Spherical twists and the center of autoequivalence groups of K3 surfaces

TL;DR

This paper explores the structure of autoequivalence groups of K3 surfaces, focusing on spherical twists, and computes their centers, linking algebraic and symplectic geometry through mirror symmetry.

Contribution

It introduces an intersection number for spherical twists and determines the center of autoequivalence groups of K3 surfaces, advancing understanding of derived category symmetries.

Findings

Computed the center of autoequivalence groups of K3 surfaces.

Introduced an intersection number relating to spherical twists.

Linked algebraic and symplectic perspectives via mirror symmetry.

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 mapping class groups of closed oriented real surfaces, we study spherical twists for derived categories of algebraic varieties. We introduce the intersection number and relate it to group-theoretic properties of spherical twists. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.