Monotone Lagrangian submanifolds of ${\mathbb{C}}^n$ and toric topology

Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside ${\mathbb{C}}^n$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$. In this paper we develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano. In some examples, we further compute the minimal Maslov numbers. Namely, let $\mathcal{N}\to T^k$ be some fibration over the $k$-dimensional torus with a fiber equal to either $S^k\times S^l$, or $S^k\times S^l\times S^m$, or ${\mathrm{\#}}_5(S^{2p-1}\times S^{n-2p-2})$. We construct monotone Lagrangian embeddings $\mathcal{N}\subset {\mathbb{C}}^n$ with different minimal Maslov number, and therefore distinct up to Lagrangian isotopy. Moreover, we show that some of our embeddings are smoothly isotopic but not Lagrangian isotopic.