Multi-valued Morse homotopy for the SYZ mirror of the complex projective plane

We propose a definition of the category $\mathit{Mo}^{\mathrm{mult}}(P)$ of multi-valued Morse homotopy on $P$ consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory $\mathit{Mo}^{\mathrm{mult}}_{\mathcal{E}}(P)$ of $\mathit{Mo}^{\mathrm{mult}}(P)$ is $A_\infty$-equivalent to a full subcategory $\mathit{DG}_{\mathcal{E}}^{\mathrm{vect}}(\mathbb{C}P^2)$ of the category $\mathit{DG}^{\mathrm{vect}}(\mathbb{C}P^2)$ consisting of holomorphic vector bundles over the complex projective plane $\mathbb{C}P^2$. As an application, we study the mirror description for global sections of the holomorphic tangent bundle over $\mathbb{C}P^2$.