Coulomb Gas Approach to Conformal Field Theory and Lattice Models

Explore the mathematical connections between Coulomb gas techniques and conformal field theory in this advanced lecture delivered at the Centre International de Rencontres Mathématiques. Delve into the historical development of conformal field theory concepts that emerged through the work of den Nijs and Nienhuis before the formal introduction by Belavin, Polyakov and Zamolodchikov. Examine how Coulomb gas techniques postulate that height functions of statistical mechanics lattice models, including percolation, Ising, and 6-vertex models, converge to the Gaussian Free Field, enabling the derivation of critical exponents and dimensions. Investigate the mysterious nature of this convergence, particularly its traditional formulation on tori and cylinders rather than in the presence of boundaries. Discover recent mathematical progress in extending these formulations to general domains and Riemann surfaces, and understand their relationships to conformal field theory, Schramm-Loewner evolution (SLE), and conformal invariance of critical lattice models. Learn about emerging objects in complex geometry and potential theory that arise from these investigations, presented by a leading expert in the field during this specialized mathematical conference focused on statistical mechanics, combinatorics and geometry.