Cyclic symmetry and adic convergence in Lagrangian Floer theory

Abstract

In this article we use a continuous family of multisections of the moduli space of pseudoholomorphic discs to partially improve the construction of the Lagrangian Floer cohomology of [11] in the case of $\mathbb{R}$ coefficient. Namely, we associate a cyclically symmetric filtered $A_{\infty }$-algebra to every relatively spin Lagrangian submanifold. We use the same trick to construct a local rigid analytic family of filtered $A_{\infty }$-structures associated to a (family of) Lagrangian submanifolds. We include the study of homological algebra of pseudoisotopy of cyclic (filtered) $A_{\infty }$-algebras.