Combinatorics of elliptic lattice models

Research Project, 2021 – 2024

We will explore new emerging connections between several different areas of mathematics, including branches of algebra, analysis, combinatorics and mathematical physics. In particular, we will study mathematical properties of elliptic lattice models in statistical mechanics, such as the eight-vertex model, the three-colour model and the XYZ spin chain. We are particularly interested in elliptic versions of the Razumov-Stroganov correspondence, which should relate ground states of the XYZ spin chain to the combinatorics of three-colourings and loop configurations. This topic has close connections to the theory of affine Lie algebras and to Painlevé equations.We will work on several specific starting points for trying to understand the elliptic Razumov-Stroganov correspondence. We will study three-colourings with various boundary conditions. We will try to obtain explicit expressions for ground states of the supersymmetric eight-vertex model, and study three-colourings corresponding to simple fixed link patterns in the description as fully packed loops. We will also try to compute the emptiness formation probability for the supersymmetric XYZ spin chain and for the free-fermionic eight-vertex-solid-on-solid model.