Journal of Symplectic Geometry

Volume 21 (2023)

Number 1

Maurer–Cartan deformation of Lagrangians

Pages: 1 – 71

DOI: https://dx.doi.org/10.4310/JSG.2023.v21.n1.a1

Author

Hansol Hong (Department of Mathematics, Yonsei University, Seoul, South Korea)

Abstract

The Maurer–Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer–Cartan algebra with the 0‑th cohomology of the Koszul dual dga of $CF(L,L;\Lambda)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer–Cartan algebra of $L$ and a suitable subspace of the completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is dual to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.

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Received 15 January 2021

Accepted 22 July 2022

Published 27 July 2023