Exceptional structure of the dilute A$_3$ model: E$_8$ and E$_7$ Rogers--Ramanujan identities

TL;DR

This paper demonstrates the presence of E8 and E7 algebraic structures in the dilute A3 model, proving related Rogers-Ramanujan identities through fermionic sums and polynomial identities, linking lattice models to exceptional Lie algebras.

Contribution

It explicitly establishes E8 and E7 structures in the dilute A3 model via configuration sums, providing new proofs of Rogers-Ramanujan identities and connecting lattice models to exceptional Lie algebra symmetries.

Findings

E8 structure in regime 2 of the dilute A3 model

Proof of E8 Rogers-Ramanujan identity from configuration sums

A1×E7 structure in regime 3 and related identities

Abstract

The dilute A$_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing the 1-dimensional configuration sums in terms of fermionic sums which explicitly involve the E$_8$ root system. In the thermodynamic limit, these polynomial identities yield a proof of the E$_8$ Rogers--Ramanujan identity recently conjectured by Kedem {\em et al}. The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by $q\to 1/q$. In this case we find an A$_1\times\mbox{E}_7$ structure and prove a Rogers--Ramanujan identity of A$_1\times\mbox{E}_7$ type. Finally, in the critical $q\to 1$ 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.