Gaussian Free Fields and Conformal Loop Ensembles Coupled with Liouville Quantum Gravity

This monograph explores the rigorous probabilistic coupling between the Gaussian Free Field (GFF), Conformal Loop Ensembles (CLE), and Liouville Quantum Gravity (LQG) surfaces. We analyze the construction of the Liouville measure h = eh d2z on simply connected domains, establishing the convergence of the regularized exponential field in the L2 sense for (0, 2). Furthermore, we detail the canonical isomorphism between the CLE nesting field and the GFF, specifically focusing on the critical case =4 where CLE loops correspond to the level lines of the zero-boundary GFF. We present the "Mating of Trees" framework, demonstrating how two correlated Brownian motions encode the geometry of an LQG sphere decorated with an SLE curve (the "Quantum Zipper"). Finally, we discuss the KPZ relation in the context of fractal dimension scaling within the quantum metric tensor.