Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops

TL;DR

This paper constructs a non-commutative crepant resolution for minimal nilpotent orbit closures of type A, linking it to crepant resolutions and derived equivalences, and introduces multi-mutation as a key operation.

Contribution

It provides an explicit construction of an NCCR as a path algebra of a double Beilinson quiver and relates it to crepant resolutions and derived category autoequivalences.

Findings

NCCR is isomorphic to the path algebra of the double Beilinson quiver.

Crepant resolutions are realized as moduli spaces of quiver representations.

Multi-mutation corresponds to P-twists and is a composition of Iyama-Wemyss mutations.

Abstract

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ in terms of an NCCR. We also show that the P-twist on the derived category of $Y$ corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama-Wemyss's mutations.