Modular categories with transitive Galois actions
aa r X i v : . [ m a t h . QA ] J u l MODULAR CATEGORIES WITH TRANSITIVE GALOISACTIONS
SIU-HUNG NG, YILONG WANG, AND QING ZHANG
Abstract.
In this paper, we study modular categories whose Galois groupactions on their simple objects are transitive. We show that such modularcategories admit unique factorization into prime transitive factors. The repre-sentations of SL ( Z ) associated with transitive modular categories are provento be minimal and irreducible. Together with the Verlinde formula, we char-acterize prime transitive modular categories as the Galois conjugates of theadjoint subcategory of the quantum group modular category C ( sl , p −
2) forsome prime p >
3. As a consequence, we completely classify transitive modularcategories. Transitivity of super-modular categories can be similarly defined.A unique factorization of any transitive super-modular category into s-simpletransitive factors is obtained, and the split transitive super-modular categoriesare completely classified. Introduction
Modular categories are spherical braided fusion categories over C whose braidingsare nondegenerate. The notion of modular categories has evolved from the studiesof rational conformal field theory [36], topological quantum field theory [52] andthe quantum invariants of knots and 3-manifolds such as the Jones polynomial[33, 47]. Moreover, unitary modular categories are the mathematical foundationto describe topological phases of matter [57] and topological quantum computing[56, 50]. Similar to the role of groups in the study of symmetries, modular categoriesare natural algebraic objects to organize “quantum symmetries”.An important family of examples of modular categories is obtained from thequantum group construction [3, 49]. In general, for any simple Lie algebra g and a suitable root of unity q ∈ C , one can construct a modular category by taking thesemisimplification of the category of tilting modules of the quantum group U q ( g )specialized at the root of unity q [1, 2]. The 3-manifold invariants [4, 51] and themapping class group representations [5, 29] associated with quantum group modularcategories are also well-studied in the literature.Modular categories have interesting arithmetic properties, such as the Verlindeformula, which are encoded in the matrices S and T (see Section 2). More precisely,let s := (cid:16) −
11 0 (cid:17) and t := (cid:16) (cid:17) be the generators of the modular group SL ( Z ).For any modular category C , the assignment ρ C : s S , t T gives a projectiverepresentation of SL ( Z ) [3, 52]. Another notable arithmetic property of C is thefact that the kernel of ¯ ρ C is a congruence subgroup whose level is equal to theorder of the T-matrix [43]. Moreover, ¯ ρ C admits liftings to linear representations ofSL ( Z ) which are also shown to have congruence kernels in [22]. In addition, these The first author was partially supported by the NSF grant DMS-1664418. liftings enjoy certain symmetries under the action of the absolute Galois groupGal( ¯ Q ). These properties of the liftings are essential to our proofs in this paper.Since the irreducible characters of the fusion ring of a modular category C can beindexed by the set Irr( C ) of isomorphism classes of simple objects of C [19, 15], theaction of Gal( ¯ Q ) on the these characters induces a permutation action on Irr( C ).The number of Galois orbits are also invariants of modular categories.It is always important to classify mathematical structures of a certain propertyin any mathematical theory. There has been literature on classifying modularcategories by rank [10, 48, 9], Frobenius-Perron dimension [8, 12] and Frobenius-Schur exponent [13, 55]. Note that there are finitely many modular categories up toequivalence for any given rank [11]. The number of Galois orbits plays prominentroles in most of these papers (see also [16, 32]), which leads to the idea of classifyingmodular categories by the number of Galois orbits.In this paper, we investigate modular categories with only one Galois orbit,which are called transitive modular categories . The smallest nontrivial exampleof a transitive modular category is the Fibonacci modular category given by theadjoint subcategory C ( sl , (0) of the quantum group category C ( sl ,
3) of sl atlevel 3. More generally, the adjoint subcategory C ( sl , p − (0) of C ( sl , p −
2) and itsGalois conjugates are prime and transitive modular categories for any prime p >
Theorem I.
Let C be a nontrivial modular category. Then C is prime and transitiveif and only if ord( T ) = p is a prime > C is equivalent to a Galois conjugateof C ( sl , p − (0) as modular categories. Theorem II.
Let C be a nontrivial modular category. Then C is transitive if andonly if C is equivalent to a Deligne product of prime transitive modular categorieswhose T-matrices have distinct orders. In particular, ord( T ) is a square-free oddinteger > C , we denote by Q ( S )the Q -extension by adjoining all the entries of the S-matrix, and denote by G C thecorresponding Galois group over Q . Our first observation is that the action of G C on Irr( C ) is fixed-point free (Proposition 3.2), and so Irr( C ) is a G C -torsor. More-over, every fusion subcategory of a transitive modular category is also transitiveand modular (Corollary 3.9). We conclude that any transitive modular categoryhas a unique factorization into a Deligne product of prime transitive modular cat-egories (Theorem 3.11). In Section 4, we study the Galois conjugates of modularcategories C ( sl , k ) (0) of odd level k . We show that for any prime p >
3, everyGalois conjugate of C ( sl , p − (0) is prime and transitive.Inspired by the Galois symmetries of the representations of SL ( Z ) associatedwith modular categories, we define the notion of minimal representations of SL ( Z )and the characteristic 2-group of a modular category in Section 5. The minimalrepresentations of SL ( Z ) associated with a modular category C are completely ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 3 determined by the eigenvalues of the images of t (Lemma 5.6). Moreover, the char-acteristic 2-group of C naturally corresponds a decomposition of any representationof SL ( Z ) associated with C in Proposition 5.11. By studying these two notions, weprove in Theorem 5.14 that any representation of SL ( Z ) associated with a tran-sitive modular category C is minimal and irreducible, and that the order of theT-matrix of C is odd and square-free.We completely classify transitive modular categories in Section 6 by character-izing the prime and transitive modular categories. By using the minimality andirreducibility of the representations of SL ( Z ) associated with transitive modularcategories, we show that the order of the T-matrix of any prime transitive modularcategory C is a prime p >
3, and it has the same fusion rules as C ( sl , p − (0) .Applying the classification result of [30], we show that C must be a Galois conjugateof C ( sl , p − (0) (cf. Theorem 6.4). We combine the unique factorization theoremto conclude the full classification in Theorem 6.5.We end this paper with a discussion on transitive super-modular categories inSection 7. We classify all the transitive split super-modular categories by using theclassification of transitive modular categories (Theorem 7.4). Moreover, a uniquefactorization of transitive super-modular categories into s-simple transitive factorsis obtained in Theorem 7.13. Then we exhibit a family of non-split transitive primecategories over sVec and conjecture that these are all of them.The paper is organized as follows. In Section 2, we give a brief review on modularcategories and notations. In Section 3, we define transitive modular categoriesand derive some fundamental properties of them. In particular, we establish theprime factorization theorem in Theorem 3.11. In Section 4, we discuss the primeand transitive modular categories obtained from the quantum group categories C ( sl , p −
2) for any odd prime p . In Section 5, we study the modular grouprepresentations associated with modular categories. We show in Theorem 5.14 thatthe representations associated with transitive modular categories are irreducibleand minimal. In Section 6, we characterize the prime transitive modular categoriesin Theorem 6.4, which implies the complete classification of transitive modularcategories in Theorem 6.5. Finally, in Section 7, transitive super-modular categoriesare introduced and studied. We classify split transitive super-modular categories inTheorem 7.4 and prove a unique factorization theorem of transitive super-modularcategories in Theorem 7.13.Throughout this paper, we tacitly use the following notations: ζ n = exp(2 πi/n ), Q n = Q ( ζ n ), and i = ζ = √−
1. A subcategory of any category is assumed to afull subcategory, unless stated otherwise.2.
Preliminaries
In this section, we recall some basic definitions and notations. The readers arereferred to [3, 27, 34] for more details.2.1.
Braided fusion categories. A fusion category is a semisimple, C -linearabelian, rigid monoidal category with finite-dimensional Hom-spaces and finitelymany isomorphism classes of simple objects including the tensor unit . For anyfusion category C , we denote by Irr( C ) the set of isomorphism classes of simpleobjects of C . When it is clear from the context, we will denote the isomorphismclass of an object X of C by the same notation X . SIU-HUNG NG, YILONG WANG, AND QING ZHANG
The Grothendieck group of C , denoted by K ( C ), admits a ring structure givenby the tensor product. More precisely, we have X ⊗ Y = P Z ∈ Irr( C ) N ZX,Y Z for any X, Y ∈ Irr( C ), where(2.1) N ZX,Y := dim C C ( X ⊗ Y, Z )are called the fusion coefficients . The collection of fusion coefficients N ZX,Y for all
X, Y, Z ∈ Irr( C ) is referred to as the fusion rules of C . The fusion matrix N X of X ∈ Irr( C ) is defined as ( N X ) Z,Y := N ZX,Y for any
Y, Z ∈ Irr( C ). The largest realeigenvalue of N X , denoted by FPdim( X ), is called the Frobenius-Perron dimensionof X . The Frobenius-Perron dimension of C is defined asFPdim( C ) := X X ∈ Irr( C ) FPdim( X ) . Let C be a fusion category. For any object X ∈ C , the left dual of X is atriple ( X ∗ , ev X , coev X ), where X ∗ is an object of C , ev X : X ∗ ⊗ X → andcoev X : → X ⊗ X ∗ are respectively the evaluation and coevaluation morphismsassociated with the left dual object X ∗ of X . A simple object X ∈ Irr( C ) is called invertible if X ⊗ X ∗ ∼ = . The pointed subcategory of C , denoted by C pt , is the fullabelian subcategory generated by the invertible objects of C . A fusion category C is called pointed if C pt = C . The adjoint subcategory of C , denoted by C ad or C (0) , isthe full abelian subcategory generated by the subobjects of X ⊗ X ∗ for any X ∈ C (cf. [28, 31]). Both C pt and C ad are fusion subcategories of C .The left duality of C can be extended to a contravariant monoidal functor ( − ) ∗ ,and so ( − ) ∗∗ defines a monoidal functor on C . A pivotal structure on a fusioncategory C is an isomorphism of monoidal functors j : id C ∼ = −→ ( − ) ∗∗ . A fusioncategory equipped with a pivotal structure is called a pivotal fusion category . If C is a pivotal fusion category, then for any X ∈ C and f ∈ End C ( X ), the (left) quantum trace of f can be defined astr j ( f ) := ev X ∗ ◦ (( j X ◦ f ) ⊗ id X ∗ ) ◦ coev X ∈ End C ( ) ∼ = C . A pivotal structure j on C is called spherical if tr j ( f ) = tr j ( f ∗ ) for any endomor-phism f of C . A spherical fusion category is a fusion category equipped with aspherical pivotal structure. When the pivotal structure is clear from the context,we will drop the subscript j . In a pivotal category C , the quantum dimension d X of any object X ∈ C is defined to be d X := tr(id X ). It has been shown in [28] thatif C is a spherical fusion category, then d X is a totally real algebraic integer for anyobject X ∈ C .The global dimension of any fusion category was introduced in [37, Def. 2.5]. If C is a spherical fusion category, its global dimension is given bydim( C ) = X X ∈ Irr( C ) d X . In particular, dim( C ) is a totally positive algebraic integer. We will denote thepositive square root of dim( C ) by p dim( C ).A braiding on a fusion category C is a natural isomorphism β X,Y : X ⊗ Y ∼ = −→ Y ⊗ X satisfying the Hexagon axioms. A fusion category equipped with a braiding is calleda braided fusion category . Let C be a braided fusion category, and D ⊂ C a collection
ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 5 of objects of C . The M¨uger centralizer of D in C (cf. [38]), denoted by C C ( D ), isthe full subcategory of C with the collection of objects given by { X ∈ C | β Y,X ◦ β X,Y = id X ⊗ Y , ∀ Y ∈ D} . It follows directly from the definition of a braiding that C C ( D ) is a fusion subcat-egory of C . In particular, the fusion subcategory C C ( C ) is called the M¨uger center of C , and is denoted by C ′ . A braided fusion category C (or its braiding β ) is called nondegenerate if C ′ is equivalent to Vec, the category of finite-dimensional vectorspaces over C . A braided fusion category is called a symmetric fusion category if C ′ = C . By Deligne’s theorems [20, 21], if C is a symmetric fusion category, thendim( C ) ∈ Z .2.2. Modular categories and arithmetic invariants. A premodular category(or a ribbon fusion category) is a spherical braided fusion category. A modularcategory C is a premodular category whose underlying braiding β is nondegenerate.The (unnormalized) S-matrix of a premodular category C is defined to be S X,Y := tr( β Y,X ∗ ◦ β X ∗ ,Y ) , X, Y ∈ Irr( C ) . In particular, S X, = S ,X = d X . It has been proved in [38] that a premodularcategory is modular if and only if its S-matrix is invertible. Moreover, when C ismodular, the fusion coefficients can be expressed in terms of the S-matrix by theVerlinde formula (see, for example, [3]):(2.2) N ZX,Y = 1dim( C ) X W ∈ Irr( C ) S X,W S Y,W S Z ∗ ,W S ,W . Let C be a modular category. A natural isomorphism θ : id C ∼ = −→ id C , called the ribbon structure of C , can be defined using the spherical pivotal structure of C andthe Drinfeld isomorphism (cf. [41, Sec. 2]). The ribbon structure is compatible withthe braiding and the duality in the following sense:(2.3) θ X ⊗ Y = ( θ X ⊗ θ Y ) ◦ β Y,X ◦ β X,Y and θ X ∗ = ( θ X ) ∗ for any objects X, Y ∈ C . If X ∈ Irr( C ), then θ X is a nonzero scalar multiple ofid X . We will use the abuse notation to denote both this scalar and the isomorphismitself by θ X whenever X is simple. The T-matrix of C is defined to be the diagonalmatrix T X,Y := δ X,Y θ X , X, Y ∈ Irr( C ) . It follows from [54] (see also [3, Thm. 3.1.19]) that θ X has finite order for any X ∈ Irr( C ), and so does the T-matrix. The pair of matrices ( S , T ) is called the (unnormalized) modular data of C . We may denote the modular data of a modularcategory C by ( S C , T C ) when the context needs to be clarified.For any m ∈ Z , the m -th Gauss sum [44] of a modular category C is defined as τ m ( C ) := X X ∈ Irr( C ) d X θ mX . If gcd( m, ord( T )) = 1, the m -th (multiplicative) central charge and the m -th anom-aly of C are defined as(2.4) ξ m ( C ) := τ m ( C ) | τ m ( C ) | and α m ( C ) := ξ m ( C ) . It is well-known that | τ ( C ) | = p dim( C ), and ξ m ( C ) is a root of unity (cf. [3, 38, 44]). SIU-HUNG NG, YILONG WANG, AND QING ZHANG
Galois actions on modular categories.
Let C be a modular category withmodular data ( S , T ). We denote by Q ( A ) the field extension of Q by adjoining theentries of a complex matrix A . It has been proved in [43] that Q ( S ) ⊂ Q ( T ) = Q N ,where N = ord( T ). In particular, Q ( S ) is an abelian extension over Q , and itsGalois group is denoted by G C . It is immediate to see that Q ( S ) = Q ( S X,Y /d Y | X, Y ∈ Irr( C )) . By the Verlinde formula, for any Y ∈ Irr( C ), the assignment χ Y : Irr( C ) → C , X S X,Y d Y defines a character of the fusion ring K ( C ), and { χ Y | Y ∈ Irr( C ) } is the set ofirreducible characters of K ( C ) (cf. [3]). Thus, for any σ ∈ G C , σ ( χ Y ) = χ ˆ σ ( Y ) forsome permutation ˆ σ on Irr( C ), and the map G C → Sym(Irr( C )) , σ ˆ σ is a group monomorphism. The set of orbits under this G C -action is abbreviated asOrb( C ). We will denote any automorphism σ ∈ G C and its associated permutationon Irr( C ) by ˆ σ .For any Galois extension E over Q ( S ), the Galois group Gal( E/ Q ) acts onIrr( C ) via the restriction on Q ( S ) or the surjection Gal( E/ Q ) res −−→ G C . Therefore,the Gal( E/ Q )-orbits in Irr( C ) are identical to the G C -orbits. In this convention,for any σ ∈ Gal( E/ Q ), we use ˆ σ C to represent the restriction of σ on Q ( S ) andalso its permutation on Irr( C ). When it is clear from the context, ˆ σ C will simply bedenoted by ˆ σ . In particular, one can take E = ¯ Q and so the absolute Galois groupGal( ¯ Q )-acts on Irr( C ). According to [19], for any σ ∈ Gal( ¯ Q ), X, Y ∈ Irr( C ), wehave(2.5) σ S X,Y p dim( C ) ! = ± S ˆ σ ( X ) ,Y p dim( C ) . If C = A ⊠ B for some modular categories A and B , then Irr( C ) = Irr( A ) × Irr( B )under the identification X ⊠ Y ( X, Y ). In this case, S C = S A ⊗ S B , the Kroneckerproduct of the S-matrices. Therefore, Q ( S C ) = Q ( S A ) Q ( S B ), the composite field of Q ( S A ) and Q ( S B ). Let F = Q ( S A ) ∩ Q ( S B ), and L = Gal( F / Q ). The restrictionson F define two epimorphisms res A : G A → L and res B : G B → L . Their fiberproduct G A • G B , defined as G A • G B := { (ˆ σ, ˆ τ ) ∈ G A × G B | res A (ˆ σ ) = res B (ˆ τ ) } , satisfies the commutative diagram: G A • G B p A / / p B (cid:15) (cid:15) G A res A (cid:15) (cid:15) G B res B / / L where p A , p B are coordinate projections. By the universal property of the fiberproduct, the restriction epimorphisms π A : G C → G A and π B : G C → G B induce agroup homomorphism(2.6) f : G C → G A • G B , f (ˆ σ C ) = (ˆ σ A , ˆ σ B ) ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 7 for any ˆ σ C ∈ G C . It follows from [24, Prop. 14.4.21] that f is an isomorphism.This proves the first part of statement (i) of the following lemma. The second partfollows directly from [24, Cor. 14.4.20]. Lemma 2.1.
Let C = A ⊠ B for some modular categories A , B , and let F = Q ( S A ) ∩ Q ( S B ) . Then: (i)
The map f : G C → G A • G B , f (ˆ σ C ) = (ˆ σ A , ˆ σ B ) , defines an isomorphism ofgroups, and (2.7) | G C | = | G A | · | G B | [ F : Q ] . (ii) For any σ ∈ Gal( ¯ Q ) , X ∈ Irr( A ) and Y ∈ Irr( B ) , we have ˆ σ C ( X ⊠ Y ) = ˆ σ A ( X ) ⊠ ˆ σ B ( Y ) . (iii) For any O A ∈ Orb( A ) and O B ∈ Orb( B ) , G C acts on O A × O B , under theidentification of Irr( C ) = Irr( A ) × Irr( B ) , and the number of G C -orbits in O A × O B is bounded by [ F : Q ] . In particular, the numbers of Galois orbitsof these categories satisfy | Orb( A ) | · | Orb( B ) | ≤ | Orb( C ) | ≤ | Orb( A ) | · | Orb( B ) | · [ F : Q ] . Proof.
The equality (2.7) follows directly from [24, Cor. 14.4.20] and the definitionof G C .The action of G C on Irr( C ) is equivalent to the action of G A • G B on Irr( A ) × Irr( B )by the definition of the Galois group actions. The statement (ii) follows immediatelyfrom this observation.To prove (iii), consider any X ∈ O A and Y ∈ O B . By definition,Stab G C ( X ⊠ Y ) ⊂ Stab G A ( X ) × Stab G B ( Y ) . Therefore, by Burnside’s lemma (see, for example, [24, Ex. 18.3.8]) and (2.7), wehave 1 ≤ number of G C -orbits in O A × O B = 1 | G C | X ( X,Y ) ∈ O A × O B | Stab G C ( X ⊠ Y ) |≤ | G C | X ( X,Y ) ∈ O A × O B | Stab G A ( X ) | · | Stab G B ( Y ) | = | G A | · | G B || G C | = [ F : Q ] . Now, we establish the last inequalities by summing over all O A × O B ∈ Orb( A ) × Orb( B ). (cid:3) Unique factorization of transitive modular categories
In this section, we introduce the definition of transitive modular categories.These modular categories have spectacular properties which provide the founda-tions for the classification. We prove in Theorem 3.11 that every fusion subcategoryof a transitive modular category is a transitive modular subcategory and the primefactorization of a transitive modular category is unique up to permutation primefactors.
SIU-HUNG NG, YILONG WANG, AND QING ZHANG
Definition 3.1.
A modular category C is said to be transitive if G C (or Gal( ¯ Q ))acts transitively on Irr( C ), i.e., | Orb( C ) | = 1.Recall that a transitive subgroup G of the symmetric group S n is called regular if the G -action on { , . . . , n } is fixed-point free (cf. [58]). Proposition 3.2. If C is a transitive modular category, then G C is regular and | G C | = | Irr( C ) | .Proof. Since C is a transitive modular category, G C is an abelian transitive subgroupof Sym(Irr( C )). By [58, Prop. 4.4], G C is regular. In particular, | G C | = | Irr( C ) | . (cid:3) Since G C is regular, for any X ∈ Irr( C ), there is a unique ˆ σ ∈ G C such that X = ˆ σ ( ). Therefore, we simply identify G C with Irr( C ) via the identificationˆ σ ˆ σ ( ). For convenience, we will use and ˆid interchangeably. In particular,the action of ˆ σ on ˆ µ is equal to the product ˆ σ ˆ µ for any ˆ σ, ˆ µ ∈ G C .Thus, for any transitive modular category C , its modular data can be indexed by G C . Moreover, the S-matrix can be expressed in terms of the dimensions of simpleobjects as in the following lemma. Lemma 3.3.
Let C be a transitive modular category. For ˆ σ, ˆ µ ∈ Irr( C ) , we have (3.8) S ˆ σ, ˆ µ = ˆ σ ( d ˆ µ ) d ˆ σ = ˆ µ ( d ˆ σ ) d ˆ µ . Consequently, all the entries of the S-matrix are totally real algebraic units, andevery simple object of C is self-dual.Proof. Recall from Section 2.3 thatˆ µ (cid:18) S ˆ σ, d (cid:19) = S ˆ σ, ˆ µ d ˆ µ , so we have S ˆ σ, ˆ µ = ˆ µ ( d ˆ σ ) d ˆ µ . Since S is symmetric, we also have S ˆ σ, ˆ µ = ˆ σ ( d ˆ µ ) d ˆ σ .According to [11, Prop. 3.6], d ˆ σ = d ˆ σ ( ) is an algebraic unit for all ˆ σ ∈ G C . Sinceboth d ˆ σ and d ˆ µ are totally real (cf. [28]), S ˆ σ, ˆ µ is a totally real unit. In particular,the matrix s = √ dim( C ) S is a unitary real symmetric matrix, and so we haveid = s = C , where C X,Y = δ X,Y ∗ is the charge conjugation matrix (cf. [3, 28]).Therefore, every simple object of C is self-dual. (cid:3) Corollary 3.4.
Let C be a transitive modular category. Then there exists a uniqueelement ˆ σ ∈ G C such that ˆ σ ( d ˆ µ ) = FPdim(ˆ µ ) for all ˆ µ ∈ G C , and ˆ σ (dim( C )) = FPdim( C ) . Proof.
Since the Frobenius-Perron dimension defines a character of the fusion ring K ( C ) and the simple objects are in one-to-one correspondence to the characters ofthe fusion ring (see Section 2.3), there exists a unique simple object ˆ σ ∈ G C suchthat χ ˆ σ (ˆ µ ) = FPdim(ˆ µ ) for all ˆ µ ∈ G C . Therefore, by Lemma 3.3, we haveFPdim(ˆ µ ) = χ ˆ σ (ˆ µ ) = S ˆ µ, ˆ σ d ˆ σ = ˆ σ ( d ˆ µ ) . The second assertion follows directly from the first statement and the definitionsof dim( C ) and FPdim( C ) . (cid:3) Now, we can prove the first major observation on transitive modular categories.
Theorem 3.5.
Let C be a transitive modular category. Then: ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 9 (i)
For any ˆ σ, ˆ µ ∈ Irr( C ) , if ˆ σ = ˆ µ , then d σ = d µ . In particular, if ˆ µ = ˆid , then d µ = 1 and ˆ µ (dim( C )) = dim( C ) . (ii) If X is an invertible object in C , then X ∼ = . In particular, C pt ≃ Vec asfusion categories. (iii) Q ( S ) = Q (dim( C )) = Q ( d X | X ∈ Irr( C )) .Proof. Suppose there exist ˆ σ = ˆ µ ∈ Irr( C ) such that d σ = d µ . Then d ˆ σ = εd ˆ µ forsome ε ∈ {± } . By Lemma 3.3, for any ˆ λ ∈ Irr( C ), we have S ˆ σ, ˆ λ = ˆ λ ( d ˆ σ ) d ˆ λ = ˆ λ ( εd ˆ µ ) d ˆ λ = ε ˆ λ ( d ˆ µ ) d ˆ λ = εS ˆ µ, ˆ λ . Consequently, the rows S ˆ σ, ∗ and S ˆ µ, ∗ of S are linearly dependent, which contradictsthe invertibility of the S-matrix. This proves the first assertion of statement (i).Note that d ˆid = d = 1. Therefore, for any ˆ µ = ˆid, we have d µ = 1. In particular,up to isomorphism, there is no other invertible object in C than , which impliesstatement (ii). Moreover, by (2.5), we find(3.9) ˆ µ (cid:18) d dim( C ) (cid:19) = ˆ µ (cid:18) C ) (cid:19) = d µ dim( C ) . Hence, dim( C )ˆ µ (dim( C )) = d µ = 1 , and we have completed the proof of statement (i).Since Q (dim( C )) is a subfield of Q ( S ), it is abelian and hence Galois over Q . By(i), there is no nontrivial element of G C fixing dim( C ). By the fundamental theoremof Galois theory, Q (dim( C )) = Q ( S ). By the definition of Q ( S ), we always have theinclusions Q (dim( C )) ⊆ Q ( d ˆ µ | ˆ µ ∈ G C ) ⊆ Q ( S ) . The equality Q (dim( C )) = Q ( S ) implies Q ( S ) = Q ( d ˆ µ | ˆ µ ∈ G C ). (cid:3) Corollary 3.6. If C is a transitive modular category, then the underlying braidedfusion category has a unique pivotal structure up to isomorphism.Proof. By [11, Lem. 2.4], there is a bijective correspondence between Irr( C pt ) andisomorphism classes of pivotal structures of the underlying fusion category of C . ByTheorem 3.5 (ii), Irr( C pt ) is trivial since C is transitive. Therefore, the underlyingpivotal structure of the modular category C is the only one up to isomorphism. (cid:3) Recall that a fusion category C is called weakly integral if FPdim( C ) ∈ Z , and issaid to be trivial if it is tensor equivalent to Vec. Corollary 3.7. If C is a transitive modular category and D ⊂ C a nontrivial fusionsubcategory, then dim( D ) Z . In particular, C does not contain any nontrivialweakly integral fusion subcategories.Proof. Suppose D is a fusion subcategory of C such that dim( D ) ∈ Z . Let ˆ σ ∈ G C be the canonical element realizing the Frobenius-Perron dimension in Corollary 3.4.Then ˆ σ (dim( D )) = FPdim( D ) and hence FPdim( D ) ∈ Z . In other words, D isweakly integral. By [28, Prop. 8.27], for any ˆ µ ∈ Irr( D ), ˆ σ ( d µ ) = FPdim(ˆ µ ) ∈ Z .Therefore, d µ ∈ Z . By Lemma 3.3, d ˆ µ is a real algebraic unit for any ˆ µ ∈ G C , and so d µ = 1. However, by Theorem 3.5, this means ˆ µ = ˆid and hence Irr( D ) = { } .This proves the first statement of the corollary.Note that every weakly integral fusion category B satisfies FPdim( B ) = dim( B ) ∈ Z (cf. [28]). Therefore, if B is a weakly integral fusion subcategory of C , then B must be trivial by the preceding assertion. (cid:3) Remark 3.8.
As a consequence of Corollary 3.7, if C is a transitive modularcategory satisfying dim( C ) ∈ Z , then C is trivial. In the following, we study fusion subcategories and Deligne products of transitivemodular categories.
Corollary 3.9.
Every fusion subcategory of a transitive modular category C is amodular subcategory of C .Proof. Let D be a fusion subcategory of C . Then D is premodular with the braidingand the spherical pivotal structure inherited from C . Now consider the M¨uger center D ′ = C D ( D ) of D . It is a symmetric fusion subcategory of D and hence of C . Then,by [20, 21], there exists a finite group H such that D ′ is tensor equivalent to Rep( H ).In particular, dim( D ′ ) = | H | is an integer. By Corollary 3.7, D ′ is equivalent Vecas a fusion category. Therefore, D is modular. (cid:3) Example 3.10.
The quantum group modular category C = C ( sl , (0) is a Fi-bonacci modular category with two isomorphism classes of simple objects, say and τ , such that τ ⊗ τ = ⊕ τ (cf. [48]). The S-matrix of C is given by S = (cid:18) d τ d τ − (cid:19) where d τ = √ . Therefore, Q ( S ) = Q ( √
5) and G C ∼ = Z with the generatorˆ σ : √
7→ −√
5. Therefore, C is transitive.Recall that a modular category C is prime if every modular subcategory is equiv-alent to C or Vec. By [39, Thm. 4.5], every modular category admits a primefactorization , i.e., it is equivalent to a finite Deligne product of prime modularcategories. Theorem 3.11.
Let C be a transitive modular category. Then: (i) Every fusion subcategory of C is a transitive modular subcategory, and (ii) the prime factorization of C is unique up to permutation of factors.Proof. Let A be an arbitrary fusion subcategory of C . It follows from Corollary3.9 that A is a modular subcategory of C . By the double centralizer theorem [39,Thm. 4.2], B = C C ( A ), the M¨uger centralizer of A in C , is modular. Moreover,there is an equivalence of modular categories C ≃ A ⊠ B . As noted in Section 2.3, for any X ⊠ Y ∈ Irr( C ) = Irr( A ⊠ B ) and any σ ∈ Gal( ¯ Q ),we have ˆ σ C ( X ⊠ Y ) = ˆ σ A ( X ) ⊠ ˆ σ B ( Y ). Since C is transitive, for any X ∈ Irr( A ),there exists σ ∈ Gal( ¯ Q ) such thatˆ σ A ( A ) ⊠ ˆ σ B ( B ) = ˆ σ C ( A ⊠ B ) = X ⊠ B . Therefore, G A acts transitively on Irr( A ). This completes the proof of (i). ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 11
It follows from [39, Thm. 4.5] that C admits a prime factorization. By Corollary3.7, C pt ≃ Vec . Therefore, by [17, Prop. 2.2], the prime factorization of C is uniqueup to permutation of factors. (cid:3) Proposition 3.12. If A , B are transitive modular categories and C = A ⊠ B , then (3.10) | Orb( C ) | = | G A | · | G B || G C | = [ Q (dim( A )) ∩ Q (dim( B )) : Q ] . In particular, if A , B are nontrivial modular categories and they are Galois conjugateto each other, then A ⊠ B is not transitive.Proof. Since A , B are transitive, as discussed at the beginning of this section, theaction of G A (resp. G B ) on Irr( A ) = G A (resp. G B = Irr( B )) is just the leftmultiplication. By Lemma 2.1, G A • G B ∼ = G C is a subgroup of G A × G B , andthe action of G C on Irr( C ) = Irr( A ) × Irr( B ) = G A × G B is equivalent to the leftmultiplication by G A • G B . Therefore, the orbits of this G A • G B -action are thecosets of G A • G B in G A × G B , which implies the first equality in (3.10). The secondequality in (3.10) is a direct application of Lemma 2.1(i) and Theorem 3.5(iii).Suppose A and B are nontrivial Galois conjugate modular categories. Then Q (dim( A )) = Q (dim( B )) and Q (dim( A )) is a proper extension of Q (cf. Theorem3.5). Therefore, we have | Orb( A ⊠ B ) | = [ Q (dim( A )) ∩ Q (dim( B )) : Q ] = [ Q (dim( A )) : Q ] > , which means A ⊠ B is not transitive. This completes the proof of the last assertion. (cid:3) The following corollary provides a necessary and sufficient condition for transi-tivity of a Deligne product.
Corollary 3.13.
Let C , D be modular categories. Then C ⊠ D is transitive if andonly if the following two conditions hold: both C , D are transitive and Q (dim( C )) ∩ Q (dim( D )) = Q . Proof. If C ⊠ D is transitive, then, by Theorem 3.11, C and D are also transitive.By Proposition 3.12, 1 = | Orb( C ⊠ D ) | = [ Q (dim( C )) ∩ Q (dim( D )) : Q ], and so Q (dim( C )) ∩ Q (dim( D )) = Q .Conversely, assume C and D are transitive modular categories and Q (dim( C )) ∩ Q (dim( D )) = Q , then [ Q (dim( C )) ∩ Q (dim( D )) : Q ] = 1. It follows from Proposi-tion 3.12 that C ⊠ D is transitive. (cid:3) Primality of transitive quantum group modular categories
A quantum group modular category C ( g , k ) can be constructed from a simple Liealgebra g and a positive integer k , which is called the level . This modular categoryis a semisimplification of the tilting module category of the quantum group U q ( g )specialized at a root of unity q determined by k and g . The readers are referred to[3, 49] and the references therein for details.In this paper, we focus on the cases when g = sl . Let k a positive integer and q = exp (cid:16) πik +2 (cid:17) . For any r ∈ Q , we define q r := exp (cid:18) πirk + 2 (cid:19) . The quantum integer [ n ] ζ for any root of unity ζ = ± n ] ζ := ζ n − ζ − n ζ − ζ − . The isomorphism classes of simple objects of C ( sl , k ) are indexed by the integers a ∈ [0 , k ]. The modular data ( S, T ) of the modular category C ( sl , k ) is given by(cf. [3], see also [47] with a different convention)(4.11) S a,b = [( a + 1)( b + 1)] q , T a,b = δ a,b q a ( a +2) / , ≤ a, b ≤ k . One can replace q by any Galois conjugate q ′ = q l for some l relatively prime to2( k + 2) to get another modular category C ( sl , k, q l ). The simple objects of thismodular category are also indexed by the integers in [0 , k ] and its modular data isalso given by (4.11) with q replaced by q l .For the discussions of the remainder of this paper, we will simply write A k,l forthe modular category C ( sl , k, q l ) where gcd( l, k + 2)) = 1. Let V a denote theisomorphism class of the simple objects of A k,l indexed by the integer a ∈ [0 , k ].Then V is the isomorphism class of the tensor unit . The fusion rules of A k,l arethe same for any possible integer l , and they are given by (cf. [3])(4.12) N ca,b = , if | a − b | ≤ c ≤ min( a + b, k − a − b )and c ≡ a + b (mod 2) ;0 , otherwise.One can observe directly from the fusion rules that A k,l is Z -graded, where thehomogeneous component A ( j ) k,l , j ∈ { , } , is the C -linear subcategory (additively)generated by the simple objects V a satisfying a ≡ j (mod 2) for any integer a ∈ [0 , k ]. Moreover, the adjoint fusion subcategory of A k,l is A (0) k,l , which is a modularsubcategory of A k,l if and only if k is odd (cf. [6, 51]), and(4.13) Irr( A (0) k,l ) = (cid:26) V j | ≤ j ≤ k − (cid:27) . In particular, when k = 1, A (0)1 ,l is tensor equivalent to Vec , and when k = 3, A (0)3 ,l is a Fibonacci modular category (see Example 3.10).For any fusion category C , we say that a simple object X ∈ Irr( C ) tensor gener-ates C if every simple object of C is isomorphic to a summand of a tensor power of X . Lemma 4.1.
For any positive odd integer k and l ∈ ( Z / k + 2) Z ) × , every non-trivial simple object of A (0) k,l tensor generates A (0) k,l . In particular, A (0) k,l is a primemodular category.Proof. Since A (0)1 ,l is trivial, the statements are true for k = 1. We assume k ≥ k = 3, V ⊗ V = V ⊕ V ; when k > ≤ j ≤ k − , we have V j ⊗ V = V j − ⊕ V j ⊕ V j +2 . Therefore, V tensor generates A (0) k,l . Moreover, for any 1 ≤ j ≤ k − , we have2 ≤ min(4 j, k − j ), so N j, j = 1, which means V is a direct summand of V j ⊗ V j . Therefore, V j tensor generates A (0) k,l . (cid:3) ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 13
For any odd integer k and l ∈ ( Z / k + 2) Z ) × , the modular data ( S (0) , T (0) ) of A (0) k,l is indexed by j = 0 , . . . , k − , and is given by(4.14) S (0) j,m = [(2 j + 1)(2 m + 1)] q l , T (0) j,m = δ m,j q lj ( j +1) , ≤ j, m ≤ k − . It is well-known that the first central charge of A k, is given by (cf. [3]) ξ ( A k, ) = exp (cid:18) kπi k + 2) (cid:19) . By definition (see (2.4)), the first anomaly of A k, is α ( A k, ) = ξ ( A k, ) = exp (cid:18) kπi k + 2) (cid:19) . By the fusion rules, Irr(( A k, ) pt ) = { V , V k } , and ( A k, ) pt is a modular subcategoryof A k, . By [23, Cor. 3.27], C A k, (( A k, ) pt ) = A (0) k, . Therefore, A k, ≃ A (0) k, ⊠ ( A k, ) pt as modular categories by the double centralizer theorem [39, Thm. 4.2]. Conse-quently, by [44, Lemma 3.12], α ( A k, ) = α ( A (0) k, ) · α (( A k, ) pt ). Following (4.11),we have α (( A k, ) pt ) = 1 + i k − i k = i k . Therefore,(4.15) α ( A (0) k, ) = α ( A k, ) α (( A k, ) pt ) = exp (cid:18) (1 − k ) kπi k + 2) (cid:19) . In the literature, A (0) k, is often referred to as the quantum group modular category“SO(3) at level k ” or “PSU(2) at level k ”. The ribbon categories with these fusionrules for odd k are completely classified in [30, Cor. 8.2.7], with a slightly differentparametrization. Lemma 4.2.
For any positive odd integer k , the modular categories A (0) k,l , l ∈ (cid:18) Z k + 2) Z (cid:19) × form a complete list of inequivalent ribbon categories with the fusion rules of SO(3) at level k . If k + 2 = p > is a prime, each of these modular categories is equivalentto a Galois conjugate of A (0) p − , = C ( sl , p − (0) .Proof. The first part follows directly from [30, Cor. 8.2.7]. If k + 2 = p > | ( Z / p Z ) × | = p − k . Note that all Galois conjugates of A (0) k, have the same fusion rules. Hence, they are equivalent to the modular categoriesin the list. According to (4.15), α ( A (0) p − , ) = exp (cid:16) (3 − p )( p − πi p (cid:17) ∈ Q p , which is aroot of unity of order p or 2 p for p >
3. By definition, the first anomalies of theGalois conjugates of A (0) k, are the Galois conjugates of α ( A (0) k, ). Therefore, thereare at least ϕ ( p ) = p − A (0) k, ,and we are done by the first assertion. (cid:3) Now, we can show a family of these quantum group modular categories are primeand transitive in the following proposition.
Proposition 4.3.
Let p be any odd prime and l ∈ ( Z / p Z ) × . Then the modularcategory A (0) p − ,l is prime and transitive.Proof. By Lemma 4.1, A (0) p − ,l is prime. Therefore, it suffices to show that A (0) p − ,l istransitive.The underlying root of unity q l is the primitive 2 p -th root of unity. Since p isodd, Q p = Q ( q l ) = Q ( q l ), it suffices to show that Gal( Q p / Q ) acts transitively onIrr( A (0) p − ,l ).For any nonnegative integer m ≤ p − , gcd(2 m + 1 , p ) = 1. So there exists σ ∈ Gal( Q p / Q ) such that σ ( q ) = q m +1 . Thus, we have σ S (0) j, S (0)0 , ! = σ ([(2 j + 1)] q l ) = [(2 j + 1)(2 m + 1)] q l [2 m + 1] q l = S (0) j,m S (0)0 ,m for all nonnegative integer j ≤ p − . Therefore, ˆ σ ( V ) = V m and hence Gal( Q p / Q )acts transitively on Irr( A (0) p − ,l ). (cid:3) Proposition 4.4.
Let p , . . . , p ℓ > be distinct primes. For any ( l , . . . , l ℓ ) ∈ ( Z / p Z ) × × · · · × ( Z / p ℓ Z ) × , the Deligne product C = A (0) p − ,l ⊠ · · · ⊠ A (0) p ℓ − ,l ℓ is a transitive modular category.Proof. We proceed to prove the statement by induction on ℓ . The statement obvi-ously holds for ℓ = 1 by Proposition 4.3. Now we assume p , . . . , p ℓ > l , . . . , l ℓ ) ∈ ( Z / p Z ) × × · · · × ( Z / p ℓ Z ) × for some integer ℓ >
1. Bythe induction assumption, C = ⊠ ℓ − a =1 A (0) p a − ,l a is a transitive modular category. Notethat dim( C ) = ℓ − Y a =1 dim( A (0) p a − ,l a ) ∈ Q p ′ ℓ and dim( A (0) p ℓ − ,l ℓ ) ∈ Q p ℓ , where p ′ ℓ = p · · · p ℓ − . Since Q p ′ ℓ ∩ Q p ℓ = Q , it follows from Corollary 3.13 andProposition 4.3 that C ⊠ A (0) p ℓ − ,l ℓ is transitive. (cid:3) Representations of SL ( Z ) associated with modular categories In this section, we show that the representations of SL ( Z ) associated with tran-sitive modular categories are irreducible and minimal. As a consequence, the orderof the T-matrix of any nontrivial transitive modular category is square-free and itsprime factors are greater than 3.Let C be a modular category with modular data ( S, T ). We denote by GL C ( C )the group of all invertible matrices over C indexed by Irr( C ), and V C = K ( C ) ⊗ Z C with the standard basis E C = { e X | X ∈ Irr( C ) } . Note that S, T ∈ GL C ( C ),and the group GL C ( C ) acts on V C via the standard basis E C , namely A ( e Y ) = P X ∈ Irr( C ) A XY e X for any A ∈ GL C ( C ) and Y ∈ Irr( C ). We often identify GL( V C )with GL C ( C ) in this manner. ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 15
Recall that s := (cid:18) −
11 0 (cid:19) and t := (cid:18) (cid:19) are generators of the group SL ( Z ),subjected to the relations s = id and ( st ) = s , and the assignment(5.16) ¯ ρ C : SL ( Z ) → PGL( V C ) , s S, t T defines a group homomorphism (cf. [3, 53]). This projective representation ρ C canbe lifted to an ordinary representation ( ρ, V C ) such that the diagramSL ( Z ) ρ C % % ❑❑❑❑❑❑❑❑❑❑ ρ / / GL( V C ) η (cid:15) (cid:15) PGL( V C )commutes, where η : GL( V C ) → PGL( V C ) is the natural surjection. Any lifting( ρ, V C ) of ρ C , called a representation of SL ( Z ) associated with C , yields an actionof SL ( Z ) on V C given by a · e Y := ρ ( a )( e Y ) = X X ∈ Irr( C ) ρ ( a ) X,Y ( e X )for any a ∈ SL ( Z ). We call V C an SL ( Z ) -module of C throughout this pa-per. If ( ρ, V C ) is a representation of SL ( Z ) associated with C , then the pair( s, t ) := ( ρ ( s ) , ρ ( t )), called the normalized modular data , uniquely determines ρ ,and the matrices s, t are unitary and symmetric (cf. [28]). Moreover, the groupof 1-dimensional representations of SL ( Z ) acts transitively on representations ofSL ( Z ) associated with C by tensor product (cf. [22]).For any positive integer m , we denote by π m : SL ( Z ) → SL ( Z /m Z ) the naturalsurjection. We say that a representation φ : SL ( Z ) → GL r ( C ) is of level m if φ = ˜ φ ◦ π m for some representation ˜ φ : SL ( Z /m Z ) → GL r ( C ) and m = ord( φ ( t )).By [22, Thm. II], if ρ is a representation of SL ( Z ) associated with C , then ρ isof level n = ρ ( t ) and ρ ( a ) X,Y ∈ Q n for any a ∈ SL ( Z ) and X, Y ∈ Irr( C ). Inparticular, s, t are matrices defined over Q n . Thus, for σ ∈ Gal( Q n / Q ), ( σ ρ, V C ) isalso a representation of SL ( Z ) where σ ρ ( a ) = σ ( ρ ( a )) for any a ∈ SL ( Z ), and thecorresponding σ -twisted SL ( Z )-action on V C is denoted by(5.17) σ a · v = σ ρ ( a )( v )for any v ∈ V C .Let ( ρ, V C ) be a level n representation of SL ( Z ) associated with a modularcategory C . The action of the Galois group Gal( Q n / Q ) on the normalized modulardata ( s, t ) satisfies some interesting conditions as follows: for σ ∈ Gal( Q n / Q ), thereexists a sign function ε σ : Irr( C ) → {± } such that(5.18) σ ( s X,Y ) = ε σ ( X ) s ˆ σ ( X ) ,Y = ε σ ( Y ) s X, ˆ σ ( Y ) for any X, Y ∈ Irr( C )(cf. [15, 19]), and(5.19) σ ( t X,X ) = t ˆ σ ( X ) , ˆ σ ( X ) for any X ∈ Irr( C ) (cf. [22, Thm. II (iii)]). Moreover, the absolute Galois groupGal( ¯ Q ) acts on the normalized modular data via the restrictionres QQ n : Gal( ¯ Q ) → Gal( Q n / Q ) . The condition of the action of Gal( ¯ Q ) on s defines a Gal( ¯ Q )-action on V C . Let g σ ∈ GL( V C ) defined by ( g σ ) X,Y := ε σ ( X ) δ ˆ σ ( X ) ,Y . Then, (5.18) and (5.19) can be rewritten as(5.20) σ ( s ) = g σ s = sg − σ , σ ( t ) = g σ tg − σ , and the assignment φ ρ : Gal( ¯ Q ) → GL( V C ) , σ g σ , defines a group homomorphism (cf. [15]). Therefore, for any σ ∈ Gal( ¯ Q ), we have(5.21) σ ρ ( a ) = g σ ρ ( a ) g − σ for all a ∈ SL ( Z ) . In particular, ( ρ, V C ) ∼ = ( σ ρ, V C ) as representations of SL ( Z ).Now, Gal( ¯ Q ) acts on V C via the representation ( φ ρ , V C ) of Gal( ¯ Q ), namely(5.22) σ · e X := g σ ( e X ) = ε σ ( X ) e ˆ σ ( X ) for any σ ∈ Gal( ¯ Q ) and X ∈ Irr( C ). Thus, in view of [22, Thm. II (iii)] or (5.21),for any a ∈ SL ( Z ), v ∈ V C and σ ∈ Gal( ¯ Q ), we have(5.23) σ · ( a · v ) = g σ ρ ( a )( v ) = g σ ρ ( a ) g − σ g σ ( v ) = σ a · ( σ · v ) . By [22, Thm. II (iv)], if σ ( ζ n ) = ζ an for some integer a coprime to n , then(5.24) g σ = ρ ( t a st b st a s − ) , where b is an inverse of a modulo n . Therefore, the Gal( ¯ Q )-action on V C is uniquelydetermined by ρ , and every SL ( Z )-submodule of V C also inherits the action ofGal( ¯ Q ).5.1. Minimal representations of SL ( Z ) . To proceed, we set up the followingconventions. We will denote by spec( M ) the set of the eigenvalues of an linear oper-ator M on a finite-dimensional complex vector space. For any finite multiplicativeabelian group A , A := { a | a ∈ A } is a subgroup of A of order | A | / | Ω ( A ) | , where Ω ( A ) is the elementary 2-subgroupof A . In particular, for any positive integer m , Ω (Gal( Q m / Q )) is simply denotedby Ω m and we define ϕ ( m ) := (cid:12)(cid:12) (( Z /m Z ) × ) (cid:12)(cid:12) = (cid:12)(cid:12) Gal( Q m / Q ) (cid:12)(cid:12) . It is immediately seen that ϕ is a multiplicative function. Moreover, for any prime p , we have(5.25) ϕ ( p m ) = ( p − p m − if p is odd;2 m − if p = 2 and m ≥ p = 2 and m = 1 , . Suppose ( s, t ) is a normalized modular data of a modular category C . By (5.19),the assignment ( σ, ζ ) σ ( ζ ) defines a Gal( ¯ Q )-action on spec( t ), and the Gal( ¯ Q )-orbit of t X,X , for any X ∈ Irr( C ), is then given byGal( ¯ Q ) · t X,X = { σ ( t X,X ) | σ ∈ Gal( Q m / Q ) } where m = ord( t X,X ). In particular, | Gal( ¯ Q ) · t X,X | = ϕ ( m ). We denote byspec( t ) / Gal( ¯ Q ) the set of Gal( ¯ Q )-orbits of spec( t ). ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 17
Lemma 5.1.
Let ( ρ, V C ) be a representation of SL ( Z ) associated with a modularcategory C . If ( ρ | W , W ) is a subrepresentation of ( ρ, V C ) , then spec( ρ ( t ) | W ) is closedunder the action of Gal( ¯ Q ) on spec( ρ ( t )) . In particular, spec( ρ ( t ) | W ) / Gal( ¯ Q ) ⊆ spec( ρ ( t )) / Gal( ¯ Q ) , and every direct sum decomposition of ( ρ, V C ) as representations of SL ( Z ) deter-mines a partition of spec( ρ ( t )) / Gal( ¯ Q ) .Proof. For any ζ ∈ spec( ρ ( t )), B ζ = { e X | t · e X = ζe X } is a basis for the cor-responding eigenspace of ρ ( t ). Let ζ ∈ spec( ρ ( t ) | W ) and w ∈ W \ { } such that t · w = ζw . Then w is a C -linear combination of B ζ . Thus, for any σ ∈ Gal( ¯ Q ), wehave σ t · w = σ ( ζ ) w and σ − · w ∈ W by (5.24). It follows from (5.23) that t · ( σ − · w ) = σ − · ( σ t · w ) = σ ( ζ ) σ − · w, and so σ ( ζ ) ∈ spec( ρ ( t ) | W ). (cid:3) The minimal possible dimension of a SL ( Z )-submodule of V C of the precedingproposition inspires the following definition. Definition 5.2.
A level m representation ( φ, W ) of SL ( Z ) is called minimal ifdim( W ) = ϕ ( m ) andspec( φ ( t )) = { σ ( ζ lm ) | σ ∈ Gal( Q m / Q ) } for some l ∈ ( Z /m Z ) × . In this case, ( φ, W ) or the corresponding SL ( Z )-module issaid to be minimal of type l . Corollary 5.3.
Let ( ρ, V C ) be a representation of SL ( Z ) associated with a modularcategory C . If ( ρ | W , W ) is a minimal subrepresentation of ( ρ, V C ) , then ( ρ | W , W ) is irreducible.Proof. Since ( ρ, V C ) is of some level n = ord( t ), ker( ρ | W ) is a congruence subgroupof SL ( Z ). Let m be the level of ( ρ | W , W ). Since ( ρ | W , W ) is minimal, dim( W ) = ϕ ( m ) andspec( ρ ( t ) | W ) = { σ ( ζ lm ) | σ ∈ Gal( Q m / Q ) } for some l ∈ ( Z /m Z ) × . In particular, Gal( ¯ Q ) acts transitively on spec( ρ ( t ) | W ). If ( ρ | U , U ) is a nontrivialsubrepresentation of ( ρ | W , W ) and ζ ∈ spec( ρ ( t ) | U ), then the Gal( ¯ Q )-orbit of ζ isspec( ρ ( t ) | W ). Therefore,dim( U ) ≥ | spec( ρ ( t ) | W ) | = ϕ ( m ) = dim( W ) . Therefore, U = W and hence ( ρ | W , W ) is irreducible. (cid:3) The following examples are building blocks of all the minimal irreducible repre-sentations of SL ( Z ). Example 5.4.
For any odd prime p , there are precisely two inequivalent irreduciblerepresentations of SL ( Z ) of level p and dimension ϕ ( p ) = ( p − /
2, denoted by( η pj , C ϕ ( p ) ) or simply η pj ( j = ± a ∈ ( Z /p Z ) × , and set j = (cid:18) ap (cid:19) , the Legendre symbol of a modulo p . For any integers x, y ∈ [1 , ( p − / η pj ( s ) x,y = 2 i j √ p ∗ sin (cid:18) π axyp (cid:19) and η pj ( t ) x,y = δ x,y exp (cid:18) πi ax p (cid:19) where √ p ∗ = (cid:26) √ p if p ≡ , − i √ p if p ≡ . The representation type of η pj is independent of the choice of a with (cid:18) ap (cid:19) = j .The standard basis for C ϕ ( p ) is an eigenbasis of η pj ( t ) and the representation η pj isuniquely determined by spec( η pj ( t )), which is either { σ ( ζ p ) | σ ∈ Gal( Q p / Q ) } or { σ ( ζ ap ) | σ ∈ Gal( Q p / Q ) } where a is quadratic nonresidue modulo p . In particular, η p ± are level p minimal representations of SL ( Z ). Example 5.5.
The isomorphism classes of 1-dimensional representations of SL ( Z )form a cyclic group of order 12 under tensor product, and they are completelydetermined by the images of t . If x is a 12-th root of unity, we denote by χ x the 1-dimensional representation of SL ( Z ) such that χ x ( t ) = x . In particular, χ ± ζ = χ ζ ± = η ± , and the level of χ x is the order of x . Since ord( x ) |
12 and ϕ ( d ) = 1 for any positive integer d |
12, every 1-dimensional representation ofSL ( Z ) is minimal.We close this subsection with the following characterization of minimal irre-ducible representations of SL ( Z ) which extends the preceding examples in a generalsetting. Lemma 5.6.
Let ( φ, V ) be a level n irreducible representation of SL ( Z ) . If ( φ, V ) is minimal of type l , then n = d · p · · · p ℓ for some positive integer d | anddistinct primes p , . . . , p ℓ ≥ . In this case, there exist unique l ∈ ( Z /d Z ) × and l i ∈ ( Z /p i Z ) × such that ζ ln = ζ l d ζ l p · · · ζ l ℓ p ℓ and φ ∼ = χ x ⊗ η p j ⊗ · · · ⊗ η p ℓ j ℓ , where x = ζ l d and j i = (cid:18) l i p i (cid:19) . In particular, φ is uniquely determined by ζ ln up toequivalence.Proof. Let p be a prime factor of n , and m a positive integer such that n = p m · n ,where n is a positive integer not divisible by p . Set n = p m . By the ChineseRemainder Theorem, there exist irreducible representations φ i : SL ( Z ) → GL( V i )of level n i such that φ ∼ = φ ⊗ φ . Therefore, for any ω ∈ spec( φ ( t )), ω = ω · ω where ω i ∈ spec( φ i ( t )). Since ( φ, V ) is minimal of type l , ω = σ ( ζ ln ) for some σ ∈ Gal( Q n / Q ) is a primitive n -th root. Thus, ω i is primitive n i -th root for i = 1 ,
2. Note that the group µ n of n -th roots of unity is an internal direct productof µ n and µ n , the pair ( w , ω ) is uniquely determined by ω . More precisely, thereexists a unique l i ∈ ( Z /n i Z ) × such that l = l i n/n i in Z /n i Z . Then ζ ln = ζ l n · ζ l n and ω i = σ (cid:0) ζ l i n i (cid:1) ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 19 for i = 1 ,
2. As σ runs through Gal( Q n / Q ), we find { σ (cid:0) ζ l i n i (cid:1) (cid:12)(cid:12) σ ∈ Gal( Q n i / Q ) } is a subset of spec( φ i ( t )). Therefore, dim( V i ) ≥ ϕ ( n i ) and so ϕ ( n ) ≥ dim( V ) · dim( V ) ≥ ϕ ( n ) · ϕ ( n ) = ϕ ( n ) . This implies dim( V i ) = ϕ ( n i ) andspec( φ i ( t )) = { σ (cid:0) ζ l i n i (cid:1) (cid:12)(cid:12) σ ∈ Gal( Q n i / Q ) } . Thus, both φ and φ are minimal of type l and l respectively.The level p m irreducible representations of SL ( Z ) were classified by [45, 46] (seealso [26, Tbl. 1 - 8]). Since φ is an irreducible representation of level p m anddimension ϕ ( p m ), whose values are given by (5.25), we find m = (cid:26) p is odd;1 or 2 if p = 2 . In this case, φ ∼ = η p ± for p > φ is 1-dimensional if p ≤ p can be any prime factor of n , we obtain the factorization n = d · p · · · p ℓ for some positive integer d |
12 and p , . . . , d ℓ ≥ φ by φ p , then, by induc-tion, we have φ ∼ = φ d ⊗ O prime p> p | n φ p , where φ d = O prime p ≤ p | n φ p is 1-dimensional. There exist a unique integer l p (mod p ) satisfying l ≡ l p n/p (mod p ) for each odd prime divisor p of n , and a unique l ∈ ( Z /d Z ) × satisfying l ≡ l n/d (mod d ). Then, we have ζ ln = ζ l d Y prime p> p | n ζ l p p and ζ l d = φ d ( t ) . Therefore, φ d = χ ζ l d and φ p = η pj p , where j p = (cid:18) l p p (cid:19) (cf. Examples 5.4 and 5.5).Consequently, φ ∼ = χ ζ l d ⊗ O prime p> p | n η pj p . (cid:3) Characteristic 2-group of modular categories.
Let C be a modular cat-egory with the modular data ( S, T ). For any normalized modular data ( s, t ) of C , Q ( S ) ⊂ Q N ⊆ Q n , where N = ord( T ) and n = ord( t ) (cf. [43, 22]). Therestriction of the Galois automorphisms of Q n to Q ( S ) defines an epimorphismres Q n Q ( S ) : Gal( Q n / Q ) → G C of groups. Note that(5.27) ker(res Q n Q ( S ) ) = Gal( Q n / Q ( S )) ⊆ Ω n by [22, Prop. 6.7]. Definition 5.7.
Let ( s, t ) be a normalized modular data of a modular category C ,and n = ord( t ). The image of the elementary 2-subgroup Ω n of Gal( Q n / Q ) underthe restriction map res Q n Q ( S ) : Gal( Q n / Q ) → G C is called the characteristic 2-group of C , and denoted by H C . In view of (5.27), we have the exact sequence of abelian groups:(5.28) 1 → Gal( Q n / Q ( S )) incl −−→ Ω n Q n Q ( S ) −−−−→ H C → . Proposition 5.8.
The characteristic 2-group H C of C is independent of the choiceof the normalized modular data ( s, t ) of C . Moreover, if n = ord( t ) , then G C /H C ∼ = Gal( Q n / Q )Ω n . In particular, | G C | / | H C | = ϕ ( n ) .Proof. Let ( s, t ) and ( s ′ , t ′ ) be normalized modular data of C and let ( ρ, V C ) and( ρ ′ , V C ) be the corresponding representations of SL ( Z ) associated with C respec-tively. Then ρ ′ ∼ = χ ⊗ ρ for some 1-dimensional character of SL ( Z ). Since χ = 1, t ′ = xt for some 12-th root x of unity. Let m = ord( t ′ ), and l = lcm( m, n ). Then Q l = Q m ( x ) = Q n ( x ). Obviously, res Q l Q n (Ω l ) ⊆ Ω n . For any σ ∈ Ω n , there existsan extension τ ∈ Gal( Q l / Q ) such that τ | Q n = σ . Since x = 1, τ ( x ) = x . Thus, τ = id and hence τ ∈ Ω l . Therefore,res Q l Q n (Ω l ) = Ω n . By the same argument, we also haveres Q l Q m (Ω l ) = Ω m . Since the diagram Gal( Q l / Q ) res Q l Q m / / res Q l Q ( S ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ res Q l Q n (cid:15) (cid:15) Gal( Q m / Q ) res Q m Q ( S ) (cid:15) (cid:15) Gal( Q n / Q ) res Q n Q ( S ) / / G C of restriction maps is commutative, we haveres Q m Q ( S ) (Ω m ) = res Q l Q ( S ) (Ω l ) = res Q n Q ( S ) (Ω n ) . This proves the first assertion of the statement.By (5.28), we also have the following commutative diagram of abelian groupswith exact rows:1 / / Gal( Q n / Q ( S )) incl / / id (cid:15) (cid:15) Ω n Q n Q ( S ) / / incl (cid:15) (cid:15) H C / / incl (cid:15) (cid:15) / / Gal( Q n / Q ( S )) incl / / Gal( Q n / Q ) res Q n Q ( S ) / / G C / / . Therefore, G C /H C ∼ = Gal( Q n / Q ) / Gal( Q n / Q ( S ))Ω n / Gal( Q n / Q ( S )) ∼ = Gal( Q n / Q )Ω n . (cid:3) Corollary 5.9.
Let C be a modular category with the modular data ( S, T ) . If N = ord( T ) is not a multiple of , then H C = res Q N Q ( S ) (Ω N ) . ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 21
Proof.
Since 4 ∤ N , by [22, Lem. 2.2], there exists a level N representation ( ρ, V C )of SL ( Z ) associated with C . Therefore, ρ ( t ) = t has order N . Now, the resultfollows directly from Definition 5.7 of H C . (cid:3) Example 5.10.
Let A be a finite abelian group and q : A → C × a nondegeneratequadratic form. The pointed modular category C = C ( A, q ) has the S- and T-matrices given by S a,b = q ( a ) q ( b ) q ( ab ) , T a,b = δ a,b q ( a )for any a, b ∈ A .(i) If | A | is odd, then Q ( S ) = Q ( T ) = Q N , where N = ord( T ). Since | A | isodd, and so it N . Therefore, by Corollary 5.9, H C = Ω N is nontrivial.(ii) If A = h a i is a cyclic group of order 2 and and q ( a ) = ± i , then C is calleda semion category . In this case, ord( T ) = 4 and Q ( S ) = Q . Therefore, H C is trivial.Let ( ρ, V C ) be a level n representation of SL ( Z ) associated with C , and ( s, t ) thecorresponding normalized modular data. Since Ω n −−→ H C is an epimorphism ofelementary 2-groups, there exists a subgroup ˜ H C of Ω n such thatres Q n Q ( S ) : ˜ H C ∼ −→ H C is an isomorphism. Now, recall that Gal( ¯ Q ) acts on V C via the restriction mapres QQ n : Gal( ¯ Q ) → Gal( Q n / Q ). Therefore, ˜ H C acts on V C in the same way (cf. (5.22)),namely σ · e X = g σ ( e X ) = ε σ ( X ) e ˆ σ ( X ) for any σ ∈ ˜ H C . One can decompose this ˜ H C -module V C into its isotypic components V C = M χ ∈ Irr( ˜ H C ) V χ C , where Irr( ˜ H C ) denote the set of irreducible characters of ˜ H C , and V χ C the isotypiccomponent in V C corresponding to the irreducible character χ of ˜ H C . Proposition 5.11.
Let C be a modular category and ( ρ, V C ) a representation of SL ( Z ) associated with C . Then the isotypic components of V C are nonzero andpairwise inequivalent SL ( Z ) -submodules of V C . In particular, (5.29) V C = M χ ∈ Irr( ˜ H C ) V χ C is a decomposition of SL ( Z ) -modules.Proof. By (5.23), for any v ∈ V χ C , σ ∈ ˜ H C , and a ∈ SL ( Z ), σ · ( a · v ) = σ a · ( σ · v ) = χ ( σ ) a · v . Therefore, V χ C is an invariant subspace of V C under the SL ( Z )-action correspondingto the representation ( ρ, V C ) of SL ( Z ) .If { g σ | σ ∈ ˜ H C } is C -linearly independent, then P χ := 1 | ˜ H C | X σ ∈ ˜ H C χ ( σ ) g σ is a nonzero idempotent operator on V C and V χ C = P χ ( V C ). Therefore, to provethat V χ C = 0 for any χ ∈ Irr( ˜ H C ), it suffices to show the C -linearly independence of { g σ | σ ∈ ˜ H C } .Suppose P σ ∈ ˜ H C α σ g σ = 0 for some α σ ∈ C . Then X σ ∈ ˜ H C α σ σ ( s ) = X σ ∈ ˜ H C α σ g σ s = 0Therefore, P σ ∈ ˜ H C α σ σ = 0 as a function from Q ( s ) to C . Since Q ( S ) ⊆ Q ( s ), P σ ∈ ˜ H C α σ σ | Q ( S ) = 0 as a function from Q ( S ) to C . The bijectivity of res : ˜ H C → H C implies X ˆ σ ∈ H C α σ ˆ σ = 0 . Since { ˆ σ | ˆ σ ∈ H C } is linearly independent over C (cf. [24, Sec. 14.2]), α σ = 0 forall σ ∈ ˜ H C . Therefore { g σ | σ ∈ ˜ H C } is linear independent over C .Let χ, χ ′ be distinct irreducible characters of ˜ H C . Then, there exists σ ∈ ˜ H C suchthat χ ( σ ) = χ ′ ( σ ). By (5.24), g σ = ρ ( a ) for some a ∈ SL ( Z ), and the restrictions of ρ ( a ) on V χ C and V χ ′ C are the distinct scalars χ ( σ ) and χ ′ ( σ ) respectively. Therefore, V χ C and V χ ′ C are inequivalent representations of SL ( Z ). This completes the proofof the proposition. (cid:3) Proposition 5.12.
Let ( ρ, V C ) be a level n representation of SL ( Z ) associatedwith a modular category C . If ρ is irreducible, then H C is trivial, the S-matrix of C is real, and C is self-dual. Moreover, there exists X ∈ Irr( C ) such that ρ ( t ) X,X is aprimitive n -th root of unity.Proof. If ρ is an irreducible representation of SL ( Z ), the decomposition of ρ de-termined by the characteristic 2-subgroup H C must be trivial, that means H C istrivial. Let σ ∈ Gal( Q n / Q ) denote the complex conjugation. Then ˆ σ ( X ) = X ∗ for X ∈ Irr( C ). Since H C is trivial, σ | Q ( S ) = id and so X ∗ = ˆ σ ( X ) = X for X ∈ Irr( C ).Therefore, S C is real and C is self-dual.Let n = p n · · · p n ℓ ℓ be the prime factorization of n , where p , . . . , p ℓ are distinctprime factors of n . Since ρ is irreducible, by the Chinese Remainder Theorem, thereexists a level p n i i irreducible representation ( ρ i , V i ) of SL ( Z ) for i = 1 , . . . , ℓ suchthat ( ρ, V C ) ∼ = ( ρ , V ) ⊗ · · · ⊗ ( ρ ℓ , V ℓ ) . Since ( ρ i , V i ) is of level p n i i , there exists a nonzero eigenvector v i ∈ V i of ρ i ( t )with an eigenvalue ω i which is a primitive p n i i -th root of unity. Thus, ρ ( t ) has aneigenvalue ζ = ω · · · ω ℓ which is a primitive n -th root of unity. Since Irr( C ) is aneigenbasis for ρ ( t ), there exists X ∈ Irr( C ) such that ρ ( t ) X,X = ζ . (cid:3) The SL ( Z ) -modules of transitive modular categories. In this section,we show that the representations of SL ( Z ) associated with transitive modularcategories C are minimal and irreducible, and that the order of T C is odd andsquare-free.Let C be a transitive modular category, ( ρ, V C ) a level n representation of SL ( Z )associated with C , and ( s, t ) the corresponding normalized modular data. As before,the Galois group G C is identified with Irr( C ) via the bijection ˆ σ ˆ σ ( ). Then, we ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 23 have(5.30) spec( t ) = { t ˆ σ, ˆ σ | ˆ σ ∈ G C } = { σ ( ζ ) | σ ∈ Gal( Q n / Q ) } , where ζ = t , . Here, the last equality is a consequence of (5.19). Therefore,Gal( ¯ Q ) acts on spec( t ) transitively, and so every eigenvalue of t is a primitive n -throot of unity. In particular, Q ( t ) = Q n = Q ( ζ ) . Lemma 5.13.
The characteristic 2-group H C is given by H C = { ˆ σ ∈ G C | t ˆ σ, ˆ σ = t , } . Moreover, for any ˆ σ, ˆ τ ∈ H C , t ˆ σ, ˆ σ = t ˆ τ, ˆ τ if and only if ˆ σH C = ˆ τ H C . In particular,each eigenvalue of t has algebraic multiplicity | H C | .Proof. Let ζ = t , . Since Q ( ζ ) = Q n , we haveΩ n = { σ ∈ Gal( Q n / Q ) | σ ( ζ ) = ζ } = { σ ∈ Gal( Q n / Q ) | t ˆ σ, ˆ σ = ζ } . Thus, if ˆ σ ∈ H C , then there exists σ ∈ Ω n such that σ | Q ( S ) = ˆ σ , which means t ˆ σ, ˆ σ = ζ . Conversely, if ˆ σ ∈ G C such that t ˆ σ, ˆ σ = ζ , then there exists σ ∈ Gal( Q n / Q )such that σ | Q ( S ) = ˆ σ . By (5.19), σ ( t , ) = t ˆ σ, ˆ σ = ζ . Thus, σ ∈ Ω n , and henceˆ σ ∈ H C . This proves the first statement.Let ˆ σ, ˆ τ ∈ G C , and τ ∈ Gal( Q n / Q ) such that τ | Q ( S ) = ˆ τ . If t ˆ σ, ˆ σ = t ˆ τ, ˆ τ , then t ˆ σ, ˆ σ = τ ( ζ ) or ζ = τ − ( t ˆ σ, ˆ σ ) = t ˆ τ − ˆ σ, ˆ τ − ˆ σ . Therefore, ˆ τ − ˆ σ ∈ H C and so ˆ τ H C = ˆ σH C . Conversely, if ˆ τ H C = ˆ σH C , then ˆ σ = ˆ τ ˆ µ for some ˆ µ ∈ H C , and hence t ˆ σ, ˆ σ = t ˆ τ ˆ µ, ˆ τ ˆ µ = τ ( t ˆ µ, ˆ µ ) = τ ( t , ) = t ˆ τ, ˆ τ . (cid:3) Now, we can prove the major theorem of this section.
Theorem 5.14.
Let C be a nontrivial transitive modular category. Then everyrepresentation of SL ( Z ) associated with C is minimal and irreducible. Moreover,the order of the T-matrix T of C is odd and square-free, and every prime factor of ord( T ) is greater than 3.Proof. Let ( ρ, V C ) be a level n representation of SL ( Z ) associated with C , and H C the characteristic 2-group of C . By Proposition 5.11, V C admits a of SL ( Z )-moduledecomposition, V C = X χ ∈ Irr( ˜ H C ) V χ C , where ˜ H C denotes a subgroup of Ω n such that res Q n Q ( S ) : ˜ H C ∼ −→ H C is an isomor-phism. We proceed to determine V χ C for each χ ∈ Irr( ˜ H C ).Recall that the ˜ H C -action on V C is given by σ · e ˆ µ = g σ ( e ˆ µ ) = ε σ (ˆ µ ) e ˆ σ ˆ µ for any σ ∈ ˜ H C and ˆ µ ∈ G C . For any ˆ µ ∈ G C , the subspace V ˆ µ of V C spanned by { e ˆ σ ˆ µ | ˆ σ ∈ H C } is closed under this ˜ H C -action, and t acts as the scalar t ˆ µ, ˆ µ on V ˆ µ by Lemma 5.13. Since g id = id V C , the character ψ ˆ µ of ˜ H C afforded by V ˆ µ is givenby ψ ˆ µ ( σ ) = | H C | · δ σ, id for any σ ∈ ˜ H C . Therefore, as an ˜ H C -module, V ˆ µ is equivalent to the regular representation of ˜ H C .Consequently, dim( V χ ˆ µ ) = 1 for each χ ∈ Irr( ˜ H C ).Let Λ be a complete set of coset representatives of H C in G C . Then, V C = M ˆ µ ∈ Λ V ˆ µ is a decomposition of ˜ H C -modules. Therefore, V χ C = M ˆ µ ∈ Λ V χ ˆ µ for each χ ∈ Irr( ˜ H C ), and dim( V χ C ) = | G C | / | H C | = ϕ ( n ) by Proposition 5.8. Let( ρ χ , V χ C ) denote the corresponding subrepresentation of ( ρ, V C ). Thenspec( ρ χ ( t )) = { t ˆ σ, ˆ σ | ˆ σ ∈ Λ } = { σ ( t , ) | σ ∈ Gal( ¯ Q ) } by Lemma 5.13. Therefore, for any χ ∈ Irr( ˜ H C ), the level n representation ( ρ χ , V χ C )of SL ( Z ) is a minimal of type l ∈ ( Z /n Z ) × , where l is determined by ζ ln = t , .Hence, by Corollary 5.3, ( ρ χ , V χ C ) is irreducible for each χ ∈ ˜ H C .It follows from Lemma 5.6 that V χ C ∼ = V χ ′ C as SL ( Z )-modules for any χ, χ ′ ∈ Irr( ˜ H C ). In view of Proposition 5.11, ˜ H C must be trivial and so does H C . Therefore,( ρ, V C ) is minimal and irreducible, and n = d · p · · · p ℓ where d |
12 and p , . . . , p ℓ are distinct primes greater than 3. Moreover, ρ ∼ = χ ⊗ ρ ′ for some 1-dimensional representation χ and a level m = p · · · p ℓ representation( ρ ′ , V ′ ) of SL ( Z ). By tensoring ρ with the dual representation χ ∗ of χ , we find( ρ ′ , V ′ ) is equivalent to a representation of SL ( Z ) associated with C . By [22, ThmII (i)], we have ord( T ) | m |
12 ord( T )which implies ord( T ) = m since gcd( m,
12) = 1. (cid:3) Classification of transitive modular categories
In this section, we prove that a nontrivial prime and transitive modular categoriesmust be A (0) p − ,l for some prime p > l ∈ ( Z / p Z ) × . In view of Theorem 3.11,we complete the classification of transitive modular categories in Theorem 6.5. Theirreducibility of the representations of SL ( Z ) associated with transitive modularcategories is crucial to the characterization of the prime ones.We begin with some realizations of minimal irreducible representations of SL ( Z )as representations of SL ( Z ) associated with some modular categories. Lemma 6.1.
Let p > be a prime. Then every level p minimal irreducible rep-resentation of SL ( Z ) is equivalent to a representation of SL ( Z ) associated with A (0) p − ,l for some l ∈ ( Z / p Z ) × .Proof. Recall from Proposition 4.3 that A (0) p − ,l is a prime and transitive modularcategory any prime p > l ∈ ( Z / p Z ) × . Moreover, the order of the T-matrixof A (0) p − ,l is p . There exists a level p representation ( ρ, C ϕ ( p ) ) of SL ( Z ) associatedwith A (0) p − , , and we set t = ρ ( t ). Then, by Theorem 5.14, ( ρ, C ϕ ( p ) ) is a minimal ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 25 irreducible representation of SL ( Z ) of type a where ζ ap = t , . Therefore, byLemma 5.6, ( ρ, C ϕ ( p ) ) ∼ = ( η pj , C ϕ ( p ) ) , where j = (cid:18) ap (cid:19) . For any l ∈ ( Z / p Z ) × , let σ l ∈ Gal( Q p / Q ) defined by σ l ( ζ p ) = ζ lp . Since ρ ( a )is a matrix over Q p for any a ∈ SL ( Z ) (cf. [22, Thm. II]), ρ l ( a ) := σ l ( ρ ( a ))defines another level p representation of SL ( Z ), and ( ρ l , C ϕ ( p ) ) is a representationof SL ( Z ) associated with A (0) p − ,l . Since σ l ( t , ) = ζ alp , we have( ρ l , C ϕ ( p ) ) ∼ = ( η pj l , C ϕ ( p ) ) , where j l = (cid:18) alp (cid:19) . Therefore, every level p minimal irreducible representation of SL ( Z ) is equivalentto a representation of SL ( Z ) associated with A (0) p − ,l for some l ∈ ( Z / p Z ) × . Thisproves the lemma. (cid:3) Corollary 6.2.
Let n = p · · · p ℓ for some distinct primes p , · · · , p ℓ > . Thenevery level n minimal irreducible representation of SL ( Z ) is equivalent to a repre-sentation of SL ( Z ) associated a transitive modular category D = A (0) p − ,l ⊠ · · · ⊠ A (0) p ℓ − ,l ℓ for some l a ∈ ( Z / p a Z ) × , a = 1 , . . . , ℓ .Proof. Let ( φ, V ) be a level n minimal irreducible representation of SL ( Z ). ByLemma 5.6, there exists level p a minimal irreducible representation ( η p a j a , V a ) ofSL ( Z ) for each a = 1 , . . . , ℓ such that( φ, V ) ∼ = ( η p j , V ) ⊗ · · · ⊗ ( η p ℓ j ℓ , V ℓ )where V a = C ϕ ( p a ) . By Lemma 6.1, ( η p a j a , V a ) is equivalent to a representation( ρ a , V D a ) of SL ( Z ) associated with a transitive modular category D a = A (0) p a − ,l a for some l a ∈ ( Z / p a Z ) × . Let D = D ⊠ · · · ⊠ D ℓ . Then D is transitive byProposition 4.4 and ( ρ, V D ) = ( ρ , V D ) ⊗ · · · ⊗ ( ρ ℓ , V D ℓ )is a representation of SL ( Z ) associated with D . Now, we have( φ, V ) ∼ = ( ρ, V D ) . (cid:3) Theorem 6.3.
Let C be a nontrivial prime and transitive modular category. Thenthe order of the T-matrix is a prime number greater than 3.Proof. By Theorem 5.14, ord( T C ) = N is odd and has a prime factor p > N representation ( ρ, V C )of SL ( Z ) associated with C . Again, by Theorem 5.14, ( ρ, V C ) is minimal andirreducible.Suppose N is not a prime. Then N = pq for some odd square-free integer q not divisible by p and all the prime factors of q are greater than 3. In particular, ϕ ( q ) >
1. In view of Lemma 5.6, there exist minimal and irreducible SL ( Z )-representations ( φ , V ) and ( φ , V ) of levels p and q respectively such that(6.31) ( ρ, V C ) ∼ = ( φ , V ) ⊗ ( φ , V ) . It follows from Lemma 6.1 and Corollary 6.2 that there exist modular categories B , B such that ( φ i , V i ) is equivalent to a representation ( ρ i , V B i ) associated with B i and B = A (0) p − ,l for some l ∈ ( Z / p Z ) × . Note that ( ρ , V B ) ⊗ ( ρ , V B ) is a representation of SL ( Z ) associated with B = B ⊠ B and(6.32) ( ρ, V C ) ∼ = ( ρ , V B ) ⊗ ( ρ , V B ) . Let E i be the standard basis for V B i . Then, E i is an eigenbasis of ρ i ( t ) and E B = { x ⊗ x | ( x , x ) ∈ E × E } is an eigenbasis of ρ ( t ) ⊗ ρ ( t ) for V B = V B ⊗ V B . Since E C = { e X | X ∈ Irr( C ) } is an eigenbasis of ρ ( t ) = t for V C , the equivalence (6.32) implies there exists abijection Φ : Irr( C ) → E × E , which is defined as follows: for any X ∈ Irr( C ),there exists a unique pair ( x , x ) ∈ E × E satisfying( ρ ( t ) ⊗ ρ ( t ))( x ⊗ x ) = t X,X · x ⊗ x , and we define Φ( X ) := ( x , x ).Let Φ( ) = ( b , b ), and D := Φ − ( E × { b } ) ⊆ Irr( C ). Let D be the fullsubcategory of C additively generated by the simple objects whose isomorphismclasses are in D , i.e. D is a semisimple subcategory of C with Irr( D ) = D . Weproceed to show D is fusion subcategory of C .By [10, Lem. 3.17], there exists an intertwining operator U : ( ρ, V C ) → ( ρ ⊗ ρ , V B )such that for any X ∈ Irr( C ), U ( e X ) = U ( x ,x ) x ⊗ x for some scalar U ( x ,x ) = ± X ) = ( x , x ). Let s ( i ) = ρ i ( s ) for i = 1 , s = ρ ( s ). Then for any X, Y ∈ Irr( C ), we have s X,Y = s (1) x ,y s (2) x ,y U ( x ,x ) U ( y ,y ) , where Φ( X ) = ( x , x ) , Φ( Y ) = ( y , y ) ∈ E × E . By the Verlinde formula, forany X, Y ∈ D and Z ∈ Irr( C ), we have N ZX,Y = X W ∈ Irr( C ) s X,W s Y,W s Z,W s ,W = X ( w ,w ) ∈ B s (1) x ,w s (1) y ,w s (1) z ,w (cid:16) s (2) b ,w (cid:17) s (2) z ,w U ( x ,b ) U ( y ,b ) U ( z ,z ) U w ,w ) s (1) b ,w s (2) b ,w U ( b ,b ) U ( w ,w ) where Φ( X ) = ( x , b ) , Φ( Y ) = ( y , b ) , Φ( Z ) = ( z , z ) and Φ( W ) = ( w , w ).Since U w ,w ) = 1, we have N ZX,Y = U ( x ,b ) U ( y ,b ) U ( z ,z ) U ( b ,b ) X w ∈ B s (1) x ,w s (1) y ,w s (1) z ,w s (1) b ,w X w ∈ B s (2) b ,w s (2) z ,w = δ b ,z U ( x ,b ) U ( y ,b ) U ( z ,z ) U ( b ,b ) X w ∈ B s (1) x ,w s (1) y ,w s (1) z ,w s (1) b ,w , where the last equality is given by the fact that s (2) is symmetric and unitary.Therefore, N ZX,Y = 0 whenever Z D . Thus, D is closed under the tensor productof C and hence a fusion subcategory. ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 27
By Theorem 3.9, D is a modular subcategory of C . Since p >
3, we have | Irr( D ) | = | E | = ϕ ( p ) > D is nontrivial. Moreover, since C is prime, C = D and so ϕ ( p ) = | Irr( C ) | . Therefore, ϕ ( q ) = 1, a contradiction! Therefore, N is aprime. (cid:3) Now, we can prove our major theorem of this section.
Theorem 6.4.
Let C be a nontrivial transitive prime modular category. Then C is equivalent to A (0) p − ,l , for some prime p > and l ∈ ( Z / p Z ) × , as modularcategories. Moreover, the set {A (0) p − ,l | l ∈ ( Z / p Z ) × } is a complete set of inequivalent transitive prime modular categories whose T-matrices are of order p .Proof. Suppose C is a nontrivial transitive prime modular category, then by Theo-rem 6.3, ord( T C ) is a prime p >
3. It follows from [22, Lem. 2.2] that there existsa level p representation ( ρ, V C ) of SL ( Z ) associated with C . Let ( s, t ) denote thecorresponding normalized modular data ( s, t ) of C . By Theorem 5.14, ( ρ, V C ) isminimal and irreducible. In view of Lemma 6.1, there exists a modular category D = A (0) p − ,l for some l ∈ ( Z / p Z ) × and a level p representation ( ρ ′ , V D ) associatedwith D such that ( ρ, V C ) ∼ = ( ρ ′ , V D ) . Let ( s ′ , t ′ ) be the normalized modular data of D corresponding to ( ρ ′ , V D ). Recallthat Irr( D ) = { V d | d ∈ D } where D = { j | ≤ j ≤ ( p − / } . We simply write e a for the basis element e V a for V D , and the entry ρ ′ ( a ) V a ,V b as ρ ′ ( a ) a,b for any a ∈ SL ( Z ). As in the proof of Theorem 6.3, we have a bijection Φ : D → Irr( C ) bycomparing the eigenvalues of the images of t : for a ∈ D , we define Φ( a ) = X ∈ Irr( C )if ρ ( t ) X,X = ρ ′ ( t ) a,a .To simplify notations, we denote s Φ( a ) , Φ( b ) by s a,b for any a, b ∈ D . By [10,Lem. 3.17], there exists a diagonal matrix U , indexed by D , of order at most 2 suchthat s = U s ′ U .
Let x = Φ − ( ). Then for any a, b, c ∈ D , the Verlinde formula yields the equations N Φ( c )Φ( a ) , Φ( b ) = X j ∈ D s a,j s b,j s c,j s x,j = U a,a U b,b U c,c U x,x X j ∈ D s ′ a,j s ′ b,j s ′ c,j s ′ x,j . Since D is transitive, there exists σ ∈ Gal( Q p / Q ) such that ˆ σ ( V ) = V x and wesimply write ˆ σ (0) = x . Applying σ to the preceding equation, we find N Φ( c )Φ( a ) , Φ( b ) = U a,a U b,b U c,c U x,x X j ∈ D ε σ ( a ) ε σ ( b ) ε σ ( c ) ε σ ( x ) s ′ ˆ σ ( a ) ,j s ′ ˆ σ ( b ) ,j s ′ ˆ σ ( c ) ,j s ′ ,j = ± N V ˆ σ ( c ) V ˆ σ ( a ) ,V ˆ σ ( b ) . Since the fusion coefficients N Φ( c )Φ( a ) , Φ( b ) and N V ˆ σ ( c ) V ˆ σ ( a ) ,V ˆ σ ( b ) are nonnegative, we have N Φ( c )Φ( a ) , Φ( b ) = N V ˆ σ ( c ) V ˆ σ ( a ) ,V ˆ σ ( b ) for all a, b, c ∈ D . Therefore, the assignmentΦ( a ) V ˆ σ ( a ) , for a ∈ D, defines a Z + -based ring isomorphism between K ( C ) and K ( D ). By Lemma 4.2, C is equivalent to A (0) p − ,l as modular categories for some l ∈ ( Z / p Z ) × .The second statement is an immediate consequence of Lemma 4.2 and Proposi-tion 4.3. (cid:3) Finally, we establish the complete classification of nontrivial transitive modularcategories.
Theorem 6.5.
Let C be a nontrivial modular category. Then C is transitive if andonly if C is equivalent to a Deligne product ⊠ ℓa =1 A (0) p a − ,l a as modular categories forsome distinct primes p , . . . , p ℓ > and l a ∈ ( Z / p a Z ) × .Proof. If C is transitive, then by Theorem 5.14, ord( T C ) = p · · · p ℓ for some distinctprimes p , . . . , p ℓ >
3. It follows from Theorem 3.11, C can be uniquely factorizedinto a Deligne product of transitive prime modular categories up to the ordering ofthe factors. Therefore, by Theorem 6.4, C is equivalent to ⊠ ℓa =1 A (0) p a − ,l a as modularcategories for some l a ∈ ( Z / p a Z ) × .The converse of the statement follows directly from Proposition 4.4. (cid:3) In view of Theorem 6.5, nontrivial transitive modular categories C up to equiva-lence are uniquely parameterized by a pair ( n, l ) in which n = ord( T C ) is a square-free integer relatively prime to 6 and l is a congruence class in ( Z / n Z ) × , whichcan be determined by the anomaly α ( C ).7. Transitivity of super-modular categories
In this section, we investigate super-modular categories with transitive Galoisactions. We first recall the definition of super-modular categories and the Galoisgroup actions on their reduced
S-matrices.The tensor category of Z / Z -graded finite-dimensional vector spaces over C equipped with the super braiding is denoted by sVec. This braided fusion categorysVec is symmetric and it can be endowed with two inequivalent spherical struc-tures. The nontrivial simple object f ∈ sVec is a fermion that means f ⊗ f ∼ = and β f,f = − id f ⊗ f . The two inequivalent spherical structures on sVec are distinguishedby d f = ±
1. The corresponding premodular categories are respectively denoted bysVec ε with d f = ε .A premodular category C is called super-modular or super-modular category over sVec ε if C ′ is equivalent to sVec ε as premodular categories for some ε = ±
1. Let f be the transparent fermion of C . Then, for any X ∈ Irr( C ), we have d X ⊗ f = εd X and θ X ⊗ f = − εθ X by the twist equation (2.3). Hence, f ⊗ X = X . The transparentfermion f ∈ C may also be denoted by f C if the context needs to be clarified.The group Irr( C ′ ) = { , f } ∼ = Z / Z acts on Irr( C ) by tensor product. We denoteby X the Z / Z -orbit { X, f ⊗ X } of Irr( C ), and set of Z / Z -orbits by Irr( C ). Bythe above discussions, this Z / Z -action is fixed-point free, and so there exists acomplete set of representatives Π C of Irr( C ) such that ∈ Π C and Π C is closedunder taking duals. We call such a set Π C of simple objects of C essential or essential subset of Irr( C ), and simply denote Π C by Π when there is no ambiguity.In general, Irr( C ) = Π ∪ ( f ⊗ Π) and there is no canonical choice of Π unless C isa split super-modular category, i.e., C ≃ D ⊠ sVec ε as premodular categories forsome modular category D and some ε ∈ {± } . We call a super-modular category C non-split if C is not a split super-modular category. ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 29
With respect to the decomposition Irr( C ) = Π ∪ ( f ⊗ Π), the S-matrix of C admitsthe block form S = (cid:18) ˆ S d f ˆ Sd f ˆ S ˆ S (cid:19) , where ˆ S is a symmetric invertible matrix indexed by Π, called the reduced S -matrix of C . The reduced S-matrix ˆ S of C has the unitary normalization ˆ s = √ √ dim( C ) ˆ S which satisfies a Verlinde-like formula. Since C embeds into Z ( C ) as a premodularcategory, S is defined over Q N where N is the Frobenius-Schur exponent of C or theorder of the T-matrix of Z ( C ) (cf. [43]). The reduced S-matix of C will be denotedby ˆ S C when the clarification is necessary.It is immediate to see that Q ( ˆ S ) = Q ( S ) ⊆ Q N . Similar to modular categories,we define G C := Gal( Q ( S ) / Q ). By [40, Sec. 2.2], for any extension E over Q ( S )and σ ∈ Gal( E/ Q ), there exists a unique permutation ˆ σ on Π satisfying(7.33) σ ˆ S X,Y ˆ S ,Y ! = ˆ S X, ˆ σ ( Y ) ˆ S , ˆ σ ( Y ) for any X, Y ∈ Π (see also [14]). The permutation ˆ σ on Π induces a permutation onIrr( C ), namely ˆ σ ( X ) := ˆ σ ( X ) for X ∈ Π, and we denote this permutation on Irr( C )by the same notation ˆ σ . This gives rise to an action of Gal( E/ Q ) on Irr( C ) by therestriction to Q ( S ). In particular, Gal( ¯ Q ) acts on Irr( C ). Note that the action ofGal( ¯ Q ) on Irr( C ) is independent of the choices of Π.Since the group homomorphism ˆ · : G C → Sym(Irr( C )) injective, we will iden-tify G C with the image of ˆ · as for modular categories. In other words, for any σ ∈ Gal( ¯ Q ), we use ˆ σ to denote both the Galois automorphism on Q ( S ) and theassociated permutation on Irr( C ). Again, we denote the Gal( ¯ Q )-orbit of Irr( C ) byOrb( C ). Definition 7.1.
We call a super-modular category C transitive if the Gal( ¯ Q )-actionon Irr( C ) is transitive.We first derive some properties of the Galois actions on super-modular categories.The following lemma is an analog of [11, Prop. 3.6]. Lemma 7.2.
Let C be a super-modular category. Then for any σ ∈ Gal( ¯ Q ) , d X isa totally real algebraic unit for X ∈ ˆ σ ( ) .Proof. Let Π be an essential subset of Irr( C ). By [40, Lem. 2.2], for any σ ∈ Gal( ¯ Q ),we have(7.34) d σ ( ) = dim( C ) σ (dim( C )) . Since dim( C ) σ (dim( C )) has algebraic norm 1, and d ˆ σ ( ) is a totally real algebraic integer(see [28]), d ˆ σ ( ) is a a totally real algebraic unit. Now the statement follows fromthe fact that ˆ σ ( ) = { ˆ σ ( ) , f ⊗ ˆ σ ( ) } and d f ⊗ ˆ σ ( ) = d σ ( ) . (cid:3) On split transitive super-modular categories, we begin with the following lemma.
Lemma 7.3.
Let D be a modular category. Then the split super-modular category C = D ⊠ sVec ε for any ε = ± is transitive if and only if D is transitive. Proof.
We can take Π = Irr( D ). The Gal( ¯ Q )-action on Irr( C ) is equivalent to itsaction on Π, which coincides with the Gal( ¯ Q )-action on the modular category D .Therefore, the statement follows. (cid:3) Combining Lemma 7.3 and Theorem 6.5, we obtain the full classification of splittransitive super-modular categories.
Theorem 7.4.
Let C be a nontrivial split super-modular category. Then C istransitive if and only if C is equivalent to (cid:16) ⊠ ℓa =1 A (0) p a − ,l a (cid:17) ⊠ sVec ε as premodu-lar categories for some ε ∈ {± } , distinct primes p , . . . , p ℓ > and ( l , . . . , l ℓ ) ∈ ( Z / p Z ) × × · · · × ( Z / p ℓ Z ) × . (cid:3) Transitive super-modular categories have similar properties as transitive modularcategories. For example, the following lemma is parallel to Proposition 3.2.
Lemma 7.5. If C is a transitive super-modular category, then we have | G C | = | Irr( C ) | / .Proof. Since G C acts transitively on Irr( C ), G C is regular and so | G C | = | Irr( C ) | = | Irr( C ) | / . (cid:3) Therefore, for any transitive super-modular category C with an essential subsetΠ of Irr( C ), we can identify G C with Π via ˆ σ ˆ σ ( ). Under this identification, wewill simply denote f ⊗ ˆ σ by f ˆ σ for any ˆ σ ∈ G C . Now, we can compare the followingtheorem to Lemma 3.3 and Theorem 3.5. Theorem 7.6.
Let C be a transitive super-modular category with an essential subset Π of Irr( C ) . Let ˆ S be the reduced S-matrix of C indexed by G C according to thepreceding identification of G C and Π . Then: (i) For any ˆ σ, ˆ µ ∈ G C , we have ˆ S ˆ σ, ˆ µ = ˆ σ ( d ˆ µ ) d ˆ σ = ˆ µ ( d ˆ σ ) d ˆ µ . In particular, allentries of ˆ S and S are totally real algebraic units. (ii) For any ˆ σ, ˆ µ ∈ G C , if ˆ σ = ˆ µ , then d µ = d µ . In particular, if ˆ µ = , then d µ = 1 and ˆ µ (dim( C )) = dim( C ) . (iii) For any X ∈ Irr( C ) , if d X ∈ Z , then X ∈ { , f } . For any fusion subcategory D ⊂ C , if f ∈ D , then D is a super-modular category, otherwise, D is amodular category. In particular, C has no nontrivial Tannakian subcategory. (iv) Q ( ˆ S ) = Q (dim( C )) = Q ( d ˆ σ | ˆ σ ∈ G C ) .Proof. The first equality of statement (i) follows from (7.33) by setting X = ˆ µ , Y = , and the second equality follows from the fact that ˆ S is symmetric. Consequently,by Lemma 7.2, all entries of ˆ S and S are totally real algebraic units.Now we have (i) and (7.34), the proof of (ii) and (iv) are the same as that ofTheorem 3.5 (i) and (iii) by replacing S by ˆ S .For statement (iii), assume X ∈ Irr( C ) satisfies d X ∈ Z . By the above discussions,since C is transitive, there exists ˆ σ ∈ G C such that X = ˆ σ or X = f ˆ σ . In either case,we have d X = d σ ∈ Z . By Lemma 7.2, d ˆ σ is a real algebraic unit, so d X = d σ = 1.Consequently, by (ii), we have ˆ σ = . Therefore, X = or X = f .Let D ⊂ C be any fusion subcategory. Then D is a premodular subcategoryof C . Since the M¨uger center D ′ of D is a symmetric fusion subcategory of C , wehave d X ∈ Z for any X ∈ D ′ . Therefore, we have Irr( D ′ ) ⊂ { , f } and hence D is super-modular (resp. modular) if and only if f ∈ D (resp. f
6∈ D ). Finally, if D ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 31 is a Tannakian subcategory of C , then f / ∈ D and so D is modular. Therefore, D braided equivalent to Vec , and this completes the proof of the theorem. (cid:3) A super-modular category over sVec ε for some ε = ± A and B are super-modularcategories over sVec ε , then their tensor product A ⊠ sVec B = ( A ⊠ B ) A is a nondegeneratebraided fusion category over sVec, where A = A ⊠ B ⊕ f A ⊠ f B is a connected´etale algebra in A ⊠ B . It is immediate to see that dim( A ) = 2 and θ A = id A forany ε = ±
1. Therefore, A ⊠ sVec B admits a spherical structure inherited from A ⊠ B by [35]. Therefore, A ⊠ sVec B is a super-modular category.Let dim A ( M ) denote the categorical dimension of any object M ∈ A ⊠ sVec B , G : A ⊠ sVec B → A ⊠ B the forgetful functor, and F : A ⊠ B → A ⊠ sVec B defined by F ( X ⊠ Y ) = ( X ⊠ Y ) ⊗ A for X ∈ A , Y ∈ B . Then, by [35], F is a surjective tensorfunctor, G is a right adjoint to F , anddim A ( F ( X ⊠ Y )) = dim A ⊠ B ( X ⊠ Y ) = dim A ( X ) dim B ( Y )for any X ∈ A and Y ∈ B . Therefore, F : A ⊠ B → A ⊠ sVec B preserves the sphericalstructures (cf. [42]).Since f A ⊠ f B acts freely on Irr( A ⊠ B ), F ( X ⊠ Y ) is simple for any X ∈ Irr( A )and Y ∈ Irr( B ). The transparent fermion of A ⊠ sVec B is given by(7.35) F ( f A ⊠ B ) ∼ = f A ⊠ B ⊕ A ⊠ f B ∼ = F ( A ⊠ f B )and dim A ( F ( f A ⊠ B )) = dim A ( f A ) = ε . Therefore, A ⊠ sVec B is a super-modular category over sVec ε . This proves the firststatement of the following lemma. Lemma 7.7.
Let A and B be super-modular categories over sVec ε for some ε ∈{± } . Then: (i) C := A ⊠ sVec B is a super-modular category over sVec ε , Irr( C ) = { F ( X ⊠ Y ) | ( X, Y ) ∈ Irr( A ) × Irr( B ) } , and dim A ( F ( X ⊠ Y )) = d X d Y for any X ∈ A and Y ∈ B . (ii) Let Π A and Π B be essential subsets of Irr( A ) and Irr( B ) respectively. Then Π C = { F ( X ⊠ Y ) | ( X, Y ) ∈ Π A × Π B } is an essential subset of Irr( C ) . Moreover, the corresponding reduced S-matrix ˆ S C of C is given by the Kronecker product ˆ S C = ˆ S A ⊗ ˆ S B .Proof. We continue the preceding discussion for the proof (ii). For any X ∈ Irr( A )and Y ∈ Irr( B ), we have GF ( X ⊠ Y ) ∼ = X ⊠ Y ⊕ ( X ⊗ f A ) ⊠ ( Y ⊗ f B ) . Therefore, for any (
X, Y ) = ( X ′ , Y ′ ) ∈ Irr( A ) × Irr( B ), F ( X ⊠ Y ) ∼ = F ( X ′ ⊠ Y ′ ) if and only if X ′ ⊠ Y ′ ∼ = ( X ⊗ f A ) ⊠ ( Y ⊗ f B ) . For any (
X, Y ) , ( X ′ , Y ′ ) ∈ Π A × Π B , X ′ ⊠ Y ′ = ( X ⊗ f A ) ⊠ ( Y ⊗ f B ) by the definitionof an essential subset. Since F is a tensor functor, C ∈ Π C and Π C is closed undertaking dual. It follows from (7.35) thatIrr( C ) = Π C ∪ ( f C ⊗ Π C )where f C = F ( A ⊠ f B ) . Therefore, Π C is an essential subset of Irr( C ).By [35, Thm. 4.1], for any ( X, Y ) , ( X ′ , Y ′ ) ∈ Π A × Π B ,dim( A )( S C ) X ⊠ Y ,X ′ ⊠ Y ′ = ( S A ⊠ B ) X ⊠ Y,X ′ ⊠ Y ′ + ( S A ⊠ B ) X ⊠ Y, ( f A ⊗ X ′ ) ⊠ ( f B ⊗ Y ′ ) = 2( S A ) X,X ′ ( S B ) X ′ ,Y ′ where X ⊠ Y = F ( X ⊠ Y ). Since dim( A ) = 2, we have( S C ) X ⊠ Y ,X ′ ⊠ Y ′ = ( S A ) X,X ′ ( S B ) X ′ ,Y ′ , which is equivalent to ˆ S C = ˆ S A ⊗ ˆ S B . (cid:3) Recall the definition of the fiber product in Section 2.3.
Corollary 7.8.
Let A , B be super-modular categories over sVec ε for some ε = ± with essential subsets Π A and Π B of simple objects of A and B respectively. Let C := A ⊠ sVec B and F = Q ( S A ) ∩ Q ( S B ) . Then: (i) The map g : G C → G A • G B , g (ˆ σ C ) = (ˆ σ A , ˆ σ B ) , defines an isomorphism ofgroups, and | G C | = | G A | · | G B | [ F : Q ] . (ii) For any σ ∈ Gal( ¯ Q ) , X ∈ Π A and Y ∈ Π B , we have ˆ σ C ( F ( X ⊠ Y )) = F (ˆ σ A ( X ) ⊠ ˆ σ B ( Y )) . (iii) | Orb( A ) | · | Orb( B ) | ≤ | Orb( C ) | ≤ | Orb( A ) | · | Orb( B ) | · [ F : Q ] . (iv) If A and B are transitive, then | Orb( C ) | = | G A | · | G B || G C | = [ Q (dim( A )) ∩ Q (dim( B )) : Q ] . Proof.
In view of Lemma 7.7, by replacing Irr( A ) , Irr( B ) , Irr( C ) respectively withΠ A , Π B , and the associated Π C , the statements (i)-(iii) can be proved in the sameway as Lemma 2.1, and the proof of (iv) is similar to that of Proposition 3.12. (cid:3) Proposition 7.9.
Let C be a super-modular category over sVec ε for some ε = ± .If A is a super-modular subcategory of C , then both A and its M¨uger centralizer B = C C ( A ) are super-modular categories over sVec ε , and there is an equivalence ofpremodular categories over sVec , (7.36) C ≃ A ⊠ sVec B . Proof.
It is clear that A is a super-modular over sVec ε . Note that C ′ is a premodularsubcategory of B , which is a nondegenerate braided fusion category over sVec by[18, Prop. 4.3]. Therefore, by Lemma 7.7, B and A ⊠ sVec B are super-modular categoriesover sVec ε .By [18, Prop. 4.3], there exists a braided tensor equivalence A ⊠ sVec B ≃ C over sVec.In fact, the tensor product functor ⊗ : A ⊠ B → C , X ⊠ Y X ⊗ Y , for any X ∈ A and Y ∈ B , defines an essentially surjective braided tensor functor. This braided ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 33 tensor functor descends to a braided tensor equivalence ⊗ : A ⊠ sVec B ∼ −→ C over sVec,which satisfies the commutative diagram(7.37) A ⊠ B CA ⊠ sVec B ⊗ F ⊗ . By Lemma 7.7, any simple object in A ⊠ sVec B is isomorphic to F ( X ⊠ Y ) for some( X, Y ) ∈ Irr( A ) × Irr( B ) anddim A ( F ( X ⊠ Y )) = d X d Y = d X ⊗ Y = d ⊗ ( F ( X ⊠ Y )) . Therefore, ⊗ preserves spherical structures, and hence is an equivalence of premod-ular categories. (cid:3) Corollary 7.10.
Let C be a transitive super-modular category. Then any fusionsubcategory of C is transitive modular or super-modular.Proof. By Theorem 7.6 (iii), any fusion subcategory
A ⊂ C is either modular orsuper-modular. Assume first that A is super-modular. In view of Proposition 7.9and Corollary 7.8, the proof of transitivity of A is the same as that of Theorem 3.11with the sets Irr( A ) , Irr( B ) and Irr( C ) of irreducible objects replaced by essentialsets of simple objects Π A , Π B and the corresponding Π C . Now, we assume A ismodular. Then D := A ∨ C ′ is a super-modular subcategory of C . By the abovediscussions, D is transitive. Therefore, by Lemma 7.3, A is transitive. (cid:3) Corollary 7.11.
Let A , B be super-modular over sVec ε for some ε = ± . Then A ⊠ sVec B is transitive if and only if the following two conditions hold: both A , B aretransitive, and Q (dim( A )) ∩ Q (dim( B )) = Q .Proof. Let C = A ⊠ sVec B be transitive. Then both A and B are transitive by Corollary7.10. Therefore, by Corollary 7.8 (iv), we have | Orb( C ) | = [ Q (dim( A )) ∩ Q (dim( B )) : Q ] = 1 , and so Q (dim( A )) ∩ Q (dim( B )) = Q .Conversely, it follows immediately from Corollary 7.8 (iv) that if A and B aretransitive, and Q (dim( A )) ∩ Q (dim( B )) = Q , then C is transitive. (cid:3) The following definition generalizes the primality of modular categories.
Definition 7.12.
Let E be a symmetric fusion category, and C a nondegeneratebraided fusion category C over E . We say that C is E -prime if it has no nondegener-ate braided fusion subcategory over E except E and C . An E -prime braided fusioncategory is called E -simple if it is not pointed. For E = sVec, we simply use theterms s-prime and s-simple instead of sVec-prime and sVec-simple.Note the definition of E -simple categories is consistent with the definition ofs-simple categories introduced in [18]. We will call a super-modular category triv-ial if it is braided equivalent to sVec. In particular, sVec ± are trivial. In viewof Theorem 7.6 (iii), nontrivial s-prime transitive super-modular categories are s-simple. Now we can state and prove the prime decomposition theorem for transitivesuper-modular categories (cf. Theorem 3.11). Theorem 7.13.
Let C be a nontrivial transitive super-modular category over sVec ε for some ε = ± . Then (7.38) C ≃ C ⊠ sVec · · · ⊠ sVec C m , as premodular categories, where C , . . . , C m form the complete list of inequivalent s -simple subcategories of C . Moreover, such factorization into s-simple super-modularcategories over sVec ε of C is unique up to permutation of factors.Proof. By Theorem 7.6 (iii), C has no Tannakian subcategory other than Vec and C pt = C ′ ≃ sVec ε . According to [18, Thm. 4.13] (i) and Proposition 7.9 C ≃ C ⊠ sVec · · · ⊠ sVec C m as premodular categories for some s-simple subcategories C , . . . , C m of C . It followsfrom Corollary 7.11 that C , ..., C m are transitive and Q (dim( C i )) ∩ Q (dim( C ) / dim( C i )) = Q for any i = 1 , . . . , m . In particular, these s-simple super-modular subcategories of C have distinct global dimensions. According to [18, Thm. 4.13] (ii), C , · · · , C m are all the s-simple super-modular subcategories of C . Thus, if C ≃ D ⊠ sVec · · · ⊠ sVec D n as premodular categories for some s-simple super-modular categories D , . . . , D n over sVec ε , then they are equivalent to a complete list of inequivalent s-simplesuper-modular subcategories of C . Therefore, m = n and the statement follows. (cid:3) Now, we demonstrate a family of transitive non-split super-modular categoriesderived from quantum group modular categories.According to [7], for any k ≥ l ∈ ( Z / k + 1) Z ) × , the category C = A (0)4 k +2 ,l (see Section 4) is a super-modular with Irr( C ) = { V j | ≤ j ≤ k +1 } . The fermionof C is V k +2 . By the fusion rules (4.12), we have V j ⊗ V k +2 = V k +2 − j . In thefollowing discussions, we chooseΠ = { V j | ≤ j ≤ k } . When k = 0, C is braided equivalent to sVec, and when k ≥ C is non-split. Proposition 7.14.
For any k ≥ , the super-modular category A (0)4 k +2 ,l is s-simple.Proof. First, we show that any nontrivial fusion subcategory of C is either C or C ′ .Recall that C pt = C ′ and Irr( C ′ ) = { , V k +2 } . Assume that D is a nontrivialfusion subcategory of C and D is not pointed. Then D has a simple object X whichis not invertible, and so X ∼ = V j for some 1 ≤ j ≤ k . In particular, we have4 j ≥
4, and 2(4 k + 2) − j ≥
4. So by the fusion rules, N j, j = 1, which means D contains V . Since V tensor generates C , we have D = C . Therefore, C is s-prime.Since k ≥ C 6 = sVec, so it is s-simple. (cid:3)
Proposition 7.15.
Let C = A (0)4 k +2 ,l for some integer k ≥ and l ∈ ( Z / k + 1) Z ) × .Then C is transitive if and only if k = 2 x − for x ≥ .Proof. Recall that the quantum parameter of C is q l = exp( lπi k +1) ), and Q ( S ) isa real subfield of Q k +1) , so | G C | divides ϕ (8( k + 1)) /
2, where ϕ is the Euler phifunction. Assume C is transitive. Then | Π | = k + 1 must divide ϕ (8( k + 1)) / ODULAR CATEGORIES WITH TRANSITIVE GALOIS ACTIONS 35
We first observe that k must be odd. Suppose k is even. Then k + 1 ≥ ϕ (8( k + 1)) / ϕ ( k + 1). Therefore, k + 1 | ϕ (8( k + 1)) / k + 1 | ϕ ( k + 1). This divisibility does not hold for any k >
0. Therefore, k must be odd.Let k + 1 = 2 x w , where x ≥ w is odd. Then ϕ (8( k + 1)) / x +1 ϕ ( w ).Since k + 1 divides ϕ (8( k + 1)) /
2, we have w | ϕ ( w ) and hence w | ϕ ( w ). This canonly happen when w = 1, or equivalently, k = 2 x − k = 2 x − x ≥ C = A (0)4 k +2 ,l . With respect to ourchoice of Π , for any 0 ≤ a, b ≤ k , we haveˆ S a, b = [(2 a + 1)(2 b + 1)] q l . Following the same argument as in the proof of Proposition 4.3, one can show that C is transitive. More precisely, since Q ( S ) ⊂ Q k +1) = Q x +3 , for any 0 ≤ j ≤ k ,we have gcd(2 j + 1 , x +3 ) = 1. So there exists σ ∈ Q x +3 such that σ ( q ) = q j +1 .Therefore, σ ˆ S i, ˆ S , ! = σ ([2 i + 1] q l ) = [(2 i + 1)(2 j + 1)] q l [2 j + 1] q l = ˆ S i, j ˆ S , j . In other words, ˆ σ ( V ) = V j , and hence C is transitive. (cid:3) In light of Theorem 6.5, it is natural to ask whether there are other non-splittransitive super-modular categories that are s-simple, and we propose the followingquestion at the end this paper.
Conjecture 7.16.
The quantum group categories in Proposition 7.15 are all thes-simple non-split transitive super-modular categories up to Galois conjugates andspherical structures.
Acknowledgements
This paper is based upon work supported by the National Science Foundationunder the Grant No. DMS-1440140 while the first and the last authors were inresidence at the Mathematical Sciences Research Institute in Berkeley, California,during the Spring 2020 semester. They would also like to thank Eric Rowell forfruitful discussions.
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Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A
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