The developments of Statistical Mechanics and Quantum Field Theory are among the major achievements of the 20th century's science. In the second half of the century, these two subjects started to converge. In two dimensions, this resulted in one of the most remarkable chapters of mathematical physics: Conformal Field Theory (CFT) reveals deep structures allowing for extremely precise investigations, making such theories powerful building blocks of many subjects of mathematics and physics. Unfortunately, this convergence has remained non-rigorous, leaving most of the spectacular field-theoretic applications to Statistical Mechanics conjectural.
About 15 years ago, several mathematical breakthroughs shed new light on this picture. The development of SLE curves and discrete complex analysis has enabled one to connect various statistical mechanics models with conformally symmetric processes. Recently, major progress was done on a key statistical mechanics model, the Ising model: the connection with SLE was established, and many formulae predicted by CFT were proven.
Significant progress towards connecting Statistical Mechanics and CFT appeared possible. This was the goal of this proposal, organized along three objectives:
• (I)Establish a complete connection between the Ising model and CFT: develop a deep correspondence between the objects and structures arising in Ising and CFT frameworks.
• (II)Gather the insights of (I) to study new connections to CFT, particularly for minimal models, current algebras and parafermions.
• (III)Combine the results of (I) and (II) to go beyond conformal symmetry: link the Ising model with massive integrable field theories.
The aim was to build one of the first rigorous bridges between Statistical Mechanics and CFT and to help to close the gap between physical derivations and mathematical theorems.
This project has led to a number of achievements, some planned and some unexpected:
Part (I) led in particular to a complete description of the Ising model primary fields, both local and quasi-local ones, to an identification of a new Virasoro structure at the lattice level, to the definition of lattice local fields, and a proof of convergence to the first orders. It also led one to the proof of the long-conjectured connection between the Ising model loops and the so-called Conformal Loop Ensembles.
Part (II) led to new insights into the connection between parafermions and supersymmetric conformal field theories, to the investigation of discrete models related to quantum gravity and, surprisingly, to results on random features applied to machine learning.
Part (III) led to a proof of convergence of massive Ising observables to solutions of massive Cauchy-Riemann equations, yielding in particular a simple derivation of the Painless III equations for the massive Ising model and to the first proof of convergence of massive spin correlations in the scaling limit.
Overall, this project has revealed a new system of structures and given new insights into lattice models, that will inspire much new research, and that connects nicely with applied research.
