Construction of Gauge Theories on Curved Noncommutative Spacetime

TL;DR

This paper develops a framework for constructing gauge theories on curved noncommutative spacetimes using derivations of star-product algebras, linking noncommutative geometry with classical gauge theories.

Contribution

It introduces a method to build covariant derivatives and actions for noncommutative gauge theories on curved backgrounds, extending to arbitrary gauge groups via the Seiberg-Witten map.

Findings

Derived covariant derivatives using star-product algebra derivations

Formulated a noncommutative action reducing to scalar electrodynamics in the classical limit

Provided explicit formulas for star-products and Seiberg-Witten maps up to second order

Abstract

We present a method where derivations of star-product algebras are used to build covariant derivatives for noncommutative gauge theory. We write down a noncommutative action by linking these derivations to a frame field induced by a nonconstant metric. An example is given where the action reduces in the classical limit to scalar electrodynamics on a curved background. We further use the Seiberg-Witten map to extend the formalism to arbitrary gauge groups. A proof of the existence of the Seiberg-Witten-map for an abelian gauge potential is given for the formality star-product. We also give explicit formulas for the Weyl ordered star-product and its Seiberg-Witten-maps up to second order.