Grassmann-Grassmann conormal varieties, integrability, and plane partitions

TL;DR

This paper proposes a conjectural formula linking sheaves on conormal varieties in Grassmannians with integrable loop models and quantum KZ equations, providing geometric insights into their algebraic and combinatorial structures.

Contribution

It introduces a conjectural formula connecting sheaves on conormal varieties with integrable models and proves it in smooth cases, linking geometry, algebra, and combinatorics.

Findings

Sheaves supported on conormal varieties relate to integrable loop model partition functions.

The $K$-theoretic pushforward solves the level 1 quantum Knizhnik-Zamolodchikov equation.

Degeneration of sheaves reveals a connection to plane partitions and Razumov-Stroganov correspondence.

Abstract

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant $K$-class is given by the partition function of an integrable loop model, and furthermore their $K$-theoretic pushforward to a point is a solution of the level $1$ quantum Knizhnik-Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov-Stroganov correspondence satisfied by the loop model.