Floer homology via Twisted Loop Spaces

TL;DR

This paper introduces a new integral approach to Lagrangian Floer homology that overcomes previous restrictions, leading to stronger bounds on Lagrangian self-intersections using algebraic techniques related to loop spaces.

Contribution

It presents a novel method for defining integral Lagrangian Floer homology without the relative Pin restriction, and derives new self-intersection bounds via algebraic properties of loop space chains.

Findings

Established an integral Floer homology for oriented closed exact Lagrangians.

Derived improved self-intersection bounds for certain Lagrangians.

Connected algebraic properties of chain complexes to geometric intersection bounds.

Abstract

Answering a question of Witten, we introduce a novel method for defining an integral version of Lagrangian Floer homology, removing the standard restriction that the Lagrangians in question must be relatively Pin. Using this technique, we derive stronger bounds on the self-intersection of certain exact Lagrangians $\mathbb{RP}^2 \times L'$ than those that follow from traditional methods. We define a integral version of Lagrangian Floer homology all oriented closed exact Lagrangians $L$ in a Liouville domain and prove a general self-intersection bound coming from the algebraic properties of the diagonal bimodule of a twist of the dg-algebra of chains on the based loop space of $L$.