On the D-module and formal-variable approaches to vertex algebras

TL;DR

This paper establishes an equivalence between Beilinson-Drinfeld's chiral algebras defined via D-modules and the formal-variable approach to vertex algebras, clarifying their conceptual relationship in algebraic geometry and conformal field theory.

Contribution

It demonstrates that chiral algebras in algebraic geometry are essentially the same as vertex algebras without vacuum or grading, linking two frameworks.

Findings

Proves the equivalence of chiral algebras and vertex algebras.

Shows the relations of skew-symmetry and Jacobi identity are compatible.

Bridges algebraic geometry and vertex operator algebra theory.

Abstract

In a program to formulate and develop two-dimensional conformal field theory in the framework of algebraic geometry, Beilinson and Drinfeld have recently given a notion of ``chiral algebra'' in terms of D-modules on algebraic curves. This definition consists of a ``skew-symmetry'' relation and a ``Jacobi identity'' relation in a categorical setting. In this paper, we show directly that these chiral algebras are essentially the same as vertex algebras without vacuum vector (and without grading), by establishing an equivalence between the skew-symmetry and Jacobi identity relations of Beilinson-Drinfeld and the (similarly-named, but different) skew-symmetry and Jacobi identity relations in the formal-variable approach to vertex operator algebra theory as formulated by Borcherds, Frenkel-Lepowsky-Meurman and Frenkel-Huang-Lepowsky.