Geodesic
Fukaya categories of surfaces, spherical objects and mapping class groups
Forum of Mathematics, Sigma, Tome 9 (2021)

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We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ 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. 
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     author = {Denis Auroux and Ivan Smith},
     title = {Fukaya categories of surfaces, spherical objects and mapping class groups},
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Denis Auroux; Ivan Smith. Fukaya categories of surfaces, spherical objects and mapping class groups. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.21

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