Maurer-Cartan deformation of Lagrangians

The Maurer-Cartan algebra of a Lagrangian L is the algebra that encodes the deformation of the Floer complex CF(L,L;Λ) as an A_∞-algebra. We identify the Maurer-Cartan algebra with the 0-th cohomology of the Koszul dual dga of CF(L, L; Λ). 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.