Covariant Star Product for Exterior Differential Forms on Symplectic Manifolds

TL;DR

This paper introduces a covariant star product for exterior differential forms on symplectic manifolds, providing an explicit second-order expression and exploring the associated graded differential Poisson algebra with connection constraints.

Contribution

It presents the first explicit construction of a covariant star product for differential forms on symplectic manifolds up to second order in deformation parameter.

Findings

Explicit second-order covariant star product derived

Connection constraints identified for the graded differential Poisson algebra

Framework applicable to symplectic manifolds with torsion

Abstract

After a brief description of the $\mathbb{Z}$-graded differential Poisson algebra, we introduce a covariant star product for exterior differential forms and give an explicit expression for it up to second order in the deformation parameter $\hbar$, in the case of symplectic manifolds. The graded differential Poisson algebra endows the manifold with a connection, not necessarily torsion-free, and places upon the connection various constraints.