Window categories for a simple $9$-fold flop of Grassmannian type
Will Donovan, Wahei Hara, Michał Kapustka, Marco Rampazzo
TL;DR
The paper proves a derived equivalence for the local simple flop of type $A_4^G$ between the total spaces $X_+$ and $X_-$, realized via four distinct window categories built from tilting bundles on an ambient Artin stack. By constructing explicit tilting bundles and establishing vanishing results for Ext groups, the authors obtain four $R$-linear equivalences that commute with the flop pushforwards and yield noncommutative crepant resolutions with End algebras isomorphic on both sides. This framework not only re-derives the known derived equivalence for the flop but also applies to Calabi–Yau threefolds arising as zero loci of sections in Grassmannians, providing a new proof even in singular settings. The approach connects geometric invariant theory, window categories, and GLSM physics, revealing four fundamental window choices that align with physical phases and offering a robust method to study CY pairs that are not birationally related. Overall, the work extends the tilting-bundle/window paradigm to a broad class of simple flops beyond the classical Atiyah and Mukai cases and clarifies the landscape of NCCRs associated to these birational transitions.
Abstract
The local simple $9$-fold flop of Grassmannian type is a birational transformation between total spaces of vector bundles on the Grassmannians $\mathrm{Gr}(2, 5)$ and $\mathrm{Gr}(3, 5)$. We produce four different derived equivalences which commute with the pushforward functors for the flopping contractions. These equivalences are realized by identifying four different window categories inside the derived category of coherent sheaves on an Artin stack. As an application, our approach provides a new proof of derived equivalence for a pair of non-birational Calabi-Yau threefolds realized as zero loci of sections of homogeneous vector bundles in Grassmannians.