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

Abstract

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

37 pages, 23 figures. Several typos are fixed, and new sections are added with examples of non-trivial jagged gradient trees to explain the necessity of the jagged gradient tree. Comments are welcome!