General relativity with a topological phase: an action principle

TL;DR

The paper presents a unified action principle that couples general relativity to a topological $SO(5)$ gauge theory by promoting a symmetry-breaking element to a dynamical field $\Gamma$ constrained by $\Gamma^2=1$. Solving the resulting equations yields four disconnected sectors: a topological $F\wedge F$ theory, a genuine BF theory, and two gravitational theories corresponding to Palatini GR and Ashtekar–Sen variables, including an Immirzi parameter fixed at 1. It shows that phase transitions between these sectors are first-order, and that phase boundaries resemble horizons under appropriate boundary conditions, providing a concrete framework to study dynamical transitions between topological and gravitational phases. The work also discusses how a boundary between phases is governed by isolated-horizon-like conditions and hints at richer boundary dynamics when $\Gamma$ is allowed to vary in spacetime.

Abstract

An action principle is described which unifies general relativity and topological field theory. An additional degree of freedom is introduced and depending on the value it takes the theory has solutions that reduce it to 1) general relativity in Palatini form, 2) general relativity in the Ashtekar form, 3) $F\wedge F$ theory for SO(5) and 4) $BF$ theory for SO(5). This theory then makes it possible to describe explicitly the dynamics of phase transition between a topological phase and a gravitational phase where the theory has local degrees of freedom. We also find that a boundary between adymnamical and topological phase resembles an horizon.