Autoequivalences of derived categories via geometric invariant theory

TL;DR

This paper investigates autoequivalences of derived categories arising from GIT quotients, showing they are spherical twists linked to mutations, and extends these ideas to dg-categories, with implications for mirror symmetry and quantum monodromy.

Contribution

It demonstrates that autoequivalences from GIT variations are spherical twists derived from mutations, and generalizes this to dg-categories, connecting to mirror symmetry predictions.

Findings

Autoequivalences are spherical twists from mutations.

All spherical twists in dg-categories can be obtained via mutation.

Additional autoequivalences relate to monodromy of quantum connections.

Abstract

We study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these automorphisms are spherical twists, and describe how they result from mutations of semiorthogonal decompositions. Beyond the GIT setting, we show that all spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner. Motivated by a prediction from mirror symmetry, we refine the recent notion of "grade restriction rules" in equivariant derived categories. We produce additional derived autoequivalences of a GIT quotient and propose an interpretation in terms of monodromy of the quantum connection. We generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists.