Emergence of Gravity's Dynamical and Topological Sectors from Pre-geometry

Abstract

We identify the complete set of fundamental building blocks for a 4D pre-geometric theory of gravity. Based on a gauge theory of $SO(1,4)$ or $SO(2,3)$ coupled to a Higgs-like field under the rigid constraint of general covariance in the unbroken phase, these building blocks - $\mathcal{L}_\text{MM}$, $\mathcal{L}_\text{W}$, $\mathcal{L}_{J}$, $\mathcal{L}_θ$ and $\mathcal{L}_\vartheta$ - constitute the minimal generating set of independent field monomials from which any pre-geometric action, including arbitrary functionals thereof, can be constructed. While the most general pre-geometric theory can extend beyond linear combinations, these five irreducible invariants serve as the 'atomic' constituents of all possible pre-geometric dynamics. Upon spontaneous symmetry breaking, they collectively generate an emergent gravitational theory consisting of the Einstein-Hilbert action, the cosmological constant term and all 4D topological invariants: the Gauss-Bonnet, Pontryagin, Holst and Nieh-Yan terms. The unification of gravity's dynamical and topological sectors from a common pre-geometric source represents the central result of this work. We also uncover a see-saw mechanism linking the Planck mass and the cosmological constant, as well as several novel relations for the coupling constants of the topological sector, inclusive of the Barbero-Immirzi parameter. This framework establishes the pre-geometric foundations from which all aspects of gravitation can dynamically emerge, providing a unified starting point for quantum gravity, dark energy phenomenology and the study of topological phases in gravitational theories.