Formal Connections for families of Star Products

TL;DR

This paper introduces the concept of formal connections for families of star products on symplectic manifolds, generalizing previous work and establishing conditions for their existence and classification.

Contribution

It defines formal connections for smooth families of star products, generalizes the formal Hitchin connection, and characterizes the space of such connections.

Findings

Necessary and sufficient condition for existence of formal connections

Formal connections form an affine space modeled by derivations

Flat formal connections are related by automorphisms if the parameter space has trivial first cohomology

Abstract

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled by the derivations of the star products. Moreover we show that if the parameter space has trivial first cohomology group any two flat formal connections are related by an automorphism of the family of star products.