Noncommutative Einstein-AdS gravity in three dimensions

We present a Lorentzian version of three-dimensional noncommutative Einstein-AdS gravity by making use of the Chern–Simons formulation of pure gravity in 2+1 dimensions. The deformed action contains a real, symmetric metric and a real, antisymmetric tensor that vanishes in the commutative limit. These fields are coupled to two Abelian gauge fields. We find that this theory of gravity is invariant under a class of transformations that reduce to standard diffeomorphisms once the noncommutativity parameter is set to zero.