U(2,2) gravity on noncommutative space with symplectic structure

TL;DR

This paper develops a noncommutative gravity theory based on U(2,2) gauge symmetry on a symplectic manifold, extending classical Einstein gravity with first-order noncommutative corrections.

Contribution

It constructs a noncommutative U(2,2) gauge theory of gravity on symplectic manifolds using the covariant coordinate method and Seiberg-Witten map, including first-order noncommutative corrections.

Findings

Derived a noncommutative gravity Lagrangian with first-order corrections.

Showed consistency with previous SL(2,C) noncommutative gravity results.

Extended classical gravity formulation to noncommutative symplectic spaces.

Abstract

The classical Einstein's gravity can be reformulated from the constrained U(2,2) gauge theory on the ordinary (commutative) four-dimensional spacetime. Here we consider a noncommutative manifold with a symplectic structure and construct a U(2,2) gauge theory on such a manifold by using the covariant coordinate method. Then we use the Seiberg-Witten map to express noncommutative quantities in terms of their commutative counterparts up to the first-order in noncommutative parameters. After imposing constraints we obtain a noncommutative gravity theory described by the Lagrangian with up to nonvanishing first order corrections in noncommutative parameters. This result coincides with our previous one obtained for the noncommutative SL(2,C) gravity.