Derived autoequivalences of length 2 flops via GIT

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 Kähler 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.