UC Irvine

The necessary conditions for a quantization of a module over an algebra on a symplectic manifold to exist are investigated. Considered is a symplectic algebraic variety $M$ with a fixed deformation quantization $\mathcal{O}_{\hbar}$ of its sheaf of regular functions, and a vector bundle $E$ on M with a deformation quantization of order $k$ (as a module over $\mathcal{O}_{\hbar}$). It is found that range of cohomology classes must vanish if this order admits an extension to quantization of order $\ell>k$. For $\ell<2k+2$ these conditions are also sufficient. For $\ell\ge2k+2$ a previously unknown obstruction class is found. To construct an explicit form of the obstruction class, one employs a Gelfand-Fuks map from the Lie algebra cohomology to the de Rham cohomology of $M$. The properties of the Gelfand-Fuks map imply that if a lift of quantization from order $k$ to $\ell$ exists, then any element in the kernel of Lie algebra extension - an obstruction class - is mapped to an element in the image that is equivalent to zero. To illustrate the mechanism behind this statement the the Fedosov connection approach is generalized to realize this class via explicit expressions. The generalized Fedosov connection is treated in a manner analogous to the method employed in Tsygan and Nest (2001), wherein the quantization of complex manifolds are studied. It is shown how Gelfand-Fuks classes may be obtained as brackets of the Fedosov connection forms.