Matrix model for noncommutative gravity and gravitational instantons

Abstract

We introduce a matrix model for noncommutative gravity, based on the gauge group $U(2)\otimes U(2)$. The vierbein is encoded in a matrix $Y_{\mu }$, having values in the coset space $U(4)/(U(2)\otimes U(2))$, while the spin connection is encoded in a matrix $X_{\mu }$, having values in $U(2)\otimes U(2)$. We show how to recover the Einstein equations from the $\theta \to 0$ 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 $U(2)\otimes U(2)$ 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.

28 pages, LaTeX, no figures

References in corpus (3)

• Three-dimensional Noncommutative Gravity
• Noncommutative Gravity in two Dimensions
• ADHM/Nahm Construction of Localized Solitons in Noncommutative Gauge Theories

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• (2 + 1) noncommutative gravity and conical spacetimes