Self-Dual Gravity and the Immirzi parameter

TL;DR

This paper introduces a new gravity action in the first order formalism that unifies self and anti-self-dual curvature components, deriving a (2+1)-dimensional Immirzi parameter linked to classically equivalent actions.

Contribution

It proposes a novel gravity action combining self and anti-self-dual parts, explicitly deriving the (2+1)D Immirzi parameter and connecting it to classical action equivalences.

Findings

Derived the (2+1)-dimensional Immirzi parameter explicitly.

Linked the Immirzi parameter to standard and exotic gravity actions.

Provided a new perspective on gravity as a gauge theory.

Abstract

Working in the first order formalism of gravity, we propose an action that combines the self and anti-self-dual parts of the curvature and comprises all the diffeomorphism invariant Lagrangians that one can consider in this formalism. The action that we propose is motivated by (A)dS gauge theories of gravity. We use this action to derive the (2+1)-dimensional version of the Immirzi parameter. Our derivation relates explicitly the Immirzi parameter to the existence of two classically equivalent actions for the description of gravity in (2+1) dimensions, namely the standard and exotic actions introduced by Witten in the description of (2+1) gravity as a gauge theory. This relation had been conjectured previously in the literature, but not derived.