Universal star products

TL;DR

This paper introduces the concept of universal deformation quantization, providing a framework that constructs star products on manifolds with Poisson structures using universal polynomial expressions involving curvature and derivatives.

Contribution

It establishes the existence of universal deformation quantizations and analyzes their uniqueness at order 3 through universal Poisson cohomology calculations.

Findings

Universal deformation quantizations exist for any manifold with a Poisson structure.

The paper computes universal Poisson cohomology at order 3.

Results suggest conditions for the uniqueness of star products at third order.

Abstract

One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $\P$ on $M$ and any torsionfree linear connection $\nabla$ on $M$, a universal deformation quantization associates to this data a star product on $(M,\P)$ given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor $\P$, the curvature tensor $R$ and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poisson cohomology.