J an 1 99 6 MINIMAL MODEL FUSION RULES FROM 2-GROUPS Füsun

The fusion rules for the (p, q)-minimal model representations of the Virasoro algebra are shown to come from the group G = Z 2 in the following manner. There is a partition G = P1 ∪ . ∪ PN into disjoint subsets and a bijection between {P1, . , PN} and the sectors {S1, . , SN} of the (p, q)-minimal model such that the fusion rules Si ∗ Sj = ∑ k D(Si, Sj, Sk)Sk correspond to Pi ∗ Pj = ∑ k∈T (i,j) Pk where T (i, j) = {k|∃a ∈ Pi, ∃b ∈ Pj , a+ b ∈ Pk}. The Virasoro algebra Vir is the infinite dimensional Lie algebra with basis {c, Lm | m ∈ Z} and brackets [Lm, Ln] = (m− n)Lm+n + m −m 12 δm,−nc 1991 Mathematics Subject Classification. Primary 17B68, 81R10; Secondary 81T40. 1Partially supported by the National Security Agency under Grant number MDA904-94-H-2019. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. This paper is in final form and no version of it will be submitted for publication elsewhere. Revised: Jan. 3, 1996 Typeset by AMS-TEX 1 where c is central. A highest weight representation [KR] of Vir is a module M containing a vector v0 such that Lmv0 = 0 for all m ≥ 1, L0v0 = hv0 for some h ∈ C, and M is generated by v0. It means that M is spanned by vectors of the form L−n1L−n2 . L−nrv0 for n1 ≥ n2 ≥ . ≥ nr ≥ 1 with r ≥ 0. Since c is central, its value on M is a scalar, which we continue to denote by c. The Verma module M(c, h) is the module, determined up to isomorphism by the values of c and h, such that the above vectors form a basis. For certain values of these parameters the Verma module is not irreducible, but contains nonzero null vectors, that is, vectors which generate proper highest weight submodules. Taking the quotient of M(c, h) by its maximal proper submodule, consisting of the sum of all submodules generated by null vectors, yields the irreducible Vir-module denoted by V (c, h). Note that in order to determine if a vector v0 such that L0v0 = hv0 is null, it suffices to check whether L1 and L2 kill v0, since the bracket formula shows that brackets of these generate all Lm for m ≥ 1. In [FZ] it is shown that for any c ∈ C, V (c, 0) has the structure of a Vertex Operator Algebra (VOA) [FLM] and each V (c, h) is a module for that VOA. Various special classes of Vir-modules have been investigated and classified. A highest weight module with c and h real admits a Hermitian form (·, ·) determined by the conditions (v0, v0) = 1 and (Lmv, w) = (v, L−mw) for any v, w in the module and any integer m. An important class of highest weight modules consists of the unitary modules for which the Hermitian form is positive definite. If c ≥ 1 and h ≥ 0 are real then V (c, h) is unitary, but there is also a discrete series of unitary modules with c < 1 and only a finite number of corresponding special h values. The c values are given by the formula c = 1− 6 p(p+ 1) for p ≥ 2 and the h values are given by hm,n = (n(p+ 1)−mp) − 1 4p(p+ 1) for 0 < m < p+ 1, 0 < n < p. A larger discrete series of modules was investigated in [BPZ], and called the minimal models. For integers p, q ≥ 2 with gcd(p, q) = 1, the (p, q)-minimal model consists of the Virasoro modules V (c, hm,n) whose values of c and hm,n are given by