On odd-dimensional modular tensor categories
aa r X i v : . [ m a t h . QA ] J u l ON ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES
AGUSTINA CZENKY AND JULIA PLAVNIK
Abstract.
We study odd-dimensional modular tensor categories and maximally non-selfdual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranksof modular tensor categories in terms of the rank of the adjoint subcategory and the orderof the group of invertible objects. As an application of these results, we prove that MNSDmodular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSDtensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect. Introduction
Integral modular tensor categories have been deeply studied in the last decade, see forexample [BR], [B+], [DGNO1], [DLD], [DN], [DT], [EGO], [ENO1], [ENO2], [NR]. Onelarge class of examples is given by odd-dimensional modular tensor categories [GN]. Ngand Schauenburg proved that if the dimension of a modular tensor category is odd thenthe category is maximally non-self dual (MNSD), i.e. the only self-dual simple object isthe unit object [NS]. In the other direction, Hong and Rowell showed in [HR, Theorem 2.2]that MNSD modular tensor categories are always integral, and as a consequence they mustbe odd-dimensional.In [BR], the authors studied more in detail odd-dimensional modular tensor categories. Inparticular, they asked if odd-dimensional modular tensor categories are neccesarily group-theorerical. A negative answer to this question can be deduced from results of Larsen andJordan in [JL]. Roughly speaking, they proved that integral fusion categories of dimension pq , where p and q are distinct primes, need not be group-theoretical.Bruillard and Rowell showed in [BR] that any MNSD modular tensor category of rankat most 11 is always pointed. Recall that a fusion category is called pointed if all itssimple objects are invertible, which is equivalent to requiring that all its simple objectshave Frobenius-Perron dimension equal to 1. They also found an example of a MNSDmodular tensor category of rank 25 that is not pointed (but it is group-theoretical). Anatural follow-up question is if there exists a non-pointed MNSD modular tensor categoryof rank less than 25 [BR].In this manuscript we continue the study of odd-dimensional modular tensor categories.We prove some useful relations between the ranks of the different components of a faithfulgrading of an odd-dimensional modular tensor category. Moreover, we give some boundsof the rank of the category in terms of the order of the group of invertibles, the rank ofthe adjoint subcategory, and certain prime dividing the order of the group of invertiblesof the adjoint. We apply these general results to classify MNSD modular tensor categoriesof low rank. In particular, we give a partial answer to the question mentioned above:MNSD modular tensor categories of rank 13 and 15 are pointed, and MNSD modular tensorcategories of rank at most 23 are either pointed or perfect. By a perfect fusion category,we mean a fusion category with trivial group of invertible objects. As a consequence of our results, a non-pointed MNSD modular tensor category of rankbetween 17 and 23 cannot be weakly group-theoretical. There is an important conjecturein fusion categories that states that every weakly integral fusion category is weakly group-theoretical [ENO2, Question 2]. Hence the veracity of this conjecture would imply thatMNSD modular tensor categories of ranks between 17 and 23 are pointed. Moreover, thisleads us to the following conjecture:
Conjecture 1.1.
Odd-dimensional modular tensor categories have at least one non-trivialinvertible object, i.e. they cannot be perfect.
Notice that this is equivalent to:
Conjecture 1.2.
Odd-dimensional fusion categories are solvable.
See Section 8 for more details.The paper is organized as follows. In Section 2 we introduce the basic notions and weprove some results that we will use throughout this article. In Section 3 we study modulartensor categories whose simple objects have dimension a multiple of an odd prime number p and modular tensor categories with dimension a power of p , in particular, we prove thata modular tensor category of dimension p , with p an odd prime, must be pointed. InSection 4 we investigate the entries of the S -matrix of a modular tensor category (with norestrictions on the dimension this time); specifically, we give conditions for having (or not)zeros on it. In Section 5 we show that odd-dimensional modular tensor categories withexactly one invertible object (the unit) have no non-trivial symmetric categories. This facthas strong consequences, such as that every fusion subcategory of such category is modularand therefore the category is split. In Section 6 we find bounds for the rank of a the categoryin terms of data associated to its universal grading. We also give conditions on the rank ofthe graded components in relation to the fixed points by the action of the group of invertibleobjects of the category. In Section 7 we apply the results on the previous sections to classifyMNSD modular tensor categories of rank between 13 and 23. Lastly, in Section 8 we givesome equivalent statements to Conjectures 8.1 and 8.2. Acknowledgements
The authors specially thank A. Brugui`eres for enlightening discussions. They also thankP. Bruillard, E. Campagnolo, H. Pe˜na Polastri, G. Sanmarco and L. Villagra for usefulcomments. The authors thank C. Galindo, M. , M¨uller, V. Ostrik, E. Rowell, and A.Schopieray for helpful remarks on an early draft.Substantial portions of this project were discussed at the CIMPA Research Schools onQuantum Symmetries (Bogot´a, 2019) and Hopf algebras and Tensor Categories (C´ordoba,2019) and the authors would like to thank all the various organizers and granting agenciesinvolved. Both authors were partially supported by NSF grant DMS-0932078, administeredby the Mathematical Sciences Research Institute while the first author attended a workshopand the second author was in residence at MSRI during the Quantum Symmetries programin Spring 2020. JP was partially supported by NSF grants DMS-1802503 and DMS-1917319.JP gratefully acknowledges the support of Indiana University, Bloomington, through aProvost’s Travel Award for Women in Science.
N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 3 Preliminaries
In this paper, we will always work over an algebraically closed field k of characteristiczero. We refer to [BK] and [EGNO] for the basic theory of fusion categories and braidedfusion categories, and for unexplained terminology used throughout this paper.A fusion category C over k is a k -linear semisimple rigid tensor category with a finitenumber of simple objects and finite dimensional spaces of morphisms, and such that theendomorphism algebra of the identity object (with respect to the tensor product) is k .Let C be a fusion category over k . We shall denote by O ( C ) the set of isomorphismclasses of simple objects of C . After fixing an enumeration of O ( C ), we denote the fusioncoefficients by N kij := dim Hom( X i ⊗ X j , X k ), where X i , X i , X k ∈ O ( C ). The Frobenius-Perron dimension of an object X is denoted by FPdim X (see [ENO1]). We will use thenotation c . d . ( C ) = { FPdim( X ) : X ∈ O ( C ) } . The Frobenius-Perron dimension of C willbe denoted by FPdim( C ). When C is endowed with a pivotal structure, d X denotes thequantum dimension of an object X and dim( C ) denotes the global dimension of C . Whenthe Frobenius-Perron dimension of a fusion category is an integer, it coincides with theglobal dimension (see [ENO1, Proposition 8.24]).A fusion category is pointed if all simple objects are invertible, which is equivalent tohaving c . d . ( C ) = { } . In this case C is equivalent to the category of finite dimensional G -graded vector spaces Vec ω G , where G is a finite group and ω is a 3-cocycle on G withcoefficients in k × codifying the associativity constraint. The group of isomorphism classesof invertible objects of C will be indicated by G ( C ). The largest pointed subcategory of C will be denoted by C pt , that is the fusion subcategory of C generated by G ( C ).2.1. Fusion categories.
Nilpotent, solvable and weakly group-theoretical fusion categories.
Let C be a fusioncategory. The adjoint subcategory , indicated by C ad , is the fusion subcategory of C generatedby { X ⊗ X ∗ : X ∈ O ( C ) } . The upper central series of C is the sequence of fusion subcategoriesof C defined recursively by C (0) = C and C ( n ) = (cid:0) C ( n − (cid:1) ad for all n ≥ . The fusion category C is nilpotent if its upper central series converges to the trivial fusionsubcategory Vec of finite dimensional vector spaces; that is, there exists n ∈ N ≥ such that C ( n ) = Vec [GN], [ENO2]. Fusion categories of Frobenius-Perron dimension a power of aprime integer are nilpotent [GN]. We make repeated use of this result.More generally, C is weakly group-theoretical if it is Morita equivalent to a nilpotent fusioncategory [ENO2]. The class of weakly group-theoretical fusion categories is closed undertaking extensions and equivariantizations, Morita equivalence, tensor product, Drinfeld cen-ters and subcategories [ENO2, Proposition 4.1].A fusion category C is solvable if it is Morita equivalent to a cyclically nilpotent fusioncategory. The class of solvable categories is closed under taking extensions and equivarianti-zations by solvable groups, Morita equivalent categories, tensor products, Drinfeld center,fusion subcategories and component categories of quotient categories [ENO2, Proposition4.5].For a solvable braided fusion category C , we have that either C = Vec or G ( C ) is nottrivial [ENO2, Proposition 4.5]. It is also known that braided nilpotent fusion categories aresolvable [ENO1, Proposition 4.5]. When C is nilpotent, then FPdim( X ) divides FPdim( C ad )for all X ∈ O ( C ) [GN]. A. CZENKY AND J. PLAVNIK
The universal grading.
Let G be a finite group. A G -grading on a fusion category C is a decomposition C = ⊕ g ∈ G C g , such that ⊗ : C g × C h → C gh , ∈ C e , and ∗ : C g → C g − . A G -grading is said to be faithful if C g = 0 for all g ∈ G .If C is a fusion category endowed with a faithful grading C = ⊕ g ∈ G C g , then all the com-ponents C g have the same Frobenius-Perron dimension [ENO1, Proposition 8.20]. Hence,FPdim( C ) = | G | FPdim( C e ). By [GN], any fusion category C admits a canonical faithfulgrading C = ⊕ g ∈U ( C ) C g , called the universal grading ; its trivial component is the adjointsubcategory C ad of C . If C is equipped with a braiding, then U ( C ) is abelian. Moreover, if C is modular then U ( C ) is isomorphic to the group of (isomorphism classes of) invertibles G ( C ) [GN, Theorem 6.3].2.1.3. Pseudo-unitary fusion categories.
A pivotal fusion category C is said to be pseudo-unitary if dim( C ) = FPdim( C ). This happens to be equivalent to d X = FPdim( X ) for all X ∈ O ( C ). By [EGNO, Proposition 9.6.5], weakly integral fusion categories are pseudo-unitary. We will sometimes use the Frobenius-Perron dimension and the global dimensionof C indifferently when working with pseudo-unitary categories [EGNO, Corollary 9.6.6].2.2. Modular tensor categories.
Let C be a braided fusion category endowed with abraiding σ . A twist in C is a natural isomorphism θ : Id C → Id C such that θ X ⊗ Y = ( θ X ⊗ θ Y ) ◦ σ Y,X ◦ σ X,Y , (2.1)for all X, Y ∈ C . A twist is called a ribbon structure if ( θ X ) ∗ = θ X ∗ for all X ∈ C .A pre-modular fusion category is a fusion category endowed with a compatible ribbonstructure. Equivalently, a pre-modular tensor category is a braided fusion category equippedwith a spherical structure [Br]. That is, d X = d X ∗ for all X ∈ O ( C ).In a spherical category, for an endomorphism f ∈ End C ( X ) we have a notion of trace ,which we will denote by Tr( f ) (see [EGNO, Definition 4.7.1]). Let C be a premodulartensor category, with braiding σ X,Y : X ⊗ Y ∼ −→ Y ⊗ X . The S-matrix S of C is defined by S := ( s X,Y ) X,Y ∈O ( C ) , where s X,Y = Tr( σ Y,X σ X,Y ).In a premodular tensor category C , we can obtain the entries of the S -matrix in terms ofthe twists, fusion rules, and quantum dimensions via the so-called balancing equation s X,Y = θ − X θ − Y X Z ∈O ( C ) N ZXY θ Z d Z , (2.2)for all X, Y ∈ O ( C ) [EGNO, Proposition 8.13.7].A premodular tensor category C is said to be modular if the S -matrix S is non-degenerate.2.2.1. Centralizers in braided fusion categories.
Let C be a braided fusion category and let K be a fusion subcategory of C . The centralizer K ′ of K is the fusion subcategory of C withobjects all those Y in C such that σ Y,X σ X,Y = id X ⊗ Y , for all X ∈ K [Mu] . (2.3)In particular, if C is modular then K = K ′′ and dim( K ) dim( K ′ ) = dim( C ) [Mu, Theorem3.2]; moreover, C pt = C ′ ad and C ′ pt = C ad [GN, Corollary 6.9]. A necessary and sufficientcondition for K to be modular is K ∩ K ′ = Vec. In this case K ′ is also modular and C ≃ K ⊠ K ′ as braided fusion categories [Mu].A braided fusion category is symmetric if the square of the braiding is the identity. Hence K is symmetric if and only if K ⊆ K ′ . N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 5
Remark . If C is a braided fusion category, then ( C ad ) pt is symmetric. Proof.
First note that ( C ad ) ′ = ( C ′ ) co [DGNO2, Proposition 3.25], i.e, X ∈ ( C ad ) ′ if and onlyif X ⊗ X ∗ ∈ C ′ . In particular, C pt ⊆ ( C ad ) ′ , and the claim follows since ( C ad ) pt ⊆ C pt ⊆ ( C ad ) ′ ⊆ (( C ad ) pt ) ′ . (cid:3) Given a fusion category C , we will identify the elements in G ( C ) with the invertible objectsin C pt . Remark . Let C be a modular tensor category and K be a fusion subcategory of C . Then K ∩ K ′ is symmetric.From now on let C be a pre-modular fusion category. Lemma 2.3.
Let g be an invertible object in C ad such that θ g = 1 . Suppose g ⊗ X = X for all non-invertible simple X
6∈ C ad . Then the rows of the S-matrix corresponding to (theisomorphism classes of ) g and are equal.Proof. Let g be as above. The balancing equation (2.2) yields(2.4) s g,X = θ − g θ − X d X θ X = d X , for all non-invertible simple X
6∈ C ad , that is, for all simple X such that X
6∈ C ad ∪ C pt . Now we compute s g,X for X ∈ C pt ∪ C ad . Assume first that X ∈ C ad . Since g ∈ ( C pt ) ⊆ ( C ad ) ′ , it follows from [Mu, Proposition 2.5] that(2.5) s g,X = d X . Lastly, assume X ∈ C pt . Since g ∈ ( C ad ) pt and C pt ⊆ (( C ad ) pt ) ′ , again, by [Mu, Proposition2.5], we get(2.6) s g,X = d X . Now the result follows from equations (2.4), (2.5) and (2.6). (cid:3)
For every X ∈ O ( C ) let G [ X ] = { g ∈ C : g is invertible and g ⊗ X = X } . We define(2.7) G := \ non-invertible simple X ad G [ X ] . Note that G [ X ] is a subgroup of G ( C ad ) for all simple X ∈ C , and thus so is G . Corollary 2.4. If ( C ad ) pt is isotropic, i.e. if θ g = 1 for all g ∈ ( C ad ) pt , then the rows ofthe S-matrix that correspond to the elements of G are all equal.Remark . In particular, the previous result holds if ( C ad ) pt is odd-dimensional [DGNO1,Corollary 2.7].From Corollary 2.4 we get Corollary 2.6.
Let C be a modular tensor category. If ( C ad ) pt is isotropic, then G = { } . A. CZENKY AND J. PLAVNIK Modular tensor categories with certain irreducible degrees
In this section, we study modular tensor categories whose irreducible degrees are a mul-tiple of an odd prime number p . In particular, we study modular tensor categories that areintegral and whose adjoint subcategory has dimension a power of p . We also need some moregeneral auxiliary results that we prove in this section, (see Proposition 3.4 and Lemma 3.9). Proposition 3.1.
Let C be a not pointed modular tensor category and p be an odd primenumber such that c . d . ( C ) ⊂ { }∪ p Z . If G ( C ad ) is a non-trivial p-group, then it is not cyclic.In particular, |G ( C ad ) | ≥ p .Proof. Assume G ( C ad ) is a cyclic group of order p k , for some k ∈ N . For every i ∈ { , . . . , k } denote by H i the unique subgroup of G ( C ad ) of order p i . Note that(3.1) H = { e } ⊂ H ⊂ · · · ⊂ H k . We claim that there exists 1 ≤ l ≤ k such that H l ⊆ G [ X ] for every non-invertible simpleobject X ∈ C . As p is odd, this is a contradiction by Corollary 2.6. Indeed, taking theFrobenius-Perron dimension of both sides of X ⊗ X ∗ = M g ∈ G [ X ] g ⊕ M Y ∈C ad , d Y > N YXX ∗ Y, we obtain that p divides | G [ X ] | . Thus G [ X ] has order p j X , where 1 ≤ j X ≤ k .Define l := min X j X ≥ H l of G ( C ad ). By Equation (3.1), wehave that H l ⊆ G [ X ] for every simple non-invertible X in C and then the claim follows. (cid:3) Corollary 3.2.
Let C be a not pointed modular tensor category and p an odd prime integersuch that c . d . ( C ) ⊂ { } ∪ p Z . If G ( C ) is a cyclic p-group, then G ( C ad ) is trivial. Corollary 3.3.
Let C be a not pointed modular tensor category and p an odd prime numbersuch that c . d . ( C ) ⊂ { } ∪ p Z . If C is solvable, then G ( C ) is not a cyclic p-group.Proof. This is a direct consequence of Corollary 3.2 since C ad contains a non-trivial invertibleobject because it is a non-trivial solvable category [ENO1, Proposition 4.5]. (cid:3) In this work we are mostly interested in odd-dimensional categories, but some resultshold also for p = 2. We do not assume that p is odd unless otherwise stated. Proposition 3.4.
Let C be an integral modular tensor category and p be a prime numberthat divides FPdim( C ) . Then FPdim( C ad ) = p .Proof. Assume FPdim( C ad ) = p . Then C is a non-pointed nilpotent category. Furthermore, C ad is solvable and thus it contains a non-trivial invertible object [ENO2, Proposition 4.5].This together with the fact that every simple object has Frobenius-Perron dimension 1 or p [GN, Corollary 5.3] implies that C ad is necessarily pointed.Note that by [DGNO1, Corollary 2.7] there is a non-trivial object h ∈ C ad such that θ h = 1. We show that the rows of the S-matrix corresponding to (the isomorphism classesof) h and are equal.Given g ∈ U ( C ), let a g and b g denote the number of isomorphism classes of simple objectsof dimension 1 and p in the component C g , respectively. Since p = FPdim( C g ) = a g + b g p ,the following holds: • Either C g has exactly p simple objects, all of which are invertible, or • C g has exactly one simple object, which has dimension p . N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 7
Let X be a simple object of Frobenius-Perron dimension p and let g ∈ U ( C ) such that X ∈ C g . As X is the unique simple object in C g we have h ⊗ X = X . Using the balancingequation we obtain(3.2) s h,X = θ − h θ − X d X θ X = d X . On the other hand, since h ∈ C ad = C ′ pt it follows from [Mu, Proposition 2.5] that for allinvertible object g ∈ C we have(3.3) s h,g = d h d g = d g . Equations (3.2) and (3.3) prove our claim. This is a contradiction since S is invertible. (cid:3) Note that this yields an alternate proof of Lemma 4.11 of [DN].
Corollary 3.5.
Let p be a prime number. Modular tensor categories of Frobenius-Perrondimension p are pointed.Proof. It is easy to see that p divides FPdim( C pt ). If C is not pointed we have FPdim( C pt ) = p [DLD, Lemma 3.2]. Thus the Frobenius-Perron dimension of C ad is also p , which cannothappen by Proposition 3.4. (cid:3) Theorem 3.6.
Let p be an odd prime. Modular tensor categories of Frobenius-Perrondimension p are pointed.Proof. Let C be an integral modular tensor category of Frobenius-Perron dimension p , andsuppose that C is not pointed. Then by [DLD, Lemma 3.2] and Proposition 3.4 we get thatFPdim( C pt ) = p , and so C pt ⊆ C ad [DLD, Lemma 3.4]. Hence C pt is an odd-dimensionalsymmetric subcategory of C and thus C pt ≃ Rep( G ) for some group G of order p .Consider the de-equivariantization C G of C by the Tannakinan subcategory Rep( G ) . Let C G denote the neutral component with respect to the associated G -grading. Since C isnon-degenerate then C G is non-degenerate [DGNO2, Proposition 4.6]. Moreover, we havethat FPdim( C G ) = FPdim( C ) / | G | = p . Hence C must be the gauging of a modular tensorcategory D of dimension p by the group G . Since G is odd-dimensional, it follows from[BGPR, Proposition 2.8] that the action of G on D is the trivial action. Then, the gaugingof D by G is C = D ⊠ Z (Vec ωH ). Note that the Frobenius-Perron dimensions of D and Z (Vec ωG ) are p and p , respectively, so D and Z (Vec ωG ) are both pointed, see Corollary 3.5.Consequently C is also pointed, and we arrive at a contradiction. (cid:3) Lemma 3.7.
Let C be a fusion category and p a prime number such that FPdim( C ) = p .Then C ad is pointed.Proof. As C is nilpotent, FPdim( C ad ) = 1 , p or p . If X is a simple object in C ad , thenFPdim( X ) can be either 1 or p [GN, Corollary 5.3]. By a dimension argument, C ad mustbe pointed. (cid:3) Lemma 3.8.
Let C be an integral modular tensor category and p an odd prime number suchthat FPdim( C ad ) = p . Then one of the following is true: (1) C ad is pointed and G ( C ad ) ≃ Z p × Z p × Z p . (2) ( C ad ) ad = ( C ad ) pt and G ( C ad ) ≃ Z p × Z p .Proof. First note that the dimensions of simple objects of C are either 1 or p by [GN,Corollary 5.3]. A. CZENKY AND J. PLAVNIK
Assume C ad is not pointed. Then C ad has at least one simple object of dimension p . Onthe other hand, Lemma 3.7 implies that ( C ad ) ad must be pointed, and FPdim( X ) dividesFPdim(( C ad ) ad ) for all simple X ∈ C ad [GN, Corollary 5.3]. Therefore FPdim(( C ad ) ad ) = p .Hence, by Proposition 3.1 we get G ( C ad ) ≃ Z p × Z p .Now if C ad is pointed then G ( C ad ) ≃ Z p × Z p × Z p or Z p × Z p by Proposition 3.1. Assumethe latter holds. Let X be a simple object in C of dimension p . Since X ⊗ X ∗ ∈ C ad ,then X ⊗ X ∗ = L g ∈ G [ X ] g . Hence, | G [ X ] | = p for every non-invertible simple X . As theintersection of all subgroups of order p of Z p × Z p is non-trivial, this is a contradiction byProposition 2.6. (cid:3) Lemma 3.9.
Let C be a weakly-integral modular tensor category such that |G ( C ) | is square-free. Then gcd { FPdim( X ) / non-invertible simple X } = 1 . Proof.
Assume there exists a prime p such that p divides dim( X ) for all non-invertiblesimple X in C . By [ENO2, Theorem 2.11] we have that p divides dim( C ). Note thatFPdim( C ) = |G ( C ) | + X X ∈O ( C ) \G ( C ) FPdim( X ) , and thus p divides |G ( C ) | which is a square free number, a contradiction. (cid:3) The next proposition mimics the argument in the proof of Theorem 8.2 in [N1].
Proposition 3.10.
Let C be a modular tensor category of dimension cp q r , where p, q and r are odd prime numbers and c is a square-free odd integer such that gcd( c, pqr ) = 1 . Then C is weakly group-theoretical.Proof. By the proof of [ENO2, Lemma 9.3] C contains a nontrivial symmetric subcategory D .Since C is odd-dimensional then D is Tannakian and thus D ≃
Rep( G ) for some finite group G . Since C is non-degenerate, so is its core C G [ENO1]. Moreover, dim( C G ) = dim( C ) / | G | .Thus by [N2, Theorem 7.4] C G is weakly group theoretical, and hence so is C . (cid:3) Zeros of the S -matrix Let C be a modular tensor category and g be an invertible object in C . Note that for anysimple object X in C we have that σ g,X ∈ Hom( g ⊗ X, g ⊗ X ) ≃ k . Define ξ g ( X ) ∈ k × by σ g,X = ξ g ( X ) id g ⊗ X . Theorem 4.1.
Let C be a modular tensor category. For all X ∈ O ( C ) such that G [ X ] isnon-trivial there exists Y ∈ O ( C ) such that s X,Y = 0 . Proof.
For such X , let g ∈ G [ X ] / { } . Note that for all Y ∈ O ( C ),(4.1) Tr( σ g ⊗ X,Y ) = s g ⊗ X,Y = s X,Y = Tr( σ X,Y ) . By the hexagon axioms, σ g ⊗ X,Y = (id g ⊗ σ Y,X )( σ Y,g ⊗ id X )( σ g,Y ⊗ id X )(id g ⊗ σ X,Y ) = ξ g ( Y )(id g ⊗ σ X,Y ) . Taking trace on both sides of the previous equation we get Tr( σ g ⊗ X,Y ) = ξ g ( Y ) s X,Y . Thistogether with equation (4.1) implies (1 − ξ g ( Y )) s X,Y = 0. As g = , ξ g is non-trivial,there exists a simple Y such that ξ g ( Y ) = 1. Lastly, note that for all simple Y such that ξ g ( Y ) = 1, we have that s X,Y = 0. (cid:3)
N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 9
Corollary 4.2.
Let C be a modular tensor category. If its S -matrix has no null entries, G ( C ) acts freely over O ( C ) .Proof. Suppose g ⊗ X = h ⊗ X for some g, h ∈ G ( C ) and X ∈ O ( C ). Then h − g ∈ G [ X ]which is trivial by Theorem 4.1. Hence, h = g . (cid:3) For all X ∈ O ( C ) let G ( C ) · X := { Y ∈ O ( C ) / h ⊗ X = Y for some h ∈ G ( C ) } Lemma 4.3.
Let C be a modular tensor category and X ∈ O ( C ) . If s X,Z = 0 for some Z ∈ O ( C ) , then s Y Z = 0 for all Y ∈ G ( C ) · X. In particular, if s X,Z = 0 for Z ∈ G ( C ) · X , then s X,Y = 0 for all Y ∈ G ( C ) · X .Proof. Fix Z ∈ O ( C ) such that s X,Z = 0. For Y ∈ G ( C ) · X and h ∈ G ( C ) such that h ⊗ X ≃ Y we have that0 = s X,Z s h,Z = d Z X W ∈O ( C ) N WXh s Z,W = d Z s Z,Y . Hence, s Z,Y = 0. (cid:3) Perfect modular tensor categories
In analogy with group theory, we call a fusion category perfect if the only 1-dimensionalsimple object is the unit, that is, if G ( C ) = { } . Perfect fusion categories are sometimescalled unpointed. Lemma 5.1.
Let C be a perfect odd-dimensional braided fusion category. Then C has nonon-trivial symmetric subcategories. In particular, C is modular.Proof. Let E be a symmetric subcategory of C . As C is odd-dimensional, E is Tannakianand thus E ≃
Rep( G ) for some finite group G . Note that the unit is the only invertibleobject in C , hence G must be a perfect group. Thus, G must be trivial, as the order of everynon-trivial finite perfect group is divisible by four and C is odd-dimensional. This impliesthat E is trivial. (cid:3) Corollary 5.2.
Let C be a perfect odd-dimensional modular tensor category. Then everyfusion subcategory of C is modular. In particular, C ≃ C [ X ] ⊠ C [ X ] ′ for all X ∈ O ( C ) . Proof.
Let D be a non-trivial fusion subcategory of C , and consider its M¨uger centralizer D ′ . Note that D ∩ D ′ is a symmetric subcategory of C , and thus by Lemma 5.1 it must betrivial. That is, D ∩ D ′ ≃ Vec, and therefore D is modular by [DGNO1, Section 2.5]. (cid:3) Corollary 5.3.
Let C be a perfect odd-dimensional modular tensor category of prime rank.Then C has no non-trivial fusion subcategories. Proposition 5.4.
Let C be an odd-dimensional perfect modular tensor category If X and Y are simple objects with coprime Frobenius-Perron dimensions then s X,Y = 0 or s X,Y = d X d Y . Moreover, s X,Y = 0 if and only if X and Y centralize each other. Proof.
Fix X a simple object in C . Consider the full subcategory D Xλ of objects Z ∈ C such that σ Z,X σ X,Z = λ · id X ⊗ Z as in [DGNO1, Lemma 3.15]. Moreover, the category D X = ⊕ λ ∈ k ∗ D Xλ is a fusion subcategory of C . By [DGNO1, Proposition 3.22], D X = h Y ∈O ( C ) | Y centralizes X ⊗ X ∗ i = C C (( C [ X ]) ad ). By Corollary 5.2, C [ X ] is modular and, since C is perfect, so is C [ X ]. Therefore, C [ X ]) ad = C [ X ] and D X = C C (( C [ X ]) ad ) = (( C [ X ])) ′ , by[GN, Corollary 6.9].It follows from [ENO2, Lemma 7.1] that, since D X = (( C [ X ])) ′ , for all X , Y ∈ O ( C ) ofcoprime Frobenius-Perron dimensions we have that s X,Y = 0 or s X,Y = d X d Y as desired. (cid:3) The following Corollary extends Lemma 10.2 in [NPa] when the category is prime to thecase gcd(FPdim( X ) , FPdim( Y )) = 2. Corollary 5.5.
Let C be a prime odd-dimensional perfect modular tensor category. If X and Y are simple objects with coprime Frobenius-Perron dimensions then s X,Y = 0 . Bounds on the ranks of graded components
Lemma 6.1.
Let C be a modular tensor category and consider the universal grading C = L g ∈U ( C ) C g . Then for each prime p > that divides |G ( C ad ) | there exists h ∈ U ( C ) such that C h has at least p non-invertible simple objects of the same dimension. In particular, if C is notpointed then rank( C ) ≥ rank( C ad ) + |G ( C ) | + p − for all odd prime p that divides |G ( C ad ) | .Proof. Let p > |G ( C ad ) | and let g ∈ G ( C ad ) of order p . Consider the fusionsubcategory C [ g ] generated by g . Note that C [ g ] ⊆ ( C ad ) pt and thus C [ g ] is a symmetricsubcategory of C (see Remark 2.1). Since FPdim( C [ g ]) = p is odd then θ g = 1 [DGNO1,Corollary 2.7].For all h ∈ G ( C ) \ { e } consider the action of g on the non-invertible simple elements of C h given by left multiplication. As the order of g is p it follows that for all h ∈ G ( C ) \ { e } this action is given either by the identity or by a cycle of length p . If the former holds forall h ∈ G ( C ) \ { e } , by Lemma 2.3 the rows of the S - matrix corresponding to g and areequal, which is a contradiction as S is invertible. Thus, there must exist h = 1 such that g acts as a cycle of length p on the non-invertible simple elements of C h . Therefore there areat least p different non-invertible simple objects in C h of the same dimension.Finally recall that all the components of the universal grading have at least one simpleelement. Hence rank( C ) ≥ rank( C ad ) + |U ( C ) | + p − C ad ) + |G ( C ) | + p − (cid:3) Corollary 6.2.
Let C be an odd-dimensional modular tensor category and p an odd primethat divides |G ( C ad ) | . Then rank( C ) ≥ rank( C ad ) + |G ( C ) | + 2 p − . Proof.
Let p be an odd prime that divides |G ( C ad ) | . By Lemma 6.1 there exists h = such that C h has at least p non invertible simple objects. As C is odd-dimensional, by [NS,Corollary 8.2(ii)], it is maximally non-self-dual. Thus, C h − = C ∗ h also has at least p noninvertible simple objects, and the result follows. (cid:3) Lemma 6.3.
Let C be a modular tensor category such that ( C ad ) pt is trivial. Then rank( C ) =rank( C ad ) rank( C pt ) .Proof. Note that C ad ∩ C ′ ad ≃ C ad ∩ ( C ad ) pt ≃ Vec. Thus C ad is modular and we have abraided equivalence C ≃ C ad ⊠ C pt . Therefore rank( C ) = rank( C ad ) rank( C pt ). (cid:3) N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 11
Corollary 6.4.
Let C be a modular tensor category of prime rank such that ( C ad ) pt is trivial.Then either C is pointed or C pt is trivial. Lemma 6.5.
Let C be an odd-dimensional fusion category. Then rank( D ) ≡ dim( D ) mod 8 for every fusion subcategory D of C .Proof. Any odd integer n satisfies n ≡ C is an odd integer we getdim( D ) = X X ∈O ( D ) dim( X ) ≡ X X ∈O ( D ) D ) mod 8 . (cid:3) Remark . Let C be an odd-dimensional modular tensor category and consider the uni-versal (faithful) grading C = L g ∈U ( C ) C g . As a direct consequence of Lemma 6.5, We haverank( C ad ) ≡ rank( C g ) mod 8 , for all g ∈ U ( C ) . Proposition 6.7.
Let C be a non-pointed modular tensor category such that C pt ⊆ C ad and FPdim( C pt ) = p k for some odd prime p and k ∈ N . Consider the universal grading C = L g ∈G ( C ) C g . Then rank( C g ) ≡ p, for all g ∈ G ( C ) such that g = 1 .Proof. Let g ∈ G ( C ) such that g = 1. Since G ( C ) acts on O ( C g ) by left multiplicationand G ( C ) is a p -group, we have that the number of fixed elements by the action must becongruent to |O ( C g ) | modulo p . We show that there can be no fixed elements, and so thestatement follows.Suppose there exists an element X ∈ O ( C g ) that is fixed by the action. Since C pt ⊆ C ad and C ′ pt = C ad we have that C pt is symmetric and odd-dimensional, and thus by [DGNO1,Corollary 2.7] we get θ h = 1 for all h ∈ C pt . Hence by the balancing equation s h,X = θ − h θ − X θ X d X = d X = d h d X , for all h ∈ C pt . Therefore X ∈ C ′ pt = C ad [Mu, Proposition 2.5], a contradiction. (cid:3) Proposition 6.8.
Let C be a non-pointed modular tensor category such that C pt ⊆ C ad and FPdim( C pt ) = p k for some odd prime p and k ∈ N . Then for all g ∈ G ( C ) such that g = 1 and a ∈ C the number of simple objects in C g of dimension a is congruent to 0 modulo p .Proof. Let g ∈ G ( C ) such that g = 1 and let a ∈ C . Let C ag be the set of simple objects in C g of dimension a . If C ag is empty the statement is clear. Assume C ag is not empty. Since G ( C ) acts on C ag by left multiplication and G ( C ) is a p -group, we have that the number offixed elements by the action must be congruent to |C ag | modulo p . By the same argumentgiven in the proof of Proposition 6.7 there can be no fixed elements, and so the statementfollows. (cid:3) Application: low rank MNSD modular tensor categories
In this section we prove that MNSD modular tensor categories of rank 13 and 15 arepointed. Moreover, we show that MNSD modular tensor categories of rank 17, 19, 21 and23 are either pointed or perfect, that is, they have exactly one invertible object which is theunit object in the category.Let C be a MNSD modular tensor category such that rank( C ) ∈ { , , , , , } .Note that our claim is equivalent to showing that(1) if rank( C ) = 13 or 15, then |G ( C ) | = rank( C ),(2) if rank( C ) = 17 , ,
21 or 23, then |G ( C ) | = rank( C ) or 1.We will prove our statement discarding the different possibilities for |G ( C ) | until we areleft with the cases stated above.We start by proving the following useful Lemma. Lemma 7.1.
Assume C is not pointed. Then ( C ad ) pt is trivial if and only if C pt is trivial.Proof. It follows from Corollary 6.4 that ( C ad ) pt trivial implies C pt trivial for ranks 13 , , C ) = 15 and assume ( C ad ) pt is trivial. If C pt is not trivial, then by Corollary 6.4we have that C ad is a MNSD modular tensor category of rank 3 or 5, and thus it is pointed[HR, RSW, BR], which is a contradiction.Similarly, if rank( C ) = 21 and we assume that ( C ad ) pt is trivial but C pt is not trivial, then C ad is a MNSD modular of rank 3 or 7 [HR, RSW, BR], which is a contradiction. (cid:3) Remark . Note that |G ( C ) | must be an odd integer smaller or equal to rank( C ). ByCorollary 6.2 we must have that rank( C ) ≥ rank( C ad ) + |G ( C ) | + 2 p − p that divides |G ( C ad ) | . From this and Lemma 7.1 we conclude that the following are all thepossible options for |G ( C ) | :(1) If rank( C ) = 13, then |G ( C ) | = 3 or 1.(2) If rank( C ) = 15 , , ,
21 or 23, then |G ( C ) | = 9 , , C ) = 13 , |G ( C ) | = 1, for rank( C ) = 17 , , , Theorem 7.3.
Let C be a MNSD modular tensor category of rank 13. Then C is pointed.Proof. By Remark (7.2), it is enough to discard the possibilities |G ( C ) | = 3 and 1. Recallthat by Lemma 7.1 ( C ad ) pt is not trivial if |G ( C ) | 6 = 1.Assume |G ( C ) | = 3. Note that C pt ⊆ C ad and FPdim( C ad ) cannot be equal to 3. Hence,there must exist a simple non-invertible element in C ad , and as C is MNSD the rank of C ad isat least five. Moreover, by Lemma 6 . g ∈ G ( C ) ≃ Z such that 3 ≤ rank( C g ) =rank( C g − ). Thus, C ad has rank either 5 or 7, and both cases are discarded by Remark 6.6.Assume now that |G ( C ) | = 1. We will denote the non-invertible simple objects in C by X , X ∗ , · · · , X , X ∗ , and their respective Frobenius-Perron dimensions by d , · · · , d . Upto relabeling the simple objects, we have that d ≥ d ≥ · · · ≥ d . Hence,dim( C ) = 1 + 2 d + · · · + 2 d ≤ . (7.1)On the other hand, by [ENO2, Theorem 2.11] there exists an odd integer l such thatdim( C ) = l d . Equation (7.1) implies that l ≤
12, and therefore l = 5 (see Lemma 6.5).Consequently, N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 13 = 1 + 2 d + · · · + 2 d ≤ . (7.2)Again, by [ENO2, Theorem 2.11], we know that d divides dim( C ) = 5 d , and so thereexists an odd integer m such that d = m d . Equation (7.2) implies that m = 1, that is,d = d and d = 1 + 2 d + · · · + 2 d ≤ . (7.3)By the same argument as before, there exists an odd integer n such that d = n d , andEquation (7.3) implies that n = 1. Hence,d = 1 + 2 d + · · · + 2 d , which is a contradiction. (cid:3) Theorem 7.4.
Let C be a MNSD modular tensor category of rank 15. Then C is pointed.Proof. By Remark (7.2) it is enough to discard the possibilities |G ( C ) | = 9 , , C ad ) pt is not trivial if |G ( C ) | 6 = 1.Case |G ( C ) | = 9: since ( C ad ) pt is not trivial its Frobenius-Perron dimension must be atleast 3. As FPdim( C ad ) cannot be equal to 3, the rank of C ad is at least five. Thus, thiscase is discarded by Corollary 6.2, taking p = 3 . Case |G ( C ) | = 5: here C pt ⊂ C ad . As FPdim( C ad ) cannot be equal to five, the rank of C ad is at least seven. Thus, this case is discarded by Corollary 6.2, taking p = 5 . Case |G ( C ) | = 3: again, C pt ⊂ C ad and the rank of C ad must be at least five. Moreover,by Lemma 6 . g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Thus, C ad hasrank either 5 , C ad ) = 5. By Remark 6.6, we get that rank( C g ) = 5 for all g = 1, which is acontradiction by Proposition 6.7.Lastly, assume |G ( C ) | = 1. We will denote the simple non-invertible objects in C by X , X ∗ , · · · , X , X ∗ and their respective Frobenius-Perron dimensions by d , · · · , d . Rela-bel the simple objects so that d ≥ d ≥ · · · ≥ d . Hence,dim( C ) = 1 + 2 d + · · · + 2 d ≤ . (7.4)On the other hand, by [ENO2, Theorem 2.11], there exists an odd integer l such thatdim( C ) = l d . Equation (7.4) implies that l ≤
14, and therefore l = 7 (see Lemma 6.5).Consequently, 5 d = 1 + 2 d + · · · + 2 d +2 d ≤ . (7.5)Again, by [ENO2, Theorem 2.11], we know that d divides dim( C ) = 7 d , and so thereexists an odd integer m such that d = m d . Equation 7.5 implies that m = 1, that is,d = d and 3 d = 1 + 2 d + · · · + 2 d +2 d ≤ . (7.6) By the same argument as before, there exists an odd integer n such that d = n d , andequation (7.6) implies that n = 1. Hence,d = 1 + 2 d + · · · + 2 d +2 d ≤ , (7.7)Once again, there exists an odd integer q such that d = q d , and equation (7.7) implies q = 1. Therefore, d = 1 + 2 d + · · · + 2 d +2 d which is a contradiction (cid:3) Theorem 7.5.
Let C be a MNSD modular tensor category of rank 17. Then C is pointedor C pt ≃ Vec.Proof.
By Remark (7.2), it is enough to discard the cases |G ( C ) | = 9 , C ad ) pt is not trivial if |G ( C ) | 6 = 1.Case |G ( C ) | = 9: Since ( C ad ) pt is not trivial its Frobenius-Perron dimension must be atleast 3. As FPdim( C ad ) cannot be equal to 3, the rank of C ad is at least five. Thus, thiscase is discarded by Corollary 6.2 taking p = 3.Case |G ( C ) | = 5: here C pt ⊂ C ad . As FPdim( C ad ) cannot be equal to five, the rank of C ad is at least seven. Thus, this case is discarded by Corollary 6.2 taking p = 5.Case |G ( C ) | = 3: as C pt ⊂ C ad , the rank of C ad is at least five. Moreover, by Lemma6.1 there exists g ∈ G ( C ) ≃ Z such that 3 ≤ rank( C g ) = rank( C g − ). Thus, C ad hasrank either 5, 7 , 9 or 11. The first three cases are discarded by Remark 6.6. Assumerank( C ad ) = 11. We denote the non-invertible objects in C ad by X , X ∗ , · · · , X , X ∗ , andthe invertible ones by , g, g . Note that since |G ( C ) | = 3, the action of G ( C ) by leftmultiplication on { X , X ∗ , · · · , X , X ∗ } has 2 or 8 fixed elements.Lets consider first the case in which there are exactly 2 fixed elements by the action.That is, we have that (up to relabeling) the simple objects X and X ∗ are the only simpleobjects fixed by the action. Denote by d i the Frobenius-Perron dimensions of the objects X i and X ∗ i for all i . It is easy to see that since X , X ∗ , X , X ∗ , X , X ∗ are not fixed by theaction, we have that d := d = d = d . Thus,dim( C ) = dim( C pt ) dim( C ad ) = 9 + 6 d +18 d . Hence, gcd(d , d) = 1 ,
3. Assume gcd(d , d) = 3, and consider the decomposition X ⊗ X ∗ = ⊕ N X X X ∗ X ⊕ · · · ⊕ N X ∗ X X ∗ X ∗ . Taking dimensions on both sides, we getd = 1 + d ( N X X X ∗ + N X ∗ X X ∗ ) + d( N X X X ∗ + · · · + N X ∗ X X ∗ ) , and thus 3 divides 1, which is a contradiction. Consequently, gcd(d , d) = 1. Let Y ∈{ X , X ∗ , · · · , X ∗ } . Consider the decomposition X ⊗ Y = N X X Y X ⊕ · · · ⊕ N X ∗ X Y X ∗ . Notice that neither g nor g are subobjects of X ⊗ Y since g fixes X and Y X ∗ . Takingdimensions on both sides on the previous equation we getd d = d ( N X X Y + N X ∗ X Y ) + d( N X X Y + · · · N X ∗ X Y ) . N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 15
Thus, d divides N X X Y + · · · + N X ∗ X Y and d divides N X X Y + N X ∗ X Y . Since gcd(d , d) =1, either d = N X X Y + · · · + N X ∗ X Y and N X X Y + N X ∗ X Y = 0 or d = N X X Y + N X ∗ X Y and N X X Y + · · · + N X ∗ X Y = 0. Assume the latter is true for some Y ∈ { X , . . . , X ∗ } . Then N X X Y = · · · = N X ∗ X Y = 0. In particular, N YX Y = 0 and so by the fusion rules we have that N X Y Y ∗ = 0. Thus Y ⊗ Y ∗ = ⊕ N X Y Y ∗ X ⊕ · · · ⊕ N X ∗ Y Y ∗ X ∗ , and taking dimensions on both sides, we getd = 1 + d( N X Y Y ∗ + · · · + N X ∗ Y Y ∗ ) , and so d divides 1, a contradiction. Therefore d = N X X Y + · · · + N X ∗ X Y and N X X Y + N X ∗ X Y = 0for all Y ∈ { X , . . . , X ∗ } . Consequently, N X X Y = N X ∗ X Y = 0 for all Y ∈ { X , . . . , X ∗ } , andso by the fusion rules we get that N YX X ∗ = N Y ∗ X X ∗ = 0 for all Y ∈ { X , . . . , X ∗ } .Thus, X ⊗ X ∗ = ⊕ g ⊕ g ⊕ N X X X ∗ X ⊕ N X ∗ X X ∗ X ∗ , which implies that d = 3. Hence, dim( C ) = 9 + 3 . By [ENO2, Theorem 2.11],we get that d divides 3
7. Thus, d = 9, which is a contradiction as gcd(d , d) = 1 andd = 3.Lastly, we consider the case in which all simple non-invertible elements in C ad are fixedby Z . Recall that C = C ad ⊕ C g ⊕ C g , where rank( C g ) = rank( C g ) = 3. We denotethe simple elements of C g by Y , Y , Y and their respective Frobenius-Perron dimensionsby d Y , d Y , d Y . Note that the simple elements of C g are exactly Y ∗ , Y ∗ , Y ∗ , and thusby Corollary 2.6 the action of G ( C ) ≃ Z by left multiplication on { Y , Y , Y } must benon-trivial. We may relabel the simples so that g ⊗ Y = Y and g ⊗ Y = Y . Sod Y = d Y = d Y =: d. Now, for all i = 1 , · · · ,
4, we have that X i ⊗ Y ∈ C g , so X i ⊗ Y = N Y X i Y Y ⊕ N Y X i Y Y ⊕ N Y X i Y Y . (7.8)On the other hand, X i ⊗ Y = g ⊗ X i ⊗ Y = N Y X i Y Y ⊕ N Y X i Y Y ⊕ N Y X i Y Y , (7.9) X i ⊗ Y = g ⊗ X i ⊗ Y = N Y X i Y Y ⊕ N Y X i Y Y ⊕ N Y X i Y Y . (7.10)From equations (7.8), (7.9), (7.10) we get that N Y X i Y = N Y X i Y = N Y X i Y , hence X i ⊗ Y = N Y X i Y ( Y ⊕ Y ⊕ Y ) . Consequently, d X i = 3 N Y X i Y . So, 3 divides d X i for all i = 1 , · · · ,
4. Let c X i = d X i / . Note that dim( C ) = 3 dim( C ad ) = 3 dim( C g ) = 9 d . As d X i divides dim( C ) , we get c X i divides d . Reordering the indices so that c X ≥ c X ≥ c X ≥ c X , and letting l be an oddinteger such that d = l c X , we get that3 + 2 d X + · · · + 2 d X = dim( C ad ) = dim( C g ) = 3 d . (7.11)Dividing each side of equation (7.11) by 3, we get l c X = d = 1 + 6 c X + · · · + 6 c X ≤ c X . Hence, l ≤
24, and so l = 1 or 9. If l = 9 , then 9 divides d , and as 9 also dividesd X , · · · , d X , by equation (7.11) we have that 9 divides 3. Consequently, l = 1, i.e,d = c X , and d X = 9 d = dim( C ) = 3 dim( C ad ) = 9 + 6 d X + · · · + 6 d X , which is againa contradiction. (cid:3) Theorem 7.6.
Let C be a MNSD modular tensor category of rank 19. Then C is eitherpointed or C pt ≃ Vec.Proof.
By Remark 7.2, it is enough to discard the cases |G ( C ) | = 9 , , and 3. Recall that,by Lemma 7.1, ( C ad ) pt is not trivial if |G ( C ) | 6 = 1.Case |G ( C ) | = 5: here, C pt ⊂ C ad . As FPdim( C ad ) cannot be equal to five, the rank of C ad is at least seven. Thus, this case is discarded by Remark 6.6.Case |G ( C ) | = 3 or 9: as FPdim( C ad ) cannot be equal to 3, the rank of C ad is at least5. By Lemma 6.1 there exists g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Hence,rank( C ad ) = 5 , , ,
11 or 13, and all cases are discarded by Remark 6.6. (cid:3)
Theorem 7.7.
Let C be a MNSD modular tensor category of rank 21. Then C is eitherpointed or C pt ≃ Vec.Proof.
By Remark 7.2, it is enough to discard the cases |G ( C ) | = 15 , , , and 3. Recall that( C ad ) pt is not trivial if |G ( C ) | 6 = 1 by Lemma 7.1.Case |G ( C ) | = 5: here, C pt ⊆ C ad , and since FPdim( C ad ) cannot be equal to 5 we get thatrank( C ad ) ≥
7. By Lemma 6.1 there exists g ∈ G ( C ) such that 5 ≤ rank( C g ) = rank( C g − ).Therefore rank( C ad ) = 7 or 9, and both cases are discarded by Remark 6.6.Case |G ( C ) | = 9: since FPdim( C ad ) cannot be equal to 3 we get that rank( C ad ) ≥ g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Thereforerank( C ad ) = 5 , |G ( C ) | = 3: as FPdim( C ad ) cannot be equal to 3 we have that rank( C ad ) ≥ g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Hence,rank( C ad ) = 5 , , , ,
13 or 15, and all cases but rank( C ad ) = 7 are discarded by Re-mark 6.6. If rank( C ad ) = 7 then rank( C g ) = 7 for all g ∈ G ( C ), which is a contradiction byProposition 6.7. (cid:3) Theorem 7.8.
Let C be a MNSD modular tensor category of rank 23. Then C is eitherpointed or C pt ≃ Vec.Proof.
By Remark 7.2, it is enough to discard the cases |G ( C ) | = 15 , ,
5, and 3. Recall thatby Lemma 7.1 ( C ad ) pt is not trivial if |G ( C ) | 6 = 1.Case |G ( C ) | = 9: since FPdim( C ad ) cannot be equal to 3 we have that rank( C ad ) ≥ . ByLemma 6.1 there exists g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Hence, rank( C ad ) =5 , , |G ( C ) | = 5: since FPdim( C ad ) cannot be equal to five we get that C ad ≥ . . ByLemma 6.1 there exists g ∈ G ( C ) such that 5 ≤ rank( C g ) = rank( C g − ). Hence, rank( C ad ) =7 , |G ( C ) | = 3: since FPdim( C ad ) cannot be equal to three, the rank of C ad is at leastfive. By Lemma 6.1 there exists g ∈ G ( C ) such that 3 ≤ rank( C g ) = rank( C g − ). Hence,rank( C ad ) = 5 , , , , ,
15, and all cases but rank( C ad ) = 13 are discarded by Remark6.6. Now, if rank( C ad ) = 13 then rank( C g ) = 5 for g ∈ G ( C ) such that g = 1, which is acontradiction by Proposition 6.7. N ODD-DIMENSIONAL MODULAR TENSOR CATEGORIES 17 (cid:3) Future Directions
As a consequence of our results in Section 7, a non-pointed MNSD modular tensor cate-gory of rank between 17 and 23 cannot be weakly group-theoretical . This follows since forweakly group-theoretical fusion categories there is a version of the Feit-Thompson theorem:if a weakly group-theoretical fusion category is odd-dimensional then it is solvable [NP,Proposition 7.1]. It is known that solvable fusion categories contain non-trivial invertibleobjects [ENO2, Proposition 4.5]. This result is relevant because if there exists a non-pointedMNSD modular tensor category of odd rank between 17 and 23 then it must be non-weaklygroup-theoretical but also odd-dimensional (in particular, weakly integral). This would bea counter-example for an important conjecture in fusion categories that states that everyweakly integral fusion category is weakly group-theoretical [ENO1]. In fact, any perfectodd-dimensional modular tensor category would yield a counter-example for said conjec-ture. We conjecture:
Conjecture 8.1.
Odd-dimensional modular tensor categories have at least one non-trivialinvertible object, i.e. they cannot be perfect.
By the argument explained above using Feit-Thompson for weakly group-theoretical fu-sion categories, this conjecture would be an immediate consequence of the weakly integralconjecture.Notice that if Conjecture 8.1 is false, then we can construct an odd-dimensional modulartensor category that is neither perfect nor weakly group-theoretical via the Deligne productof an odd-dimensional perfect modular tensor category and an odd-dimensional pointedmodular tensor category.Another question is if it is possible to remove the hypothesis of weakly group-theoreticalin the fusion categorical version of Feit-Thompson’s Theorem.Equivalent statements to Conjecture 8.1 are the following . Conjecture 8.2.
Odd-dimensional fusion categories are solvable.
Conjecture 8.3.
Odd-dimensional modular tensor categories are solvable.
Note that Conjectures 8.2 and 8.3 are equivalent by [ENO2, Proposition 4.5]. In fact, if C is an odd-dimensional fusion category, then its Drinfeld center Z ( C ) is an odd-dimensionalmodular tensor category. Assuming Conjecture 8.3 this would imply that Z ( C ) is solvable,and since it is Morita equivalent to C ⊠ C op we conclude that C is also solvable [ENO2,Proposition 4.5].On the other hand, if we assume Conjecture 8.1 then Conjecture 8.3 is also true asfollows. Let C be a modular tensor category of odd dimension. We prove our implicationby induction on dim( C ) . Let g be a non-trivial invertible object in C ; we may assume thatthe order of g is prime. Consider the fusion subcategory C [ g ]. If C [ g ] is not Tannakianthen it must be modular and C = C ⊠ C [ g ] ′ . Since C [ g ] ′ is a modular tensor category ofdimension strictly less than dim( C ) then by induction it is solvable and thus C is solvable[ENO2, Proposition 4.5]. On the other hand, if C [ g ] is Tannakian then C [ g ] is equivalent toRep( G ) for G = h g i , and so the trivial component ( C G ) of the de-equivariantization C G of The equivalence of these statements was pointed out to us by C. Galindo C is a modular tensor category of dimension strictly less than dim( C ) and hence solvable byinduction. It follows that C is also solvable by [ENO2, Proposition 4.5].Lastly, Conjecture 8.1 follows from Conjecture 8.3 and [ENO2, Proposition 4.5].By an analogous reasoning, we have that Conjecture 8 . References [BK]
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