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

Periodic Reporting for period 2 - CONSTAMIS (Connecting Statistical Mechanics and Conformal Field Theory: an Ising Model Perspective)

Reporting period: 2018-09-01 to 2020-02-29

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 now appears possible. This is the goal of this proposal, which is organized in 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 is to build one of the first rigorous bridges between Statistical Mechanics and CFT. It will help to close the gap between physical derivations and mathematical theorems. By linking the deep structures of CFT to concrete models that are applicable in many subjects, it will be potentially useful to theoretical and applied scientists.
We have found an exact connection between the Ising model at the discrete level and the Conformal Field Theory 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). We have shown the long-standing conjecture that the scaling limit of the Ising model interfaces is CLE(3) (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.
Major progress on all the Problems listed in the project is expected, all will be beyond the state of the art.