EXCEPTIONAL STRUCTURE OF THE DILUTE A(3) MODEL - E(8) AND E(7) ROGERS-RAMANUJAN IDENTITIES

The dilute A(3) lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we directly establish the existence of an E(8) structure in the dilute A(3) model in this regime by expressing the one-dimensional configuration sums in terms of fermionic sums which involve the E(8) root system explicitly. In the thermodynamic limit, these polynomial identities yield a proof of the E(8) Rogers-Ramanujan identity recently conjectured by Kedem et al. The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by q --> 1/q. In this case we find an A(1) x E(7) structure and prove a Rogers-Ramanujan identity of A(1) x E(7)-type. Finally, in the critical q --> 1 limit, we give some intriguing expressions for the number of L-step paths on the A(3) Dynkin diagram with tadpoles in terms of the E(8) Cartan matrix. All our findings confirm the E(8) and E(7) structure of the dilute A(3) model found recently by means of the thermodynamic Bethe ansatz.