Invariant Actions for Noncommutative Gravity

Two main problems face the construction of noncommutative actions for gravity with star products: the complex metric and finding an invariant measure. The only gauge groups that could be used with star products are the unitary groups. I propose an invariant gravitational action in $D=2n$ dimensions based on the constrained gauge group $U(2^{n})$ broken down to $U(2^{n-1})$ for $n>2$. When $n=2$ the gauge group U(4) is broken to $U(2)\times U(2).$ No metric is used, thus giving a naturally invariant measure. For odd dimensions, $D=2n+1$, the only possible gravitational actions that can be constructed in this formalism are of the Chern-Simons type based on the group $U(2^{n})$ and are therefore topological. These actions can be easily generalized to the noncommutative case by replacing ordinary products with star products. The four dimensional noncommutative action is studied.