Geometric Aspects of Mirror Symmetry (with SYZ for Rigid CY manifolds)
arXiv:math/0204168
Abstract
In this article we discuss the geometry of moduli spaces of (1) flat bundles over special Lagrangian submanifolds and (2) deformed Hermitian-Yang-Mills bundles over complex submanifolds in Calabi-Yau manifolds. These moduli spaces reflect the geometry of the Calabi-Yau itself like a mirror. Strominger, Yau and Zaslow conjecture that the mirror Calabi-Yau manifold is such a moduli space and they argue that the mirror symmetry duality is a Fourier-Mukai transformation. We review various aspects of the mirror symmetry conjecture and discuss a geometric approach in proving it. The existence of rigid Calabi-Yau manifolds poses a serious challenge to the conjecture. The proposed mirror partners for them are higher dimensional generalized Calabi-Yau manifolds. For example, the mirror partner for a certain K3 surface is a cubic fourfold and its Fano variety of lines is birational to the Hilbert scheme of two points on the K3. This hyperkahler manifold can be interpreted as the SYZ mirror of the K3 by considering singular special Lagrangian tori. We also compare the geometries between a CY and its associated generalized CY. In particular we present a new construction of Lagrangian submanifolds.
arXiv commentTo appear in the proceedings of International Congress of Chinese Mathematicians 2001, 47 pages
References in corpus (10)
- Lectures on Special Lagrangian Submanifolds
- Summing up D-Instantons
- Higher genus symplectic invariants and sigma model coupled with gravity
- BPS States, String Duality, and Nodal Curves on K3
- Hodge integrals and degenerate contributions
- Hypergeometric functions and mirror symmetry in toric varieties
- Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds
- Enumeration of -fold tangent hyperplanes to a surface
- Lagrangian Tori Fibration of Toric Calabi-Yau manifold III: Symplectic topological SYZ mirror construction for general quintics
- Critical Strings from Noncritical Dimensions: A Framework for Mirrors of Rigid Vacau