the symmetric super-Tannakian category SVec, there are 16 modular extensions (up toequivalence). They can be classified in two classes, the first one is given by 8 sing braided modular categories parametrized by ζ , such that ζ = −
1. Abrief description of them was given in Example 3.3. The second one is givenby 8 pointed modular categories Vec ( ω k ,c k ) A where A is an abelian group oforder four, and ( ω k , c k ) is an abelian 3-cocycle. A description of this typeof categories was presented in Example 2.6. More information about thisexample can be found in [Kit06, DGNO10].In [LKW16a] the set of equivalence classes of minimal modular extensionsof a unitary braided fusion category B is denoted by M ext ( B ). In particular,if B is a symmetric fusion category, M ext ( B ) is an abelian group with unitobject Z ( B ).Let E be a symmetric fusion category; the set of minimal modularextensions of E is non-empty since Z ( E ) is always a minimal modularextension. In this case, any minimal modular extension M of E can bethought as a module category over E with action induced by tensor productof M . Then, the binary operation on M ext ( E ) is defined using the tensorproduct of module categories over E . This operation is well defined accordingto [LKW16a, Lemma 4.11]. Moreover, the associativity of this operation isproved in [LKW16a, Proposition 4.12] in a more general case.The existence of the neutral element was proved in [LKW16a, Lemma4.18]. There is shown that for any symmetric category E , the Drinfel center Z ( E ) is the neutral object in M ext ( E ). Example 7.3 (Modular extensions of Tannakian fusion categories) . For asymmetric Tannakian category Rep( G ), the group of modular extensions(up to equivalence) is isomorphic to the abelian group H ( G, C × ). For each ω ∈ H ( G, C × ), Z (Vec ωG ) is a modular extension of Rep( G ), see [LKW16a].7.2. Obstruction theory to existence of minimal modularextensions.
Following [M¨00, Bru00b], we say that a braided fusion categoryis modularizable if Z ( B ) is Tannakian. Note that if Rep( G ) = Z ( B ) then B G is a modular fusion category; therefore, the term modularizable makessense from this point of view. Definition 7.4 ([GVR17]) . Let B be a modularizable braided fusioncategory with Z ( B ) = Rep( G ). The de-equivariantization B G hasassociated a monoidal functor G e ρ / / Aut br ⊗ ( B G ) Φ / / Pic( B G ) , where e ρ is the canonical action of G on the modular fusion category B G . Wedefine the H -anomaly of B as the H -obstruction of e ρ in H ( G, C × ).According to Proposition 6.8, trivial anomaly is equivalent to saying that B G has a G -extension with induced action on it given by e ρ . In fact, if the O -obstruction is trivial, there exists a lifing of e ρ to a 2-homomorphism ee ρ : G → Pic( B G ). Thus, according to Theorem 6.7, there exists a braided G -crossed extension of B G . efinition 7.5 ([ENO11a]) . A braided fusion category B is called slightlydegenerate if Z ( B ) is braided equivalent to SVec.In [GVR17], we have mentioned some relations between slightlydegenerate fusion categories and categories over Rep( e G, z ) under fermionicactions . Specifically, if the Muger center of B is a super-Tannakian categoryRep( e G, z ), the category B G is a slightly degenerate fusion category where G = e G/ h z i .Theorem 7.6 tells us that the study of minimal modular extensions for anon-modularizable fusion category can be reduced to the study of certainassociate slightly degenerate fusion category.If B is non-modularizable, that is Z ( B ) ∼ = Rep( e G, z ), the maximal centraltannakian subcategory of B is braided equivalent to Rep( G ) with G ∼ = e G/ h z i . Theorem 7.6 ([GVR17]) . Let B be a braided fusion category with non-trivial maximal central Tannakian subcategory Rep( G ) ⊆ Z ( B ) . (a) If B is modularizable, B admits a minimal modular extension if andonly if the H -anomaly of B vanishes. (b) If B is non-modularizable with Z ( B ) = Rep( e G, z ) , B admits a minimalmodular extension if and only if the following conditions hold: (i) the slightly degenerate braided fusion category B G has a minimalmodular extension S , (ii) there exists a fermionic action of ( e G, z ) on S such that B G is G -stable, and the restriction to B G coincides with the canonicalaction of G on B G , (iii) the anomaly of S G vanishes.Remark . (a) Another way of expressing part (b) of Theorem 7.6 is thefollowing: a non-modularizable braided fusion category B has a minimalmodular extension if and only if the slightly degenerate fusion category B G has a minimal modular extension S which in turn has a braided( e G, z )-crossed extension.(b) Theorem 7.6 was used in [GVR17] to show examples of braided fusioncategories without minimal modular extensions.The de-equivariantization defines a well-defined map D : M ext ( B ) → M ext ( B G ) M → ( M G ) e . For any minimal modular category M of B the map D sends M to the trivialcomponent of the de-equivariantization M G , see [LKW16a] and [GVR17].In particular, for a super-Tannakian category Rep( e G, z ) with maximalcentral Tannakian subcategory Rep( G ), D : M ext (Rep( e G, z )) → M ext (SVec)(7)is a group homomorphism. orollary 7.8 ([GVR17]) . Let
Rep( e G, z ) be a finite super-group. The map D : M ext (Rep( e G, z )) → M ext (SVec) is surjective if and only if ( e G, z ) is a trivial super-group. Theorem 7.9 ([GVR17]) . Let ( B , f ) be a spin-braided fusion categoryof dimension four, and G be a finite group with a group homomorphism ξ : G −→ Aut br ⊗ ( B , f ) . Then ξ can be extended to a bosonic action and to a2-homomorphism ee ξ : G −→ Pic( B ) . The group M ext (Rep( e G, z )) . We use the homomorphism D , Corollary6.10, and Theorem 6.14 to describe in cohomology terms the minimalmodular extensions of a super-Tannakian category Rep( e G, z ).In this subsection, we denote by B a pointed braided fusion categoryof dimension four presented in Example 2.6, by I an Ising category (seeExample 3.3), and by D : M ext (Rep( e G, z )) → M ext (SVec)the group homomorphism defined in (7). We fix the following notation,( e G, z ) is a finite super-group, G is the quotient group e G/ h z i , and the pair( G, α ) is identified with the super-group ( e G, z ) where α ∈ Z ( G, Z / Z ) is theunique 2-cocycle (up to cohomology) associated to the super-group ( e G, z ).By definition of D , for a minimal modular extension M of Rep( e G, z ), D ( M ) is a minimal modular extension of SVec. In fact, we havethat Rep( e G, z ) G ∼ = SVec, and D ( M ) = ( M G ) e is modular withFPdim(( M G ) e ) = 4.On the other hand, note that every minimal modular extension C of SVechas a natural structure of spin-braided fusion category where the naturalfermion corresponds to the inclusion of SVec in C . Proposition 7.10.
Let C be a minimal modular extension of SVec . Thespin-braided fusion category C is in the image of D if and only if C has abraided ( e G, z ) -crossed extension.Proof. Consider a minimal modular extension C of SVec in the image of D ,then there is M ∈ M ext (Rep( e G, z )) such that D ( M ) = C . By Theorem 7.6, M G is a braided ( e G, z )-crossed extension of C .If C ∈ M ext (SVec) has a braided ( e G, z )-crossed extension L , the modularfusion category L G is a category on Rep( e G, z ) according to Corollary6.10. Moreover, by Theorem 7.6, L G is a minimal modular extension ofRep( e G, z ). (cid:3) Corollary 7.11.
Let C be a minimal modular extension of SVec . The pre-image of C with respect to D is in correspondence with 2-homomorphisms ee ρ : G → Pic( C , f ) such that the truncation e ρ is a fermionic action of ( e G, z ) . roof. Using Proposition 7.10, for each
C ∈
Im( D ), there is an M in M ext (Rep( e G, z )) such that M G is a braided ( e G, z )-crossed extension of C .By Theorem 6.14, braided ( e G, z )-crossed extensions of C correspond to 2-group homomorphisms ee ρ : G → Pic( C , f ) such that e ρ is a fermionic actionof the super-group ( e G, z ). (cid:3) We know that braided G -crossed extensions of a non-degenerate fusioncategory C are in correspondence with 2-homomorphisms ee ρ : G → Pic( C ).If e ρ : G → Aut br ⊗ ( C ) is the truncation of ee ρ then liftings of e ρ are a torsoron H ( G, C × ). In a similar way, if ρ is the truncation of e ρ , liftingsof ρ are a torsor on H ρ ( G, \ K ( C )). Therefore, any 2-homomorphismassociated by truncation to ρ can be parametrized by an element in H ρ ( G, \ K ( C )) × H ( G, C × ). Conversely, if we start with a group homorphism ρ and consider the obstruction theory in order to obtain a 2-homomorphism,we can conclude that any lifting to a 2-homomorphism can be parametrizedby pairs ( µ, ϕ ) ∈ H ρ ( G, \ K ( C )) × H ( G, C × ) such that the obstructions O ( ρ ) and O ( ρ, µ ) vanish. Thus, every 2-homomorphism ee ρ : G → Pic( C )can be parametrized by triples ( ρ, µ, ϕ ), where ρ : G → Aut br ⊗ ( C , f ) is agroup homomorphism, µ belongs to a certain torsor over H ρ ( G, \ K ( C )), and ϕ belongs to a certain torsor over H ( G, C × ) such that O ( ρ ) and O ( ρ, µ )vanish.In Theorem 7.12 below, we characterize the image of the grouphomomorphism D in terms of group homomorphisms and group cohomology. Theorem 7.12.
Consider a minimal modular extension C of SVec . Thepre-image of C under D is parametrized by triples ( ρ, µ, ϕ ) , where ρ : G → Aut br ⊗ ( C , f ) is a group homomorphism, µ belongs to a certain torsor over Ker( r ∗ : H ρ ( G, \ K ( C ) → H ρ ( G, \ K (SVec)) , and ϕ belongs to a certaintorsor over H ( G, C × ) . The data µ and ϕ must satisfy the conditions thatobstruction O ( ρ, α ) and O ( ρ, µ ) vanish.Proof. Let C be a minimal modular extension of SVec. By Corollary 7.11,the pre-image of C is in correspondence with 2-homomorphisms ee ρ : G → Pic( C , f ) such that e ρ is a fermionic action. These 2-homomorphisms are incorrespondence with data ( ρ, µ, ϕ ) where ρ : G → Aut br ⊗ ( C , f ), µ belongsto certain torsor over Ker( r ∗ : H ( G, \ K ( C ) → H ( G, Z / Z ))) meaningthe fermionic condition, and ϕ belong to certain torsor over H ( G, C × ).Moreover, the data satisfies that the obstructions O ( ρ, α ) and O ( ρ, µ )vanish.Conversely, We know that data ( ρ, µ, ϕ ) with obstructions O ( ρ ) and O ( ρ, µ ) vanish parametrize 2-homomorphisms ee ρ : G → Pic( C ). The onditions O ( ρ, α ) = 0 and µ ∈ Ker( r ∗ : H ( G, \ K ( C )) → H ( G, Z / Z ))) isequivalente to say that the 2-homomrphisms ee ρ have values in Pic( C , f ) withtruncation to a ferminic action. (cid:3) Remark . The Proposition 7.12 tells us that the image of D correspondto spin-braided categories ( C , f ) in M ext (SVec) with at least one fermionicaction that can be extended to 2-homomorphisms ee ρ : G → Pic( C , f ). Corollary 7.14.
The kernel of D is parametrized by triples ( ρ, µ, ϕ ) where ρ : G → Aut br ⊗ (Vec ( ω ,c ) Z / Z × Z / Z , f ) is a group homomorphism, µ belongs toa certain torsor over Ker( r ∗ : H ρ ( G, \ K ( C )) → H ρ ( G, \ K (SVec)) , and ϕ belongs to a certain torsor over H ( G, C × ) such that O ( ρ, µ ) vanishes.Proof. The result follows from Theorem 7.12, given that the trivial elementof M ext (SVec) is Z (SVec) ∼ = Vec ( ω ,c ) Z / Z × Z / Z as we explain in Section 7.1. (cid:3) Next, we use Proposition 7.12 and Corollary 7.14 to determine the orderof M ext (Rep( Z /m Z × Z / Z , (0 , m an odd number. Theorem 7.15 (Minimal modular extensions of ( Z /m Z , α ≡ m odd) . Consider the trivial super-group Z /m Z × Z / Z where m is an odd number.The group M ext (Rep( Z /m Z × Z / Z , ([0] , [1]))) has order m .Proof. We have that(a) By Corollary 7.8 the group homomorphism D is surjective.(b) Given that m is an odd number, the unique group homomorphism Z /m Z → Z / Z × Z / Z is the trivial homomorphism.(c) H ( Z /m Z , Z / Z × Z / Z ) ∼ = 0,(d) H ( Z /m Z , C × ) = Z /m Z .Using items ((b)), ((c)), and ((d)), the Corollary 7.14 implies thatkernel of D is correspondence with triples ( ρ, µ, ϕ ) where ρ : Z /m Z → Aut br ⊗ (Vec ω ,c Z / Z × Z / Z , f ) is the trivial homomorphism, µ ∈ H ρ ( G, Z / Z × Z / Z ) = 0, and ϕ belongs to a certain torsor over H ( Z /m Z , C × ). Asthere are m of that triples, the order of M ext (Rep( Z /m Z × Z / Z , ([0] , [1]))is 16 m . (cid:3) Example 7.16 (Minimal modular extensions of Z / Z ) . The group M ext (Rep( Z / Z , [3])) has order 48. We consider Z / Z ∼ = Z / Z × Z / Z and apply Theorem 7.15. Then the group of minimal modular extensions ofRep( Z / Z , [3]) has order 48.This result agrees with [LKW16b, Table XX]. Example 7.17 (Minimal modular extensions of Z / Z ) . we can deduce thatthere are exactly 32 minimal modular extensions of Rep( Z / Z , [2]). Thisinformation agrees with the result presented by Ostrik. In this case, weprove that ker( D ) has order 4 and that the image of D consist of the pointedmodular extensions of SVec. a) The H -obstruction that we need to consider is H ( Z / Z , C × ) = 0, soeach action Z / Z → Aut br ⊗ ( C , f ) has a lifiting to a 2-homomorphism.(b) Z / Z is not a trivial super-group, so no Ising category can be in theimage of D .(c) The kernel of D is parametrized by triples ( ρ, µ, ϕ ) where ρ : Z / Z → Aut br ⊗ (Vec ( ω ,c ) Z / Z × Z / Z , f ) is the trivial homomorphism, µ ∈ H ( Z / Z , Z / Z × Z / Z ) such that r ∗ ( µ ) is non-trivial, and ϕ ∈ H ( Z / Z , C × ) ∼ = Z / Z . Then, there are 4 such triples, which impliesthat Ker( D ) has order 4. A similar analysis shows that every pointedfusion category with fusion rules given by Z / Z × Z / Z is in the imageof D .(d) Moreover, every pointed fusion category with fusion rules given by Z / Z is in the image of D . In fact, the triple ( ρ, µ, ϕ ) satisfies the conditionsin Proposition 7.12; where ρ : Z / Z → Aut br ⊗ (Vec ( ω k ,c k ) Z / Z , f ) is the trivialhomomorphism, µ is the unique non-trivial object in H ( Z / Z , Z / Z ),and ϕ ∈ H ( Z / Z , C × ). Proposition 7.18.
Let D : M ext (Rep( e G, z )) → M ext (SVec) be the grouphomomorphism defined above. We have that D is non-trivial and the imageof D has at least 4 elements. Specifically, the pointed fusion categories withfusion rules given by Z / Z × Z / Z is always in the image of D .Proof. In general, consider B one of the pointed braided fusion categories inExample 2.6. Remember that B is non-degenerate and the restriction map r : \ K ( B ) ∼ = Inv( B ) → \ K (SVec) can be consider as r ( X )( f ) = c f,X ◦ c X,f , foreach X ∈ Inv( B ). According to the braided structure of each B the grouphomomorphism r is the same for each B with fusion rules Z / Z × Z / Z .This implies that the group homomorphisms and the group cohomology inTheorem 7.12 are the same. (cid:3) Proposition 7.19.
Let B be a slightly degenerate pointed braided fusioncategory, then B has a minimal modular extension.Proof. By [ENO11b, Proposition 2.6]
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