We introduce a matrix model for noncommutative gravity, based on the gauge group U(2) X U(2) . The vierbein is encoded in a matrix Y_\mu , having values in the coset space U(4)/(U(2) X U(2)) , while the spin connection is encoded in a matrix X_\mu , having values in U(2) X U(2) . We show how to recover the Einstein equations from the \theta \rightarrow 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) X 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.
Matrix model for noncommutative gravity and gravitational instantons / P. VALTANCOLI. - In: INTERNATIONAL JOURNAL OF MODERN PHYSICS A. - ISSN 0217-751X. - STAMPA. - 19:(2004), pp. 227-248.
Matrix model for noncommutative gravity and gravitational instantons
VALTANCOLI, PAOLO
2004
Abstract
We introduce a matrix model for noncommutative gravity, based on the gauge group U(2) X U(2) . The vierbein is encoded in a matrix Y_\mu , having values in the coset space U(4)/(U(2) X U(2)) , while the spin connection is encoded in a matrix X_\mu , having values in U(2) X U(2) . We show how to recover the Einstein equations from the \theta \rightarrow 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) X 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.File | Dimensione | Formato | |
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