Finite Group Modular Data

In a remarkable variety of contexts appears the modular data associated to finite groups. And yet, compared to the well-understood affine algebra modular data, the general properties of this finite group modular data has been poorly explored. In this paper we undergo such a study. We identify some senses in which the finite group data is similar to, and different from, the affine data. We also consider the data arising from a cohomological twist, and write down explicitly the modular S and T matrices for a general twist, for what appears to be the first time in print. ∗ UMR 8627 Chercheur qualifié FNRS 1 Mathematical and Physical Origin Throughout this paper, let G denote any finite group (good references to finite group and character theory are provided by [1, 2, 3]). Physically, the modular data we will describe in the next section arise in several ways. It is a (2+1)-dimensional Chern-Simons theory with finite gauge group G [4, 5]. As Witten showed, 3-dimensional topological field theory corresponds to 2-dimensional conformal field theory (CFT), and the corresponding CFTs here are orbifolds by symmetry group G of a holomorphic CFT [6, 4, 7, 8] (e.g. the E8 level 1 WZW orbifolded by any finite subgroup of the compact simply-connected Lie group E8(R)). This CFT incarnation is important to us, as it provides some motivation for specific investigations we will perform. Nevertheless, both of these incarnations are probably of direct value only as toy models. Note that more generally, to each G we will obtain a finite-dimensional representation of the mapping class group Γg,n for the genus g surfaces with n punctures [9]. In this paper though we will consider only the modular group SL2(Z), corresponding to Γ1,0, and in particular the matrices S and T . It is clear that many different CFTs can realise the same modular data – e.g. all 71 of the c = 24 holomorphic theories have S = T = (1). So any given G will correspond to several different CFTs. Likewise, all we can say about the central charge c associated to G is that it will be a positive multiple of 8. Associated with the RCFT we expect to have some sort of quantum-group which captures the modular data and representation theory of the chiral algebra (so e.g. the fusion coefficients are tensor product coefficients of irreducible modules). This was done for arbitrary finite G in [7], and is the quantum-double of the group algebra C[G] (see also [10, 11]). There should also be a vertex operator algebra (VOA) interpretation to this data, assigning a VOA to each finite group. One way to do this is to start with the VOA associated with any even self-dual Euclidean lattice Λ for which G ≤ AutΛ. Orbifolding it by G should yield our data (with the value c = dimΛ). For example, any finite subgroup of SU2(C) or SU3(C) works with the Λ = E8 root lattice. By Cayley’s Theorem, such a lattice Λ can always be found for a given group G: e.g. embed G in some Sn and take Λ to be the orthogonal direct sum of n copies of E8. In [12, 10], this VOA interpretation is addressed. Although these holomorphic orbifolds are perhaps too artificial to be of direct interest, it can be expected that they provide a good hint of the behaviour of more general orbifolds. Indeed this is the case – e.g. they can be seen in the theory of permutation orbifolds [13]. Perhaps the most physical incarnation of this modular data is in (2+1)-dimensional quantum field theories where a continuous gauge group has been spontaneously broken to a finite group [14]. Non-abelian anyons (i.e. particles whose statistics are governed by the braid group rather than the symmetric group) arise as topological excitations. The effective field theory describing the long-distance physics is governed by the quantum-group of [7]. Adding a Chern-Simons term corresponds to the cohomological twist to be discussed shortly. Actually, this modular data arose originally in mathematics, in an important but technical way in Lusztig’s determination of the irreducible characters of the finite groups of Lie type [15, 16]. In describing some of these, the so-called ‘unipotent’ characters, he was led to consider this modular data for the groups G ∈ {Z2 ×· · ·×Z2 ,S3 ,S4 ,S5 }. For example, S5 arises in groups of type E8. Our primary fields Φ parametrise the unipotent characters