The closed-open string map for S1–invariant Lagrangians

Tonkonog, Dmitry

doi:10.2140/agt.2018.18.15

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The closed-open string map for $S^1$–invariant Lagrangians

Dmitry Tonkonog

Algebraic & Geometric Topology 18 (2018) 15–68

DOI: 10.2140/agt.2018.18.15

Abstract

Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit.

Our applications include split-generation and nonformality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the $2$ –sphere carries a nonformal Fukaya $A_{\infty}$ algebra in characteristic $2$ .

Keywords

circle action, Lagrangian submanifold, Floer homology, closed-open map, Fukaya category, split-generation, formality

Mathematical Subject Classification 2010

Primary: 53D37, 53D40, 57R58

Secondary: 53D45

References

Publication

Received: 8 April 2015

Revised: 19 May 2017

Accepted: 11 June 2017

Published: 10 January 2018

Authors

Dmitry Tonkonog
Department of Pure Mathematics and Mathematical Statistics University of Cambridge Cambridge United Kingdom

Volume 18, issue 1 (2018)