Grassmannian twists, derived equivalences and brane transport

TL;DR

This paper constructs derived equivalences for Grassmannian flops inspired by brane transport in gauged linear sigma models, using staircase complexes and defining autoequivalences as twists and cotwists.

Contribution

It introduces a novel method to produce derived equivalences for Grassmannian flops using grade restriction rules and staircase complexes, connecting physics-inspired ideas with algebraic geometry.

Findings

Derived equivalences for Grassmannian flops are explicitly constructed.

Autoequivalences are described as twists and cotwists around spherical functors.

The approach utilizes staircase complexes and extends to several examples.

Abstract

This note is based on a talk given at String-Math 2012 in Bonn, on a joint paper with Ed Segal. We exhibit derived equivalences corresponding to certain Grassmannian flops. The construction of these equivalences is inspired by work of Herbst-Hori-Page on brane transport for gauged linear sigma models: in particular, we define 'windows' corresponding to their grade restriction rules. We then show how composing our equivalences produces interesting autoequivalences, which we describe as twists and cotwists about certain spherical functors. Our proofs use natural long exact sequences of bundles on Grassmannians known as twisted Lascoux complexes, or staircase complexes. We give a compact description of these. We also touch on some related developments, and work through some extended examples.