Higher central charges and Witt groups
aa r X i v : . [ m a t h . QA ] A p r HIGHER CENTRAL CHARGES AND WITT GROUPS
SIU-HUNG NG, ERIC C. ROWELL, YILONG WANG, AND QING ZHANG
Abstract.
In this paper, we introduce the definitions of signatures of braided fusion cate-gories, which are proved to be invariants of their Witt equivalence classes. These signatureassignments define group homomorphisms on the Witt group. The higher central charges ofpseudounitary modular categories can be expressed in terms of these signatures, which areapplied to prove that the Ising modular categories have infinitely many square roots in theWitt group. This result is further applied to prove a conjecture of Davydov-Nikshych-Ostrikon the super-Witt group: the torsion subgroup generated by the completely anisotropic s-simple braided fusion categories has infinite rank. Introduction
Fusion categories can be viewed as categorical generalizations of finite groups. From thisperspective modular (tensor) categories are generalizations of metric groups (
G, q ), i.e., finiteabelian groups G equipped with a non-degenerate quadratic form q : G → C × . Indeed, anypointed modular category (i.e., with each simple object tensor-invertible) can be constructedfrom a metric group, and conversely [15]. Many structures and properties of metric groupscan be generalized to the modular category setting. Two important examples are Gauss sumsand Witt equivalence.The quadratic Gauss sum τ ( G, q ) = P x ∈ G q ( x ) of a metric group ( G, q ) is known to havethe form ζ p | G | for some 8 th -root of unity ζ . In particular, the modulus of any Galoisconjugate of the quadratic Gauss sum of a metric group is always equal to p | G | . Thecategorical dimension dim( C ) of a fusion category C plays the role of the order of G , andthe counterpart of the quadratic Gauss sum for modular categories is the first Gauss sum τ ( C ) = P X ∈ Irr( C ) ( d X ) θ X where d X are the categorical dimensions and θ X are the twists.While the categorical dimension dim( C ) of a modular category C may not be an integer, itis a totally positive cyclotomic integer (cf. [19]). This means σ (dim( C )) is a positive realcyclotomic integer for any automorphism σ of Q . It has been shown that the first Gauss sum τ ( C ) is equal to ξ ( C ) p dim( C ) where ξ ( C ) is a root of unity, and called the central charge of C (cf. [1]). However, the Galois conjugates of the Gauss sum τ ( C ) can have moduli differentfrom p dim( C ). This apparent discrepancy between the Gauss sum in the categorical setting Key words and phrases.
Higher central charge; Quantum group modular category; Signature; Witt group.The first author was partially supported by the NSF grant DMS-1664418.The second author was partially supported by a Texas A&M Presidential Impact Fellowship, a SimonsFellowship and the NSF grant DMS-1664359. and the metric group setting inspired the notions of higher Gauss sums and higher centralcharges for modular categories introduced in [31].The concept of Witt equivalence and the Witt group W for non-degenerate braided fu-sion categories was introduced in [9], generalizing the concept for metric groups. For metricgroups, Witt equivalence is defined modulo groups with a hyperbolic quadratic form, wherethe operation is the usual direct product of metric groups. For non-degenerate braided fusioncategories one uses the Deligne product ⊠ and considers equivalence classes modulo Drin-feld centers. It is worth noting that Witt classes do not depend on, or assume, any pivotalstructure. Moreover, the classical Witt group W pt corresponding to metric groups appearsas a subgroup of W , as the Witt classes of pointed modular categories. Witt equivalence forslightly degenerate braided fusion categories was introduced in [10], and the correspondingWitt group s W is called the super-Witt group in this paper. The study of the Witt groupfor non-degenerate braided fusion categories leads to many interesting questions about itsstructure (see [10, 9]). While it is known that the torsion subgroup Tor( W ) of W is a 2-groupwith exponent 32, it was not previously known whether Tor( W ) / W pt has infinite cardinalityor not. Another interesting open problem is to find a set of generators for the Witt group W .One of the reasons these problems are difficult is that there are very few known invariants ofthe Witt group.Witt classes also have physical significance: symmetry gauging [8, 2] and the reverse pro-cess boson condensation preserve Witt class. Both of these are topological phase transitionsin the theory of topological phases of matter [2, 6]. Each Witt class has a unique completelyanisotropic representative [9], and all members of that class can be reduced to this represen-tative by anyon condensation. Thus, distinguishing Witt classes can be regarded as analogousto determining allotropy classes in chemistry.It is known [19] that any pseudounitary braided fusion category has a unique canonicalspherical structure so that the categorical dimensions d X > X . Moreover, theWitt classes with a pseudounitary representative form a subgroup W un of W [9]. For apseudounitary non-degenerate braided fusion category C we may define its higher centralcharges to be those obtained from the modular category by endowing it with the canonicalspherical structure yielding d X > X . It has been proved in [31] that the higher centralcharges of degrees coprime to the Frobenius-Schur exponents of any two Witt equivalentpseudounitary modular categories are equal, generalizing the case for the first central chargewhich was proved in [9]. In particular, the higher central charges are Witt invariants on W un . The higher central charges of C with degrees coprime to the Frobenius-Schur exponentcan be reformulated as a function Ψ C : Gal( Q ) → µ ∞ , where µ ∞ ⊆ C is the group of rootsof unity in C . This central charge function Ψ C of C can be expressed in terms of the firstcentral charge ξ ( C ), and the signature ε C : Gal( Q ) → {± } of C , which is a function givenby ε C ( σ ) = σ ( √ dim( C )) | σ ( √ dim( C )) | . Since dim( C ) is totally positive, p dim( C ) is a totally real algebraicinteger. Therefore, σ ( p dim( C )) is a nonzero real number, which can only be either positiveor negative. It is proved in Section 3 that the signature of a pseudounitary modular category IGHER CENTRAL CHARGES AND WITT GROUPS 3 is an invariant of its Witt class. Moreover, both the central charges and the signatures ofpseudounitary modular categories can be extended to group homomorphisms Ψ and ε fromthe Witt subgroup W un to the group U ∞ of functions from Gal( Q ) to µ ∞ (cf. Section 4)The signatures of nonzero totally real algebraic numbers, especially those of algebraic units,are studied in number theory [17, 16]. It is simple to determine the signatures of real quadraticnumbers, but it is a difficult task for higher degree totally real algebraic number. However,we are able to determine the signatures of an infinite sequence L of quantum group modularcategories C r := so (2 r + 1) r +1 , r ≥
1, by a formula for p dim( C r ) derived in Proposition5.3. This formula is expressed in terms of a product of the sine of rational angles. Thedetermination of the signatures of these modular categories is inspired by the computation ofthe quadratic Gauss sum P k − j =0 ζ j k in [20] (see also [34]).Since the first central charge of these categories are well-known, we prove in Theorem 6.7that the restriction of central charge homomorphism Ψ on the subgroup W L of W un , generatedby the Witt classes of the categories in L , is injective. Moreover, W L ∼ = Z / ⊕ ( Z / ⊕ N . Sincethe square of each category in L is Witt equivalent to an Ising modular category, the Isingmodular category su (2) has infinitely many square roots in W .In s W , there is a subgroup s W generated by the Witt classes of completely anisotropics-simple braided fusion categories of finite order, and s W is of exponent 2 (cf. [10]). It isconjectured [10, Conjecture 5.21] that s W has infinite rank. By applying Theorem 6.7, weprove this conjecture in Theorem 7.2.The paper is organized as follows: In Section 2, we briefly review the key concepts forfusion categories and the Witt group W ( E ) over a symmetric fusion category E for later use.In Section 3, we define two versions of signatures for fusion categories over E and show thatthey are Witt invariants. In Section 4, we derive a formula of the higher central charges interms of the categorical dimension signature, and we define the central charge homomorphismΨ on W un in terms of the signature ε . In Section 5, we study the quantum group modularcategories so (2 r + 1) r +1 , with an emphasis on their signatures. In Section 6, we prove ourfirst main results Theorem 6.7, Corollaries 6.8 and 6.9. We finally prove the conjecture ofDavydov-Nikshych-Ostrik in Theorem 7.2 of Section 7.Throughout this paper, we use the notation ζ a := exp (cid:0) πia (cid:1) , and Q a := Q ( ζ a ) for any a ∈ N .In particular, i := exp (cid:0) πi (cid:1) = √−
1. 2.
Preliminaries
Fusion categories and their global dimensions. A fusion category is a semisimple, C -linear abelian, rigid monoidal category with finite-dimensional Hom-spaces and finitelymany simple objects which include the tensor unit (cf. [19]). For any fusion category C , wedenote by Irr( C ) the set of isomorphism classes of simple objects of C . If the context is clear,we will use the abuse notation to denote an object in an isomorphism class X of C by X . Thetensor product endows K ( C ), the Grothendieck group of C , a ring structure. More precisely, SIU-HUNG NG, ERIC C. ROWELL, YILONG WANG, AND QING ZHANG we have X ⊗ Y = P Z ∈ Irr( C ) N ZX,Y Z for any X, Y ∈ Irr( C ), where N ZX,Y := dim C C ( X ⊗ Y, Z ) . For any X ∈ Irr( C ), let N X be the square matrix of size | Irr( C ) | such that ( N X ) Y,Z = N ZX,Y for any
Y, Z ∈ Irr( C ). The Frobenius-Perron dimension (or FP-dimension) of X ∈ Irr( C ),denoted by FPdim( X ), is the largest positive eigenvalue for N X (cf. [19]). The Frobenius-Perron dimension of C is defined to beFPdim( C ) = X X ∈ Irr( C ) FPdim( X ) . It is shown in [19] that for any fusion category C , FPdim( C ) is a totally positive cyclotomicinteger.Let C be a fusion category. For any object V of C , the left dual of V is denoted by thetriple ( V ∗ , ev V , coev V ), where ev V : V ∗ ⊗ V → and coev V : → V ⊗ V ∗ are respectively theevaluation and coevaluation morphisms for the left dual V ∗ of V (cf. [18]). A simple object X of C is called invertible if X ⊗ X ∗ ∼ = . For any morphism f : X → X ∗∗ , its left quantumtrace is defined to betr X ( f ) := ev X ∗ ◦ ( f ⊗ id X ∗ ) ◦ coev X ∈ End C ( ) ∼ = C . Since C is a fusion category, V ∼ = V ∗∗ for any V ∈ ob( C ) (cf. [26, 19]). If X ∈ Irr( C ) and h : X → X ∗∗ is a nonzero morphism, the squared norm of X is defined as | X | := tr X ( h ) · tr X ∗ (( h − ) ∗ ) ∈ C , which is independent of the choice of h . The global dimension (or categorical dimension) of C is defined as dim( C ) = X X ∈ Irr( C ) | X | . By [19, Theorem 2.3], | X | > X ∈ Irr( C ), and dim( C ) ≥ C .Moreover, by [19, Remark 2.5], dim( C ) is a totally positive algebraic integer. We will denotethe positive square root of dim( C ) by p dim( C ).One can extend the left duality of C to a contravariant functor ( − ) ∗ . Then ( − ) ∗∗ definesa monoidal functor on C . A pivotal structure on a fusion category C is an isomorphism ofmonoidal functors j : id C ∼ = −→ ( − ) ∗∗ . A fusion category equipped with a pivotal structure iscalled a pivotal fusion category , in which the quantum dimension of any object V ∈ ob( C ) isdefined to be dim j ( V ) := tr V ( j V ) ∈ End C ( ) ∼ = C . A pivotal structure j on C is called spherical if dim j ( V ) = dim j ( V ∗ ) for all V ∈ ob( C ).A pivotal fusion category C is called a spherical fusion category if its pivotal structure isspherical. We will simply denote the quantum dimension of an object V ∈ ob( C ) by d V ordim( V ) when the pivotal structure is clear from the context. IGHER CENTRAL CHARGES AND WITT GROUPS 5 If C is a spherical fusion category, then d X = | X | for any X ∈ Irr( C ) (cf. [26, 19]).Therefore, d X = 0 for X ∈ Irr( C ) anddim( C ) = X X ∈ Irr( C ) d X . By [30, Proposition 5.7], if C is a spherical fusion category, then dim( C ) ∈ Q N , where N = FSexp( C ) is the Frobenius-Schur exponent of C (cf. [29]). If C is not spherical, then wecan consider the “sphericalization” ˜ C of C defined in [19, Remark 3.1]. In particular, ˜ C is aspherical fusion category such that dim( ˜ C ) = 2 dim( C ) (cf. [19, Proposition 5.14]). The sameargument as above implies that dim( C ) ∈ Q ˜ N , where ˜ N = FSexp( ˜ C ). The upper bound ˜ N can be determined by dim( C ). In particular, dim( C ) is a totally positive cyclotomic integerfor any fusion category C . Remark 2.1.
The cyclotomicity of dim( C ) can also be derived from [32, Corollary 1.4] and[19, Theorem 8.51].A fusion category C is called pseudounitary if dim( C ) = FPdim( C ). It is shown in [19, Propo-sition 8.23] that a pseudounitary fusion category C admits a canonical spherical structure suchthat d V = FPdim( V ) for any V ∈ ob( C ). In this paper, we assume that any pseudounitaryfusion category is spherical and is equipped with its canonical spherical structure.2.2. Braided fusion categories and the square root of dimension.
Let C be a fusioncategory. A braiding on C is a natural isomorphism β V,W : V ⊗ W ∼ = −→ W ⊗ V satisfying the Hexagon axioms (cf. [18]). A fusion category equipped with a braiding is calleda braided fusion category .The degeneracy of a braiding β on C is characterized by double braidings. More precisely,let C ′ denote the M¨uger center of C , which is the full subcategory C determined by objects V ∈ ob( C ) such that β W,V ◦ β V,W = id V ⊗ W for all W ∈ ob( C ). The M¨uger center C ′ is a fusionsubcategory of C . A braided fusion category C is called non-degenerate if Irr( C ′ ) = { } , i.e. C ′ is equivalent to the category Vec of finite-dimensional vector spaces over C . From any fusioncategory A , one can construct a non-degenerate braided fusion category [22, 23, 24], which iscalled the Drinfeld center of A and is denoted by Z ( A ).In contrast to non-degenerate braided fusion categories, a braided fusion category C iscalled a symmetric fusion category if C ′ = C . For any finite group G , let Rep( G ) be the fusioncategory of finite-dimensional complex representations of G , equipped with the usual braiding.If z ∈ G is a central element of order 2, let Rep( G, z ) be the fusion category Rep( G ) equippedwith the braiding given by the universal R -matrix R = 1 ⊗ ⊗ z + z ⊗ − z ⊗ z . Both Rep( G )and Rep( G, z ) are symmetric fusion categories, and the theorems of Deligne imply that anysymmetric fusion category is braided equivalent to Rep( G ) or Rep( G, z ) for some finite group G (cf. [12, 13]). A symmetric fusion category is called Tannakian (resp. super-Tannakian ) if
SIU-HUNG NG, ERIC C. ROWELL, YILONG WANG, AND QING ZHANG it is braided equivalent to Rep( G ) (resp. Rep( G, z )) for some finite group G . In particular, if C is symmetric, then dim( C ) ∈ Z .In general, if C is a braided fusion category, then C ′ is either Tannakian or super-Tannakian.If C ′ is braided equivalent to Rep( Z , C , then C is called slightly degenerate .A premodular category is a spherical braided fusion category. A premodular category C is called modular if C is non-degenerate. The (unnormalized) S-matrix of a premodularcategory C is defined to be S X,Y := tr X ⊗ Y ∗ ( β Y ∗ ,X ◦ β X,Y ∗ ) , X, Y ∈ Irr( C ) . An alternative criterion for modularity of a premodular category is that the S-matrix isinvertible (cf. [27]).Let C be a premodular category. A natural isomorphism θ : id C ∼ = −→ id C , called the ribbonstructure of C , can be defined using the spherical pivotal structure of C and the Drinfeldisomorphism (cf. [18]). The ribbon structure satisfies(2.1) θ V ⊗ W = ( θ V ⊗ θ W ) ◦ β W,V ◦ β V,W and(2.2) θ V ∗ = θ ∗ V for any V, W ∈ ob( C ). In particular, for any X ∈ Irr( C ), θ X is equal to a non-zero scalar timesid X . By an abuse of notation, we denote both the scalar and the isomorphism itself by θ X forall simple X . The T-matrix of a premodular category C is defined to be the diagonal matrix T X,Y := θ X · δ X,Y , X, Y ∈ Irr( C ) . It is well-known that if C is modular, then the S- and the T-matrices give rise to a projectiverepresentation of SL ( Z ) (cf. [1, 35]).A premodular category C is called super-modular if C is a slightly degenerate braided fusioncategory. The non-trivial simple object in C ′ , denoted by f , is an invertible object suchthat β f,f = − id f ⊗ f . This implies that θ f d f = −
1, so that θ f = − d f = ±
1. It is readilyseen by (2.1) and the dimension equation d f ⊗ X = d f d X that tensoring with f gives rise to apermutation on Irr( C ) without any fixed point, and X ∗ = f ⊗ X for any X ∈ Irr( C ). Therefore,Irr( C ) can be written as a disjoint unionIrr( C ) = Π ∪ ( f ⊗ Π )for some subset Π of Irr( C ) containing and closed under taking duals. With respect to thisdecomposition of Irr( C ), the S-matrix of C takes the form S = (cid:18) ˆ S d f ˆ Sd f ˆ S ˆ S (cid:19) . IGHER CENTRAL CHARGES AND WITT GROUPS 7
For any
X, Y ∈ Π , by [28, Lemma 2.15], we have(2.3) 2 X Z ∈ Π ˆ S X,Z ˆ S Z,Y = X Z ∈ Irr( C ) S X,Z S Z,Y = dim( C ) X W ∈ Irr( C ′ ) N WX,Y d W = dim( C ) δ X,Y ∗ , where the last equality is guaranteed by the assumption that Π is closed under taking duals.In particular, ˆ S is a non-zero multiple of the charge conjugation matrix of Π , and so ˆ S isinvertible.Let P denote the free abelian group over Π . For any X, Y, Z ∈ Π , letˆ N ZX,Y := N ZX,Y + d f · N f ⊗ ZX,Y . One can verify directly that the bilinear map • : P × P → P given by X • Y = P Z ∈ Π ˆ N ZX,Y Z defines a commutative ring structure on P with the identity . Moreover, by [28, Lemma 2.4],for any X, Y, Z ∈ Π , we have(2.4) ˆ S X,Y d Y · ˆ S Z,Y d Y = X W ∈ Irr( C ) N WX,Z S W,Y d Y = X W ∈ Π (cid:18) N WX,Z S W,Y d Y + N f ⊗ WX,Z S f ⊗ W,Y d Y (cid:19) = X W ∈ Π N WX,Z ˆ S W,Y d Y + d f · N f ⊗ WX,Z ˆ S W,Y d Y ! = X W ∈ Π ˆ N WX,Z ˆ S W,Y d Y . Therefore, the function χ Y : P → C defined by χ Y ( X ) := ˆ S X,Y /d Y for X ∈ Π is a C -linearcharacter of P . Now, (2.3) implies that { χ Y : Y ∈ Π } is a set of C -linearly independentcharacters P , and hence it is the set of all the C -linear characters of P . Moreover, we havethe following Verlinde-like formula(2.5) ˆ N ZX,Y = 2dim( C ) X W ∈ Π ˆ S X,W ˆ S Y,W ˆ S Z ∗ ,W ˆ S ,W . Recall that S is a submatrix of the S-matrix of the Drinfeld center of C . It follows from[30, Proposition 5.7] that S is a matrix defined over a certain cyclotomic field, and so is ˆ S .The above discussion on the characters on P implies that the absolute Galois group acts onΠ by permutation. More precisely, let Gal( Q ) be the absolute Galois group, then for any σ ∈ Gal( Q ), and for any Y ∈ Π , σ ( χ Y ) is another character of P . Hence, there exists a unique SIU-HUNG NG, ERIC C. ROWELL, YILONG WANG, AND QING ZHANG ˆ σ ( Y ) ∈ Π such that σ ( χ Y ) = χ ˆ σ ( Y ) . Thus, for any σ ∈ Gal( Q ) and for any X, Y ∈ Π , wehave(2.6) σ ˆ S X,Y d Y ! = σ ( χ Y )( X ) = χ ˆ σ ( Y ) ( X ) = ˆ S X, ˆ σ ( Y ) d ˆ σ ( Y ) . Note that the above Galois property of a super-modular category is similar to that of amodular category (cf. [11, 7, 19]).The proof of the following lemma is identical to the proof for modular categories as in[11, 7, 19]. However, we provide the proof for the sake of completeness.
Lemma 2.2.
Let C be a super-modular category, D the positive square root of dim( C ) . (1) For any σ ∈ Gal( Q ) , there exists a function g σ : Π → {± } such that for any X, Y ∈ Π , σ ˆ S X,Y D ! = g σ ( X ) ˆ S ˆ σ ( X ) ,Y D = g σ ( Y ) ˆ S X, ˆ σ ( Y ) D . (2)
The positive real number D is a cyclotomic integer.Proof. Recall that for any X ∈ Π , d X = 0 (cf. Section 2.1). By Eqs. (2.3) and (2.6), for any σ ∈ Gal( Q ), we have σ (cid:18) dim( C ) d X (cid:19) = σ X Y ∈ Π ˆ S X,Y ˆ S Y,X ∗ d X = 2 X Y ∈ Π σ ˆ S X,Y d X ! σ ˆ S Y,X ∗ d X ∗ ! = 2 X Y ∈ Π ˆ S ˆ σ ( X ) ,Y d ˆ σ ( X ) ˆ S Y, ˆ σ ( X ∗ ) d ˆ σ ( X ∗ ) = δ ˆ σ ( X ) ∗ , ˆ σ ( X ∗ ) dim( C ) d σ ( X ) . Therefore, we have ˆ σ ( X ∗ ) = ˆ σ ( X ) ∗ and σ (cid:18) Dd X (cid:19) = g σ ( X ) Dd ˆ σ ( X ) for some g σ ( X ) ∈ {± } . Again by (2.6), for any X, Y ∈ Π , we have σ ˆ S X,Y D ! = σ ˆ S X,Y d Y ! σ (cid:18) d Y D (cid:19) = g σ ( Y ) ˆ S X, ˆ σ ( Y ) d ˆ σ ( Y ) d ˆ σ ( Y ) D = g σ ( Y ) ˆ S X, ˆ σ ( Y ) D .
IGHER CENTRAL CHARGES AND WITT GROUPS 9
Since dim( C ) = D is an algebraic integer, so is D . Note that ˆ S , /D = 1 /D and ˆ S issymmetric. For any σ, τ ∈ Gal( Q ), στ (cid:18) D (cid:19) = g σ ( ) g τ ( ) ˆ S ˆ σ ( ) , ˆ τ ( ) D ! = τ σ (cid:18) D (cid:19) . Therefore, Gal( Q ( D ) / Q ) is abelian. By the Kronecker-Weber Theorem, Q ( D ) is contained ina cyclotomic field. In particular, D is a cyclotomic integer. (cid:3) Now we are ready to prove the following theorem.
Theorem 2.3.
For any pseudounitary braided fusion category C , p dim( C ) is a totally realcyclotomic integer.Proof. Let D = p dim( C ). Since dim( C ) is a totally positive algebraic integer, D is a totallyreal algebraic integer. We are left to show the cyclotomicity of D .We have the following two cases.(1) If C ′ is Tannakian, then the pseudounitarity of C implies θ X = id X for X ∈ Irr( C ′ ).By [4, 25], the de-equivariantization on C with respect to C ′ gives rise to a modular category M ( C ) with dim( M ( C )) = dim( C )dim( C ′ ) . By [30, Theorem 7.1], p dim( M ( C )) ∈ Q m as FSexp( M ( C )) | m , where m = FSexp( C ).Note that dim( C ′ ) ∈ Z as C ′ is symmetric. Therefore, p dim( C ′ ) is a cyclotomic integer, andso is D = p dim( M ( C )) dim( C ′ ) . (2) If C ′ is super-Tannakian, then C ′ has a maximal fusion subcategory C ′ + which is Tan-nakian and dim( C ′ ) = 2 dim( C ′ + ) . De-equivariantizing C with respect to C ′ + gives rise to a super-modular category S ( C ) (cf. [25,Section 5.3]) and dim( S ( C )) = dim( C )dim( C ′ + ) . By Lemma 2.2 (2), p dim( S ( C )) is a cyclotomic integer. Using similar argument as in Case(1), D is a cyclotomic integer. (cid:3) Remark 2.4. (i) The pseudounitary condition in the previous theorem could be removedbut some technicality is required. However, this technicality can be circumvented if everysuper-modular category admits a minimal modular extension or every fusion category has aspherical structure.(ii) In the proof of Theorem 2.3, the conductor of p dim( C ) can be shown to be bounded by12 · FSexp( C ) if C ′ is Tannakian by using the Cauchy Theorem [5]. It is unclear a similarbound can be obtained when C ′ is super-Tannakian. The Witt group W ( E ) . In this section, we follow [10] to study the Witt group ofnon-degenerate braided fusion categories over symmetric fusion categories.Let E be a symmetric fusion category. Throughout this paper, a fusion category over E isa fusion category A equipped with a braided tensor functor T A : E → Z ( A ) such that thecomposition of T A and the forgetful functor Z ( A ) → A is fully faithful.A tensor functor F : A → B between two fusion categories over E is called a tensor functorover E if F is compatible with the embeddings T A and T B . For details, see [10, Section 2].Let A , B be two fusion categories over E , and R : E → E ⊠ E be the right adjoint functorto the tensor product functor ⊗ : E ⊠ E → E . Then A := ( T A ⊠ T B ) R ( ) is a connected ´etalealgebra in Z ( A ⊠ B ). The tensor product A ⊠ E B of A and B over E is defined to be ( A ⊠ B ) A ,the fusion category over E of right A -modules. By [9, Lemma 3.11], we have(2.7) FPdim( A ⊠ E B ) = FPdim( A ) FPdim( B )FPdim( A ) = FPdim( A ) FPdim( B )FPdim( E ) . Recall that the M¨uger center C ′ of any braided fusion category C is a symmetric fusioncategory. A braided fusion category C equipped with a braided tensor equivalence T : E → C ′ is called a non-degenerate braided fusion category over E . In particular, with this terminology,non-degenerate braided fusion categories are non-degenerate over Vec, and slightly degeneratebraided fusion categories are non-degenerate over sVec.For any fusion category A over E , the M¨uger centralizer of T A ( E ) in Z ( A ) is denoted by Z ( A , E ), which is a typical example of non-degenerate braided fusion categories over E (cf. [28,Theorem 3.2], [15, Theorem 3.10]). Since Z ( A ) is non-degenerate over Vec, by [19, Theorem2.5] and [15, Theorem 3.14], we have(2.8) FPdim( Z ( A , E )) = FPdim( Z ( A ))FPdim( E ) = FPdim( A ) FPdim( E ) . Two non-degenerate braided fusion categories C and D over E are called Witt equivalent ifthere exist fusion categories A and B over E and a braided equivalence over E such that(2.9) C ⊠ E Z ( A , E ) ∼ = D ⊠ E Z ( B , E ) . According to [10], the Witt equivalence is an equivalence relation among braided fusioncategories over E , and the Witt equivalence classes form a group whose multiplication is givenby ⊠ E . We call this group the Witt group over E , and we denote it by W ( E ). We denotethe Witt class of a braided fusion category C over E by [ C ]. In case E = Vec or sVec, wesimply denote by W for W (Vec) and s W for W (sVec). The Witt group s W is also called the super-Witt group in this paper.By [10, Proposition 5.13], the assignment(2.10) S : W → s W ; [ C ] [ C ⊠ sVec]is a group homomorphism, and it is shown in loc. cit. that ker( S ) is a cyclic group of order16 generated by the class of any Ising braided category. An Ising category is a non-pointedfusion category C of FPdim( C ) = 4. There are 2 Ising categories up to tensor equivalence,and each of them admits 4 inequivalent braidings and they are all non-degenerate. Since IGHER CENTRAL CHARGES AND WITT GROUPS 11
Ising categories are pseudounitary (cf. [19]), these 8 inequivalent Ising braided categories aremodular and they are classified by their central charges.It is shown in [10] that the group W has only 2-torsion, and the maximal finite order of anelement of W is 32. We have seen in the above paragraph that the classes of pseudounitaryIsing modular categories are of order 16, but less is known about elements in W of order 32.In Sections 5 and 6, we will show that a pseudounitary Ising modular category has infinitelymany square roots in W .3. The E -signatures of the Witt group W ( E )Let α = 0 be a totally real algebraic number. For each σ ∈ Gal( Q ), σ ( α ) is either positiveor negative. The sign of σ ( α ) is 1 if it is positive, and -1 otherwise. This assignment ε ( α ) ofsigns(3.11) ε ( α )( σ ) := sgn( σ ( α ))for each σ ∈ Gal( Q ) is called the signature of α .Let µ n ⊆ C × denote the group of the n th -roots of unity and µ ∞ = S ∞ n =1 µ n . Then the set U ∞ of functions from Gal( Q ) to µ ∞ is an abelian group under pointwise multiplication, and U n = µ Gal( Q ) n is a subgroup of U ∞ . Thus, if F is a totally real subfield of C , ε : F × → U is group homomorphism.Recall that both FPdim( C ) and dim( C ) of any fusion category C are totally positive cy-clotomic integers. Therefore, the positive square roots p FPdim( C ) and p dim( C ) are totallyreal. Definition 3.1.
Let C be a fusion category. We define the signature ε C of C as ε ( p FPdim( C ))and the categorical dimension signature ε ′C of C as ε ( p dim( C )).By Theorem 2.3, for pseudounitary braided fusion categories, we can change Gal( Q ) toGal( Q ab ) in the definition of the categorical dimension signature. Remark 3.2.
For any pseudounitary fusion category C , ε C = ε ′C . Lemma 3.3.
Let C , D be fusion categories. (a) ε C ⊠ D = ε C · ε D , and ε ′C ⊠ D = ε ′C · ε ′D , i.e., both signatures respect the Deligne tensorproduct of fusion categories. (b) Both ε Z ( C ) and ε ′Z ( C ) are the constant function 1.Proof. Statement (a) follows from the multiplicativity of the FP-dimension and the categoricaldimension with respect to the Deligne tensor product.Statement (b) follows from p FPdim( Z ( C )) = FPdim( C ), p dim( Z ( C )) = dim( C ) (cf. [19],[27]) and the total positivity of both the FP-dimension and the categorical dimension. (cid:3) Theorem 3.4.
For any symmetric fusion category E , the assignment I E : W ( E ) → U ; [ C ] ε C · ε E is a well-defined group homomorphism.Proof. We first show that the assignment I E is well-defined. Indeed, for any braided fusioncategories C and D over E which are Witt equivalent over E , there exist fusion categories A , B over E such that C ⊠ E Z ( A , E ) ∼ = D ⊠ E Z ( B , E ). Therefore, by Eqs. (2.7) and (2.8), we haveFPdim( C ) FPdim( A ) FPdim( E ) = FPdim( D ) FPdim( B ) FPdim( E ) . This implies that p FPdim( C ) FPdim( A ) = p FPdim( D ) FPdim( B ). As mentioned in theprevious subsection, FPdim( A ) and FPdim( B ) are totally positive, so for any σ ∈ Gal( Q ), wehave ε C ( σ ) = sgn( σ (cid:16)p FPdim( C ) (cid:17) )= sgn( σ (cid:16)p FPdim( C ) FPdim( A ) (cid:17) )= sgn( σ (cid:16)p FPdim( D ) FPdim( B ) (cid:17) )= sgn( σ (cid:16)p FPdim( D ) (cid:17) )= ε D ( σ )which means ε C = ε D , and hence I E is well-defined.Again by (2.7), for any σ ∈ Gal( Q ), we have I E ([ C ⊠ E D ])( σ ) = ε C ⊠ E D ( σ ) · ε E ( σ )= sgn( σ s FPdim( C ) FPdim( D )FPdim( E ) ! ) sgn( σ (cid:16)p FPdim( E ) (cid:17) )= sgn( σ (cid:16)p FPdim( C ) FPdim( D ) (cid:17) )= ( ε C · ε E · ε D · ε E )( σ )= ( I E ([ C ]) · I E ([ D ]))( σ )as desired. (cid:3) Theorem 3.5.
For E = Vec or sVec , the assignment I ′E : W ( E ) → U ; [ C ] ε ′C · ε ′E is a well-defined group homomorphism.Proof. By Theorem 2.5 and Theorem 3.10 (i) of [15], for any fusion category A over E ,(3.12) dim( Z ( A , E )) = dim( A ) dim( E ) . IGHER CENTRAL CHARGES AND WITT GROUPS 13
For E = Vec or sVec, let A and B be fusion categories over E . In this case, there existsa finite group G such that Rep( G ) embeds into E ⊠ E as a braided fusion subcategory. Infact, G is trivial when E = Vec, and G = Z / Z when E = sVec. Note that | G | = dim( E )in both cases. Moreover, the image of the regular algebra of Rep( G ) under the compositionRep( G ) ֒ → E ⊠ E T A ⊠ T B −−−−→ Z ( A ⊠ B ) coincides with the algebra A in the definition of thetensor product over E (cf. Section 2.3). Therefore, A ⊠ E B is braided equivalent to the de-equivariantization ( A ⊠ B ) G , and by [15, Proposition 4.26],(3.13) dim( A ⊠ E B ) = dim( A ) dim( B ) | G | = dim( A ) dim( B )dim( E ) . Having established Eqs. (3.12) and (3.13), we are done by repeating the proof of Theorem3.4 with all the FP-dimensions changed into categorical dimensions. (cid:3)
Remark 3.6.
Let E be an arbitrary symmetric fusion category. If (3.13) is true for anytwo fusion categories over E , then the assignment I ′E in Theorem 3.5 is a well-defined grouphomomorphism. Definition 3.7.
We call the group homomorphism I E : W ( E ) → U defined in Theorem 3.4the E -signature on W ( E ), and I E ([ C ]) the E -signature of [ C ]. For E = Vec or sVec, we callthe group homomorphism I ′E : W ( E ) → U defined in Theorem 3.5 the categorical dimension E -signature on W ( E ), and I ′E ([ C ]) the categorical dimension E -signature of [ C ].For simplicity, I Vec , I sVec , I ′ Vec and I ′ sVec are denoted by I , sI , I ′ and sI ′ respectively. Corollary 3.8.
The following diagrams of group homomorphisms are commutative W s WU S I sI , W s WU S I ′ sI ′ . Proof.
The statement follows immediately from Theorems 3.4, 2.3 and the definition (2.10)of S . (cid:3) Higher central charges and signatures
Let C be a modular category. The n th Gauss sum τ n ( C ) of C introduced in [31] is definedas τ n ( C ) = X X ∈ Irr( C ) d X θ nX . If τ n ( C ) = 0, the n th central charge ξ n ( C ) is defined by ξ n ( C ) = τ n ( C ) | τ n ( C ) | . In particular, if N is the Frobenius-Schur exponent of C and n is coprime to N , by [31,Theorem 4.1], τ n ( C ) = 0 and ξ n ( C ) is a root of unity. When there is no ambiguity, we simplywrite τ n and ξ n for the n th Gauss sum and the n th central charge of C .Recall that there is a group homomorphism ˆ · : Gal( Q ) → Sym(Irr( C )) from the absoluteGalois group to the permutation group Sym(Irr( C )) of Irr( C ). By [14, Proposition 4.7], forany third root γ of ξ , we have(4.14) θ ˆ σ ( ) = γσ ( γ ) . The following theorem shows the relation between higher central charges of C and thesignature of p dim( C ) (cf. (3.11)). Theorem 4.1.
Let C be a modular category with Frobenius-Schur exponent N . Then for anyinteger n coprime to N , ξ n = ε ′C ( σ ) · σ ( ξ ) · γ n σ ( γ n ) where γ is any third root of ξ , σ ∈ Gal( Q ) such that σ − ( ζ N ) = ζ nN .Proof. By [31, Theorem 4.1], τ n = σ ( τ ) dim( C ) σ (dim( C )) θ n ˆ σ ( ) . Since dim( C ) is totally positive and θ ˆ σ ( ) is a root of unity, we find | τ n | = | σ ( τ ) | · dim( C ) σ (dim( C )) . Therefore, ξ n = σ ( τ ) | σ ( τ ) | θ n ˆ σ ( ) . Since τ = | τ | · ξ and dim( C ) = | τ | , we have σ ( τ ) = σ ( D ) · σ ( ξ ) , where D = p dim( C ). Thus, | σ ( τ ) | = | σ ( D ) | . Therefore, ξ n = σ ( D ) | σ ( D ) | σ ( ξ ) θ n ˆ σ ( ) = ε ′C ( σ ) · σ ( ξ ) · θ n ˆ σ ( ) . Now, the formula follows from (4.14). (cid:3)
Note that since both σ ( ξ ) and γ/σ ( γ ) are completely determined by ξ and σ , ξ n iscompletely determined by σ , ξ and ε ′C ( σ ). IGHER CENTRAL CHARGES AND WITT GROUPS 15
Remark 4.2.
Consider S , the unnormalized S-matrix of a modular category C . Similar toLemma 2.2, there exists a sign function ǫ σ : Irr( C ) → {± } such that σ (cid:18) S X,Y D (cid:19) = ǫ σ ( X ) S ˆ σ ( X ) ,Y D = ǫ σ ( Y ) S X, ˆ σ ( Y ) D for any X, Y ∈ Irr( C ) (cf. [18, 14]). In particular, σ ( S , /D ) = ǫ σ ( ) S ˆ σ ( ) , /D . This implies σ ( D ) = ǫ σ ( ) Dd ˆ σ ( ) . Therefore, ε ′C ( σ ) = sgn( σ ( D )) = ǫ σ ( ) · sgn( d ˆ σ ( ) ) . If d X > X ∈ Irr( C ) or C is pseudounitary, then ε ′C ( σ ) = ǫ σ ( ).For the remaining discussion, it would be more convenient to define the multiplicativecentral charges of degrees coprime to the Frobenius-Schur exponent of C as a function in U ∞ .For any N ∈ N , and k coprime to N , we use σ k to denote the element in Gal( Q N / Q ) suchthat σ k ( ζ N ) = ζ kN . Definition 4.3.
Let C be a modular category and N = ord ( T C ). We define the central chargefunction Ψ C ∈ U ∞ of C as follows: for any σ ∈ Gal( Q ), if σ | Q N = σ k , thenΨ C ( σ ) := ξ k ( C ) . In this convention, Theorem 4.1 can be restated as follows.
Theorem 4.4.
Let C be a modular category. Then for any σ ∈ Gal( Q ) , Ψ C ( σ ) = ε ′C ( σ − ) · σ − ( ξ ( C )) · σ ( γ ) σ − ( γ ) where γ is any third root of ξ ( C ) .Proof. The theorem is a direct consequence of Theorem 4.1 and Definition 4.3. (cid:3)
This formula of higher central charges allows us to define the function Ψ, in the followingdefinition, on the subgroup W un of W generated by the pseudounitary modular categories.This function Ψ will be shown to be a group homomorphism in the subsequent proposition. Definition 4.5.
Let W un be the subgroup of W generated by the pseudounitary modularcategories. The function Ψ : W un → U ∞ , called the central charge homomorphism , is definedby Ψ([ C ]) = Ψ C for any pseudounitary modular category C . Proposition 4.6.
The central charge homomorphism
Ψ : W un → U ∞ is a well-defined grouphomomorphism. Proof. If C and D are Witt equivalent pseudounitary modular categories, then ξ ( C ) = ξ ( D )and ε ′C = ε ′D by Theorem 3.5. Let γ ∈ C be any 3 rd root of ξ ( C ). Then, for any σ ∈ Gal( Q ),we haveΨ C ( σ ) = ε ′C ( σ − ) · σ − ( ξ ( C )) · σ ( γ ) σ − ( γ ) = ε ′D ( σ − ) · σ − ( ξ ( D )) · σ ( γ ) σ − ( γ ) = Ψ D ( σ ) . By [31, Lemma 3.1] or Theorem 4.4, we also haveΨ C ⊠ D ( σ ) = Ψ C ( σ ) · Ψ D ( σ )for any σ ∈ Gal( Q ) and pseudounitary modular categories C , D . Therefore, Ψ is a grouphomomorphism and this completes the proof of the statement. (cid:3) We close this section with the following proposition which will be useful for the last twosections.
Proposition 4.7.
The kernel of Ψ consists of the Witt classes [ C ] ∈ W un such that ξ ( C ) = 1 and ε C is the constant function 1.Proof. If ξ ( C ) = 1 and ε C ≡
1, one can take γ = 1 to be the third root of ξ ( C ). Then, for any σ ∈ Gal( Q ), Ψ C ( σ ) = 1 by Theorem 4.4. Therefore, [ C ] ∈ ker(Ψ). Conversely, if [ C ] ∈ ker(Ψ),then Ψ C ( σ ) = 1 for all σ ∈ Gal( Q ). In particular, ξ ( C ) = Ψ C (id) = 1. Again, take γ = 1 tobe the third root of ξ ( C ). Then, by Theorem 4.4, we have1 = Ψ C ( σ − ) = ε C ( σ ) · σ ( ξ ( C )) · σ − ( γ ) σ ( γ ) = ε C ( σ )for all σ ∈ Gal( Q ). Thus, ε C ≡ (cid:3) The modular tensor categories so (2 r + 1) r +1 In this section, we provide some basic facts for the quantum group modular categories C r := so (2 r + 1) r +1 for r ≥
1. The readers are referred to [1, 33] for more details on these categories. In particular,by [37], C r is pseudounitary. We prove a formula for the higher central charges of C r in Lemma5.1 and a formula for D r = p dim( C r ) in Proposition 5.3 which are essential to the proof ofour major result.5.1. Notations and formulas.
Some basic facts of the categories C r can be extracted fromthe underlying Lie algebras so (2 r + 1) and their root/weight datum. The conventions andnotations of roots, weights and information of C r are adopted from [1, 3, 21, 33]. We list belowsome of the datum we will use in the next few sections.Let n = 2 r + 1 for some r ≥
1. The Lie algebra so ( n ) = so (2 r + 1) is of type B r . Weconsider the quantum group modular category C r of so ( n ) at level n , and use the followingnotations for r ≥ • Orthonormal basis for the inner product space ( R r , ( · | · )): { e , . . . , e r } . IGHER CENTRAL CHARGES AND WITT GROUPS 17 • Normalized inner product such that any short root α has squared length 2: h e j , e k i = 2 δ j,k = 2( e j | e k ) . • The set of positive roots: ∆ + . Its contains the following elements e j , j = 1 , . . . , r ; e j − e k , ≤ j < k ≤ r ; e j + e k , ≤ j < k ≤ r . In particular, | ∆ + | = r . • Fundamental weights: ω j = e + · · · + e j , j = 1 , . . . , r − ω r = 12 ( e + · · · + e r ) . • The set of dominant weights: Φ + . • Root lattice: Q . • Coroot lattice: Q ∨ = { α ( α | α ) | α ∈ Q } . • Weight lattice: P . Note that the index of Q ∨ in P is given by(5.15) | P/Q ∨ | = | P/Q | · |
Q/Q ∨ | = 4 . • Half sum of positive roots: ρ = 12 (cid:16) (2 r − e + (2 r − e + · · · + 3 e r − + e r (cid:17) . • Highest root: ϑ = e + e . • Dual Coxeter number:(5.16) h ∨ = 2 r − n − . • The fundamental alcove:(5.17) C r = { λ ∈ Φ + | ( λ + ρ | ϑ ) < n + h ∨ } = { λ ∈ Φ + | ( λ + ρ | ϑ ) < r } = { λ ∈ Φ + | ( λ | ϑ ) ≤ n } . Note that the isomorphism classes of simple objects of C r are indexed by C r , and sowe identify C r and Irr( C r ). • Quantum parameter: q = exp (cid:18) πi n + h ∨ ) (cid:19) = exp (cid:18) πi n − (cid:19) = exp (cid:18) πi r (cid:19) . • Quantum integer: [ m ] = q m − q − m q − q − . • Twist: θ λ = q λ | λ +2 ρ ) for λ ∈ C r . • Quantum dimension: d λ = Y α ∈ ∆ + [2( λ + ρ | α )][2( ρ | α )] for λ ∈ C r . • First central charge:(5.18) ξ ( C r ) = exp (cid:18) πi · n dim C ( so ( n )) n + h ∨ (cid:19) = exp (cid:18) πi · (2 r + 1) · ( r (2 r + 1))4 r (cid:19) = exp (cid:18) πi (2 r + 1) (cid:19) . For r = 1, C r = so (3) . All the above notations are the same except that ϑ = e .5.2. Higher central charge of C r . Let D r = p dim( C r ) be the positive square root ofdim( C r ). Let N r be the Frobenius-Schur exponent of C r , and T r the T-matrix of C r . By [29,Theorem 7.7], N r = ord ( T r ) = lcm { ord ( θ λ ) | λ ∈ C r } . Lemma 5.1.
Let r be a positive integer. Then (a) lcm { , r } | N r | r . (b) D r ∈ Q N r , and for any σ ∈ Gal( Q ) , we have (5.19) Ψ([ C r ])( σ ) = ε C r ( σ − ) · σ ( ξ ( C r )) σ − ( ξ ( C r )) . Proof.
Since 2( λ | λ + 2 ρ ) ∈ Z , θ λ = q λ | λ +2 ρ ) is a 32 r th -root of unity. Therefore, T rr = idor N r | r .By (5.17), we have ω r = ( e + · · · + e r ) ∈ C r . Therefore,2( ω r | ω r + 2 ρ ) = r (2 r + 1)2 . Therefore, θ ω r = q ω r | ω r +2 ρ ) = exp (cid:18) πi r · r (2 r + 1)2 (cid:19) = exp (cid:18) (2 r + 1) πi (cid:19) , which implies that ord ( θ ω r ) = 32. Thus, it suffices to consider r ≥ IGHER CENTRAL CHARGES AND WITT GROUPS 19
Note that by (5.17), we have 2 e ∈ C r for r ≥
3. We have2(2 e | e + 2 ρ ) = 8 + 4(2 r −
1) = 8 r + 4 . Therefore, θ e = exp (cid:18) πi r · (8 r + 4) (cid:19) = − exp (cid:18) πi r (cid:19) which implies that ord ( θ e ) = 4 r . The above computations imply the first divisibility ofstatement (a).By (5.18), γ r = ξ ( C r ) is a 3 rd root of ξ ( C r ), and γ r = 1. By (a), γ r ∈ Q N r . This impliesthat D r ∈ Q N r by [14, Theorem II (ii)]. The remaining statement of (b) follows immediatelyfrom Theorem 4.4. (cid:3) Remark 5.2.
The preceding lemma implies that ord ( T r ) is non-decreasing with respect to r .5.3. A formula for D r .Proposition 5.3. The square root of the global dimension of C r is given by (5.20) D r = √ r r r − r − r Y ℓ =1 sin (cid:18) (2 ℓ − π r (cid:19) r − Y j =1 sin (cid:18) jπ r (cid:19) m r ( j ) − where m r ( j ) , ≤ j ≤ r − , is given by m r ( j ) = ( , if r = 1 ; r − l j m , if r ≥ . Proof.
According to [1, Theorem 3.3.20],(5.21) D r = p | P/ ( n + h ∨ ) Q ∨ | Y α ∈ ∆ + (cid:18) (cid:18) ( α | ρ ) n + h ∨ · π (cid:19)(cid:19) − . By (5.15), (5.16) and the fact that | ∆ + | = r , we have(5.22) D r = p r ) r r Y α ∈ ∆ + (cid:18) sin (cid:18) ( α | ρ )4 r π (cid:19)(cid:19) − = √ r r r − r − Y α ∈ ∆ + (cid:18) sin (cid:18) ( α | ρ )4 r π (cid:19)(cid:19) − Recall that ∆ + = { e ℓ , e a ± e b | ≤ ℓ ≤ r, ≤ a < b ≤ r } , then ( α | ρ ) can be easily givenas follows:When α = e ℓ for some 1 ≤ ℓ ≤ r , ( α | ρ ) = ( r − ℓ ) + 1 / α = e a + e b for some 1 ≤ a < b ≤ r , ( α | ρ ) = 2 r − ( a + b ) + 1 which is an integer satisfying2 ≤ ( α | ρ ) ≤ r − when α = e a − e b for some 1 ≤ a < b ≤ r , ( α | ρ ) = b − a which is also an integer satisfying1 ≤ ( α | ρ ) ≤ r − . Let m r ( j ) = { α ∈ ∆ + | ( α | ρ ) = j } for 1 ≤ j ≤ r −
2. We can now rewrite (5.22) as D r = √ r r r − r − r Y ℓ =1 sin (cid:18) (2 ℓ − π r (cid:19) r − Y j =1 sin (cid:18) jπ r (cid:19) m r ( j ) − When r = 1, ( α | ρ ) is not an integer for α ∈ ∆ + and hence the lemma follows directlyfrom (5.22). We proceed to show that m r ( j ) = r − l j m for r ≥ ≤ j ≤ r − r .Note that the equations j = b − a and j = 2 r − ( b + a ) + 1 have no common integer solution( a, b ) for any integer j . Thus m r ( j ) = | M r ( j ) | for 1 ≤ j ≤ r −
2, where M r ( j ) := { ( a, b ) | ≤ a < b ≤ r such that ( b − a − j )(2 r − ( b + a ) + 1 − j ) = 0 } . One can check directly that m ( j ) = 2 − l j m for 1 ≤ j ≤
2. Assume that m r ( j ) = r − l j m for all 1 ≤ j ≤ r − r . Note that if 1 ≤ j ≤ r −
2, then M r ( j ) + (1 ,
1) = { ( a, b ) ∈ M r +1 ( j ) | a ≥ } . Thus, M r +1 ( j ) = ( M r ( j ) + (1 , ∪ { (1 , j + 1) } if 1 ≤ j ≤ r, ( M r ( j ) + (1 , ∪ { (1 , r + 2 − j ) } if r + 1 ≤ j ≤ r − . Therefore, m r +1 ( j ) = m r ( j ) + 1 for 1 ≤ j ≤ r −
2. It is easy to see that M r +1 (2 r −
1) = { (1 , } , M r +1 (2 r ) = { (1 , } and (cid:24) r − (cid:25) = (cid:24) r (cid:25) = r . Thus, m r +1 ( j ) = | M r +1 ( j ) | = 1 = r + 1 − (cid:24) j (cid:25) for 2 r − ≤ j ≤ r . Therefore, we have m r +1 ( j ) = r + 1 − l j m for 1 ≤ j ≤ r . (cid:3) IGHER CENTRAL CHARGES AND WITT GROUPS 21 Witt subgroups generated by C r Let I := su (2) be a fixed Ising modular category (cf. [15]). It is well-known (cf. [9]) thatfor any n = 2 r + 1, the conformal embedding so ( n ) n × so ( n ) n ⊆ so ( n ) implies C r ⊠ C r isWitt equivalent to an Ising modular category. By comparing the first central charges, theWitt class [ C r ] of C r satisfies(6.23) [ C r ] = [ C r ⊠ ] = ( [ I ] if r ≡ , [ I ] , if r ≡ . Since the first central charge of C r is a primitive 32 nd root of unity (cf. (5.18)), the cyclicsubgroup h [ C r ] i of W generated by [ C r ] is of order 32 for any positive integer r .In this section, we study the subgroup of the Witt group W generated by the Witt classes[ C r ] for r ≥ The signature of C r . In this subsection, we compute some values of the signatures ofan infinite subset of {C r | r ∈ N } . Lemma 6.1.
For any integer r, j, k with k ≡ and gcd( k, r ) = 1 , (6.24) σ k (cid:18) sin (cid:18) jπ r (cid:19)(cid:19) = sin (cid:18) kjπ r (cid:19) in Q r . Proof. σ k (cid:18) sin (cid:18) jπ r (cid:19)(cid:19) = σ k e jπi r − e − jπi r i ! = e kjπi r − e − kjπi r i ! = sin (cid:18) kjπ r (cid:19) . (cid:3) Lemma 6.2.
Let r be an odd positive integer. For any integer k coprime to r , σ k ( √ r ∗ ) = (cid:18) kr (cid:19) √ r ∗ , where r ∗ = (cid:18) − r (cid:19) r , and (cid:18) • r (cid:19) is the Jacobi symbol. If, in addition, k ≡ , then, in Q r , we have σ k ( √ r ) = (cid:18) kr (cid:19) √ r . Proof.
For any prime factor p of r , √ p ∗ ∈ Q p ⊆ Q r (cf. [36]). Note that Gal( Q p / Q ) is cyclicof order p −
1. Thus, if a is a primitive root of ( Z /p ) × , then σ a is a generator of Gal( Q p / Q )and σ a ( √ p ∗ ) = −√ p ∗ . Therefore, σ ja ( √ p ∗ ) = ( − j √ p ∗ for any integer j . Hence, we have σ k ( √ p ∗ ) = (cid:18) kp (cid:19) √ p ∗ . If r = p · · · p ℓ is the prime factorization of r , then r ∗ = p ∗ · · · p ∗ ℓ and so σ k ( √ r ∗ ) = σ k ( p p ∗ ) · · · σ k ( p p ∗ ℓ ) = (cid:18) kp (cid:19) · · · (cid:18) kp ℓ (cid:19) √ r ∗ = (cid:18) kr (cid:19) √ r ∗ . If, in addition, k ≡ Q r , we have σ k ( √ r ) = σ k √ r ∗ s(cid:18) − r (cid:19)! = (cid:18) kr (cid:19) √ r ∗ s(cid:18) − r (cid:19) = (cid:18) kr (cid:19) √ r. (cid:3) Proposition 6.3.
For any integers l > , w > and x , let a = 2 l + 1 + w (8 l + 2) and k = 8 xa + 4 l + 1 . Then, gcd( k, a ) = 1 and ε C a ( σ k ) = ( − x . Proof.
The first assertion follows directly by the Euclidean algorithm. We proceed to computethe sign of each component of σ k ( D a ) in the right hand side of (5.20). The sign of the secondsine component is 1 by the following lemma. Lemma 6.4.
We have the following equality: sgn σ k a − Y j =1 sin (cid:18) jπ a (cid:19) m a ( j ) = 1 . Proof of Lemma 6.4.
Since k ≡ σ k a − Y j =1 sin (cid:18) jπ a (cid:19) m a ( j ) = a − Y j =1 sgn (cid:18) sin (cid:18) kjπ a (cid:19)(cid:19) m a ( j ) . For each j = 1 , . . . , a −
2, we havesin (cid:18) kjπ a (cid:19) = sin (cid:18) (8 xa + 4 l + 1) jπ a (cid:19) = sin (cid:18) (4 l + 1) jπ a (cid:19) . Moreover, by the definition of a and the assumption that w >
0, we have(4 l + 1)(2 a − a − l > . Therefore, 2 l < (4 l + 1)(2 a − a < (4 l + 1)(2 a )4 a = 2 l + 12 . Consequently, for j = 1 , . . . , a −
2, sgn (cid:16) sin (cid:16) (4 l +1) jπ a (cid:17)(cid:17) = − q − < (4 l + 1) j a < q for some 1 ≤ q ≤ l . IGHER CENTRAL CHARGES AND WITT GROUPS 23
For any q = 1 , . . . , l , (6.25) is equivalent to(8 w + 2)(2 q −
1) + 4 q − l + 1 < j < (8 w + 2)(2 q ) + 4 q l + 1 . Since 1 ≤ q ≤ l , we have 0 < q − l + 1 < q l + 1 < . So (6.25) is equivalent to (8 w + 2)(2 q −
1) + 1 ≤ j ≤ (8 w + 2)(2 q ) . There are exactly 8 w + 2 integers between (8 w + 2)(2 q −
1) + 1 and (8 w + 2)(2 q ) inclusively.They can be written in pairs (8 w + 2)(2 q −
1) + (2 t − , (8 w + 2)(2 q −
1) + 2 t for 1 ≤ t ≤ w + 1.However, for each such t , we have m a ((8 w + 2)(2 q −
1) + (2 t − a − (cid:24) (8 w + 2)(2 q −
1) + (2 t − (cid:25) = a − ((4 w + 1)(2 q −
1) + t )= a − (cid:24) (4 w + 1)(2 q −
1) + (2 t )2 (cid:25) = m a ((8 w + 2)(2 q −
1) + (2 t )) . Thus we have(6.26) a − Y j =1 sgn (cid:18) sin (cid:18) kjπ a (cid:19)(cid:19) m a ( j ) = l Y q =1 4 w +1 Y t =1 ( − m a ((8 w +2)(2 q − t − · ( − m a ((8 w +2)(2 q − t ) =1 . as desired. (cid:3) The sign of the first sine component of the right hand side of (5.20) is computed in thefollowing lemma.
Lemma 6.5.
We have the following equality: sgn σ k a Y j =1 sin (cid:18) (2 j − π a (cid:19) = ( − x + l . Proof of Lemma 6.5.
Since k ≡ σ k a Y j =1 sin (cid:18) (2 j − π a (cid:19) = a Y j =1 sin (cid:18) k (2 j − π a (cid:19) . Thus, sgn σ k a Y j =1 sin (cid:18) (2 j − π a (cid:19) = a Y j =1 sgn (cid:18) sin (cid:18) k (2 j − π a (cid:19)(cid:19) . Moreover, by definition, we havesin (cid:18) k (2 j − π a (cid:19) = sin (cid:18) (8 xa + 4 l + 1)(2 j − π a (cid:19) = ( − x sin (cid:18) (4 l + 1)(2 j − π a (cid:19) . Therefore, since a is odd, we have(6.27) sgn σ k a Y j =1 sin (cid:18) (2 j − π a (cid:19) = ( − x a Y j =1 sgn (cid:18) sin (cid:18) (4 l + 1)(2 j − π a (cid:19)(cid:19) . Since (4 l + 1)(2 a − a − l > , we have l < (4 l + 1)(2 a − a < (4 l + 1)(2 a )8 a = l + 14 . Now we consider two cases.
Case 1. If l is even, then for any j = 1 , . . . , a , sgn(sin( (4 l +1)(2 j − a π )) = − q − < (4 l + 1)(2 j − a < q for some 1 ≤ q ≤ l/
2. Note that for any q = 1 , . . . , l/
2, (6.28) is equivalent to(8 w + 2)(2 q −
1) + 4 q − l + 1 + 12 < j < (8 w + 2)(2 q ) + 4 q l + 1 + 12 . Since 1 ≤ q ≤ l/
2, we have 0 < q − l + 1 < q l + 1 < , (6.28) is equivalent to(8 w + 2)(2 q −
1) + 1 ≤ j ≤ (8 w + 2)(2 q ) = 16 wq + 4 q . IGHER CENTRAL CHARGES AND WITT GROUPS 25
Hence, we have a Y j =1 sgn (cid:18) sin (cid:18) (4 l + 1)(2 j − π a (cid:19)(cid:19) = l/ Y q =1 wq +4 q Y j =(8 w +2)(2 q − ( − = 1as there are 8 w + 2 terms in each of the product corresponding to q . Case 2. If l is odd, then for any j = 1 , . . . , a , sgn(sin( (4 l +1)(2 j − a π )) = − j satisfies(6.29) 2 q − < (4 l + 1)(2 j − a < q for some 1 ≤ q ≤ ( l − /
2, or j satisfies(6.30) l < (4 l + 1)(2 j − a < l + 14 . By the same argument as in Case 1, the sign for the j ’s satisfying (6.29) is equal to ( l − / Y q =1 wq +4 q Y j =(8 w +2)(2 q − ( − = 1 . Note that (6.30) is equivalent to8 wl + 2 l + 2 l l + 1 + 12 < j < a + 12 . Since l l +1 < , and by definition of a , (6.30) is equivalent to a − w ≤ j ≤ a . Note that there are 2 w + 1 such j ’s, and so their sign contribution is -1. Therefore, a Y j =1 sgn (cid:18) sin (cid:18) (4 l + 1)(2 j − π a (cid:19)(cid:19) = ( l − / Y q =1 wq +4 q Y j =(8 w +2)(2 q − ( − · a Y j = a − w ( −
1) = − . This completes Case 2, and we have a Y j =1 sgn (cid:18) sin (cid:18) (4 l + 1)(2 j − π a (cid:19)(cid:19) = ( − l . for any positive integer l .Combining with (6.27), we obtainsgn σ k a Y j =1 sin (cid:18) (2 j − π a (cid:19) = ( − x + l as claimed. This completes the proof of Lemma 6.5. (cid:3) Now we are ready to prove Proposition 6.3. By definition and the quadratic reciprocity ofJacobi symbols, we have(6.31) (cid:18) ka (cid:19) = (cid:18) l + 1 a (cid:19) = (cid:18) a l + 1 (cid:19) = (cid:18) l + 14 l + 1 (cid:19) = (cid:18) l + 12 l + 1 (cid:19) = (cid:18) − l + 1 (cid:19) = ( − l . Therefore, by Lemma 6.2, σ k (cid:16) √ a a (cid:17) = σ k (cid:16) a a − √ a (cid:17) = (cid:18) ka (cid:19) a a − √ a = ( − l a a − √ a , i.e., sgn( σ k ( √ a a )) = ( − l . The proposition follows directly from Lemmas 6.4, 6.5 and (5.20). (cid:3) The central charge homomorphism Ψ on W un . For any z ∈ N ∪ { } , let l = 4 z + 2and a z, = 2 l + 1. We define the sequence aaa z = { a z,n } ∞ n =0 inductively by letting a z,n +1 to bethe smallest positive integer such that a z,n +1 ≡ l + 1 (mod 8 l + 2)and gcd( a z,n +1 , a z,j ) = 1 for all j = 0 , . . . , n . Then the sequence aaa z is infinite for any z by theDirichlet prime number theorem. For example, the sequence { a ,n } ∞ n =0 begins with5 , , , , , , , , , , . . . . Recall that for any r ∈ N , C r is pseudounitary, and so [ C r ] ∈ W un . In particular, by Remark3.2, ε C r = ε ′C r . For any infinite subsequence fff = { f j } ∞ j =0 ⊆ aaa z , let G fff,n be the subgroup of W un generated by { [ C f j ] | j = 0 , . . . , n } , and let G fff = ∞ [ n =0 G fff,n . In particular, G fff is a direct limit of { G fff,n } . For any positive integer r and σ ∈ Gal( Q ), wesimply set ε r ( σ ) := ε C r ( σ ) = ε ′C r ( σ ) . Lemma 6.6.
For any z, n ∈ N ∪ { } and any infinite subsequence fff of aaa z , consider anarbitrary element A = Q nj =0 (cid:2) C f j (cid:3) b j ∈ G fff,n . If there exists a nonnegative integer m ≤ n suchthat b m is odd, then A / ∈ ker(Ψ) . In particular, A = [Vec] .Proof. Note that [ C r ] = [Vec] for all r ∈ N . Therefore, we may assume that b j > A = [ A ] where A = C ⊠ b f ⊠ · · · ⊠ C ⊠ bn f n .Assume that A ∈ ker(Ψ). Then1 = Ψ( A )(id) = Ψ A (id) = ξ ( A ) , IGHER CENTRAL CHARGES AND WITT GROUPS 27 and we may take γ in Theorem 4.4 to be 1. By Theorem 4.4, for any τ ∈ Gal( Q ),(6.32) Ψ( A )( τ ) = τ − ( ξ ( A )) τ ( γ ) τ − ( γ ) n Y j =0 ε f j ( τ − ) b j = n Y j =0 ε f j ( τ − ) b j = 1 . Now let N := lcm { ord ( T f j ) | ≤ j ≤ n and j = m } . Then by Lemma 5.1 (a),32 | N, and N | Y ≤ j ≤ nj = m f j . By the definition of fff , the integer f m is coprime to 32 and f j for all j = m , and hencegcd( f m , N ) = 1. Therefore, there exist x, y ∈ Z such that xf m + yN = 1 . Set k := − z + 1) xf m + 4 l + 1 = 8(2 z + 1) yN + 1. Then k ≡ N ). Moreover, byProposition 6.3, gcd( k, f m ) = 1 with l = 2(2 z + 1) and a = f m . Therefore, gcd( k, N f m ) = 1.Let σ ∈ Gal( Q ) such that σ | Q Nfm = σ k . It follows from (6.32) thatΨ( A )( σ − ) = n Y j =0 ε f j ( σ ) b j = 1 . For any j = 0 , . . . , n and j = m , we have D f j ∈ Q ord ( T fj ) ⊆ Q N by Proposition 5.3 or [14].Since k ≡ N ), σ ( D f j ) = D f j and hence ε f j ( σ ) = 1. Also, by letting l = 4 z + 2 inProposition 6.3, we have ε f m ( σ ) = ( − (2 z +1) x = ( − x . Note that xf m + yN = 1 implies that x has to be odd, and so we have n Y j =0 ε f j ( σ ) b j = ( − x = − , which contradicts (6.32). (cid:3) Theorem 6.7.
For any z, n ∈ N ∪ { } and any infinite subsequence fff of aaa z , the restriction Ψ | G fff,n of the central charge homomorphism on G fff,n is injective, Ψ( G fff,n ) ∼ = Z / ⊕ ( Z / n and G fff,n / h [ I ] i ∼ = ( Z / n +1 . Moreover, Ψ | G fff is injective, G fff ∼ = Z / ⊕ ( Z / ⊕ N and G fff / h [ I ] i ∼ =( Z / ⊕ N .Proof. Suppose ker(Ψ) ∩ G fff,n contains a non-trivial class A = Q nj =0 (cid:2) C f j (cid:3) b j for some integers b , . . . , b n . We claim that one of the b j ’s must be odd. Recall that for any r , [ C r ] is a powerof [ I ]. If b j are even for all j = 0 , . . . , n , then A = [ I ] t for some t ∈ N . Since [ I ] t ∈ ker(Ψ),1 = Ψ([ I ] t )(id) = ξ ( I ⊠ t ) = ξ ( I ) t . However, according to [15], ξ ( I ) t = 1 if and only if [ I ] t = [Vec], which contradicts theassumption. Therefore, there exists a nonnegative integer m ≤ n such that b m is odd. In this case, wereach a contradiction to Lemma 6.6. Therefore, ker(Ψ) ∩ G fff,n is trivial. Consequently, Ψ | G fff,n is injective, and so Ψ | G fff is also injective since G fff = lim −→ G fff,n .For any fff , we prove that Ψ( G fff,n ) ∼ = Z / ⊕ ( Z / ⊕ n by induction on n . When n = 0, by(6.23), [ C f ] = [ I ] or [ I ] . By (5.18), ξ ( C l +1 ) = ζ (2 l +1) , so G fff, ∼ = Z /
32 (cf. the beginningof Section 6).Suppose the statement is true for n = v . Since [ C r ] is a power of [ I ] ∈ G fff, ⊆ G fff,v ⊆ G fff,v +1 , then G fff,v +1 /G fff ,v is either trivial or isomorphic to Z / C f v +1 ] ∈ G fff,v or not. If [ C f v +1 ] ∈ G fff,v , then there exists b , . . . , b v ∈ Z such that (cid:2) C f v +1 (cid:3) − Q vj =0 (cid:2) C f j (cid:3) b j =[Vec], which contradicts to Lemma 6.6. Hence, G fff,v +1 /G fff,v ∼ = Z / → G fff,v → G fff,v +1 π −→ G fff,v +1 /G fff,v → , where π is the natural surjection. Set A = C f if f v +1 ≡ f (mod 4) and A = C ⊠ f if f v +1 f (mod 4). Then [ C f v +1 ] = [ A ] by (6.23) and the fact that [ I ] is of order 16in W . Consequently, [ C f v +1 ][ A ] − is an order 2 element in G fff,v +1 .The homomorphism ι : G fff,v +1 /G fff,v → G fff,v +1 defined by ι ([ C f v +1 ] G fff,v ) = [ C f v +1 ][ A ] − is a section of π . Thus theexact sequence splits and we have G fff,v +1 ∼ = G fff,v ⊕ G fff,v +1 /G fff,v ∼ = G fff,v ⊕ Z / . The isomorphism G fff,n / h [ I ] i ∼ = ( Z / n +1 for all n ∈ N follows immediately from the precedingresult. The last statement follows from the fact that G fff is a direct limit of G fff,n . (cid:3) Corollary 6.8.
The Witt class of the Ising modular category I has infinitely many squareroots in W .Proof. By Theorem 6.7, G aaa ∼ = Z / ⊕ ( Z / ⊕ N . Since A ∈ h [ I ] i for all A ∈ G aaa . Thestatement follows. (cid:3) The subgroup generated by {S ([ C r ]) | r ≥ } in s W is an abelian group of exponent 2. It isconjectured in [10, Conjecture 5.21] that {S ([ C r ]) | r ≥ } is linearly independent in s W . Thefollowing corollary prove that this holds for infinitely many subsequences of {S ([ C r ]) | r ≥ } ,but we do not know whether they are in s W or not. The group s W will be further discussedin Section 7. Corollary 6.9.
For any nonnegative integer z , the sequence sasasa z = {S ([ C a z,j ]) | j ≥ } is linearly independent in s W .Proof. Suppose sasasa z is dependent in s W . There exist distinct nonnegative integers 0 ≤ j < · · · < j n such that S ([ C r ⊠ · · · ⊠ C r n ]) = [sVec] IGHER CENTRAL CHARGES AND WITT GROUPS 29 where r i = a z,j i for i = 1 , . . . , n . Since h [ C r ] i = h [ I ] i , which is the kernel of S by [10],[ C r ] b · [ C r ] · · · [ C r n ] = [Vec]for some positive odd integer b . However, this contradicts Lemma 6.6. (cid:3) It is not difficult to show that the central charge homomorphism Ψ is injective on W pt ( p ),the Witt subgroup of W pt ( p ) generated by the Witt classes of the pointed modular categories C ( H, q ) where H is a finite abelian p -group. However, the kernel of Ψ | W pt is not trivial.For any odd prime p , let A p be the unique Witt class in W pt ( p ) of trivial signature. Inparticular, A p has order 2 and can be represented by an abelian group of order p . Let A := [ I ] , which is the unique class of W pt (2) of order 2 with trivial signature. Therefore, A p is the unique element of W pt ( p ) of order 2 with trivial signature for all prime p . Remark 6.10.
Note that the first central charge of A p is − p , and so A p ker(Ψ). However, for any primes p and p ′ , A p A p ′ ∈ ker(Ψ) by Proposition 4.7. Proposition 6.11.
The intersection ker(Ψ) ∩ W pt is generated by A p A p ′ , where p and p ′ aredistinct primes.Proof. Let B ∈ ker(Ψ) ∩ W pt be a non-trivial Witt class. Then there exists an anisotropicmetric group ( H, q ) such that B = [ C ( H, q )]. In particular, Ψ( B )(id) = ξ ( C ( H, q )) = 1.Hence, by Theorem 4.4, I ( B )( σ ) = 1 for any σ ∈ Gal( Q ).Let h = Q nj =1 p e j j be the order of H , where p , . . . , p n are distinct primes. Then B = B p · · · B p n where B p j ∈ W pt ( p j ) is nontrivial for j = 1 , . . . , n .We claim that e j has to be even for all j = 1 , ..., n . In particular, I ( B p j ) = 1. Otherwise, √ h Q and there exists σ ∈ Gal( Q ) such that σ ( √ h ) = −√ h , that means I ( B )( σ ) = −
1, acontradiction.To complete the proof, it suffices to show n is even and B p j = A p j for j = 1 , . . . , n . Since e j is even for any j = 1 , ..., n , B p j = A p j if p j is odd. Thus, if h is odd, so are p j for all j .Then n Y j =1 A p j = n Y j =1 B p j = B ∈ ker(Ψ)implies that n must be even by Remark 6.10.We now consider the case when h is even. Then h = 2 e Q nj =2 p e j j for some even positiveinteger e . Suppose n is odd. Then n Y j =2 B p j = n Y j =2 A p j ∈ ker(Ψ)by Remark 6.10. Since B ∈ ker(Ψ), and so B ∈ ker(Ψ). However, this contradicts thatΨ is injective on W pt (2) (cf. [31, Example 6.2]). Therefore, n is also even in this case. Consequently, B has the same first central charge as Q nj =2 A − p j , which is -1. Therefore, B = A as I ( B ) = 1. (cid:3) Theorem 6.7 and Proposition 6.11 inspire the following question.
Question 6.12.
Is the intersection between ker(Ψ) and the torsion subgroup Tor( W un ) con-tained in W pt ? 7. The group s W Let s W pt be the image S ( W pt ) in s W . By [10, Proposition 5.18], the super-Witt group s W can be decomposed into a direct sum s W = s W pt ⊕ s W ⊕ s W ∞ where s W (resp. s W ∞ ) is the subgroup of s W generated by the Witt classes of completelyanisotropic s-simple fusion categories of Witt order 2 (resp. of infinite Witt order). In partic-ular, the torsion part of s W is s W pt ⊕ s W . It is conjectured [10, Conjecture 5.21] that s W has infinite rank. In this section, we give a proof for this conjecture in Theorem 7.2.Fix z ∈ N ∪ { } and l = 4 z + 2 as before. Consider the subsequence ppp = { p j } ∞ j =0 of aaa z consisting of all prime number terms. Again by the Dirichlet prime number theorem, ppp is aninfinite sequence. Proposition 7.1.
We have G ppp ∩ W pt = h [ I ] i .Proof. Since [ I ] ∈ W pt and h [ I ] i = h [ C p ] i , h [ I ] i ⊆ G ppp ∩ W pt . Suppose A ∈ G ppp ∩ W pt isnontrivial. Then A = Q nj =0 (cid:2) C p j (cid:3) b j for some integers b , . . . , b n . As is illustrated in the proof ofLemma 6.6, we can assume that all the b j ’s are nonnegative, and we let A = C ⊠ b p ⊠ · · · ⊠ C ⊠ bn p n .We first show that all the b j ’s are even. If not, there exists a nonnegative integer m ≤ n such that b m is odd. Since [ C p j ] ∈ h [ C p ] i = h [ I ] i for all j , we may simply assume m = n .Since A ∈ W pt , there exists a finite abelian group H and a non-degenerate quadratic form q : H → C × such that the corresponding pseudounitary modular category H = C ( H, q )satisfies [ H ] = A in W un . By Theorem 3.5,(7.33) ε A ( σ ) = ε H ( σ )for all σ ∈ Gal( Q ).Let h denote the order of H . Since p n is an odd prime different from p , . . . , p n − , we canwrite h as h = h h , where gcd( h , p n ) = 1, and h = p sn for some s ∈ N ∪ { } . Let M := 32 · h · n − Y j =0 p j . Again, Lemma 5.1 (a) implies that ord ( T p j ) | M for j = 0 , . . . , n −
1. Moreover, by construc-tion, gcd( p n , M ) = 1. Hence, there exist x, v ∈ Z such that xp n + vM = 1 . IGHER CENTRAL CHARGES AND WITT GROUPS 31
Note that since M is even, x has to be odd.Set k := − z +1) xp n +4 l +1 = 8(2 z +1) vM +1. Then k ≡ M ) and gcd( k, p n ) = 1by Proposition 6.3. Therefore, we have k ≡ h ), and gcd( k, p n M ) = 1.Let σ ∈ Gal( Q ) be such that σ | Q N = σ k where N = M p n . Then by Proposition 5.2 or[14], for j = 0 , . . . , n − D p j ∈ Q ord ( T pj ) ⊆ Q M . Since k ≡ M ), for j = 0 , .., n − σ ( D p j ) = D p j , and hence ε p j ( σ ) = 1. Apply Proposition 6.3 for l = 4 z + 2, we have ε p n ( σ ) = ( − (2 z +1) x = − . Therefore, since b n is odd, ε A ( σ ) = n Y j =0 ε p j ( σ ) b j = ( − b n = − . By the definition of M , we have √ h ∈ Q N and √ h ∈ Q M . On one hand, since k ≡ M ), σ ( √ h ) = √ h . On the other hand, k ≡ (cid:18) kp n (cid:19) = ( − l = ( − z +2 = 1 . Therefore, by Lemma 6.2 and h = p sn , we have ε H ( σ ) = σ ( √ h ) √ h σ ( √ h ) √ h = (cid:18) kh (cid:19) = (cid:18) kp n (cid:19) s = 1 , contradicting (7.33). Therefore, b , . . . , b n are all even, and hence A ∈ h [ I ] i .Since h [ I ] i ∩ W pt = h [ I ] i , A ∈ h [ I ] i . Therefore, G ppp ∩ W pt = h [ I ] i . (cid:3) Theorem 7.2.
The group s W has infinite rank.Proof. Let ppp = { p j } ∞ j =0 be the prime number subsequence of aaa z for any nonnegative integer z .Then S ( G ppp ) is an elementary 2-group, and so S ( G ppp + W pt ) ⊆ s W pt ⊕ s W by [10, Proposition5.18]. By [10, Proposition 5.18], s W pt = S ( s W pt ). Thus, S ( G ppp + W pt ) S ( W pt )is isomorphic to a subgroup of s W . By Proposition 7.1, we have G ppp ∩ W pt = h [ I ] i . Sinceker( S ) = h [ I ] i , by Theorem 6.7, we have S ( G ppp + W pt ) S ( W pt ) ∼ = G ppp + W pt h [ I ] i + W pt ∼ = G ppp + W pt W pt (cid:30) h [ I ] i + W pt W pt ∼ = G ppp G ppp ∩ W pt (cid:30) h [ I ] i G ppp ∩ W pt ∼ = G ppp h [ I ] i ∼ = ( Z / ⊕ N . Therefore, s W , is an infinite group. (cid:3) Acknowledgements
The authors would like to thank Zhenghan Wang for his valuable discussions on the squareroots of Ising modular categories. The first and the third authors would like to thank LingLong for many fruitful discussions on the signatures of real algebraic integers. This paper isbased upon work supported by the National Science Foundation under the Grant No. DMS-1440140 while the first two and the last authors were in residence at the Mathematical SciencesResearch Institute in Berkeley, California, during the Spring 2020 semester.
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