Homological Lagrangian monodromy for some monotone tori

Abstract

Given a Lagrangian submanifold L in a symplectic manifold X, the homological Lagrangian monodromy group HL 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 HL 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.