Complexified gravity in noncommutative spaces

The presence of a constant background antisymmetric tenser for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang-Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent;l;nd one can obtain complexified gravity by gauging the symmetry U(1, D - 1) instead of the usual SO(1, D - 1). The final theory depends on a Hermitian tenser containing both the symmetric metric and antisymmetric tenser. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces.