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Connecting Statistical Mechanics and Conformal Field Theory: an Ising Model Perspective

Periodic Reporting for period 4 - CONSTAMIS (Connecting Statistical Mechanics and Conformal Field Theory: an Ising Model Perspective)

Reporting period: 2021-09-01 to 2022-02-28

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.
We have found an exact connection between the Ising model at the discrete level and the Conformal Field Theory (CFT) Virasoro structures, answering a long-standing question (work with Kytölä and Viklund). We have constructed explicit bridges between local fields of the Ising model and Conformal Field Theory, obtaining new exact formulae and introducing new methods (work with Gheissari and Park); further down the line all correlation functions for Ising model primaries, including quasi-local ones, for general multiply connected domains were shown to have conformally covariant scaling limits, proving a number of long-standing conjectures (with Chelkak and Izyurov). Concerning the random curves of the Ising model, the long-standing conjecture that the scaling limit of the Ising model interfaces is CLE(3) was solved (work with Benoist).
The study of the massive Ising model has been developed by Park for his PhD thesis, and a new derivation of the Painlevé equations for the two-spin correlations in the plane has been obtained. In a related line of work, new exact finite-size formulae for the Ising model for layered half-plane setup (generalizing the arbitrary temperature case) were obtained, showing interesting connections with orthogonal polynomials (with Chelkak and Mahfouf).
The study of generalized models and field theories led to surprising insights in the mathematical study of Machine Learning methods: for instance, random matrix theory tools, such as the Stieltjes transform could be used to relate random feature models (counterparts of lattice models) and the corresponding kernel methods (counterparts of field theories), in a way that reminds the study of CFT correlation functions.
Also, this project has led to the obtention of new results linking lattice models and quantum gravity field theories. This work lays down a novel formalism of discrete geometry whose generality and naturality make it possible to consider a very general notion of field among which models of Statistical Mechanics.
Finally, this work has led to the writing of a large monograph synthesizing and clarifying a number of insights obtained during the project about the connections between lattice models and CFTs, constructing a dictionary between lattice models and their CFT counterparts: this includes among other things, a clarification of the connection between local fields and their lattice counterparts, of the geometric nature of the so-called Ward identities, of the origin of unitarity from reflection-positivity, and on the classification of the unitary minimal models.
This work has been presented at various venues: conferences, summer schools, colloquia, and seminar talks. It has led to two PhD theses.
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