Homological mirror symmetry and torus fibrations

Kontsevich, Maxim; Soibelman, Yan

Homological mirror symmetry and torus fibrations

Nov, 2000

63 pages

Part of Symplectic geometry and mirror symmetry. Proceedings, 4th KIAS Annual International Conference, Seoul, South Korea, August 14-18, 2000, 203-263

Contribution to:

KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry, 203-263

e-Print:

math/0011041 [math.SG]

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Abstract:

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than the one in terms of D-branes. Also we make an attempt to prove the homological mirror conjecture using the torus fibrations. In the case of abelian varieties, and for a large class of Lagrangian submanifolds, we obtain an identification of Massey products on the symplectic and holomorphic sides. Tools used in the proof are of a mixed origin: not so classical Morse theory, homological perturbation theory and non-archimedean analysis.

transformation: mirror
homology
field theory: conformal
dimension: 2
field theory: torus
boundary condition
differential geometry
fibre bundle
Morse theory

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