Fukaya categories of surfaces, spherical objects, and mapping class groups

TL;DR

This paper characterizes spherical objects in the Fukaya category of surfaces, showing they correspond to simple closed curves with local systems, and establishes a connection between autoequivalence groups and mapping class groups.

Contribution

It proves that certain spherical objects are quasi-isomorphic to simple closed curves with local systems, answering a key question and linking Fukaya categories to mapping class groups.

Findings

Spherical objects correspond to simple closed curves with rank one local systems.

Autoequivalence group surjects onto the mapping class group.

Application to high-dimensional symplectic mapping class groups.

Abstract

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least two whose Chern character represents a non-zero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank one local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions, and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.