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

TL;DR

This paper computes a specific part of the closed-open string map for $S^1$-invariant Lagrangians, revealing new insights into their algebraic structures and applications in symplectic geometry.

Contribution

It introduces a method to compute the closed-open string map for invariant Lagrangians, leading to new results on split-generation and non-formality in symplectic topology.

Findings

The equatorial circle on the 2-sphere has a non-formal Fukaya A-infinity algebra in characteristic two.

Applications include split-generation results for real Lagrangians in toric varieties.

Demonstrates the non-formality of certain Lagrangian submanifolds.

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 non-formality 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 non-formal Fukaya A-infinity algebra in characteristic two.