$\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology

Question

Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$. This is fairly easy to prove using the fact that $\mathbb{C}\mathbb{P}^n$ is monotone itself. But it is also mentioned that the positive generator of the abelian subgroup $[\mu|_{\pi_2(P,L)}]\subset \mathbb{Z}$ is $\Sigma_{\mathbb{R}\mathbb{P}^n}=n+1$. However I am not sure how one can check this last result and the author does not seem to provide any references for it.

So my question is if anyone knows any references for this or how can one go about proving this result ? Thanks in advance.

Answer 1

The relative homology long exact sequence puts this group in between $H_2(\mathbb{CP}^n)=\mathbb{Z}$ and $\mathbb{Z}/2$. It maps surjectively to $\mathbb{Z}/2$. Let D be a generator for relative homology. Then $2D$ generates $H_2(\mathbb{CP}^n)$. The Maslov number of a relative class coming from $H_2(\mathbb{CP}^n)$ is twice the Chern class; for the positive generator $2D$ that gives $2(n+1)$. Therefore the Maslov number of D is $n+1$.