The Seiberg-Witten map for non-commutative pure gravity and vacuum Maxwell theory

TL;DR

This paper explores the Seiberg-Witten map in non-commutative gauge theories, specifically pure gravity and vacuum Maxwell theory, revealing difficulties in finding solutions at first order in non-commutativity for certain classical solutions.

Contribution

It provides the second-order Seiberg-Witten map for pure gravity with constant non-commutativity and analyzes solution existence for vacuum solutions in gravity and Maxwell theory.

Findings

No solutions at first order for certain vacuum gravity solutions.

Solutions in Maxwell theory only exist under special conditions.

Highlights limitations of the Seiberg-Witten map in non-commutative gravity and electromagnetism.

Abstract

In this paper the Seiberg-Witten map is first analyzed for non-commutative Yang-Mills theories with the related methods, developed in the literature, for its explicit construction, that hold for any gauge group. These are exploited to write down the second-order Seiberg-Witten map for pure gravity with a constant non-commutativity tensor. In the analysis of pure gravity when the classical space-time solves the vacuum Einstein equations, we find for three distinct vacuum solutions that the corresponding non-commutative field equations do not have solution to first order in non-commutativity, when the Seiberg-Witten map is eventually inserted. In the attempt of understanding whether or not this is a peculiar property of gravity, in the second part of the paper, the Seiberg-Witten map is considered in the simpler case of Maxwell theory in vacuum in the absence of charges and currents. Once more, no obvious solution of the non-commutative field equations is found, unless the electromagnetic potential depends in a very special way on the wave vector.