Deformation quantization of gerbes (Q2642050)

This is the first in a series of articles devoted to deformation quantization of gerbes. Here the authors give basic definitions and interpret deformations of a given gerbe as Maurer-Cartan elements of a differential graded Lie algebra (DGLA). The authors classify all deformations of a given gerbe on a symplectic manifold, as well as they provide a deformation-theoretic interpretation of the first Rozansky-Witten class. More precisely, the main results of this paper are as follows. (1) The authors prove that deformations of every stack are classified by Maurer-Cartan elements of a DGLA. This generalizes some results of Gerstenhaber for associative algebras and of V. Hinich for sheaves of associative algebras. (2) The authors show that the DLGA controlling deformations of a gerbe on a manifold is equivalent to the Hochschild complex of this manifold twisted by the cohomology class of the gerbe. (3) The authors classify deformation quantizations of all gerbes on a symplectic manifold. This generalizes the classification results for deformation quantizations of \(C^{\infty}\) symplectic manifolds. (4) The authors show that the first Rozansky-Witten class of a holomorphic symplectic manifold is an obstruction for a canonical stack deformation quantization to be a sheaf of algebras.