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. [email protected] [email protected] [email protected] [email protected] The field theory limit of open string amplitudes in the presence of a Bμν background gives rise to gauge theories formulated in a noncommutative geometry [1]. The parameter θ which encodes the noncommutativity properties is essentially given by B μν . In a field theory perturbative framework it naturally produces an effective cutoff, ∆x ≡ θpν , that modifies the UV behaviour of the theory [2]. As long as θ is treated as an external, given background, this amounts to an ad hoc procedure. In order to gain true insights one would like to see noncommutativity/nonlocality entering the game as a dynamical object. One way to make progress in this direction might be to understand how to formulate gravity in a noncommutative geometry. While noncommutative gauge theories have been studied extensively at the perturbative and nonperturbative level, not much is known about the corresponding formulation of a gravitational theory. One of the main obstacle to overcome is related to the fact that the Moyal product, which implements the noncommutativity, does not maintain reality. In the noncommutative formulation of a gauge theory this does not represent a real problem since the deformed gauge transformations are such that they produce new, real gauge fields. One possible way to preserve reality of the gravitational field is to use explicitely the Seiberg-Witten map [3]. Otherwise it seems that one is forced to complexify the fields [4, 5]. However, complex gravity may be plagued by inconsistencies already at the commutative level [6, 7]. In two and in three dimensions however one can take advantage of the fact that a theory of gravity can be formulated as a gauge theory. In fact since we know how to deform gauge transformations and since the metric does not appear in the volume form, in order to obtain the corresponding noncommutative version of the theory it is sufficient to introduce the ⋆-product and extend the gauge group appropriately. In this letter we study noncommutative gravity in three dimensions, while the twodimensional theory will be treated in a forthcoming paper [8]. It is well-known that in three dimensions pure Einstein gravity can be written as a ChernSimons theory. More precisely the Einstein-Hilbert action with negative cosmological constant is equivalent to the difference of two SO(2, 1) Chern-Simons actions, modulo boundary terms [9, 10]. This was used in [11] in order to obtain a Euclidean version of three-dimensional noncommutative gravity, based on the gauge groupGL(2,C). We follow a similar approach and define a three-dimensional Lorentzian version of noncommutative gravity. One advantage of our formulation is given by the fact that all the fields are real and the metric is naturally identified. We find a set of transformations which are invariances of the action and reduce to standard diffeomorphisms in the commutative limit. Here we present the main results; more details and insights will be given in [8]. Let us introduce the action For extensions cf. [12].