SL(2,C) Gravity with Complex Vierbein and Its Noncommutative Extension

TL;DR

This paper develops a gravity theory using a complex vierbein based on SL(2,C) gauge invariance, extending it to noncommutative spaces with a star product, resulting in a bigravity-like model.

Contribution

It introduces a novel formulation of gravity with a complex vierbein and extends it to noncommutative geometry using the Seiberg-Witten map.

Findings

Formulation of gravity with complex vierbein based on SL(2,C) gauge invariance.

Derivation of a bigravity-like theory with massless and massive gravitons.

Extension of the formalism to noncommutative space with explicit deformed action.

Abstract

We show that it is possible to formulate gravity with a complex vierbein based on SL(2,C) gauge invariance. The proposed action is a four-form where the metric is not introduced but results as a function of the complex vierbein. This formulation is based on the first order formalism. The novel feature here is that integration of the spin-connection gauge field gives rise to kinetic terms for a massless graviton, a massive graviton with the Fierz-Pauli mass term, and a scalar field. The resulting theory is equivalent to bigravity. We then show that by extending the gauge group to GL(2,C} the formalism can be easily generalized to apply to a noncommutative space with the star product. We give the deformed action and derive the Seiberg-Witten map for the complex vierbein and gauge fields.