Deformation Quantization over a Z-graded base

Abstract

We investigate the problem how to describe the equivalence classes of formal deformations of a symplectic manifold $M$ in the case when we have several deformation parameters $\ve_1, \ve_2, \dots, \ve_g$ of non-positive degrees. We define formal deformations of $M$ over the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ as Maurer-Cartan elements of the differential graded Lie algebra $(\ve, \ve_1, \dots, \ve_g) \sPD^\bullet(M)[[\ve, \ve_1, \dots, \ve_g]]$ where $\sPD^\bullet(M)$ denotes the algebra of polydifferential operators on $M$. The interesting feature of such deformations is that, if at least one formal parameter carries a non-zero degree, then the resulting Maurer-Cartan element corresponds to a $\bbC[[\ve, \ve_1, \dots, \ve_g]]$-multilinear $A_\infty$-structure on the graded vector space $\cO(M)[[\ve, \ve_1, \dots, \ve_g]]$ with the zero differential, where $\cO(M)$ is the algebra of smooth complex-valued functions $M$. This dissertation focuses on formal deformations of $\cO(M)$ with the base ring $\bbC[[\ve, \ve_1, \dots, \ve_g]]$ such that corresponding MC elements $\mu$ satisfy these two conditions: The Kodaira-Spencer class of $\mu$ is $\ve \al$ and $\mu$ satisfies the equation $\mu \rvert_{\ve=0} =0$. The main result of this study gives us a bijection between the set of isomorphism classes of such deformations and the set of all degree 2 vectors of the graded vector space \, $\bigoplus_{q \geq 0} \, (\ve, \ve_1, \dots, \ve_g) \, H^q(M, \bbC)[[\ve, \ve_1, \dots, \ve_g]]$ where $H^\bullet(M, \bbC)$ is the singular cohomology of $M$ with coefficients in $\bbC$ and every vector of $H^q(M, \bbC)$ carries degree $q$.