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

Ali H. Chamseddine
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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.

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Ali H. Chamseddine. 2003-11-17. SL(2,C) Gravity with Complex Vierbein and Its Noncommutative Extension. https://doi.org/10.1103/physrevd.69.024015

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