Infinitesimal deformations of a formal symplectic groupoid

TL;DR

This paper introduces the concept of infinitesimal deformations of formal symplectic groupoids over Poisson manifolds, relating them to pairs of star products and providing explicit formulas and algorithms for their analysis.

Contribution

It defines infinitesimal deformations of formal symplectic groupoids and connects them to pairs of star products, offering explicit formulas and computational algorithms.

Findings

Defined infinitesimal deformations of formal symplectic groupoids.

Related deformations to pairs of star products with separation of variables.

Provided an algorithm for principal symbols of the logarithm of a Berezin transform.

Abstract

Given a formal symplectic groupoid $G$ over a Poisson manifold $(M, \pi_0)$, we define a new object, an infinitesimal deformation of $G$, which can be thought of as a formal symplectic groupoid over the manifold $M$ equipped with an infinitesimal deformation $\pi_0 + \epsilon \pi_1$ of the Poisson bivector field $\pi_0$. The source and target mappings of a deformation of $G$ are deformations of the source and target mappings of $G$. To any pair of natural star products $(\ast, \tilde\ast)$ having the same formal symplectic groupoid $G$ we relate an infinitesimal deformation of $G$. We call it the deformation groupoid of the pair $(\ast, \tilde\ast)$. We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kaehler- Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.