This article studies the geometry of moduli spaces of G 2 -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 G 2 -manifolds.We also discuss similar structures and transformations for Spin(7)manifolds.

In this paper we study the geometry of manifolds with vector cross product 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 G2manifolds.Second we classify Kähler manifolds with the complex analog of vector cross product, namely they are Calabi-Yau manifolds and hyperkähler 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 π/2 in Calabi-Yau manifolds.Third we prove that, given any Calabi-Yau manifold, 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.

No abstract

We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F n 1 ,n 2 ,••• ,n k ;n via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including complex Grassmannian Gr(2, n) and complete flag variety F 1,2,3;4 . Contents 18 4.4. Transversal intersections in Gr(2, n) 19 4.5. Discussions for complete flag varieties 20 5. Appendix: an anti-canonical divisor −K X 23 Acknowledgements 24 References 24

No abstract