Fiber bundle structure in Ashtekar-Barbero-Immirzi formulation of general relativity

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(2) 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(2)-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(2)-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.