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# 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

#### 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.

Received 15 January 2021

Accepted 22 July 2022

Published 27 July 2023