Plebanski-like action for general relativity and anti-self-dual gravity

TL;DR

This paper introduces a new $BF$-type action for complex general relativity that simplifies the formulation and unifies it with anti-self-dual gravity, offering advantages over the traditional Plebanski approach.

Contribution

It presents a novel $BF$-type action where the symmetric matrix can be integrated out, unifying general relativity and anti-self-dual gravity within a single framework.

Findings

The new action can be reduced to a pure $BF$-type form for GR.

The canonical analysis aligns with the Ashtekar formalism.

A specific parameter choice yields anti-self-dual gravity.

Abstract

We present a new $BF$-type action for complex general relativity with or without a cosmological constant resembling Plebanski's action, which depends on an SO(3,$\mathbb{C}$) connection, a set of 2-forms, a symmetric matrix, and a 4-form. However, it differs from the Plebanski formulation in the way that the symmetric matrix enters into the action. The advantage of this fact is twofold. First, as compared to Plebanski's action, the symmetric matrix can now be integrated out, which leads to a pure $BF$-type action principle for general relativity; the canonical analysis of the new action then shows that it has the same phase space of the Ashtekar formalism up to a canonical transformation induced by a topological term. Second, a particular choice of the parameters involved in the formulation produces a $BF$-type action principle describing conformally anti-self-dual gravity. Therefore, the new action unifies both general relativity and anti-self-dual gravity.