Chiral vector bundles

TL;DR

This paper introduces the concept of chiral vector bundles as sheaves of vertex algebras associated with smooth G-vector bundles, extending the chiral de Rham sheaf framework.

Contribution

It constructs sheaves of vertex algebras called chiral vector bundles, incorporating differential geometric structures and Lie algebra actions, linking to the chiral de Rham sheaf.

Findings

Defines sheaves of vertex algebras for G-vector bundles

Establishes connections with the chiral de Rham sheaf

Provides a new geometric framework for chiral structures

Abstract

Given a smooth $G$-vector bundle $E \to M$ with a connection $\nabla$, we propose the construction of a sheaf of vertex algebras $\mathcal{E}^{ch(E,\nabla)}$, which we call a \textit{chiral vector bundle}. $\mathcal{E}^{ch(E,\nabla)}$ contains as subsheaves the sheaf of superalgebras $\Omega \otimes \Gamma (SE \otimes \Lambda E)$ and the sheaf of Lie algebras generated by certain endomorphisms of these superalgebras: $\nabla$, the infinitesimal gauge transformations of $E$, and the contraction operators $\iota_X$ on differential forms $\Omega$. Another subsheaf of primary importance is the chiral vector bundle $\mathcal{E}^{ch(M\times \C,d)}$, which is closely related to the chiral de Rham sheaf of Malikov et alii.