# Discrete and continuous symmetries in monotone Floer theory

## TL;DR

This paper explores how symmetries influence the self-Floer theory of monotone Lagrangian submanifolds, revealing constraints on differentials and analyzing special cases with homogeneous Lagrangians in projective spaces.

## Contribution

It introduces methods to incorporate symmetries into Floer theory calculations and applies these to specific homogeneous Lagrangians, uncovering new properties.

## Key findings

- Constraints on Floer differentials due to symmetries
- Vanishing of differentials in certain symmetric cases
- Identification of unusual properties in homogeneous Lagrangians

## Abstract

This paper studies the self-Floer theory of a monotone Lagrangian submanifold $L$ of a symplectic manifold $X$ in the presence of various kinds of symmetry. First we suppose $L$ is $K$-homogeneous and compute the image of low codimension $K$-invariant subvarieties of $X$ under the length-zero closed-open string map. Next we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, incorporating its action on the classes of holomorphic discs. This imposes constraints on the differentials which force them to vanish in certain situations. These techniques are combined to study a family of homogeneous Lagrangians in products of projective spaces, which exhibit some unusual properties.