Annihilating fields of standard modules of sl(2,C)~ and combinatorial identities

TL;DR

This paper constructs vertex operator algebras from affine Lie algebras, identifies annihilating fields of standard modules, and derives combinatorial identities akin to Rogers-Ramanujan, linking algebraic structures to combinatorics.

Contribution

It introduces a method to determine annihilating fields of standard modules and connects these to combinatorial identities for affine Lie algebra of type A1^{(1)}.

Findings

Set of local admissible fields generates a vertex algebra.

Constructed bases of standard modules parameterized by colored partitions.

Derived Rogers-Ramanujan type combinatorial identities.

Abstract

We show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $\tilde\goth g$ we construct the corresponding level $k$ vertex operator algebra and we show that level $k$ highest weight $\tilde\goth g$-modules are modules for this vertex operator algebra. We determine the set of annihilating fields of level $k$ standard modules and we study the corresponding loop $\tilde\goth g$ module---the set of relations that defines standard modules. In the case when $\tilde\goth g$ is of type $A_1^{(1)}$, we construct bases of standard modules parameterized by colored partitions and, as a consequence, we obtain a series of Rogers-Ramanujan type combinatorial identities.