Annihilating Fields of Standard Modules of $\mathfrak {sl}(2, \mathbb {C})^\sim $ and Combinatorial Identities

In this volume, the authors show that a set of local admissible fields generates a vertex algebra. For an affine Lie algebra $\tilde{\mathfrak g}$, they construct the corresponding level $k$ vertex operator algebra and show that level $k$ highest weight $\tilde{\mathfrak g}$-modules are modules for this vertex operator algebra. They determine the set of annihilating fields of level $k$ standard modules and study the corresponding loop $\tilde{\mathfrak g}$-module--the set of relations that defines standard modules. In the case when $\tilde{\mathfrak g}$ is of type $A^{(1)}_1$, they construct bases of standard modules parameterized by colored partitions, and as a consequence, obtain a series of Rogers-Ramanujan type combinatorial identities.