Noncommutative gravity at second order via Seiberg-Witten map

TL;DR

This paper develops a method to express noncommutative gravity actions in terms of commutative fields using a geometric Seiberg-Witten map, providing explicit second-order corrections and preserving symmetries.

Contribution

It introduces a recursive scheme to relate noncommutative gravity actions to commutative ones via the Seiberg-Witten map, including explicit second-order expressions.

Findings

Explicit second-order noncommutative gravity action in terms of commutative fields.

Recursive solution for the Seiberg-Witten map at all orders.

Manifest invariance under local Lorentz and coordinate transformations.

Abstract

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map (SW map) between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a SW differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter \theta. We thus express the action at order \theta^n+2 in terms of noncommutative fields of order at most \theta^n+1 and, iterating the procedure, in terms of noncommutative fields of order at most \theta^n. This in particular provides the explicit expression of the action at order \theta^2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.