Virasoro Algebra, Dedekind η-function and Specialized Macdonald Identities

Abstract

We motivate and prove a series of identities which form a generalization of the Euler pentagonal number theorem, and are closely related to specialized Macdonald identities for powers of the Dedekind η-function. More precisely, we show that what we call “denominator formula” for the Virasoro algebra has “higher analogue” for all c_s,t-minimal models. We obtain one identity per series which is in agreement with features of conformal field theory such as fusion and modular invariance that require all the irreducible modules of the series. In particular, in the case of c_2,2k+1-minimal models we give a new proof of a family of specialized Macdonald identities associated with twisted affine Lie algebras of type A⁽²⁾ _2k k ≥ 2 (i.e., BC_k affine root system) which involve (2k² - k)-th powers of the Dedekind η-function.