Floer cohomology of real Lagrangians in the Fermat quintic threefold

TL;DR

This paper computes Floer cohomology between a family of real Lagrangian submanifolds in the Fermat quintic threefold, developing new techniques to analyze low energy moduli spaces and their differentials.

Contribution

It introduces methods for calculating Floer cohomology of real Lagrangians in the Fermat quintic, including formulas for Maslov index and obstruction bundles, leveraging their real structure.

Findings

Successfully computed Floer cohomology for most pairs of real Lagrangians.

Developed formulas for Maslov index and obstruction bundles.

Established relations between holomorphic strips, discs, and spheres.

Abstract

Let X be the Fermat quintic threefold. The set of real solutions L forms a Lagrangian submanifold of X. Multiplying the homogeneous coordinates of X by various fifth roots of unity gives automorphisms of X; the images of L under these automorphisms defines a family of 625 different Lagrangian submanifolds, called real Lagrangians. In this paper we try to calculate the Floer cohomology between all pairs of these Lagrangians. We are able to complete most of the calculations, but there are a few cases we cannot do. The basic idea is to explicitly describe some low energy moduli spaces and then use this knowledge to calculate the differential on the E_2 page of the standard spectral sequence for Floer cohomology. It turns out that this is often enough to calculate the cohomology completely. Several techniques are developed to help describe these low energy moduli spaces, including a formula for the Maslov index, a formula for the obstruction bundle, and a way to relate holomorphic strips and discs to holomorphic spheres. The real nature of the Lagrangians is crucial for the development of these techniques.