Grassmann-Grassmann conormal varieties, integrability, and plane partitions
Abstract
We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant -class is given by the partition function of an integrable loop model, and furthermore their -theoretic pushforward to a point is a solution of the level 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.
Cite
@article{arxiv.1612.04465,
title = {Grassmann-Grassmann conormal varieties, integrability, and plane partitions},
author = {A. Knutson and P. Zinn-Justin},
journal= {arXiv preprint arXiv:1612.04465},
year = {2016}
}