Maurer-Cartan deformation of Lagrangians

TL;DR

This paper explores the Maurer-Cartan algebra of a Lagrangian, linking it to wrapped Floer cohomology and mirror symmetry, with explicit calculations demonstrating the theoretical results.

Contribution

It establishes a natural isomorphism between the Maurer-Cartan algebra and a subspace of wrapped Floer cohomology for dual Lagrangians, connecting deformation theory and mirror symmetry.

Findings

Identifies Maurer-Cartan algebra with the 0-th cohomology of the Koszul dual dga.

Proves an isomorphism between Maurer-Cartan algebra and wrapped Floer cohomology for dual Lagrangians.

Provides explicit examples illustrating the theoretical framework.

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 \emph{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.