Homological mirror symmetry and torus fibrations

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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.
version accepted for publication

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Symplectic Geometry, High Energy Physics - Theory, Mathematical Physics, Algebraic Geometry, Differential Geometry, Quantum Algebra, 14J32

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http://salesiana.dossiersoluciones.com/handle/123456789/60293

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0011041
 http://arxiv.org/abs/math/0011041

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arXiv