Classification of Deformation Quantization Algebroids on Complex Symplectic Manifolds

Abstract

A (holomorphic) deformation quantization algebroid over a complex symplectic manifold $X$ is a stack locally equivalent to the ring of WKB operators, that is, microdifferential operators with an extra central parameter $\tau$. In this paper, we will show that the (holomorphic) deformation quantization algebroids endowed with an anti-involution are classified by $H^2(X;k_X^{\ast })$, where $k^{\ast }$ is a subgroup of the group of invertible series in $C[[{\tau }^{-1}]]$. In the formal case, the analogous classification is given by $H^2(X;C_X)[[\hslash {]]}^{\mathrm{odd}}$ , where one sets $\hslash ={\tau }^{-1}$.