A topological limit of gravity admitting an SU(2) connection formulation

TL;DR

This paper proves that a certain limit of gravity theory, characterized by an SU(2) connection, is topological and explores its potential implications for understanding black hole entropy in loop quantum gravity.

Contribution

It demonstrates that the topological limit of gravity with an SU(2) connection formulation is equivalent to a topological field theory, providing new insights into quantum gravity.

Findings

The theory is shown to be topological in both covariant and Dirac formulations.

In the time gauge, the phase space matches that of Ashtekar-Barbero variables.

Quantization of this theory could illuminate the origin of black hole entropy.

Abstract

We study the Hamiltonian formulation of the generally covariant theory defined by the Lagrangian 4-form L=e_I e_J F^{IJ}(\omega) where e^I is a tetrad field and F^{IJ} is the curvature of a Lorentz connection \omega^{IJ}. This theory can be thought of as the limit of the Holst action for gravity for the Newton constant G goes to infinity and Immirzi parameter goes to zero, while keeping their product fixed. This theory has for a long time been conjectured to be topological. We prove this statement both in the covariant phase space formulation as well as in the standard Dirac formulation. In the time gauge, the unconstrained phase space of theory admits an SU(2) connection formulation which makes it isomorphic to the unconstrained phase space of gravity in terms of Ashtekar-Barbero variables. Among possible physical applications, we argue that the quantization of this topological theory might shed new light on the nature of the degrees of freedom that are responsible for black entropy in loop quantum gravity.