This document summarizes the activities and outcomes of the Long Program “Geometry, Statistical Mechanics, and Integrability,” held at the Institute of Pure and Applied Mathematics (IPAM) from March 11 to June 14, 2024. We also briefly explore current open questions and future directions in the field.

In the last 20-30 years, probability theory and statistical mechanics have been revitalized with the introduction of various tools, notably conformal geometry and discrete analyticity, as well as algebraic geometry and integrable systems.

Many familiar statistical mechanics models, such as the Ising model, dimer model, spanning tree model, and their cousins, have an intrinsic underlying geometric structure. For example, discrete analytic geometry was used by Kenyon, Lawler, Schramm, Werner, Smirnov, and Chelkak in their proofs of conformal invariance of scaling limits. Other work on the dimer model has led to connections with hyperbolic geometry, Lorentzian geometry, and symplectic geometry.

Recent connections between classical and discrete geometric structures on surfaces and combinatorial models such as the dimer and six-vertex models have revealed a significant connection to integrable systems and discrete geometry.

There are well-known connections between statistical mechanics models, algebraic combinatorics and representation theory: they have a common interest in Young tableaux, Gelfand–Tsetlin patterns, Knutson–Tao puzzles, and Littlewood–Richardson coefficients and their generalizations, for example. The Bethe Ansatz and Yang–Baxter equations were developed for the six-vertex model but are now fundamental tools in combinatorial representation theory, also giving explicit connections with integrability.

The application of conformal field theory (CFT) to statistical mechanics has been another pivotal area of research. CFT has proven to be a powerful tool for describing the scaling limits of critical lattice models. Schramm Loewner Evolution (SLE) and its variants were instrumental in understanding conformal invariance and scaling limits. Extending these methods to higher-rank models remains a complex and open question.

The program brought together many researchers from this somewhat disparate realm of ideas, united by the underlying themes of geometry and statistical mechanics. Activities included introductory tutorials, four workshops, and eight working groups. The groups studied recent literature and open problems in areas such as adapted geometric embeddings, vertex models, and asymptotic algebraic combinatorics.

Multiple directions for future research emerged from the program. Higher rank versions of statistical mechanics models have mysterious and unexplored connections with more sophisticated CFTs including W-algebras, as well as deeper connections with representation theory. Recent progress connecting the six-vertex model with symmetric polynomials gives a new direction of exploration towards a better understanding of asymptotic behavior of structure constants and other algebraic quantities.

The progress connecting geometries to free fermionic models leads us to analogous questions for the six-vertex model and beyond: is there an appropriate discrete geometry underlying any critical statistical mechanics model? There is clearly much to be explored!

Read the full report.