Monotone Lagrangians in ${\mathbb{C}\mathbb{P}}^n$ of minimal Maslov number $n+1$

We show that a monotone Lagrangian $L$ in ${\mathbb{C}\mathbb{P}}^n$ of minimal Maslov number $n+1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}\mathbb{P}}^n$. To prove this we use Zapolsky’s canonical pearl complex for $L$ with coefficients in $\mathbb{Z}$, and various twisted versions thereof, where the twisting is determined by connected covers of $L$. The main tool is the action of the quantum cohomology of ${\mathbb{C}\mathbb{P}}^n$ on the resulting Floer homologies.