Self-dual manifolds and mirror symmetry for the quintic threefold (Q2565971)

The author describes a way to geometrically interpolate between the large Kähler structure limit of the Kähler moduli space of the anticanonical divisor in \(\mathbb{P}^n\) and a large complex structure limit of the complex structure moduli space of the mirror partner given by the \textit{B. R. Greene, M. R. Plesser} orbifold construction [Duality in Calabi-Yau moduli space, Nucl. Phys. B338, 15--37 (1990)]. This is achieved by constructing a two-dimensional family of smooth manifolds \(\mathbb{X}_{\rho_1,\rho_2}\) of real dimension \((3(n-1) + 2)\) endowed with a ``weakly self-dual'' (WSD) structure. A WSD structure consists of three closed 2-forms and a Riemannian metric satisfying certain integrability and compatibility conditions. Taking appropriate limiting values for \(\rho_1\) and \(\rho_2\), the manifolds \(\mathbb{X}_{\rho_1,\rho_2}\) approach the large Kähler structure limit of the anticanonical divisor and the large complex structure limit of the mirror in a normalized Gromov-Hausdorff sense. The construction starts with the fiber product \((\mathbb{C}^*)^{n+1} \times_\mu (\mathbb{C}^*)^{n+1}\) over \(\mathbb{R}^{n+1}\), where \(\mu: (\mathbb{C}^*)^{n+1} \rightarrow \mathbb{R}^{n+1}\) is the usual \(T^{n+1}\)-moment map given by rotating the coordinates. The WSD manifolds arise from a sort of ``polysymplectic reduction'' of \((\mathbb{C}^*)^{n+1} \times_\mu (\mathbb{C}^*)^{n+1}\) by a group action arising from the reflexive polytope construction of \(\mathbb{P}^n\) and its toric dual.