Chiral homology of the projective line and extensions of vertex algebra modules

TL;DR

This paper establishes a mathematical link between module extensions over vertex algebras and first chiral homology of the projective line, deepening understanding of algebraic structures in conformal field theory.

Contribution

It introduces an isomorphism connecting module extension spaces to chiral homology, providing a new perspective in vertex algebra theory.

Findings

Established an isomorphism between extension spaces and chiral homology.

Connected algebraic module extensions with geometric homology concepts.

Enhanced the mathematical framework for studying vertex algebras and their modules.

Abstract

The purpose of this note is to establish an isomorphism from the vector space of extensions between two modules over a vertex algebra, to an appropriate first chiral homology of one dimensional projective space with coefficients in the corresponding chiral algebra.