2000/11/07 by Maxim Kontsevich, Kontsevich, Maxim, Yan Soibelman +1 · 6 citations
Mathematics · Physics and Astronomy · #14J32 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Symplectic Geometry (math.SG) #hep-th #math-ph #math.AG #math.DG #math.MP #math.QA #math.SG #msc:14J32
paper · pdf · doi:10.48550/arxiv.math/0011041
version accepted for publication
arxiv created 2001/06/04 · arxiv updated 2009/11/30
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.