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

Abstract

We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathbb{RP}^n$. 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 $\mathbb{CP}^n$ on the resulting Floer homologies.