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Fukaya categories of surfaces, spherical objects and mapping class groups

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  18 March 2021

Denis Auroux  and
Ivan Smith
Denis Auroux
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA02138, USA; E-mail: auroux@math.harvard.edu.
Ivan Smith
Affiliation:
Centre for Mathematical Sciences, University of Cambridge, CB3 0WB, England; E-mail: is200@cam.ac.uk.

Abstract

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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.

MSC classification

Primary: 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Type
Topology
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e26
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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