The A_{2n}^{(2)} Rogers-Ramanujan identities

TL;DR

This paper generalizes classical Rogers-Ramanujan identities within a broad family linked to affine Kac-Moody algebras, providing new identities for even moduli related to C_n^{(1)} and D_{n+1}^{(2)}.

Contribution

It introduces a unified framework for Rogers-Ramanujan-type identities associated with affine Lie algebras, extending known identities to new algebraic structures and moduli.

Findings

Embedded identities for fixed m and n relate to affine algebra characters.

Product formulas involve theta functions of specific moduli.

New identities for even moduli linked to C_n^{(1)} and D_{n+1}^{(2)}.

Abstract

The famous Rogers-Ramanujan and Andrews--Gordon identities are embedded in a doubly-infinite family of Rogers-Ramanujan-type identities labelled by positive integers m and n. For fixed m and n the product side corresponds to a specialised character of the affine Kac-Moody algebra A_{2n}^{(2)} at level m, and is expressed as a product of n^2 theta functions of modulus 2m+2n+1, or by level-rank duality, as a product of m^2 theta functions. Rogers-Ramanujan-type identities for even moduli, corresponding to the affine Lie algebras C_n^{(1)} and D_{n+1}^{(2)}, are also proven.