Classification of deformation quantization algebroids on complex symplectic manifolds

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 τ. In this paper, we will show that the deformation quantization algebroids endowed with an anti-involution are classified by H^2(X; k_X^∗ ), where k^∗ is a subgroup of the group of invertible series in C[[τ−1]]. In the formal case, the analogous classification is given by H^2(X;C_X)[[h]]^odd, where one sets h = τ^−1.