Higher Gauss Sums of Modular Categories Are Closely Related to the Frobenius-Schur In- Dicators

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In the late 20th century, a notion of Gauss sum incidentally emerged in quantum invariants of 3-manifolds derived from modular categories introduced by Reshetikhin and Turaev (cf. [54, 59, 42, 53, 29, 25, 33]). The modulus of a Gauss sum τ and the finiteness of the order of τ/τ reemerge in the contexts of invariants of 3-manifolds and rational conformal field theory (RCFT) (cf. [54] and [55]). Similar to its classical counterpart, the Gauss sums characterize modular categories with T-matrices of order 2, up to equivalence [57]. Moreover, the classification of relations among the Witt classes [ (sl2, k)] [8] and [ (sl3, k)] [46] were proven in part using first central charge arguments. C C In this paper, we define higher Gauss sums τn and anomalies αn of a premodular category as generalizations of the Gauss sum of a quadratic form of a finite abelian group G and the 1 Jacobi symbol −G . A natural choice of the square root of the anomaly αn is defined as the | | higher (multiplicative ) central charge ξn for n Z which is motivated by rational conformal field theory. ∈ Frobenius-Schur indicators are arithmetic invariants of premodular categories. They were first introduced for the representations of finite groups a century ago. Their recent generalizations to Hopf algebras [30] and rational conformal field theory [3] inspired the development of Frobenius-Schur indicators of pivotal categories [35]. Similar to their classical counterparts, the higher Gauss sums of modular categories are closely related to the Frobenius-Schur indicators. In particular, the modulus of the higher Gauss sums of a modular category are completely determined by the Frobenius-Schur indicators. More relations between these arithmetic invariants and examples are demonstrated in Section 3. One subtlety to the definitions of higher anomaly or central charges for a premodular category is that they are only well-defined when τ ( ) = 0. For a modular category , n C 6 C τ1( )τ 1( ) = dim( ) = 0 (cf. [54, 2, 32, 15]). Our Theorem 4.1 shows that for any modular C − C C 6 category , τn( ) = 0 when n is coprime to the order of the T-matrix of . A d-numberC isC an6 algebraic integer which generates in the ring of algebraicC integers an ideal fixed by the absolute Galois group [38]. The formal codegrees of spherical fusion categories are d-numbers and they have been used to prove the non-existence of fusion categories associated with specific fusion graphs/rules (cf. [56, 5, 41]). In Corollary 4.2, higher Gauss sums τ ( ) n C are shown to be d-numbers and the higher central charge ξn( ) are root of unity under the assumption that is modular and n is coprime to the orderC of the T-matrix of . In particular, when isC the Drinfeld center of a spherical fusion category, the coprime higherC central charges areC 1 (Theorem 4.4), which may not hold for other n by Example 4.6. Higher Gauss sums and central charges behave well under standard constructions such as de-equivariantizations of premodular categories and categories of local modules over certain connected étale algebras (also known as simple current extensions or condensations). In 0 0 0 particular, Theorem 5.6 proves that G τn( G) = τn( ) and ξn( G) = ξn( ) where G is the condensation of the premodular category| | Cby a TannakianC subgroupC G ofC for anyC integer n coprime to the order of the T-matrix ofC . In addition, if is modular (TheoremC 5.8) the C 0 C preceding statement holds for the category A of local modules over a ribbon algebra A of (defined in Definition 2.4). C C As an application of Sections 4 and 5, we prove in Theorem 6.1 that two Witt equivalent pseudounitary modular categories must have the same higher central charge ξn for any integer n coprime to the orders of their T-matrices. We use this result to distinguish Witt equivalence classes of pseudounitary modular categories which are indistinguishable using the first central charge alone. The higher central charges which are well-defined for any two Witt HIGHERGAUSSSUMSOFMODULARCATEGORIES 3 equivalent pointed modular categories are invariants, and we conjecture this statement could be generalized for all Witt equivalence classes. The organization of this paper is as follows: Section 2 describes the notation, basic definitions, and necessary concepts while introducing fundamental examples. Section 3 motivates our definitions of higher Gauss sums and higher (multiplicative) central charges for premodular categories. The relations between higher Gauss sums and Frobenius-Schur indicators, and a relation between the first and second Gauss sums are shown in this section. A broad range of examples are given to illustrate their properties. Section 4 describes the Galois action on the modular data of modular categories which is the key to prove our main result Theorem 4.1. Section 5 consists of a sequence of technical lemmas proven by graphical calculus. These lemmas and Theorem 4.1 are then applied to prove Theorems 5.6 and 5.8. Some examples of computing higher Gauss sums and central charges are illustrated. Section 6 proves the Witt invariance of certain higher central charges (Theorem 6.1). Applications of higher central charges to differentiate Witt equivalence classes which are indistinguishable by the first central charge alone are demonstrated. The paper ends with some open questions to stimulate conversation and future research.

2 Preliminaries 2.1 Premodular and modular categories In this section, we recall some basic definitions and results on fusion and modular categories. The readers are referred to [14] and [32] for the details. Throughout this paper, a fusion category is a C-linear, abelian, semisimple, rigid monoidal category with finite-dimensional Hom-spaces,C finitely-many isomorphism classes of simple

objects and simple monoidal unit ½. In particular, we abbreviate the dimension of the Hom- space (X,Y ) over C by C [X : Y ] := dimC( (X,Y )) C C for any X,Y , and the set of isomorphism classes of simple objects of by ( ). ∈C C O C The duality of can be extended to a contravariant monoidal equivalence ( )∗, and so ( ) defines a monoidalC equivalence on . A pivotal structure on is a natural isomorphism− − ∗∗ C C j : id ( )∗∗ of monoidal functors. For any given pivotal structure j on , one can define a (left)C →trace− Tr (f) C for any endomorphism f : V V in (see for exampleC [34]). In C ∈ → C particular, Tr (idV ) is called the (left pivotal) dimension of V and denoted by dim (V ), or dim(V ) whenC the category is clear from the context. The pivotal structure j is saidC to be spherical if dim(V ) = dim(V ∗) for all V ( ). In this case, dim(V ) is a real cyclotomic integer for V (cf. [15]). In particular,∈ O dim(C V ) is totally real. The global dimension dim( ) of is∈ given C by C C dim( )= dim(V )2. C V ( ) ∈OXC Note that dim( ) can be defined without using any pivotal structure of [32, 15]. Recall that an algebraicC number a is called totally positive if every Galois conjugateC of a is a positive real number. Since dim(V ) is totally real, dim( ) is a totally positive cyclotomic integer. A fusion category is called pseudounitary ifC dim( ) agrees with FPdim( ), the Frobenius- Perron dimension ofC . In this case, admits a uniqueC spherical pivotal structureC such that dim(V ) is the Frobenius-PerronC dimensionC of V for all V (cf. [15]). Throughout this paper, we assume that any pseudounitary fusion category is∈ equipped C with such a canonical spherical pivotal structure. 4 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

By virtue of [35, Theorem 2.2] we may assume without loss of generality that any pivotal category in this paper is strict. In other words, is a strict monoidal category such that C C ( )∗ is a strict monoidal functor, ( )∗∗ = id , and the pivotal structure j : id ( )∗∗ is the− identity. Under this assumption,− we can performC computations using graphicalC → calculus− with the conventions of [2, 26] for any premodular category . For a braided fusion category with braiding c, the M¨ugerC centralizer or relative commu- tant of a full fusion subcategoryC of is the full subcategory generated by D′ D C X ( ) cY,X cX,Y = idX Y for all Y ( ) . { ∈ O C | ◦ ⊗ ∈ O D } Note that is always a fusion subcategory of . A braided fusion category is said to be D′ C nondegenerate if ( ′)= ½ . Note that the Drinfeld center ( ) of a fusion category is a nondegenerate braidedO C fusion{ } category. Z C C A premodular category is a braided spherical fusion category. The (unnormalized) S- matrix of a premodular categoryC is deﬁned as C SX,Y = Tr (cY,X∗ cX∗,Y ) for X,Y ( ) .

C ◦ ∈ O C ½ In particular, S½, = 1. Nondegeneracy of the braiding of any premodular category can be characterized by the invertibility of its S-matrix (cf. [32]). In this case, is called aCmodular category. C If is a spherical fusion category, then its Drinfeld center ( ) inherits the spherical structureD of , and hence a modular category. Throughout thisZ paper,D the Drinfeld center ( ) of a sphericalD fusion category is always assumed to be spherical with the inherited ZsphericalD structure from . D In a premodular categoryD , the underlying braiding c determines the Drinfeld isomorphism C u : id ( )∗∗ of C-linear functors (cf. [34, p38]). The twist or ribbon isomorphism is defined as θ =C →u 1−j where j is the spherical structure of . For each X ( ), θ = λ id for − C ∈ O C X · X some scalar λ C. We will use the abuse notation to denote λ by θX . By [1, 55], θX is a root of unity and∈ so the T-matrix of , which is defined as C T = (δX,Y θX )X,Y ( ) , C ∈O C has finite order. The order N = ord(T ( )) is called the Frobenius-Schur exponent of Z C C (cf. [34]), which is generally greater than ord(T ). Since the S-matrices of and ( ) are C C Z C defined over the cyclotomic field Q(e2πi/N ) (see [37]), N is also the conductor of . If is C C modular, then ord(T ) = ord(T ( )). Therefore, we will simply call ord(T ) the conductor or the FS-exponent of C when it isZ modular.C C If is a premodularC category with braiding c, then we define rev to be identical to as C C 1 Crev spherical fusion categories but with the reversed braiding c˜ := c− for all X,Y . X,Y Y,X ∈ C ˜ ˜ 1 rev In this case the associated twist θ satisfies θX = θX− for all X . For any finite group G, the category Rep(G) of finite-dimensional∈C complex representations of G is a braided fusion category with the braiding inherited from Vec, the category of finite-dimensional C-spaces. In particular, = Rep(G) is symmetric, i.e., ( ′) = ( ). Moreover, Rep(G) is pseudounitary and theC corresponding pivotal dimensionOs coincideC O withC the dimensions of vector spaces over C. Therefore, Rep(G) admits a natural premodular category structure and its ribbon isomorphism θ is the identity. For the purpose of this paper, a premodular category is called Tannakian if is equivalent to Rep(G) for some finite group G. In particular, theC ribbon isomorphism of aC Tannakian category is the identity natural isomorphism. HIGHERGAUSSSUMSOFMODULARCATEGORIES 5

Example 2.1. Let A be a finite abelian group and q : A C× a quadratic form. By the results of [12, 13], there exists an Eilenberg-MacLane 3-co→cycle (ω, c) of A, where ω is a 3-cocycle and c is a 2-cochain of A such that c(a, a) = q(a) for all a A. In particular, c ω ∈ defines a braiding on the fusion category VecA, the category of finite-dimensional A-graded vector spaces over C with the associativity isomorphism given by ω. The fusion category ω VecA is pseudounitary with the canonical pivotal dimensions given by the usual dimension of ω C-spaces. Therefore, VecA with the braiding c is a premodular category and we denote it by (ω,c) (ω,c) VecA . The quadratic form q completely determines the equivalence class of VecA . We denote by (A, q) any of these equivalent premodular categories. This premodular category (A, q) is modularC if and only if the corresponding quadratic form q is nondegenerate. The C category of super vector spaces sVec can be defined as the premodular category (Z2,q) with q(1) = 1. C − Example 2.2 (premodular categories from Lie theory). Another family of premodular categories can be realized by a construction based on the representation theory of the q-deformed universal enveloping algebra q(g) (cf. [31, 16]) for a complex finite-dimensional simple Lie algebra g and a complex parameterU q. These categories have a long history in mathematical physics. The readers are referred to [43] for a general survey of the subject. The properties of the resulting categories depend heavily on q. For certain roots of unity, modular tensor categories are produced while other choices may result in premodular categories, or ones with infinitely-many isomorphism classes of simple objects. For the illustrative examples in this ∨ paper we will only consider those roots of unity of the form q = eπi/(m(k+h )) where k N is ∈ the level, h∨ is the dual Coxeter number of g and m 1, 2, 3 is a scaling factor dependent on g. The categories (g, k) in this smaller collection are∈ { known} to be unitary modular tensor categories with modularC data accessible via the Kac-Petersen formulas [24]. Given two premodular categories and , their (Deligne) tensor product ⊠ has simple objects X ⊠ Y for X ( ) andCY D( ), and is again premodularC withD the ribbon ∈ O C ∈ O D structure given by θX⊠Y = θX θY . For any braided fusion category , it follows from a well-known result of Müger [32]⊗ that C (2.1) ( ) ⊠ rev Z C ≃C C is a braided equivalence if and only if is nondegenerate. C 2.2 Local modules and the Witt group There are several constructions of new modular categories from a given one in the context of rational conformal field theory. One method is to consider a certain subcategory of the tensor category of modules over a commutative algebra object in a modular category. The readers are referred to [26] for more details. When one adds various restrictions to the commutative algebras under consideration, the resulting tensor category is fusion, braided, premodular, and sometimes modular. Here we review the basics of this theory assuming the reader is familiar with the rudimentary definitions of modules over algebras in fusion categories which can be found in [26] and [14, Chapter 8] when needed. Definition 2.3. Let be a premodular category. An algebra A of (or simply A ) is called connected étale Cif the multiplication morphism m : A A AC satisfies the following∈ C conditions: ⊗ →

(a) [½ : A] = 1 (connected), (b) m c C = m, (commutative) and ◦ A,A 6 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

(c) m splits as a morphism of A-bimodules (separable). A connected étale algebra A of is called ribbon if C (d) dim A = 0 and θ = id . 6 A A The conditions (a), (b), (c) and (d) for a ribbon algebra ensure the category of left A- modules A is a spherical fusion category equipped with the spherical structure inherited from (cf. [26]).C However, the braiding of can only be passed onto some fusion subcategories ofC in general. The following definitionC is due to Pareigis [40]. CA Definition 2.4. Let be a premodular category with a connected étale algebra A . Denote by 0 the fullC subcategory of local (or dyslectic) M , i.e., ∈ C CA ∈CA (2.2) ρ c c = ρ, ◦ M,A ◦ A,M where ρ : A M M is the left A-module action of M. ⊗ → 0 It is shown in [26] that if a connected étale algebra A is in addition ribbon, then A is 0 C premodular. Moreover, A is modular if is [26, Theorem 4.5]. It is worth to note that the same results could be establishedC for theC category of right local modules of A. For a ribbon algebra A we denote the dimension of a left A-module M by ∈ C ∈ CA dimA(M). By [26, Theorem 1.18], we have the relations

dim (M) 0 dim( ) (2.3) dim (M)= C , and dim( )= C . A dim (A) CA dim (A)2 C C Let be a Tannakian subcategory of . Suppose is equivalent to Rep(H) for some finite D C D group H as premodular categories. The dual group algebra A = C[H]∗ Rep(H), called the regular algebra of , is a ribbon algebra of Rep(H), and hence of . ∈ D C Example 2.5. Let q be a nondegenerate quadratic form on a finite abelian group G. Then (G, q) is a modular category of rank G . Assume in addition that the metric group (G, q) Chas an isotropic subgroup H G (i.e.| | q(x) = 1 for all x H). Then (G, q) contains a Tannakian subcategory equivalent⊂ to Rep(H) as premodular∈ categories.C Let A be the regular algebra of . ThenD dim( (G, q)0 )= G / H 2. D C A | | | | Suppose the premodular category contains a Tannakian subcategory which is equivalent to Rep(H) for a finite group HC as premodular categories. Then D and is a D ⊂ D′ D Tannakian subcategory of ′. We denote by ( ′)H the de-equivariantization of ′ by H. Let A be the regular algebra ofD . By [11, PropositionD 4.56(i)], the local A-moduleD category 0 D CA is equivalent to ( ′)H as premodular categories. Another applicationD of local module categories lies in the study of the Witt equivalence of nondegenerate braided fusion categories introduced in [7]. Definition 2.6. Nondegenerate braided fusion categories and are Witt equivalent if there C D exist fusion categories 1, 2 such that ( 1) ⊠ ( 2) ⊠ is a braided equivalence. The Witt equivalence classA A of is denotedZ byA [ ]. C≃Z A D C C Two nondegenerate braided fusion categories and are Witt equivalent if there exist C 0D 0 connected étale algebras A and B such that A B as braided fusion categories (cf. [7, Proposition 5.15]). ∈ C ∈ D C ≃ D The Witt equivalence classes of nondegenerate braided fusion categories form an abelian 1 rev Witt group under the Deligne tensor product with [ ]− = [ ] (see Equation 2.1). The Witt group W of nondegenerate braided fusion categoriesC is a generalizaC tion of the Witt group of nondegenerateW quadratic forms of finite abelian groups. In particular, the subgroup WQ HIGHERGAUSSSUMSOFMODULARCATEGORIES 7

pt(p) of generated by the Witt equivalence classes of the pointed modular categories W(G, q), whereW p is a prime and G is a finite abelian p-group, is canonically isomorphic to the Cclassical Witt group (p) of nondegenerate quadratic forms of finite abelian p-groups (see [7, Section 5.3]). WQ 3 Higher Gauss sums and central charges 3.1 Definition and motivation Let be a modular category. The Gauss sums τ ( ) of are defined as C ± C C 1 2 (3.1) τ ±( )= θ± dim(X) . C X X ( ) ∈OXC Their properties can be found in the literature such as [2], [32], [14] and [10]. It is well- + + τ ( ) known that τ ( )τ −( ) = dim( ) 1, and τ −(C) is a root of unity [55]. In particular, C C C ≥ C τ +( ) = dim( ). From this, the definition of the (multiplicative) central charge of can | C | C C be defined as ξ( ) := τ +( )/ dim( ), which is also a root of unity. p The same definitionC (3.1)C of GaussC sums τ ( ) can be extended easily to a premodular p ± C category . However, τ ±( ) could be zero which can be demonstrated in the category sVec C C + of super vector spaces where τ (sVec) = τ −(sVec) = 1 1 = 0 (cf. Example 2.1). Therefore, ξ(sVec) is undefined with this definition. Nevertheless,− the notion of higher of Gauss sums can be generalized and studied for arbitrary premodular categories. Definition 3.1. Let be a premodular category and n Z. The nth Gauss sum of is defined as C ∈ C τ ( ) := θn dim(X)2. n C X X ( ) ∈OXC

Note that τn( )= τ n( ) as dim(X) R for all X ( ) (cf. [15]). C − C ∈ ∈ O C 3.2 Gauss sums and Frobenius-Schur indicators For any object X in a spherical fusion category and n Z, the nth Frobenius-Schur C ∈ indicator (FS-indicator) of X, denoted by νn(X), is a scalar introduced in [35, 37]. The notion was also previously defined in the contexts of Hopf algebras and rational conformal field theory (cf. [30], [3]). We will simply define the nth FS-indicator of as C (3.2) ν ( ) := dim(X)ν (X) . n C n X ( ) ∈OXC

Remark 3.2. By virtue of [37, Corollary 5.6], ν n(X) = νn(X) for n Z and X . − ∈ ∈ C Therefore, ν n( )= νn( ) for n Z. Moreover, if = Rep(H) for any semisimple quasi-Hopf − C C ∈ C algebra H over C, then νn( ) = νn(H) where H is considered as the regular representation of H. C

If is modular, the modulus of τn( ) can be expressed in terms of νn( ) as stated in the followingC proposition, which is essentiallyC proved in [22, Proposition 5.5]C with a diﬀerent emphasis in the statement. Proposition 3.3. ([22, Proposition 5.5]) Let be a modular category. Then for any integer n, we have C τ ( ) 2 = dim( )ν ( ) . | n C | C n C 8 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

2 In particular, dim( ) divides τn( ) in the ring of algebraic integers, and νn( ) is a totally non-negative cyclotomicC integer| forC | any n Z. C ∈ The preceding proposition can be further refined if is a Drinfeld center. The following refinement is essentially proved in [50, Lemma 3.8]. C Proposition 3.4. ([50, Lemma 3.8]) Let be a spherical fusion category and = ( ). Then, for n Z, D C Z D ∈ τ ( ) = dim( )ν ( ). n C D n D Thus, we have the immediate corollary for the case of semisimple quasi-Hopf algebras. Corollary 3.5. Let = (Rep(H)) for some semisimple quasi-Hopf algebra H. Then for any n Z, C Z ∈ (3.3) τ ( ) = dim(H)ν (H). n C n If in addition H is a Hopf algebra, then (3.4) τ ( ) = dim(H) Tr(S ) Z 2 C H ∈ where SH is the antipode of H. Proof. Equation (3.3) follows directly from Remark 3.2 and Proposition 3.4, and Equation (3.4) is an immediate consequence of (3.3) and [30, Theorem 2.7(3)]. Now, we can apply Corollary 3.5 to compute the higher Gauss sums of the Drinfeld double any finite group in the following corollary. Corollary 3.6. Let G be a finite group. The nth Gauss sum of (Rep(G)) is Z n τn = τ n = G x G x = 1 > 0 − | | · |{ ∈ | }| for all n Z. ∈ Proof. Recall that ν (Rep(G)) = ν (C[G]) = x G xn = 1 for any non-negative integer n n |{ ∈ | }| n. Since νn(Rep(G)) is a positive integer, ν n(Rep(G)) = νn(Rep(G)). The equation follows immediately from Corollary 3.5. − Example 3.7. In general, the higher Gauss sums of modular categories could be zero. Here, we list two examples of integral modular categories of even dimension with zero second Gauss sums. (i) If = (Z ,q) where q(1) = i for the nontrivial simple object X, then τ ( )=0 and C C 2 2 C so the second central charge is undefined despite (Z2,q) being modular. (ii) It has been shown in [49, Section 5] that there existsC a semisimple Hopf algebra H of even dimension with Tr(SH ) = 0. Therefore, if = (Rep(H)), then τ2( )=0 by Corollary 3.5. C Z C

Another important consequence of Proposition 3.3 is the invariance of νn( ) of Morita equivalence classes of any pseudounitary fusion category . D D Corollary 3.8. Let be a pseudounitary fusion category, and a fusion category Morita equivalent to , i.e., A( ) and ( ) are equivalent braided fusionB categories. Then is also pseudounitaryA and Z A Z B B ν ( )= ν ( ) for all n Z, n A n B ∈ where the underlying spherical structures of and are the canonical ones determined by their pseudounitarity. A B HIGHERGAUSSSUMSOFMODULARCATEGORIES 9

Proof. Since is pseudounitary, so is ( ). Therefore, ( ) is pseudounitary and so A Z A Z B dim( )2 = dim( ( )) = FPdim( ( )) = FPdim( )2 = FPdim( )2. B Z B Z B B A Since dim( ), FPdim( ) and FPdim( ) are positive real numbers, dim( ) = FPdim( ) = dim( ) andB hence isB pseudounitary.A Assuming both and are equippedB with the canon-B ical sphericalA structuresB determined by their pseudounitaA rity,B ( ) and ( ) are equivalent Z A Z B modular categories by [35, Corollary 6.2]. Therefore, τn( ( )) = τn( ( )) for all n Z. Since dim( ) = dim( ), it follows from Proposition 3.4 thatZνA( )= ν Z( )B for all n Z.∈ A B n A n B ∈ 3.3 Anomaly and central charge

The quotient τ1( )/ dim( ) of a modular category , called the central charge of , is a C C C 2 C square root of α = τ1( )/τ 1( ) since dim( )= τ1( )τ 1( )= τ1( ) . The choice of positive p − − square root of dim( )C determinesC a squareC root ofC α, whichC is| naturalC | but not particularly a canonical one. OneC can easily extend these notions to higher degrees for premodular categories. Definition 3.9. Let be a premodular category. For n Z, we respectively define the nth anomaly and the nth C(multiplicative) central charge of as∈ C τn( ) τn( ) (3.5) αn( ) := C and ξn( ) := C C τ n( ) C τn( ) − C | C | provided τ ( ) = 0. n C 6 The first anomaly appears as an important quantity in the 3-dimensional topological quantum field theory defined by a modular category (see for example [54]). The motivation for our definitions of higher Gauss sums and centralC charge is closely related to the Reshetikhin- Turaev invariants of links and 3-manifolds arising from modular categories. Let be a modular category and D the positive square root of dim( ). Denote the RT- invariantC of a 3-manifold M corresponding to by RT(M). Let n CN. By [54, Section II.2.2], the RT-invariant of the lens space L(n, 1)C associated to is given∈ by C 3 (3.6) RT(L(n, 1)) = D− τ 1τn − where the category in the notation of the Gauss sums has been suppressed for brevity. The same manifold withC reversed orientation, denoted by L(n, 1), has RT-invariant − 1 1 (3.7) RT( L(n, 1)) = D− (τ 1)− τ n. − − − Observe that RT( L(n, 1)) = RT(L(n, 1)). If any one of the invariants is nonzero, we have − RT(L(n, 1)) 2 2 αn (3.8) = D− (τ 1) αn = . RT( L(n, 1)) − α − 1

3.4 Basic properties and examples Some straightforward observations about Deﬁnitions 3.1 and 3.9 are collected here for future use. Lemma 3.10. For all premodular categories and integers n such that τ ( ) = 0, C n C 6 rev (i) ξn( )= ξ n( )= ξn( ), (ii) ξ (C) and all− ofC its GaloisC conjugates have modulus 1, n C 10 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

(iii) ξ ( ) is a root of unity if and only if ξ ( ) is an algebraic integer. In this case, if n C n C N is the FS-exponent of , then ord(αn( )) N if N is even, and ord(αn( )) 2N otherwise. C C | C | rev Proof. The ﬁrst claim follows from τ n( ) = τ n( ) and τn( ) = τ n( ). Note that τn( ) 2πi/N ± C ∓ C C − C α ( ) = C Q(e ). Therefore, all the Galois conjugates of α ( ) have modulus 1, n C τ n( ) ∈ n C and so do ξ C( ). If ξ ( ) is an algebraic integer, then ξ ( ) is a root of unity by a theorem n C n C n C of Kronecker (see [20]). Hence, α ( ) is also a root of unity. Since α Q(e2πi/N ), if α ( ) n C n ∈ n C is a root of unity, then ord(αn( )) N or ord(αn( )) 2N respectively depends on whether N is even or odd. C | C |

Example 3.11. Accessible formulas for the dimensions and twists of (g2, 3) can be found in Sections 2.3.4 of [47], and one computes C 1 ξ3( )= (√7 i), C 2√2 − which has minimal polynomial 2x4 3x2 + 2 and thus is not a root of unity. Corollary 4.2 below describes when such a phenomenon− is possible. The higher Gauss sums and central charges also respect the Deligne tensor product of premodular categories. Lemma 3.12. If and are premodular, then for all n Z C D ∈ (3.9) τ ( ⊠ )= τ ( )τ ( ). n C D n C n D If τ ( )τ ( ) = 0, then we also have ξ ( ⊠ )= ξ ( )ξ ( ). n C n D 6 n C D n C n D Proof. The result follows from the fact that dimensions and twists are multiplicative with respect to ⊠, i.e. dim(X ⊠ Y ) = dim (X)dim (Y ) and θX⊠Y = θX θY for any X ( ) and Y ( ). Hence C D ∈ O C ∈ O D n 2 τ ( ⊠ )= θ ⊠ dim(X ⊠ Y ) n C D X Y X⊠Y ( ⊠ ) ∈OXC D n n 2 2 = θX θY dim (X) dim (Y ) C D X ( ) Y ( ) ∈OXC ∈OXD = τ ( )τ ( ). n C n D The last statement follows directly from the definition of ξn. Corollary 3.13. If is modular, then ξ ( ( )) = 1 for all n Z such that τ ( ) = 0. C n Z C ∈ n C 6 Proof. Apply Lemma 3.10 (ii) to ( ) ⊠ rev (see Equation (2.1)). Z C ≃C C Example 3.14. One can easily see that there are families of inequivalent premodular categories which have the same higher multiplicative central charges. The first of such an example can be obtained from finite groups. Recall from Corollary 3.6 that for any finite group G the higher Gauss sums of (Rep(G)) are positive integers. Therefore, all the higher central charges of (Rep(G)) areZ all equal to 1. In fact, the same property holds for Rep(G). Since τ (Rep(G))Z = G , ξ (Rep(G)) = 1 for all n Z. n | | n ∈ The modular categories in Example 3.14 are all contained in the trivial Witt equivalence class [Vec] (Section 2.2), but the following example illustrates that higher central charges of could be different from 1, and they are the first central charges of the Galois conjugates of C. C HIGHER GAUSS SUMS OF MODULAR CATEGORIES 11

Example 3.15. Let p N be an odd prime and qa : Zp C× a nondegenerate quadratic ∈2πia/p → form such that qa(1) = e for some integer a not divisible by p. Then the Gauss sum of = (Z ,q ) is identical to the classical quadratic Gauss sum of q , which is given by Ca C p a a p 1 − a 1 τ1( a)= qa(j)= − p C p s p Xj=0 a th where p is the Legendre symbol. Thus, for p ∤ n, the n Gauss sum and multiplicative charge of a are respectively C p 1 − an 1 an 1 τn( a)= qan(j)= − p and ξn( a)= − . C p s p C p s p Xj=0 Therefore, τ ( ) = τ ( ) and ξ ( ) = ξ ( ). Since is equivalent to if and only n Ca 1 Cna n Cα 1 Cna Ca Cb if τ1( a) = τ1( b), there are only two inequivalent modular categories among a for a given C C a C prime p which are determined by the Legendre symbol p . These two inequivalent modular categories, which can be distinguished by their central charges ξ1, are also generators of the Witt group (p). Wpt In light of Propositions 3.3 and 3.4, one can see the higher Gauss sums are closely related to the higher Frobenius-Schur indicators, and they are invariants of premodular categories. The following example is an application of the higher Gauss sums to distinguish the Drinfeld centers of the Tambara-Yamagami categories. Example 3.16. A Tambara-Yamagami ( -)category is a fusion category with ( ) = A m where A is a finite abelian groupTY and the fusionC rules are given by O C ⊔ { } m m = a, a m = m a = m, a b = ab for a, b A. ⊗ ⊗ ⊗ ⊗ ∈ a A X∈ A -category is completely determined by the abelian group A, a symmetric nondegen- TY 1 erate bicharacter χ of A, and a square root r of A − , and is denoted by (A, χ, r) (cf. [51]). Every -category is pseudounitary (cf. [15])| | and its Drinfeld centerTY is consequently TY 2 a modular category. The -categories defined by the abelian group A = Z2 are representation categories of quasi-HopfTY algebras and they are completely distinguished by their higher 2 Frobenius-Schur indicators (cf. [36]). The group Z2 admits two inequivalent nondegenerate symmetric bicharacters, namely the standard symmetric bicharacter χsym and the alternat- ing bicharacter χalt. Using the higher FS-indicators computed in [36] or [49], we have the following table of higher Gauss sums for the Drinfeld centers of these -categories. TY H τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8 C 1 (A, χsym, 2 ) C[D8] 8 48 8 64 8 48 8 64 TY 1 (A, χalt, −2 ) C[Q8] 8 16 8 64 8 16 8 64 TY 1 (A, χsym, 2 ) K 8 48 8 32 8 48 8 64 TY 1 (A, χ , − ) K 8 16 8 32 8 16 8 64 TY alt 2 u Here is equivalent to Rep(H) as spherical fusion categories, τ is the nth Gauss sum of ( ), C n Z C K is the Kac algebra of dimension 8, and Ku is a twist of K defined in [36]. Since these four sequences higher Gauss sums are different, the Drinfeld centers of these -categories are inequivalent modular categories. However, all the higher central chargesTY of these modular categories are 1. 12 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

The repetition of higher central charges is also revealed by the following proposition.

Proposition 3.17. Let be a premodular category. Then τ1( )τ 2( ) R. In particular, if C C − C ∈ τ1( )τ 2( ) = 0, then α1( )= α2( ) or ξ1( )= ξ2( ). C − C 6 C C C ± C Proof. Using the graphical calculus conventions of [2], we have

(3.10) τ1( )τ 2( )= τ1( ) θ 2 = τ1( ) θ 1 θ 1 C − C C − C − −

where the line is labeled by the pseudo-object X ( ) dim (X)X. Note that [2, Lemma C 3.1.5] holds for premodular categories by the same proof.∈O C Applying [2, Lemma 3.1.5] to the P last term of (3.10), we have

1 1 (3.11) τ1( )τ 2( )= θ θ = θ− θ , C − C −

where the second equality follows from the rigidity of . Since is spherical, we have C C

1 (3.12) τ1( )τ 2( )= θ− θ = τ 1( )τ2( ), C − C − C C

where the last equality follows from [2, Lemma 3.1.5]. The second assertion follows directly from the deﬁnition.

4 Arithmetic properties of higher Gauss sums: modular case In this section, we study the action of the Galois group of Q on the higher Gauss sums of a modular category . We obtain a relation of the higher Gauss sums in terms of the action of automorphismsC of Q in Theorem 4.1. In particular, we prove in Corollary 4.2 that those Gauss sums τn( ) with n relatively prime to ord(T ) are nonzero d-numbers, and the corresponding centralC charges are roots of unity. We alsoC extend a result of M¨uger on the ﬁrst Gauss sum of the Drinfeld center of a spherical fusion category to higher Gauss sums in Theorem 4.4. HIGHER GAUSS SUMS OF MODULAR CATEGORIES 13

Let be a modular category with the unnormalized S- and T-matrices S and T respectively. Set s :=C 1 S. Then, s2 = dim(X)2/ dim( ). For any n N, denote ζ := e2πi/n. It

½ n √dim( ) X, C C ∈ was quite well known (see [9, 6, 15]) that all entries of S and T are cyclotomic integers. It has been recently shown (see [37, Proposition 5.7]) that they are elements of Q(ζord(T )). Thus, the Galois group Gal(Q(ζ )/Q) acts on the modular data of . The reader is referred to ord(T ) C [10] or [6] for more details of Galois group actions on the modular data. For any automorphism σ of Q, there exists a unique permutationσ ˆ on ( ) such that O C s sX,σˆ(Y )

(4.1) σ X,Y = ½

s s½ ,Y ,σˆ(Y ) for all X,Y ( ). Moreover, σ gives rise to a sign function ǫ : ( ) 1 such that ∈ O C σ O C → {± } (4.2) σ(sX,Y )= ǫσ(X)sσˆ(X),Y = ǫσ(Y )sX,σˆ(Y ) for all X,Y ( ). In particular, for any X ( ), we have ∈ O C ∈ O C

(4.3) σ(s2 )= s2 . ½ X, σˆ(X),½ Fix any γ C such that ∈ 3 τ1( ) (4.4) γ = ξ1( )= C , C dim( ) C 1 deﬁne t := γ− T . Then the assignment p 0 1 1 1 (4.5) ρ : − s, t 1 0 7→ 0 1 7→ uniquely determines a linear representation of SL(2, Z) (cf. [2, Remark 3.1.9]). Note that t is a diagonal matrix with entries indexed by X ( ). We will denote the ∈ O C 1 diagonal entry of t corresponding to an X ( ) by tX . In other words, tX = θX γ− for any X ( ). According to [10, Theorem II],∈ O C ∈ O C 2 (4.6) σ (tX )= tσˆ(X).

Because θ½ = 1, Equation (4.6) states

1 θσˆ(½) (4.7) σ2 = , γ γ which implies [10, Proposition 4.7] γ

(4.8) = θ ½ . σ2(γ) σˆ( )

For any n Z relatively prime to ord(T ), there exists an automorphism σn of Q such that σ : ζ ∈ ζn (cf. [27, Theorem VI.3.1]). In particular, σ (θ ) = θn for all n ord(T ) 7→ ord(T ) n X X X ( ). Letn ˜ denote the multiplicative inverse of n modulo ord(T ). Then ∈ O C 1 n (4.9) σn−˜ (θX )= σn(θX )= θX

for all X ( ). Note that such σn is not unique but its restriction on Q(ζord(T )) is unique. Now we can∈ O stateC our theorem of Galois group action on the higher Gauss sums. 14 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

Theorem 4.1. Let be a modular category, N = ord(T ), and n Z relatively prime to N. Then, for a Z, C C ∈ ∈ dim( ) (4.10) τ ( )= σ(τ ( )) C θan and an C a C σ(dim( )) σˆ(½) C dim( ) (4.11) ν ( )= σ(ν ( )) C , an C a C σ(dim( )) C where n˜ is the multiplicative inverse of n modulo N, and σ is any automorphism of Q such that σ(ζ )= ζn˜ . In particular, τ ( ) = 0 and ξ ( ) is well-deﬁned. N N n C 6 n C rd Proof. Fix a 3 root γ of ξ1( ) (cf. Equation (4.4)). By Equations (4.3), (4.6) and (4.9), we have C

τa( ) 2 a

σ C = σ s ½ t dim( )γa  X, X  X ( ) C ∈OXC =  σ(s2 )σ(ta) X,½ X X ( ) ∈OXC 2 1 2 a = s σ− (σ (t )) by (4.3) σˆ(X),½ X X ( ) ∈OXC 2 1 a = s σ− (t ) by (4.6) σˆ(X),½ σˆ(X) X ( ) ∈OXC 2 1 a dim(ˆσ(X)) σ− (θσˆ(X)) = dim( )σ 1(γa) X ( ) − ∈OXC C 1 = dim(ˆσ(X))2θan by (4.9) dim( )σ 1(γa) σˆ(X) − X ( ) C ∈OXC τ ( ) = an C , dim( )σ 1(γa) C − where the last equality is based on the fact thatσ ˆ is a permutation on ( ). Therefore, with M := σ(τ ( )) dim( )/σ(dim( )) for brevity, O C a C C C 1 a a σ− (γ ) 1 γ 1 a an

(4.12) τ ( )= M = Mσ− = Mσ− (θ )= Mθ ½ an C σ(γa) σ2(γa) σˆ(½) σˆ( ) by Equations (4.8) and (4.9). This proves (4.10). The equality (4.11) is then a consequence 2 of Proposition 3.3. The last statement follows immediately from the fact that τ1( ) = dim( ) > 1 as is modular. | C | C C Recall from [38] that a d-number is deﬁned as an algebraic integer whose principal ideal in the ring of algebraic integers is invariant under the action of the absolute Galois group. It is immediate from the deﬁnition that the subset of d-numbers in the ring of algebraic integers is closed under multiplication and taking square roots, and that any algebraic unit is a d- number. Moreover it is shown in [38, Corollary 1.4] that if is a spherical fusion category, then dim( ) is a d-number. C C HIGHER GAUSS SUMS OF MODULAR CATEGORIES 15

Corollary 4.2. Let be a modular category, N = ord(T ), n Z with gcd(n, N) = 1 and n˜ a C C ∈ n˜ multiplicative inverse of n modulo N, and σ is any automorphism of Q such that σ(ζN )= ζN . Then, (a) the nth anomaly of is given by C (4.13) α ( )= σ(α ( )) θ2n . n C 1 C · σˆ(½) In particular, α ( ) and ξ ( ) are both roots of unity such that ξ ( )4N = 1. n C n C n C (b) The nth Gauss sum of is a d-number. C Proof. By Theorem 4.1, we have n σ(τ1( ))θ τ ( ) σˆ(½) α ( )= n C = C = σ(α ( )) θ2n . (4.14) n n 1 σˆ(½) C τ n( ) σ(τ 1( ))θ− C · − C − C σˆ(½) It is known (for example, [2, Section 3]) that α Q(ζ ) is a root of unity. Therefore, α ( ) 1 ∈ N n C is a root of unity in Q(ζN ). As a square root of αn( ), ξn( ) is also a root of unity, this completes the proof of part (a). C C Now, by Proposition 3.3, we have (4.15) τ ( )2 = τ ( ) 2α ( ) = dim( )ν ( )α ( ). n C | n C | n C C n C n C By Proposition 3.6 and Proposition 3.8 of [4], νq( ) is an algebraic unit for all prime q ∤ N. By the Dirichlet prime number theorem, there existsC a prime number q such that q n (mod N). ≡ 2 Hence, νq( )= νn( ) and so νn( ) is an algebraic unit. Since dim( ) is a d-number, τn( ) is a d-numberC and soC are its squareC roots. C C For the Drinfeld center of a spherical fusion category, we have more explicit descriptions

of its Gauss sums and anomalies. We ﬁrst obtain the twists of the Galois orbit of ½. Lemma 4.3. Let be a spherical fusion category, and θ the ribbon isomorphism of ( ). D Z D Then, for any automorphism σ of Q,

θσˆ(½) = 1 .

Proof. By [32, Theorem 1.2], τ1( ( )) = τ 1( ( )) = dim( ), and so ξ1( ( )) = 1. By Z D − Z D rdD Z D

[10, Proposition 4.7] or (4.8), we ﬁnd θ ½ = 1 since 1 is a 3 root of ξ ( ( )). σˆ( ) 1 Z D Now, we can prove our second main theorem of this section, which extends a result of M¨uger [32, Theorem 1.2] to higher Gauss sums.

Theorem 4.4. Let be a spherical fusion category, N = ord(T ( )), n Z with gcd(n, N)= D Z D ∈ 1, n˜ a multiplicative inverse of n modulo N, and σ is any automorphism of Q such that σ(ζ )= ζn˜ . Then, for a Z, N N ∈ dim( )2 (4.16) τ ( ( )) = σ(τ ( ( ))) D and an Z D a Z D σ(dim( ))2 D dim( ) (4.17) ν ( )= σ(ν ( )) D . an D a D σ(dim( )) D Moreover, σ(ν ( )) (4.18) ξ ( ( )) = a D an Z D σ(ν ( )) | a D | 16 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG whenever νa( ) = 0. In particular, ξan( ( )) = 1 whenever σ(νa( )) is positive. For a = 1, we always haveD 6 Z D D dim( )2 (4.19) ξn( ( )) = 1 and τn( ( )) = τ n( ( )) = D . Z D Z D − Z D σ(dim( )) D Proof. By Theorem 4.1 and Lemma 4.3, we have dim( ( )) dim( )2 τ ( ( )) = σ(τ ( ( ))) Z D = σ(τ ( ( ))) D . an Z D a Z D σ(dim( ( )) a Z D σ(dim( ))2 Z D D Now, Proposition 3.4 implies Equations (4.17) and (4.18) whenever νa( ) = 0 since dim( ) is totally positive. Thus, if σ(ν ( )) > 0, ξ ( ( )) = 1. By [32, TheoremD 6 1.2], τ ( ( ))D = a D an Z D 1 Z D τ 1( ( )) = dim( ). Therefore, the second equation of (4.19) follows directly from (4.16). Now,− Z weD ﬁnd τ ( D( )) is also totally positive, and so ξ ( ( )) = 1. n Z D n Z D It has been shown in Example 3.14 that the higher central charges of Tannakian categories and their Drinfeld centers are all 1. Using the same notations as in the above theorem, the following examples of non-integral modular categories indicate that the case when n is not relatively prime to N is more subtle, even when the higher central charges are well-deﬁned. Example 4.5. Let be Haagerup-Izumi fusion category of rank 6 [17]. The unitary fusion category is tensorH generated by one simple object ρ, and ( ) = G Gρ where G is a multiplicativeH group of order 3. The fusion rules of are givenO H by the group⊔ multiplication of G together with H 1 1 a ρ = aρ, a bρ = (ab) ρ = bρ a− and aρ bρ = ab− cρ ⊗ ⊗ ⊗ ⊗ ⊗ ⊕ c G M∈ 3+√13 for all a, b G. By unitarity, dim(aρ) = 2 for any a G. The Drinfeld center ( ) has rank 12,∈ and its modular data can be found in [17], which∈ in principle enablesZ usH to compute all the central charges. One particularly useful information given by the modular data is that ord(T ( )) = 39. Therefore, by Theorem 4.4, if n is not a multiple of 3 or 13, then ξ ( ( )) = 1.Z H n Z H For any multiple n of 3 or 13, we compute ξn( ( )) by using νn( ). Note that in Z39, every multiple of 3 can be written as 3k for someZkHsuch that gcd(k,H39) = 1 (for example, 9 48 = 3 16 mod 39), and the same is true for multiples of 13. Therefore, by Theorem ≡ × 4.4, we only have to compute ν3( ) and ν13( ), and use Galois actions to get the indicators at other multiples of 3 and 13. H H By [52, Theorem 5.4], ν (aρ) = 1, and ν (aρ)= 1+√13 for any a G. Since G generates a 3 13 2 ∈ fusion subcategory equivalent to Rep(G) as spherical fusion categories, ν3(Rep(G)) = 3 and ν13(Rep(G)) = 1. Therefore, 3+ √13 15 + 3√13 ν ( )= ν (Rep(G)) + ν (aρ) dim(aρ) = 3 + 3( )= 3 H 3 3 2 2 a G X∈ and similarly 3+ √13 1+ √13 ν ( ) = 1 + 3( )( ) = 13 + 3√13. 13 H 2 2 Consequently, by Proposition 3.4, τ ( ( )) and τ ( ( )) are positive real numbers, hence 3 Z H 13 Z H ξ3( ( )) = ξ13( ( )) = 1. Note that ν3 and ν13 are totally positive. By Theorem 4.4, if n is aZ multipleH of 3Z orH 13, then ξ ( ( )) = 1. In summary, ξ ( ( )) = 1 for any integer m. n Z H m Z H HIGHER GAUSS SUMS OF MODULAR CATEGORIES 17

However, there are Drinfeld centers of spherical fusion categories whose ξn is not equal to 1 when n is not relatively prime to N. In fact, as we will see in the following example, ξn is not even a root of unity.

Example 4.6. Let H be the 27-dimensional Hopf algebra H27(1,ζ3) in [50, Table 1] where rd ζ3 is a primitive 3 root of unity. Let = (Rep(H)). It is shown in loc. cit. that 2 C Z ν3(H) = 3(5 + 4ζ3 ). Therefore, by Corollary 3.5, we have τ ( ) = dim(H)ν (H) = 81(5 + 4ζ2). 3 C 3 3 Since the minimal polynomial of 2 81(5 + 4ζ3 ) α3( )= C 81(5 + 4ζ3) 2 is 7x + 2x + 7, α3 is not an algebraic integer. Hence, ξ3 cannot be a root of unity. Note that dim( ) = dim(Rep(H))2 = 36. Therefore, by the Cauchy Theorem [23, 34, 4], ord(T ) is also a powerC of 3. C

5 De-equivariantization and local modules Let be a premodular category with the ribbon isomorphism θ, and let A be a ribbon C ∈C algebra of , i.e. a connected étale algebra with dim (A) = 0 and θA = idA (cf. Definition 2.3). In theC language of [26][7, Remark 3.4], A isC a rigid6 -algebra. In this section, we will derive expressions for the higher Gauss sums and centralC charges of the local A-module 0 category A in terms of those of in two different settings (Theorems 5.6 and 5.8). RecallC the notations and conceptsC related to A-modules in from Section 2.2. We will use the same convention of graphical calculus as in [26] to prove theC following lemmas which are essential to our main results of this section. Note that Lemmas 5.1 and 5.2 are in the spirit of [39, Theorem 2.5] and its proof. Lemma 5.1. Let be a premodular category and A a ribbon algebra of . Then, for any C C M A and f End (M), ∈C ∈ C

(5.1) f = dim (A) Tr (f). C C

Proof. By the rigidity of A [26, Figure 10], we have

(5.2) f = f .

M M 18 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

By the naturality of the braiding of and the associativity of the A-action on M, we have C

(5.3) f = f = f .

M M M Now [26, Lemma 1.14] and the axioms of an A-module imply

(5.4) f = dim (A) f = dim (A) f = dim (A) Tr (f). C C C C

M M M

For any M ( ), deﬁne ∈ O CA 1 (5.5) PM := (ρM cM,AcA,M (idA ρM )(iA idM )) End (M), dim (A) ⊗ ⊗ ∈ C C

where i : ½ A A is as in [26, Deﬁnition 1.11], c is the brading of and ρ : A M M A → ⊗ C M ⊗ → is the A-module structure of M. Note that PM is the same as Pπ in [26, Lemma 4.3] (with M = Xπ) where is assumed to be modular. However, the same proof of [26, Lemma 4.3] can be used for aC premodular category , and so we have C id if M ( 0 ) , (5.6) P = M A M 0 otherwise∈ O C . Lemma 5.2. Let be a premodular category with the ribbon isomorphism θ and A a ribbon algebra of . Then,C C 0 (a) for any M ( A) ( A), Tr (θM ) = 0, and ∈ O C0 \ O C Cn n (b) for any M ( A) and n Z, θM = λ idM , where λ = θX for any simple subobject ∈ O C ∈ n n X of M in . In particular, Tr (θM )= λ dimA(M)dim (A). C C C 1 Proof. It suﬃces to show Tr (θ− ) = 0 for the statement (a) since C M 1 1 (5.7) Tr (θ− )= θ− dim (X)[M : X] = Tr (θM ), C M X C C C X ( ) ∈OXC 1 where the second equality follows from θX− = θX and dim (X) R. By [26, Figure 16] and Lemma 5.1, we have C ∈

1 1 1 1 (5.8) Tr (θ− PM )= θ− = Tr (θ− ). C M dim (A) C M C

M HIGHER GAUSS SUMS OF MODULAR CATEGORIES 19

0 1 1 Therefore, by Equation (5.6), if M ( A) ( A), then 0 = Tr (θ− PM ) = Tr (θ− ). ∈ O C \ O C C M C M For (b), let ϑ be the twist of 0 and M ( 0 ). Then ϑ = λ id for some λ C . By CA ∈ O CA M M ∈ × [26, Theorem 1.7], the twist in 0 is inherited from that of . We have ϑ = θ . Therefore, CA C M M λ = θX for any X ( ) such that [M : X] = 0. Thus, by [26, Theorem 1.18], we have ∈ O C C 6

n n Tr (θM )= θX [M : X] dim (X) C C C X ( ) ∈OXC (5.9) = λn [M : X] dim (X) C C X ( ) ∈OXC n n = λ dim (M)= λ dimA(M)dim (A). C C Lemma 5.3. Let be a premodular category with ribbon isomorphism θ. If A is a ribbon algebra of , then C ∈C C n (a) τn( )= dimA(M) Tr (θM ) for all n Z, C C ∈ M ( A) ∈OXC τ ( ) (b) τ ( 0 )= 1 C , 1 CA dim (A) dim(C ) (c) dim( A)= C and dim (A) is totally positive. C dim (A) C C Proof. By [26, Theorem 1.18], dim (M)= dimC(M) for any object M . Since the forgetful A dimC(A) ∈CA functor is a left adjoint of F : ; X A X, we have CA →C C→CA → ⊗ n 2 n τn( )= θX dim (X) = θX dim (X)dimA(A X) C C C ⊗ X ( ) X ( ) ∈OXC ∈OXC n = θX dim (X) [A X : M] A dimA(M) C ⊗ C X ( ) M ( A) ∈OXC ∈OXC n = dimA(M) θX [M : X] dim (X) C C M ( A) X ( ) ∈OXC ∈OXC n = dimA(M) Tr (θM ), C M ( A) ∈OXC and this proves part (a). For the statement (b), we consider n = 1. Then, Lemma 5.2 implies

τ1( )= dimA(M) Tr (θM )+ dimA(M) Tr (θM ) C C C 0 0 M ( ) M ( A) ( ) ∈OXCA ∈O CX\O CA 2 = dim (A) θM dimA(M) + 0 C M ( 0 ) ∈OXCA 0 = dim (A)τ1( A). C C Now, we consider n = 0 for equation of part (a). We have

dim( )= τ0( )= dimA(M)dim (M) C C C M ( A) ∈OXC 2 = dim (A) dimA(M) = dim (A) dim( A) . C C C M ( A) ∈OXC 20 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

Since the global dimensions dim( ) and dim( A) are totally positive, so is dim (A), and the proof is completed. C C C The following corollary, which is a generalization of [26, Theorem 4.5] to premodular categories, is now an immediate consequence of the preceding lemma.

Corollary 5.4. Let be a premodular category with τ1( ) = 0. If A is a ribbon algebra of , then ξ ( 0 )= ξC( ). C 6 ∈C C 1 CA 1 C Let be a premodular category. Recall that a fusion full subcategory of is a Tannakian subcategoryC if is equivalent Rep(G) for some finite group G, as premodularA C categories. Let A = Fun(G)A be the regular algebra (the algebra of complex valued functions on G). In this case, A is a ribbon algebra of with dimension dim (A) = G , and A is the de- C C | | C equivariantization G of (cf. Definition 2.3 and [11]). The corresponding category of local C 0C 0 modules is denoted by G(= A). Let G be a finite subgroupC C of the group of the isomorphism classes of invertible objects of . We simply call G a Tannakian subgroup of if the full subcategory of generated by GC is a Tannakian subcategory of . In this case,C G is abelian, ( )= GAand C is equivalent Cˆ ˆ O A A to the premodular category Rep(G), where G (∼= G) is the character group of G. Therefore, the regular algebra A of is given by A (5.10) A = g. g G M∈ Thus, A g = g A = A in for g G. Hence, for any M ( A) and for any X ( ), we have ⊗ ⊗ C ∈ ∈ O C ∈ O C

(5.11) [M : g X] = [A g X : M] A = [A X : M] A = [M : X] . ⊗ C ⊗ ⊗ C ⊗ C C Now ﬁx an M ( ) and let X ( ) be a simple subobject of M in . In other words, ∈ O CA ∈ O C C [M : X] = 0. Since A X = g G g X and M is a direct summand of A X in , every C 6 ⊗ ⊗ ⊗ C simple subobject Y of M in is∈ of the form g X for some g G. In particular, for any Y ( ) such that [M : Y ] CL= 0, we have dim⊗(Y ) = dim (X)∈ and [M : Y ] = [M : X] by∈ (5.11). O C Hence, for any n C Z6 , we have C C C C ∈ n n Tr (θM )= θY [M : Y ] dim (Y ) C C C Y ( ) ∈OXC n = θY [M : Y ] dim (Y ) C C (5.12) Y ( ) ∈O C [MX:Y ]C=0 6 n = [M : X] dim (X) θY . C C Y ( ) ∈O C [MX:Y ]C=0 6 Lemma 5.5. Let be a premodular category with ribbon isomorphism θ, and G a Tannakian C 0 subgroup of . Then for any M ( G) ( G) and n Z relatively prime to ord(T ), C ∈ O C \ O C ∈ C n Tr (θM ) = 0 . C 0 Proof. By Lemma 5.2 (a), for any M ( G) ( G), we have Tr (θM ) = 0. Therefore, by Equation (5.12), we have ∈ O C \ O C C

θY = 0, Y ( ) ∈O C [M:XY ]C=0 6 HIGHER GAUSS SUMS OF MODULAR CATEGORIES 21 as the dimension of a simple object in a fusion category is not 0 and [M : Y ] = 0 by C 6 assumption. For any integer n relatively prime to ord(T ), there exists σn Gal(Q/Q) such that σ (θ )= θn for all X ( ). We then have C ∈ n X X ∈ O C

n θX = σn  θX  = 0, X ( ) X ( ) ∈OXC  ∈OXC  [M:X]C=0 [M:X]C=0  6  6  which, together with Equation (5.12), proves the lemma. Theorem 5.6. Let be a premodular category, and G a Tannakian subgroup of . Then for all n Z relatively primeC to ord(T ), C ∈ C τ ( )= G τ ( 0 ). n C | | n CG In addition, if τ ( ) = 0, then ξ ( 0 )= ξ ( ). n C 6 n CG n C Proof. Let θ be the ribbon isomorphism of , and A the regular algebra of the Tannakian subcategory equivalent to Rep(G). LemmasC 5.3 and 5.5 imply A n τn( )= dimA(M) Tr (θM ) C C M ( A) ∈OXC n n = dimA(M) Tr (θM )+ dimA(M) Tr (θM ) C C 0 0 M ( ) M ( A) ( ) ∈OXCA ∈O CX\O CA n 2 = dim (A) θM dimA(M) + 0 C M ( 0 ) ∈OXCA 0 0 = dim (A)τn( A)= G τn( G). C C | | C The last assertion follows directly from the deﬁnition of higher central charges as G is a positive integer. | |

With the notations as above, recall that in this case, the de-equivariantization ( ′)G of the centralizer of is the same as the category of local A-modules 0 . A A CA Example 5.7. Consider the category := (so5, 4) of rank 15 containing the Tannakian C C 0 subcategory := Rep(Z2) and has ord(T ) = 28. The de-equivariantization ( ′)Z2 = Z2 A C A C factors as ⊠2, where := ( (sl , 5) )rev is a premodular category of rank 3. Let d := D D C 2 pt′ 2 cos(π/7) and ζ = eπi/7. The dimensions of the nontrivial simple objects of are d and 2 2 4 D d 1, and their twists are ζ− and ζ respectively (cf. [48]). There are 12 integers m such − 2 that 1 m < 28 and gcd(m, 28) = 1. Therefore, τm( ) = 2τm( ) by Theorem 5.6, and we have ≤ C D ζ4 if m = 5, 13, 19, 27; ζ6 if m = 3, 17; (5.13) ξ ( )=  m C ζ8 if m = 11, 25;   ζ10 if m = 1, 9, 15, 23.  One would expect Theorem 5.6 could be generalized to any ribbon algebra A of a premodular category . However, we are only able to do this for modular categories as the Galois group actionsC on their modular data can be applied. C 22 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG

Theorem 5.8. Let be a modular category, A a ribbon algebra, and N = ord(T ).

C ∈ C 0 C ½ Suppose ½ and A are respectively the monoidal units of and A. If n Z is relatively prime to NC , then C C ∈

0 0 σ(dim (A))

C ½ (5.14) θ ½ = θ , ξ ( )= ξ ( ) and τ ( )= τ ( ) = 0 σˆ( C) σˆ( A) n CA n C n CA dim (A)2 n C 6 C n˜ where σ is any automorphism of Q such that σ(ζN )= ζN and n˜ is the multiplicative inverse of n modulo N as in Section 4. Proof. By Theorem 4.1 and Lemma 5.4,

dim( 0 ) τ ( 0 )= σ(τ ( 0 )) CA θn n CA 1 CA σ(dim( 0 )) σˆ(½A) CA σ(τ ( )) dim( 0 ) = 1 C CA θn σ(dim (A)) σ(dim( 0 )) σˆ(½A) C CA 0 n σ(dim( )) dim( A) θσˆ(½A) (5.15) = τn( ) C C 0

C dim( ) σ(dim (A) dim( )) θσˆ(½C ) C C CA 2 n

σ(dim( )) dim( )dim (A)− θσˆ(½A) C = τn( ) C C 1

C dim( ) σ(dim( )dim (A)− ) θσˆ(½C) C C C n

σ(dim (A)) θσˆ(½A) C = 2 τn( ) .

dim (A) C θσˆ(½C ) C 3 3 0 For any γ C such that γ = ξ1( ), we also have γ = ξ1( A) by Corollary 5.4. Thus by Equation (4.8),∈ C C

γ ½ θ ½ = = θ ,

σˆ( C ) σ2(γ) σˆ( A) ½ hence θσˆ(½C )/θσˆ( A) = 1 which can be substituted into (5.15) to yield the third equality of the 0 statement. Since is modular, τn( ) = 0 by Theorem 4.1 and hence τn( A). Since dim (A) is C C 0 C σ6 (dimC(A)) C 0 totally positive, we have τ ( ) = 2 τ ( ) . Therefore, the equality ξ ( )= ξ ( ) | n CA | dimC(A) | n C | n CA n C follows directly from definition. Theorem 5.8 could be viewed as a generalization of the last statement of Theorem 4.4. Indeed, if = ( ) for some spherical fusion category , then the Lagrangian algebra C Z D D A = I(½ ) is a ribbon algebra of where I is the left adjoint of the forgetful functor from D 2 C 0 0 to . Since dim (A) = dim( ), dim( A)=1 and so A Vec (cf. [7, Proposition 5.8]). HenceC D Theorem 5.8C implies C C C ≃ dim (A)2 dim( )2 τ ( ( )) = C τ (Vec) = D . n Z D σ(dim (A)) n σ(dim( )) C D 6 Witt relations and central charges Recall that each Witt equivalence class in has a completely anisotropic (contains no nontrivial connected étale algebras) representativeW which is unique up to braided equivalence [7, Theorem 5.13]. For Witt equivalence classes in the unitary subgroup un , generated by equivalence classes of pseudounitary nondegenerate braided fusion categories,W ⊂W this completely anisotropic representative is unique up to ribbon equivalence, as there exists a unique spherical structure with nonnegative dimensions for all objects (cf. [15]). By [7, Lemma 5.27], HIGHER GAUSS SUMS OF MODULAR CATEGORIES 23 if two pseudounitary modular categories and are Witt equivalent, then ξ1( ) = ξ1( ). The goal of this section is to extend thisC resultD to higher central charges with theC followingD theorem. Theorem 6.1. Let and be pseudounitary modular tensor categories such that [ ] = [ ]. C D C D If n Z is coprime to ord(T ) ord(T ), then ξn( )= ξn( ). ∈ C · D C D Proof. Recall [7, Corollary 5.9] that is Witt equivalent to if and only if there exists a fusion category such that ( ) C ⊠ rev is a braided equivalence.D Since ( ) is the Deligne productA of pseudounitaryZ A categories,≃ C D it is also pseudounitary. Moreover,Z sinceA dim( )2 = dim( ( )) = FPdim( ( )) = FPdim( )2, A Z A Z A A is also pseudounitary. The assumption gcd(n, ord(T )ord(T )) = 1 implies ξn( ) and A C D C ξn( ) are well-defined by Theorem 4.1. By Theorem 4.4, ξn( ( )) = 1 as ord(T ( )) = D Z A Z A lcm(ord(T ), ord(T )). Thus, by Lemmas 3.10 and 3.12, we find C D ξ ( )ξ ( ) = 1 . n C n D By Corollary 4.2, ξ ( ) and ξ ( ) are root of unity, and the result follows. n C n D The structure theorem for the classical Witt group of quadratic form [45] coincides with that of the pointed Witt group [7, Proposition 5.17]. In particular,WQ Wpt (6.1) = (p) Wpt ∼ Wpt primesM p where (p) consists of the equivalence classes of all pointed modular categories with the Wpt fusion rules of abelian p-groups. It is well-known that (2) = Z Z , (p) = Z for Wpt ∼ 8 ⊕ 2 Wpt ∼ 4 p 3 (mod 4), and pt(p) ∼= Z2 Z2 for p 1 (mod 4). ≡In Example 3.15,W we have demonstrated⊕ ≡ the application of the first central charge to distinguish the generator of (p) when p is an odd prime. In fact, the higher central charges Wpt can distinguish all element of pt(p) for any prime p. We demonstrate this application in the following example. W

Example 6.2. The group pt(2) is generated by = (Z4,q) and = (Z4,q) ⊠ (Z2,q′), W 1 C C D C C where q(1) = ζ8 and q′(1) = ζ4− . One can compute directly that ξ ( )= ζ , ξ ( )= ζ3, ξ ( ) = 1, ξ ( )= 1. 1 C 8 3 C 8 1 D 3 D − ⊠a ⊠b Denote by a,b = ⊠ for any non-negative integers a, b, and ′ the subgroup of pt(2) generated byE theC Witt equivalenceD classes of and . We find W W C D ξ([ ]) := (ξ ( ),ξ ( )) = (ζa, ( 1)bζ3a) . Ea,b 1 Ea,b 3 Ea,b 8 − 8 In particular, ξ : ′ ζ8 ζ8 defines a group homomorphism with its image isomorphic to Z Z . SinceW →(2) h isi of × horderi 16, [ ] and [ ] are generators of (2) by Theorem 6.1. 8 ⊕ 2 Wpt C D Wpt Outside of pt one can use higher central charges to differentiate various Witt equivalence classes. W

Example 6.3. In this example, we use the formulas for the modular data of (g2, k) as in [47, Sections 2.3.4]. Set := (g , 8)⊠5 and := (g , 11)⊠10. One computes C C C 2 D C 2 (6.2) ξ ( ) = exp( πi/3) = ξ ( ), 1 C − 1 D but and are not Witt equivalent. To see this, note that ord(T ) = 36, and ord(T ) = 45 [47].C By directD computation, we have ξ ( ) = 1 and ξ ( )C = 1. Hence [ ]D = [ ] 13 C 13 D − C 6 D 24 SIU-HUNGNG,ANDREWSCHOPIERAY,ANDYILONGWANG by Theorem 6.1. All Witt group relations amongst classes [ (g2, k)] were classified in [48] corroborating this result. C The following proposition implies the converse of Theorem 6.1 does not hold. Proposition 6.4. Let be a modular category and N := ord(T ). For any integer k relatively prime to N, C C ⊠ ξ ( 4N ) = 1. k C Proof. By Theorem 4.1, αk( ) and ξk( ) are well-defined. By Lemma 3.12, Corollary 4.2, and Lemma 3.10, we have e haveC C ⊠ ⊠ ⊠ ξ ( 4N )= ξ2( 2N )= α ( 2N )= α2N ( )= σ(α2N ( ))θ2N = 1. k C k C k C k C 1 C σˆ(½) Example 6.5. Let be a modular category such that its Witt equivalence class [ ] is of infinite order. TheC existence of such categories is discussed in [7, Example 6.4].C ∈W By the ⊠4N ⊠4N above proposition, for any k coprime to N, ξk( )=1= ξk(Vec), but [ ] = [Vec] in by our assumption on . Hence, the above propositionC provides counter-examplesC 6 to the Wconverse of Theorem 6.1. C 7 Questions The higher central charges of pointed modular categories can be computed by using the structure of (Section 6). There is an explicit formula for ξ ( (g, k)) based only on Wpt 1 C dim(g), k, and the dual Coxeter number of g [2, Equation 7.4.5]. As ξn is often undefined when n = 1, 0 it is unclear whether a similar general formula (i.e. without reference to Galois automorphisms6 ± in Theorem 4.1) for higher central charges exists.

Question 7.1. Does an explicit formula exist for ξn( (g, k)) for n = 1, 0 using only the input data of n, g, and k? C 6 ± By Example 3.7, there exist modular categories with some of its higher Gauss sums being 0. Thus, if is a modular category containingD a modular subcategory such that τ ( ) = 0 for someC integer a> 1, then τ ( )=0 as ⊠ . D a D a C C ≃ D D′ Question 7.2. Are there necessary and sufficient conditions for higher Gauss sums of a premodular (or modular) category to vanish (hence higher central charges are undefined)? Acknowledgement. The second author would like to thank Victor Ostrik for his suggestion to explore the notion of higher Gauss sums. Both the second and third authors would like to thank MSRI (Summer Graduate School 791) for providing the opportunity to initiate this collaboration. The third author would like to thank Thomas Kerler and James Cogdell, and the first author would like to thank Ling Long for fruitful discussions. References

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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA E-mail address: [email protected]

School of Mathematics and Statistics, UNSW Sydney, NSW 2052, Australia E-mail address: [email protected]

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA E-mail address: [email protected]