MATRIX MODEL FOR NONCOMMUTATIVE GRAVITY AND GRAVITATIONAL INSTANTONS

Abstract

We introduce a matrix model for noncommutative gravity, based on the gauge group U (2)⊗ U (2). The vierbein is encoded in a matrix Yμ, having values in the coset space U (4)/( U (2)⊗ U (2)), while the spin connection is encoded in a matrix Xμ, having values in U (2)⊗ U (2). We show how to recover the Einstein equations from the θ→0 limit of the matrix model equations of motion. We stress the necessity of a metric tensor, which is a covariant representation of the gauge group in order to set up a consistent second order formalism. We finally define noncommutative gravitational instantons as generated by U (2)⊗ U (2) valued quasi-unitary operators acting on the background of the matrix model. Some of these solutions have naturally self-dual or anti-self-dual spin connections.