Homological Lagrangian monodromy for some monotone tori

TL;DR

This paper investigates the homological Lagrangian monodromy group for monotone Lagrangian tori, providing a complete description for monotone toric fibers, advancing classification for dimension two, and proposing a conjecture for higher dimensions.

Contribution

It offers a systematic study of the homological Lagrangian monodromy group for monotone tori, including a complete classification for toric fibers and progress towards understanding the general case.

Findings

Complete description of the group for monotone toric fibers

Progress towards classification in dimension two

Conjecture for the general case in higher dimensions

Abstract

Given a Lagrangian submanifold $L$ in a symplectic manifold $X$, the homological Lagrangian monodromy group $\mathcal{H}_L$ describes how Hamiltonian diffeomorphisms of $X$ preserving $L$ setwise act on $H_*(L)$. We begin a systematic study of this group when $L$ is a monotone Lagrangian $n$-torus. Among other things, we describe $\mathcal{H}_L$ completely when $L$ is a monotone toric fibre, make significant progress towards classifying the groups than can occur for $n=2$, and make a conjecture for general $n$. Our classification results rely crucially on arithmetic properties of Floer cohomology rings.