The closed-open string map for $S^1$–invariant Lagrangians

Abstract

Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit.

Our applications include split-generation and nonformality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the $2$–sphere carries a nonformal Fukaya $A_{\infty }$ algebra in characteristic $2$.