Rogers-Ramanujan identities: A century of progress from mathematics to physics

Abstract

In this talk we present the discoveries made in the theory of Rogers-Ramanujan identities in the last five years which have been made because of the interchange of ideas between mathematics and physics. We find that not only does every minimal representation $M(p,p^{\prime })$ of the Virasoro algebra lead to a Rogers-Ramanujan identity but that different coset constructions lead to different identities. These coset constructions are related to the different integrable perturbations of the conformal field theory. We focus here in particular on the Rogers-Ramanujan identities of the $M(p,p^{\prime })$ models for the perturbations ${\varphi }_{1,3},\sim {\varphi }_{2,1},\sim {\varphi }_{1,2}$ and ${\varphi }_{1,5}.$