A note on mapping class group actions on derived categories

TL;DR

This paper explores how the pure mapping class group of a punctured symplectic torus acts on the derived category of a cycle of projective lines, linking geometric group actions to algebraic categories via homological mirror symmetry.

Contribution

It defines a new action of the pure mapping class group on the derived category of a degenerate elliptic curve using spherical twists, advancing the understanding of mirror symmetry.

Findings

Established an explicit group action on derived categories

Connected geometric symmetries with algebraic autoequivalences

Contributed to the study of homological mirror symmetry for degenerate elliptic curves

Abstract

Let $X_n$ be a cycle of $n$ projective lines, and $\bT_n$ a symplectic torus with $n$ punctures. Using the theory of spherical twists introduced by Seidel and Thomas (2001), I will define an action of the pure mapping class group of $\bT_n$ on $D^b(Coh(X_n))$. The motivation comes from homological mirror symmetry for degenerate elliptic curves, which was studied by the author with Treumann and Zaslow in arXiv:1103.2462 .