Derived autoequivalences of length 2 flops via GIT

TL;DR

This paper uses GIT and the theory of windows to explicitly construct derived autoequivalences for length 2 flops, connecting the stringy K"ahler moduli space with stability conditions and monodromy actions.

Contribution

It provides a new GIT-based method to derive autoequivalences for length 2 flops and links the SKMS with stability and monodromy frameworks.

Findings

Derived autoequivalences explicitly constructed via GIT.

The SKMS matches the description from stability manifold quotients.

Fundamental group actions recover known monodromy phenomena.

Abstract

We obtain the derived autoequivalences of a flopping rational curve of length 2 using GIT and the theory of windows applied to the universal length 2 flop. We show that the stringy K\"ahler moduli space (SKMS) associated to the GIT problem, as constructed by Halpern-Leistner--Sam, matches the description of the space obtained for length 2 threefolds by Hirano--Wemyss as a quotient of a Bridgeland stability manifold. Furthermore, we show that its fundamental group acts via contraction algebra and fibre algebra twists, hence recovering the monodromy action described by Donovan--Wemyss. In particular, this shows that the two approaches to building the SKMS agree in this setting.