Generalized *-Products, Wilson Lines and the Solution of the Seiberg-Witten Equations

TL;DR

This paper explores the structure of higher order terms in noncommutative gauge theories, focusing on generalized *-products and Wilson lines, and provides solutions to Seiberg-Witten equations up to third order in the gauge field.

Contribution

It presents explicit solutions to Seiberg-Witten equations in U(1) gauge theories, incorporating all orders of the noncommutative parameter while neglecting fourth-order gauge field terms.

Findings

Generalized *-products appear in gauge-invariant quantities.

Explicit solutions to Seiberg-Witten equations are obtained.

Wilson lines are shown to relate to these generalized *-products.

Abstract

Higher order terms in the effective action of noncommutative gauge theories exhibit generalizations of the *-product (e.g. *' and *-3). These terms do not manifestly respect the noncommutative gauge invariance of the tree level action. In U(1) gauge theories, we note that these generalized *-products occur in the expansion of some quantities that are invariant under noncommutative gauge transformations, but contain an infinite number of powers of the noncommutative gauge field. One example is an open Wilson line. Another is the expression for a commutative field strength tensor in terms of the noncommutative gauge field. Seiberg and Witten derived differential equations that relate commutative and noncommutative gauge transformations, gauge fields and field strengths. In the U(1) case we solve these equations neglecting terms of fourth order in the gauge field but keeping all orders in the noncommutative parameter.