Tensor calculus on noncommutative spaces

TL;DR

This paper develops a covariant tensor calculus framework for noncommutative spaces, reducing gauge ambiguities in star products on symplectic manifolds and exploring implications for noncommutative gravity.

Contribution

It extends star products to tensor fields covariantly on symplectic manifolds, significantly reducing gauge freedom and analyzing effects on noncommutative gravity theories.

Findings

Gauge ambiguities are nearly eliminated for tensor fields on symplectic manifolds.

Remaining ambiguities are limited to constant renormalizations or equivalent action maps.

Framework allows incorporation of Riemannian metrics in noncommutative geometry.

Abstract

It is well known that for a given Poisson structure one has infinitely many star products related through the Kontsevich gauge transformations. These gauge transformations have an infinite functional dimension (i.e., correspond to an infinite number of degrees of freedom per point of the base manifold). We show that on a symplectic manifold this freedom may be almost completely eliminated if one extends the star product to all tensor fields in a covariant way and impose some natural conditions on the tensor algebra. The remaining ambiguity either correspond to constant renormalizations to the symplectic structure, or to maps between classically equivalent field theory actions. We also discuss how one can introduce the Riemannian metric in this approach and the consequences of our results for noncommutative gravity theories.