Loop Models and $K$-Theory

TL;DR

This paper reviews the connection between exactly solvable two-dimensional loop models in statistical mechanics and the equivariant K-theory of Grassmannian cotangent bundles, providing geometric interpretations and explicit formulas.

Contribution

It introduces a geometric perspective on integrable systems concepts and offers explicit K-theory formulas for sheaves related to Schubert varieties.

Findings

Explicit formulas for K-classes of coherent sheaves

Geometric interpretation of R-matrix and partition functions

Connections between integrable models and algebraic geometry

Abstract

This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems ($R$-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for $K$-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties.