Topological limit of gravity admitting anSU(2)connection formulation

Abstract

We study the Hamiltonian formulation of the generally covariant theory defined by the Lagrangian 4-form $\mathcal{L}={e}_{I}\ensuremath{\wedge}{e}_{J}\ensuremath{\wedge}{F}^{IJ}(\ensuremath{\omega})$, where ${e}^{I}$ is a tetrad field and ${F}^{IJ}$ is the curvature of a Lorentz connection ${\ensuremath{\omega}}^{IJ}$. This theory can be thought of as the limit of the Holst action for gravity for the Newton constant $G\ensuremath{\rightarrow}\ensuremath{\infty}$ and Immirzi parameter $\ensuremath{\gamma}\ensuremath{\rightarrow}0$, while keeping the product $G\ensuremath{\gamma}$ 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.