Deformation quantization of gerbes

Abstract

This is the first in a series of articles devoted to deformation quantization of gerbes. We introduce basic definitions, interpret deformations of a given stack as Maurer-Cartan elements of a differential graded Lie algebra (DGLA), and classify deformations of a given gerbe in terms of Maurer-Cartan elements of the DGLA of Hochschild cochains twisted by the cohomology class of the gerbe. We also classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.

Classification of Deformation Quantization Algebroids on Complex Symplectic Manifolds

“…Note that this is compatible with the classification of the deformation quantization algebroids (not assumed endowed with an anti-involution) given in [8].…”

“…An invertible transformation between equivalences of (W, * )-algebroids is a filtered invertible transformation of functors α : [8], and are studied in [13].…”

“…We refer to [22] (see also [8]) for the analogous classification in the complex setting, where these algebras are constructed under suitable hypothesis.…”

*-Quantizations of Fourier–Mukai Transforms

“…It is likely that one can combine this construction with the observation (see e.g. [BGNT05, Theorem 7.1.2 and Proposition 3. 6.1]) that the quantizations of a complex space X are classified by the Maurer-Cartan elements in the dg Lie algebra of de Rham-Sullivan forms…”

Local index theorem for projective families

“…Related questions of deformation theory are considered in [9,10]. This paper is a byproduct of a joint project with E. Leichtnam.…”