Fiber bundle structure in Ashtekar-Barbero-Immirzi formulation of General Relativity

Abstract

We aim to provide a rigorous geometric framework for the Ashtekar-Barbero-Immirzi formulation of General Relativity. As the starting point of this formulation consists in recasting General Relativity as an $SU\left(2\right)$ gauge theory, it naturally lends itself to interpretation within the theory of principal bundles. The foundation of our framework is the spin structure, which connects the principal $SU\left(2\right)$-bundle construction with the Riemannian framework. The existence of the spin structure enlightens the geometric properties of the Ashtekar-Barbero-Immirzi-Sen connection and the topological characteristics of the manifold. Within this framework, we are able to express the constraints of the physical theory in a coordinate-free way, using vector-valued forms that acquire a clear geometric interpretation.

Using these geometric concepts, we analyze the phase space of the theory and discuss the implementation of symmetries through the automorphism group of the principal $SU\left(2\right)$-bundle. In particular, we demonstrate that the description of the kinematical constraints as vector-valued forms provides a natural implementation as momentum maps for the automorphism group action.