Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at (p, q)-central charge

Abstract

Abstract Let O c be the category of finite-length modules for the Virasoro Lie algebra at central charge c whose composition factors are irreducible quotients of reducible Verma modules. For any c ∈ C , this category admits the vertex algebraic braided tensor category structure of Huang–Lepowsky–Zhang. Here, we begin the detailed study of O c p , q where c p , q = 1 − 6 p − q 2 pq for relatively prime integers p, q ≥ 2; in conformal field theory, O c p , q corresponds to a logarithmic extension of the central charge c p,q Virasoro minimal model. We particularly focus on the Virasoro Kac modules K r , s , r , s ∈ Z ≥ 1 , in O c p , q defined by Morin-Duchesne–Rasmussen–Ridout, which are finitely-generated submodules of Feigin–Fuchs modules for the Virasoro algebra. We prove that K r , s is rigid and self-dual when 1 ≤ r ≤ p and 1 ≤ s ≤ q, but that not all K r , s are rigid when r > p or s > q. That is, O c p , q is not a rigid tensor category. We also show that all Kac modules and all simple modules in O c p , q are homomorphic images of repeated tensor products of K 1,2 and K 2,1 , and we determine completely how K 1,2 and K 2,1 tensor with Kac modules and simple modules in O c p , q . In the process, we prove some fusion rule conjectures of Morin-Duchesne–Rasmussen–Ridout.