Classification of transitive vertex algebroids

TL;DR

This paper classifies transitive vertex algebroids on smooth varieties, computes their stack class, and uses this to describe deformations of twisted chiral differential operators, leading to new insights in representation theory.

Contribution

It provides the first classification of transitive vertex algebroids and applies this to classify deformations of twisted chiral differential operators.

Findings

Classified transitive vertex algebroids on smooth varieties.

Computed the class of the stack of these algebroids.

Described deformations of sheaves of twisted chiral differential operators.

Abstract

We present a classification of transitive vertex algebroids on a smooth variety X carried out in the spirit of Bressler's classification of Courant algebroids. In particular, we compute the class of the stack of transitive vertex algebroids. We define deformations of sheaves of twisted chiral differential operators introduced in \cite{AChM} and use the classification result to describe and classify such deformations. As a particular case, we obtain a localization of Wakimoto modules at non-critical level on flag manifolds.