The Lagrangian flux-monodromy morphism

We define a new morphism on the fundamental group of the space of Lagrangian submanifolds, which takes into account the Lagrangian flux and monodromy. We study the discreteness of its image and relate this discreteness to the ($C^{\mathrm{\infty }}$) topology of the Hamiltonian orbit of the Lagrangian in question. We completely describe this new morphism for Lagrangian tori in ${\mathbb{R}}^4$ and $\mathbb{C}{P}^2$ and give explicit constructions. Similar constructions allow us to study the shape invariant of the Clifford torus in $\mathbb{C}{P}^n$. Finally, we upgrade our results on Lagrangian tori in ${\mathbb{R}}^4$ from the $C^{\mathrm{\infty }}$ topology to the Hausdorff one.