Cosmology in Loop Quantum Gravity: symmetry reduction preserving gauge degrees of freedom

In this manuscript, we address the issue of the loss of SU(2) internal symmetry in Loop Quantum Cosmology and its relationship with Loop Quantum Gravity. Drawing inspiration from Yang-Mills theory and employing the framework of fiber bundle theory, we propose a new gauge-invariant symmetry-reduction approach. Using this method, we successfully identify a cosmological sector of General Relativity in terms of Ashtekar variables that preserves the SU(2) structure of the theory as well as part of the diffeomorphism gauge symmetry. Additionally, we analyze the properties of cylindrical functions and the classical constraint algebra, revealing that certain cylindrical functions exhibit distinctive symmetry features.