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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 hyperk¨ahler 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.

Vector cross product structures on manifolds include symplectic, volume, G2- and Spin(7)-structures. We show that the knot spaces of such manifolds have natural symplectic structures, and relate instantons and branes in these manifolds to holomorphic disks and Lagrangian submanifolds in their knot spaces. For the complex case, the holomorphic volume form on a Calabi–Yau manifold defines a complex vector cross product structure. We show that its isotropic knot space admits a natural holomorphic symplectic structure.We also relate the Calabi–Yau geometry of the manifold to the holomorphic symplectic geometry of its isotropic knot space.

In this article, we define a new class of middle dimensional submanifolds of a Hyperkahler manifold which contains the class of complex Lagrangian submanifolds, and show that this larger class is invariant under the mean curvature flow. Along the flow, the complex phase map satisfies the generalized harmonic map heat equation, tt is also related to the mean curvature vector via a first order differential equation. Moreover, we proved a result on nonexistence of Type I singularity.

In this paper we study the geometry of manifolds with vector cross products and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and coassociative submanifolds in G2-manifolds. Second we classify K¨ahler manifolds with the complex analog of the vector cross product, namely they are Calabi-Yau manifolds and hyperk¨ahler manifolds. Furthermore we study instantons, Neumann branes and Dirichlet branes on these manifolds. For example they are special Lagrangian submanifolds with phase angle zero, complex hypersurfaces and special Lagrangian submanifolds with phase angle pi/2 in Calabi-Yau manifolds.

It is known that the hard Lefschetz action, together with K¨ahler identities for K¨ahler (resp. hyperk¨ahler) manifolds, determines a su(1, 1)_sup (resp. sp(1, 1)_sup) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also generalize it to manifolds with other holonomy groups. For semi-flat Calabi-Yau (resp. hyperk¨ahler) manifolds, these symmetries can be enlarged to a so(2, 2)_sup (resp. su(2, 2)_sup) action.

We study the moduli space M of torsion-free G2-structures on a fixed compact manifold M7, and define its associated universal intermediate Jacobian J . We define the Yukawa coupling and relate it to a natural pseudo-K¨ahler structure on J . We consider natural Chern–Simons-type functionals, whose critical points give associative and coassociative cycles (calibrated submanifolds coupled with Yang–Mills connections), and also deformed Donaldson–Thomas connections. We show that the moduli spaces of these structures can be isotropically immersed in J by means of G2-analogues of Abel–Jacobi maps.

This article studies the geometry of moduli spaces of G2-manifolds, associative cycles, coassociative cycles and deformed Donaldson–Thomas bundles. We introduce natural symmetric cubic tensors and differential forms on these moduli spaces. They correspond to Yukawa couplings and correlation functions in M-theory. We expect that the Yukawa coupling characterizes (co-)associative fibrations on these manifolds. We discuss the Fourier transformation along such fibrations and the analog of the Strominger–Yau–Zaslow mirror conjecture for G2-manifolds. We also discuss similar structures and transformations for Spin(7)- manifolds.

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No abstract uploaded!