Homological Projective Duality for the Pl\"ucker embedding of the Grassmannian

TL;DR

This paper establishes a homological projective duality framework for the Grassmannian's Pl"ucker embedding by describing its Kuznetsov component via matrix factorizations on a noncommutative crepant resolution, extending previous results.

Contribution

It constructs and describes the NCCR of the affine cone of the Grassmannian and extends homological projective duality to this setting, linking categorical resolutions and matrix factorizations.

Findings

Constructed an NCCR for the affine cone of the Grassmannian.

Proved derived equivalence between different NCCRs, demonstrating Hori duality.

Extended HPD framework to the Grassmannian's Pl"ucker embedding.

Abstract

We describe the Kuznetsov component of the Pl\"ucker embedding of the Grassmannian as a category of matrix factorizations on an noncommutative crepant resolution (NCCR) of the affine cone of the Grassmannian. We also extend this to a full homological projective dual (HPD) statement for the Pl\"ucker embedding. The first part is finding and describing the NCCR, which is also of independent interest. We extend results of \v{S}penko and Van den Bergh to prove the existence of an NCCR for the affine cone of the Grassmannian. We then relate this NCCR to a categorical resolution of Kuznetsov. Deforming these categories to categories of matrix factorizations we find the connection to the Kuznetsov component of the Grassmannian via Kn\"orrer periodicity. In the process we prove a derived equivalence between two different NCCR's; this shows Hori duality for the group $SL$. Finally we put this all into the HPD framework.