Rationality of Virasoro Vertex Operator Algebras

Vertex operator algebras (VOA) were introduced by Borcherds ( [B] ) as an axiomatic description of the ‘holomorphic part’ of a conformal field theory ( [BPZ] ). An account of the theory of vertex operator algebras may be found in [FLM]. One of the most important examples of VOAs ( [FZ] ) is the Virasoro VOAs, i.e.VOAs corresponding to the representations of the Virasoro algebra L, denoted by Vc in this paper, where Vc is the (unique) irreducible highest weight representation of L with highest weight (c, 0). It is conjectured ( [FZ] ) that Vc is rational if and only if c = cp,q = 1− 6(p− q)/pq, where p, q ∈ {2, 3, 4, ...}, and p, q are relatively prime. ( The definition of rationality of VOAs is given in Section 2 ). In this paper we prove this conjecture , and show that when c = cp,q, p, q ∈ {2, 3, 4, ...}, and (p, q) = 1, all the irreducible representations of the Virasoro VOAs are precisely those which correspond to irreducible minimal modules of the Virasoro algebra. Then we will prove the fusion rules in the minimal series cases which were stated implicitly in [FF2]. Y.Zhu in [Z] constructed an associative algebra A(V ) for a general VOA V and established a 1-1 correspondence between irreducible representations of V and irreducible representations of A(V ). This construction enabled I.Frenkel and Y.Zhu to prove the rationality of VOAs associated to the representations of affine Kač-Moody algebras with positive integral level. Here we will prove the rationality of the Virasoro VOAs with the help of Zhu’s construction.