Modular categories, integrality and Egyptian fractions

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank $n$. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A na\"ive computer search approach to the classification of rank $n$ integral modular categories using this bound quickly overwhelms the computer's memory (for $n\geq 7$). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2,3,4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories.


Introduction
The problem of classifying low-rank modular categories has its roots in the classification problem for rational conformal field theories going back to the 1980s (predating the definition of modular category [24]). Currently, the most complete results are in [22] where unitary modular categories of rank at most 4 are classified. More generally, Ostrik classified ribbon fusion categories of rank at most 3 [20] and fusion categories of rank at most 2 [19].
By a generalized form of Ocneanu rigidity [7,Prop. 2.31] one may classify modular categories of a given rank up to finite ambiguity by classifying their Grothedieck semirings. In [13] such an approach yielded a classification (up to Gr.-semirings) of modular categories of rank at most 5 with the property that some object is not isomorphic to its dual object.
Wang has conjectured that there are only finitely many inequivalent modular categories of each rank (see [22,Conjecture 6.1]), and the results mentioned above bear this out for rank at most 4 and for rank at most 5 in case some object is non-self-dual. The most general class of modular categories for which Wang's conjecture has been verified is for weakly integral categories, that is, categories C with FPdim(C) ∈ N [7,Prop. 8.38]. The proof relies upon a classical result of Landau [14] that the diophantine equation: has finitely many solutions with x i ∈ N. Such solutions are Egyptian fraction representations of 1 which are of independent interest in combinatorial number theory (see [12, Section D11] and [18, Seq. A002966]). Moreover the number of solutions to (1.1) is at least exponential in n since 1 x = 1 x+1 + 1 x 2 +x so that each (non-constant) n-term Egyptian fraction representation of 1 leads to at least 2 (n + 1)-term representation. In [13] a computational approach to classifying integral modular categories (that is, with FPdim(X) ∈ N for all objects X) of rank n is suggested using two facts: (1) [6, Lemma 1.2]: [3]: dim(C) ≤ u n where u n is inductively defined by u 1 := 1 and u k := u k−1 (u k−1 + 1).
As u n is double exponential in n, a direct search for solutions by computer quickly becomes infeasible. We attempt to circumvent this computational obstacle by finding interesting conditions on modular categories that imply integrality, and then use these conditions to simplify the classification problem for these classes of categories. Specifically, we have: Theorem. Suppose C is a modular category such that either: (a) the twist matrix T satisfies T N = I for N ∈ {2, 3, 4, 6} or (b) the only simple object in C satisfying X ∼ = X * is the unit object 1 then C is integral.
Statement (a) is proved below in Theorem 3.1 and was inspired by Davydov who posed the question for N = 2 to the second author. Statement (b) is [13, Theorem 2.2] and follows from a Galois theory argument.
We are interested in classifying categories with one of these two properties (a) or (b). We obtain a fairly explicit description of modular categories with property (a) in Theorem 3.2. Modular categories with property (b) of rank at most 11 are shown to be pointed in Theorem 4.5, that is, FPdim(X i ) = 1 for each simple object X i . While property (b) may seem to be a rare condition at first glance one can show (see Prop. 4.3) that it is equivalent to the condition that FPdim(C) is odd.
The remainder of the paper is structured as follows: in Section 2 we collect together some notation and useful facts about modular categories. We address the classification problem for modular categories having property (a) or (b) in Sections 3 and 4 respectively, and give some perspectives and futher directions in Section 5.

Notation
A modular category C is a non-degenerate braided, balanced fusion category (see [1] or [21] for the complete axiomatic definition). In this section we establish notation and describe some of the algebraic data and relations coming from the axioms of modular categories.
We shall typically adopt the notation and normalizations of [17] for the data of a modular category. The fusion coefficients are N k i,j := dim Hom(X i ⊗ X j , X k ) where 1 = X 0 , . . . , X n−1 are the (isomorphism classes of) simple objects the number of which (n) is called the rank of C. The diagonal twist matrix T ij := δ ij θ i has finite order and the S-matrix is normalized so that S 00 = 1. We will denote by d i the dimension of the simple object X i , i.e. d i = S i0 = dim(X i ). Defining the Gauss sums by is the (involutive) charge conjugation matrix, which commutes with T . In particular (S, T ) give rise to a (projective) representation of the modular group SL(2, Z). We define the fusion matrices (N i ) jk := N j ik and denote by FPdim If FPdim(C) = dim(C) then C is said to be pseudo-unitary. The Verlinde formula relates the fusion coefficients to the Smatrix entries: A pair of matrices (S, T ) satisfying the above relations such that the right-handside of (2.1) is a non-negative integer for all i, j, k is called a modular datum ([9]). A modular category C with corresponding S and T matrices is called a categorification of (S, T ). A category C is called integral if FPdim(X i ) ∈ N for all simple objects X i . All categories encountered in this work will be integral and hence pseudo-unitary so that we may assume that FPdim(X i ) = d i and FPdim(C) = dim(C) by [7,Prop. 8.23,8.24].
The entries of N i , S and T satisfy further relations: Since dim is a character of the Grothendieck semiring of C we have: The (second) FS-indicator is defined to be and satisfies: More generally the n-th FS-indiactor ν

Low-order Twist Matrices
In this section we study modular categories with twist matrix of order 2, 3, 4 or 6 of arbitrary rank. Our first result is: Proof. First we observe that by [17,Prop. 5.7] the entries of the S-matrix for C must lie in Q (θ 1 , . . . , θ n−1 ) = Q (ζ N ) where ζ N is a primitive Nth root of unity. Since s ij are algebraic integers and ϕ(N) ≤ 2 (Euler's totient ϕ), if s ij ∈ R then s ij ∈ Z. In particular some column of the S-matrix must be an integer multiple of the vector of F P -dimensions (since C is modular) and so FPdim(X i ) are rational integers.
We now characterize modular categories with T N = I for N ∈ {2, 3, 4, 6} in the following: where G is an abelian 2-group of exponent 2 and dim(C) = 2 2s (in particular, C is pointed), where G is a 2-group of exponent 2 or 4. (d) If N = 6 then C is solvable and hence weakly integral.
Proof. If T 2 = I then each X k is self-dual since the entries of the S-matrix are real. Computing the FS-indicator we have: for all k. But θ i θ j 2 = 1 by assumption and i N k ij d i = d k d j (as N k ij is totally symmetric in the self-dual case) so we may simplify: But ν k = ±1 and d k > 0 so this implies d k = 1 for all simple objects X k . We must then have X ⊗2 k ∼ = 1 as well. Thus C is pointed and has the same fusion rules as the group Z m 2 where 2 m is both the rank and global dimension of C. Now since p + ∈ R we have p + = p − ∈ Z so that by (2.2) we must have dim(C) = 2 m = (p + ) 2 and hence m = 2s. Since C is braided tensor equivalent to a subcategory of Z(C) ∼ = Rep(D ω G) we must have G an elementary abelian 2-group.
For (b) and (c) with N = 3 (resp. 4) we use [16,Theorem 8.4] to conclude that dim(C) = p t where p = 3 (resp. 2). In particular C is braided and nilpotent and hence group theoretical by [4, Theorem 6.10]. Thus we conclude that C is braided tensor equivalent to a subcategory of Z(C) ∼ = Rep(D ω G) where G is a p-group and since FSexp(C) = FSexp(Z(C)) = 3 (resp. 4) by [16,Corollary 7.8] G must have exponent 3 (resp. 2 or 4 as the exponent of G must divide that of Rep(D ω G)).
For (d) with N = 6 [16,Theorem 8.4] implies that dim(C) = 2 a 3 b with a, b > 0 and so by [8, Theorem 1.6] C is solvable, and in particular weakly group-theoretical. integer k ≥ 0 is described with S ij ∈ Z for all i, j and T 2 = I. Theorem 3.2(a) shows that this family is not categorifiable for k ≥ 1.

Integrality and Egyptian Fractions
In this section we classify integral modular categories of rank at most 6 and maximally non-self-dual modular categories of rank at most 11.
We remark that there is a non-pointed integral modular category of rank 8, namely Rep(D(S 3 )), the representation category of the double of the symmetric group S 3 . We expect that this is the smallest possible rank for non-pointed integral modular categories. Unfortunately for n = 7 one has u 7 ≈ 10 13 and attempts to implement Algorithm 4.1 quickly overwhelms a computer's memory. This motivates passing to a restricted class of modular categories that enjoy integrality.
With a view towards improving the upper bound u n above, suppose that C is a maximally non-self dual (MNSD) modular category, i.e. each non-trivial simple object X i satisfies X i ∼ = X * i . Then C is integral with rank n = 2k + 1 by [13, Theorem 2.2]. Since d i = dim(X i ) = dim(X * i ) we may label the dimensions of simple objects: The maximal non-self-dual condition may seem somewhat exceptional, but it follows from [  (1) C is maximally non-self-dual where A is an odd-dimensional, semisimple quasi-Hopf algebra.
The following generalization of [13, Lemma 2.3] gives a linear improvement of the upper bounds in Algorithm 4.1 for a restricted class of integral modular categories and includes the MNSD setting as the special case ℓ = 2: Lemma 4.4. Suppose that C is an integral modular category of rank n and there exist k (weakly decreasing) integers p 1 ≥ p 2 ≥ · · · ≥ p k such that the non-trivial (isomorphisms classes of ) simple objects can be partitioned into k sets P 1 , . . . , P k such that X ∈ P i has dim(X) = p i and |P i | = ℓ for all 1 ≤ i ≤ k. Then: (i) The numbers x i := dim C p 2 i form a weakly increasing sequence of integers such Proof. The proof of (i) proceeds exactly as in [13, Lemma 2.3] so we focus on (ii). First define x ′ i = x i for 1 ≤ i ≤ k and x ′ k+1 = x k+1 /ℓ so that: Since the x ′ i are weakly increasing the lower bounds are clear: , a contradiction. For the upper bounds follow the same strategy as in [13]: we use Takenouchi's [23] bound on the largest denominator x ′ k+1 in eqn. (4.2) and Landau's [14] estimate. In particular, if with y 1 ≤ y 2 ≤ · · · ≤ y k then Landau's result says y i ≤ (k−i+1)/r i−1 where r 0 = r and r i = r i−1 − 1/y i . Takenouchi's result says that the maximum denominator y i of a solution to 1  Proof. The only non-trivial solution the modified algorithm produces is in rank 11, consisting of 9 simple objects of dimension 1 and two simple objects of dimension 3.
To eliminate this possibility we use a result in [11]: any integral modular category C is faithfully graded by its universal grading group U(C), which is isomorphic to the group of invertible objects in C. Since each component of a faithful grading must have the same F P -dimension we see that this is impossible as |U(C)| = 9 and any component C g of the grading containing a simple object X with FPdim(X) = 3 must have FPdim(C g ) ≥ 9.
Remark 4.6. An obvious source of MNSD modular categories are those of the form Rep(D ω G) where |G| is odd (that is, the representation category of the twisted double of a finite group of odd order). The smallest rank example of a non-pointed MNSD modular category we are aware of is the rank 25 category Rep(D(Z 3 ⋉ Z 7 )) of dimension 441 where Z 3 ⋉ Z 7 is a non-abelian semidirect product. One might speculate that any MNSD modular category of rank at most 23 is pointed.
We close with the following related question that we find interesting: Question 4.7. Is there an odd dimensional modular category that is non-grouptheoretical?

Conclusions and Future Directions
We have made significant progress towards classifiying modular categories with twist matrices of order 2, 3, 4 or 6. It is not clear how to extend these techniques to other orders. One might hope that one could generalize [16,Theorem 8.4] to non-integral categories to get some statement about the possible prime divisors of FPdim(C) in terms of those of the order of the twist matrix. However, the relevant number field would no longer be Z so that other complications would arise particularly concerning units.
For rank greater than 11 even the improved bounds for MNSD modular categories become too large for our computational techniques. However, if one considers MNSD modular categories with: (1) rank r For any fixed r ≤ 23 and subsets of {d : 3 ≤ d ≤ 45, d ∈ 2Z + 1} of size at most 5 we convert these congruences to a system of linear diophantine equations and then use the Smith normal form of the corresponding matrix to solve for the k i . Applying further classification theorems and ad hoc techniques produces the following partial result: Theorem 5.1. If C is a non-pointed maximally non-self dual modular category with 13 ≤ rank (C) ≤ 23 then either C has: (a) FPdim(X i ) ≥ 47 for some simple X i or This is further evidence that perhaps a non-pointed MNSD modular category of smallest rank is indeed the category of rank 25 and dimension 441 described above.
More generally, one can also obtain partial classification results for integral modular categories of small rank by bounding the F P -dimension further (below the double-exponential bound of u n in Algorithm 4.1). For example, using a streamlined version of Algorithm 4.1 provided by K. Rusek and some classification theorems we can prove: Theorem 5.2. If a non-pointed modular category C has rank 7 then FPdim(C) > 10 5 .