Gauge connection formulations for general relativity

TL;DR

This paper introduces a new class of gauge connection formulations for general relativity that are invariant under $SO(3, ext{C})$ and diffeomorphisms, providing pure connection actions and analyzing their canonical structure.

Contribution

It presents novel gauge connection formulations for GR depending on a complex 4-form and a holomorphic function, including pure connection actions for nonzero cosmological constant.

Findings

Members have two physical degrees of freedom per point.

Only the Hamiltonian constraint is modified from Ashtekar formulation.

Provides action principles and canonical analysis for the class.

Abstract

We report a new class of $SO(3,\mathbb{C})$ and diffeomorphism invariant formulations for general relativity with either a vanishing or a nonvanishing cosmological constant, which depends functionally on a $SO(3,\mathbb{C})$ gauge connection and a complex-valued 4-form via a holomorphic function of the trace of a symmetric $3\times3$ matrix that is constructed from these variables. We present two members of this class, one of which results from the implementation of a method for obtaining action principles belonging to the class. For the case of a nonvanishing cosmological constant, we solve for the complex-valued 4-form and get pure connection action principles. We perform the canonical analysis of the class. The analysis shows that only the Hamiltonian constraint is modified with respect to the Ashtekar formulation and that the members of the class have two physical degrees of freedom per space point.