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

We propose a definition of the category ${Mo}^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}(P)$ of multi-valued Morse homotopy on $P$ consisting of multi-valued functions associated to Lagrangian multi-sections. We then show that a full subcategory ${Mo}_{\mathcal{E}}^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}(P)$ of ${Mo}^{\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}}(P)$ is $A_{\mathrm{\infty }}$-equivalent to a full subcategory ${DG}_{\mathcal{E}}^{\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}}(\mathbb{C}{P}^2)$ of the category ${DG}^{\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}}(\mathbb{C}{P}^2)$ 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 $\mathbb{C}{P}^2$.