Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 069, 48 pages      arXiv:1612.05361
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

Loop Models and $K$-Theory

Paul Zinn-Justin
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Received November 28, 2017, in final form June 27, 2018; Published online July 13, 2018

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.

Key words: quantum integrability; loop models; $K$-theory.

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