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Homological Perturbation Theory and Homological Mirror Symmetry

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Higher Structures in Geometry and Physics

Part of the book series: Progress in Mathematics ((PM,volume 287))

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Abstract

In this article, we discuss an application of homological perturbation theory (HPT) to homological mirror symmetry (HMS) based on Kontsevich and Soibelman’s proposal [Kontsevich, M., Soibelman, Y. (2001) Homological mirror symmetry and torus fibrations]. After a brief review of Morse theory, Morse homotopy and the corresponding Fukaya categories, we explain the idea of deriving a Fukaya category from a DG category via HPT, which is expected to give a solution to HMS, and apply it to the cases of \({\mathbb{R}}^{2}\) discussed in [Kajiura, H. (2007) An A ∞ -structure for lines in a plane] and then T 2. A finite dimensional A ∞ -algebra obtained from the Fukaya category on T 2 is also presented.

Mathematics Subject Classification: 18G55, 53D37

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Notes

  1. 1.

    Originally, these degrees in Fukaya categories are defined by the Maslov indices; which consequently coincide with the one defined via the Morse indices [10].

  2. 2.

    A general approach to such a modification h ε is discussed in [36].

  3. 3.

    For the relation of the noncommutative complex torus description and the usual complex torus description, for instance see [42, 23].

  4. 4.

    The number ν a corresponds to α a ∕ q a in the general setting [22]. The effect of flat connections β a there is set to be zero in this article for simplicity.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Chiba University, Chiba, 263-8522, Japan

    Hiroshige Kajiura

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  1. Hiroshige Kajiura
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Correspondence to Hiroshige Kajiura .

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Editors and Affiliations

  1. Universität Zürich, Institut für Mathematik, Winterthurerstr. 190, Zürich, CH-8057, Switzerland

    Alberto S. Cattaneo

  2. Loyola University, Chicago, Department of Mathematics, Chicago, 60626, Illinois, USA

    Anthony Giaquinto

  3. Pennsylvania State University, Department of Mathematics, Eberly College of Science, University Park, 16802, Pennsylvania, USA

    Ping Xu

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Kajiura, H. (2011). Homological Perturbation Theory and Homological Mirror Symmetry. In: Cattaneo, A., Giaquinto, A., Xu, P. (eds) Higher Structures in Geometry and Physics. Progress in Mathematics, vol 287. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4735-3_10

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