Kontsevich product and gauge invariance

TL;DR

This paper investigates gauge invariance in non-commutative spaces with a position-dependent non-commutativity parameter, showing necessary modifications to gauge transformations and derivatives, and analyzing the implications for gauge theories.

Contribution

It extends the understanding of gauge invariance to non-commutative spaces with variable non-commutativity, including explicit constructions of gauge-invariant actions and the validity of the Seiberg-Witten map.

Findings

Gauge transformations and covariant derivatives must be modified for gauge invariance.

Explicit gauge-invariant actions are constructed up to second order in .

The Seiberg-Witten map remains valid despite modifications.

Abstract

We analyze the question of $U_{\star} (1)$ gauge invariance in a flat non-commutative space where the parameter of non-commutativity, $\theta^{\mu\nu} (x)$, is a local function satisfying Jacobi identity (and thereby leading to an associative Kontsevich product). We show that in this case, both gauge transformations as well as the definitions of covariant derivatives have to modify so as to have a gauge invariant action. We work out the gauge invariant actions for the matter fields in the fundamental and the adjoint representations up to order $\theta^{2}$ while we discuss the gauge invariant Maxwell theory up to order $\theta$. We show that despite the modifications in the gauge transformations, the covariant derivative and the field strength, Seiberg-Witten map continues to hold for this theory. In this theory, translations do not form a subgroup of the gauge transformations (unlike in the case when $\theta^{\mu\nu}$ is a constant) which is reflected in the stress tensor not being conserved.