  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.
