Homological mirror symmetry and torus fibrations Export

@article{citeulike:2277921, 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.}, archivePrefix = {arXiv}, author = {Kontsevich, Maxim and Soibelman, Yan}, citeulike-article-id = {2277921}, citeulike-linkout-0 = {http://arxiv.org/abs/math.SG/0011041}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.SG/0011041}, day = {4}, eprint = {math.SG/0011041}, keywords = {kontsevich, massey, mirror}, month = {Jun}, posted-at = {2008-01-22 22:54:53}, priority = {2}, title = {Homological mirror symmetry and torus fibrations}, url = {http://arxiv.org/abs/math.SG/0011041}, year = {2001} }

TY - JOUR ID - citeulike:2277921 L3 - citeulike-article-id:2277921 N2 - 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. TI - Homological mirror symmetry and torus fibrations KW - kontsevich KW - massey KW - mirror AU - Kontsevich, Maxim AU - Soibelman, Yan PY - 2001/Jun/04/ UR - http://arxiv.org/abs/math.SG/0011041 ER -