Homological mirror symmetry and torus fibrations

By M. Kontsevich, Y. Soibelman. 2000

641 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/62c3a2464adb5ce4353e5b769af87809e1e0a7dc

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Citations in the corpus

Lagrangian 3-torus fibrations (2006, methodology): “Kontsevich and Soibelman [24] approach the complex reconstruction problem using non-Archimedean analytic spaces.”
The dual complex of Calabi–Yau pairs (2015, neutral): “Is D(Y0) Sn? For n = 2 a positive answer is given by [31]; the general case is proposed in [30].”
Large Complex Structure Limits of K3 Surfaces (2000, result): “A similar conjecture was also made by Kontsevich, Soibelman and Todorov (see [22], [25]).”
TRANSFERRING A^ (STRONGLY HOMOTOPY ASSOCIATIVE) STRUCTURES (2006, methodology): “We already mentioned in the Abstract that the formulas for vn and ^ n were given, without explicit signs, in [2] (non-inductive formulas) and also in [9] (inductive formulas).”
Derived Categories (2016, neutral): “See [48], [86], [87], [141] and [88].”
On basic concepts of tropical geometry (2011, background): “group around the concept of Newton polytope (see the monograph by Gelfand, Kapranov, and Zelevinsky [10]), Berkovich’s studies on p-adic analytic spaces (see [2]), and studies on homological mirror symmetry in the style of Kontsevich and Soibelman [14].”
Morse homology, tropical geometry, and homological mirror symmetry for toric varieties (2006, methodology): “A∞ pre-categories We recall some elementary definitions concerning A∞ pre-categories, due to Kontsevich and Soibelman [28].”
Gromov-Hausdorff Distance for Quantum Metric Spaces (2000, neutral): “O A ] 2 0 Fe b”
Tropical Geometry and its applications (2006, neutral): “[13] for a conjectural picture in the special case of Calabi–Yau n-folds.”
Lifting harmonic morphisms II: tropical curves and metrized complexes (2014, background): “For some conjectural relations between the two approaches see [Kontsevich and Soibelman 2001; 2006].”