Bimodule deformation of fibered manifolds and the HKR theorem

TL;DR

This paper investigates the formal deformation of fibered manifolds as bimodules over a star product algebra, identifying curvature as an obstruction and computing relevant Hochschild cohomology.

Contribution

It introduces a framework for deforming fibered manifolds as bimodules and computes Hochschild cohomology in this context, linking curvature to deformation obstructions.

Findings

Curvature acts as an obstruction to bimodule deformation.

Hochschild cohomology is computed for smooth maps with closed submanifold images.

Provides conditions for the existence of bimodule deformations.

Abstract

We first want to consider the formal deformation of a fibered manifold $P \rightarrow M$ as a (bi-)module or subalgebra, where $M$ has a given differential star product. Consequently we want to find obstructions for the existence of a bimodule or subalgebra, which turns out to be the curvature of the fiber bundle. Since the order by order construction of this structures amounts to solving equations in the Hochschild cohomology of the smooth functions on $M$ with values in the differential operators on $P$, we proceed to computing this cohomology for the case of a smooth map $P \xrightarrow{p} M$ such that $p(P)$ is a closed submanifold of $M$.