Loop Models and $K$-Theory

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.

References in corpus (5)

• Lectures on K-theoretic computations in enumerative geometry
• Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety
• On the K-theory stable bases of the Springer resolution
• Singularities of closures of spherical $B$-conjugacy classes of nilpotent orbits
• Parabolic Kazhdan-Lusztig basis, Schubert classes, and equivariant oriented cohomology