Homological Lagrangian monodromy for some monotone tori

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 $H_{\ast }(L)$. 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.