Generalized Rogers Ramanujan Identities Motivated by AGT Correspondence

Abstract

AGT correspondence and its generalizations attracted a great deal of attention recently. In particular, it was suggested that U(r) instantons on ${R^4/Z_p}$ describe the conformal blocks of the coset ${{A}(r,p)_n=U(1)\times sl(p)_r\times {sl(r)_p\times sl(r)_n\over sl(r)_{n+p}}}$ , where n is a parameter. It has been shown that the representations of algebra A(r,p) _n for generic values n possesses the distinguished geometrical bases. It is interesting to consider the case when the parameter n is integer. We will concentrate on this case and describe Generalized Rogers Ramanujan (GRR) identities for these cosets, which expresses the characters as certain q series. We propose that such identities exist for the coset A(r,p) _n for all positive integers n and all r and p. We treat here the case of n = 1 and r = 2, finding GRR identities for all the characters.