Anchored Lagrangian submanifolds and their Floer theory

TL;DR

This paper introduces anchored Lagrangian submanifolds to develop a new version of Floer homology that better connects with spectral invariants and variational principles, offering a refined framework for Lagrangian Floer theory.

Contribution

It defines graded anchored Lagrangian submanifolds and constructs an anchored Floer homology, providing a more natural link to spectral invariants and variational methods in symplectic topology.

Findings

Developed an anchored Floer homology theory for Lagrangian submanifolds.

Established the relation between anchored Floer theory and spectral invariants.

Discussed rationality conditions and coefficient ring reductions in Floer cohomology.

Abstract

We introduce the notion of (graded) anchored Lagrangian submanifolds and use it to study the filtration of Floer' s chain complex. We then obtain an anchored version of Lagrangian Floer homology and its (higher) product structures. They are somewhat different from the more standard non-anchored version. The anchored version discussed in this paper is more naturally related to the variational picture of Lagrangian Floer theory and so to the likes of spectral invariants. We also discuss rationality of Lagrangian submanifold and reduction of the coefficient ring of Lagrangian Floer cohomology of thereof.