Spherical twists, relations and the center of autoequivalence groups of K3 surfaces

TL;DR

This paper explores the properties of spherical twists in derived categories, introduces an intersection number concept, and applies these to compute the center of autoequivalence groups of K3 surfaces, linking algebraic and symplectic geometry.

Contribution

It introduces an intersection number for spherical twists, relates it to group properties, and computes the autoequivalence center for K3 surfaces, advancing understanding of derived category symmetries.

Findings

Established an inequality for intersection numbers under spherical twist iterations

Classified subgroups generated by two spherical twists

Computed the center of autoequivalence groups for K3 surfaces

Abstract

Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg-enhanced triangulated categories. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. We show an inequality analogous to a fundamental inequality in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists. We also classify the subgroups generated by two spherical twists using the intersection number. In passing, we prove a structure theorem for finite dimensional dg-modules over the graded dual numbers and use this to describe the autoequivalence group. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.