Complexified Gravity in Noncommutative Spaces

TL;DR

This paper develops a consistent framework for complexified gravity in noncommutative spaces by gauging a U(1,D-1) symmetry, resulting in a gauge-invariant action that unifies metric and antisymmetric tensors.

Contribution

It introduces a novel approach to noncommutative gravity using complexified metrics and gauges the U(1,D-1) symmetry, extending previous theories.

Findings

The metric becomes complex in noncommutative gravity.

The theory is gauge invariant and unique.

The approach generalizes to noncommutative spaces.

Abstract

The presence of a constant background antisymmetric tensor 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 and 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 tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces.