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

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.