Noncommutative gauge theory using covariant star product defined between Lie-valued differential forms

TL;DR

This paper develops a noncommutative gauge theory on a torsionful symplectic manifold using a covariant star product, showing that torsion must be covariantly constant for associativity, which constrains the connection.

Contribution

It introduces a covariant star product for Lie-valued differential forms on symplectic manifolds with torsion, establishing conditions for associativity and connection determination.

Findings

Torsion must be covariantly constant for star product associativity.

The connection on the manifold is fully determined by the covariant star product conditions.

An explicit example illustrates the theoretical framework.

Abstract

We develop an internal gauge theory using a covariant star product. The space-time is a symplectic manifold endowed only with torsion but no curvature. It is shown that, in order to assure the restrictions imposed by the associativity property of the star product, the torsion of the space-time has to be covariant constant. An illustrative example is given and it is concluded that in this case the conditions necessary to define a covariant star product on a symplectic manifold completely determine its connection.