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

TL;DR

This work analyzes the topological limit of gravity obtained from the Holst action by sending $G\to\infty$ and $\gamma\to0$ with $G\gamma$ fixed, yielding $I_0=\int e^I\wedge e^J\wedge F_{IJ}$. Through covariant phase-space and Dirac canonical analyses, the authors show the theory is topological in the bulk (no local degrees of freedom) but can host boundary degrees of freedom when a boundary is present; in the time gauge, the theory reduces to an $SU(2)$ connection formulation with variables $(A^i_a,E^a_i)$, mirroring Ashtekar-Barbero gravity. The constraint structure comprises first- and second-class sectors that, after reduction and gauge fixing, expose a reduced action $I_{red}[A,E]$ describing a background-independent $SU(2)$ gauge theory with no local DOF. These results suggest a possible boundary-origin of black hole entropy in loop quantum gravity and provide a simplified arena to test LQG quantization techniques on a theory sharing the same field content as GR in first-order form.

Abstract

We study the Hamiltonian formulation of the generally covariant theory defined by the Lagrangian 4-form L=e_I e_J F^{IJ}(ω) where e^I is a tetrad field and F^{IJ} is the curvature of a Lorentz connection ω^{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.