Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras

TL;DR

This paper establishes conditions for constructing vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras, using solutions to Knizhnik-Zamolodchikov equations, extending previous work for fsl_2.

Contribution

It provides new sufficient conditions for intertwining operator construction from fgh-module homomorphisms, employing a novel method involving KZ equations.

Findings

Constructed intertwining operators via KZ equations.

Identified obstructions related to series solutions.

Extended results for fsl_2 to general affine Lie algebras.

Abstract

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When $\mathfrak{g}=\mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $\hat{\mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.