Researchers have established a quantum-level connection between the SU(2) gauge theory of Ashtekar variables, which describes the classical cosmological sector, and the standard minisuperspace formulation. This advance is achieved by implementing gauge-fixing conditions at the quantum level, thereby linking Loop Quantum Gravity (LQG) with the homogeneous Bianchi I and Bianchi IX cosmological models. The key lies in imposing the divergence constraint, which fixes the gauge to a homogeneous one, and the diagonal constraint, which ensures metric diagonality and has been extensively studied.

The work focuses on constructing quantum operators for general cosmological models, using an appropriate regularization in terms of holonomies and fluxes. It is shown that a special class of homogeneous spin-network states, describing the Bianchi I and Bianchi IX models, satisfy the quantum gauge-fixing constraints. This is accomplished using techniques developed within Reduced Quantum Loop Gravity, highlighting the compatibility between these approaches.

This research provides a fundamental quantum link between cosmological theory with full SU(2) gauge symmetry and standard Loop Quantum Cosmology (LQC). The connection is established due to the effective Abelian structure of the selected states, simplifying the transition between descriptions. This result is crucial for understanding how quantum gravity can describe the universe at large scales and in its earliest stages, offering a pathway to explore the physics of the Big Bang and beyond from a rigorous quantum perspective.