Fukaya categories of surfaces, spherical objects and mapping class groups

Abstract We prove that every spherical object in the derived Fukaya category of a closed surface genus at least $2$ whose Chern character represents nonzero Hochschild homology class is quasi-isomorphic to simple curve equipped with rank $1$ local system. (The homological hypothesis necessary.) This largely answers question Haiden, Katzarkov and Kontsevich. It follows there natural surjection from autoequivalence group mapping group. The proofs appeal illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying category, equivariant Floer theory finite continuous actions mirror symmetry. An application high-dimensional symplectic groups included.