The application of star-products to noncommutative geometry and gauge theory

TL;DR

This paper develops a formalism for star-products in noncommutative geometry, enabling the construction of noncommutative gauge theories and extending their analysis with new tools like covariant derivatives and open Wilson lines.

Contribution

It introduces a method to calculate star-products using commuting vector fields and applies noncommutative differential geometry to gauge theories, extending the Seiberg-Witten formalism.

Findings

Constructed Weyl-ordered star-products for function algebras.

Extended noncommutative gauge theories with covariant derivatives.

Analyzed observables and generalized open Wilson lines.

Abstract

We develop a formalism to realize algebras defined by relations on function spaces. For this porpose we construct the Weyl-ordered star-product and present a method how to calculate star-products with the help of commuting vector fields. Concepts developed in noncommutative differential geometry will be applied to this type of algebras and we construct actions for noncommutative field theories. Derivations of star-products makes it further possible to extend noncommutative gauge theory in the Seiberg-Witten formalism with covariant derivatives. In the commutative limit these theories are becoming gauge theories on curved backgrounds. We study observables of noncommutative gauge theories and extend the concept of so called open Wilson lines to general noncommutative gauge theories.