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, mi-crodifferential operators with an extra central parameter τ. In this paper, we will show that the (holomorphic) deformation quantization algebroids endowed with an anti-involution are classified by H2(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 H2(X;CX)[[~]] odd, where one sets ~ = τ−1