Solvable Lattice Models

Solvable lattice models arose in statistical mechanics, where they were used to model systems such as ice and (via a relationship with quantum mechanics) Heisenberg spin chains. But beyond this origin in physics, solvable lattice models have emerged as a powerful multi-purpose tool. They have applications to algebraic combinatorics and related areas in algebraic geometry, to representations of p-adic groups, integrable probability and other areas. So this topic will be of interest beyond its origins in physics.

Baxter noticed that there is some machinery behind solvability based on the Yang-Baxter equation, and others, particularly Drinfeld, showed how the representation theory of quantum groups explains the Yang-Baxter equation. Exactly the same machinery underlies knot invariants such as the Jones polynomial.

We will try to strike a balance between examining the underlying theory, which comes from the representation theory of quantum groups, and looking at examples and applications.

We will discuss the instructive example of the six-vertex model in detail. There are two main kinds of integrable six-vertex models, corresponding to quantum affine $\mathfrak{gl}(2)$ and the superalgebra $\mathfrak{gl}(1|1)$. These are the field-free and free-fermionic six vertex models. Their theories are instructively different. The field free case is connected with realistic physical systems and also led to Kuperberg's famous proof of the alternating sign matrix conjecture. On the other hand, the free-fermionic case is the prototype of many applications in algebraic combinatorics. We will study both, and we will also study related bosonic models (including stochastic models). We will also look at the eight-vertex model. We will then study colored variants which are related to quantum affine $\mathfrak{gl}(n)$ and $\mathfrak{gl}(m|n)$. There we will see representations of affine Hecke algebras through Demazure-Lusztig operators. These same representations appear in other areas of mathematics, implying surprising connections.

Lecture Notes

Here are all the lectures as a single file: Here are the individual lectures.



San Francisco

In January 2024, there will be a Special Session on Solvable Lattice Models at the Joint Mathematical Meetings.