Derived autoequivalences of length 2 flops via GIT

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.