Noncommutative gauge fields coupled to noncommutative gravity

TL;DR

This paper develops a noncommutative gravity and gauge field theory framework using twisted star products, deriving a deformed Einstein-Yang-Mills action with explicit first-order corrections related to noncommutativity.

Contribution

It introduces a novel noncommutative gravity coupled to gauge fields with a twisted star product and derives a first-order corrected action using the Seiberg-Witten map.

Findings

First-order correction proportional to gauge field strength cubic products.

Deformed action invariant under diffeomorphisms and gauge transformations.

Explicit form of noncommutative correction involving the anomaly tensor D_{IJK}.

Abstract

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted star product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and twisted gauge transformations. The Seiberg-Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor D_{IJK}.