Chiral vector bundles

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.