Chiral vector bundles

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