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Prepotentials of Yukawa Couplings of Certain Calabi-Yau 3-Folds and Mirror Symmetry

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The Arithmetic and Geometry of Algebraic Cycles

Part of the book series: NATO Science Series ((ASIC,volume 548))

Abstract

In this note, we will give a rather naive mathematical approach to verifying the Mirror Symmetry Conjecture for certain Calabi-Yau 3-folds. Though the origin of Mirror symmetry is superstring theory in mathematical physics ([16], [37], [40], [42]), we will not discuss any background material from physics. Instead, we will focus our attention on prepotentials of A-model and B-model Yukawa couplings for certain Calabi-Yau 3-folds. The prepotential of the A-model Yukawa coupling is related to the number of holomorphic maps from a curve of genus g with n-marked points into a Calabi-Yau manifold. In this note, we will only consider the prepotential in the case when (g, n) = (0, 3) which is the simplest non-trivial case. However it is still far from complete mathematical understanding at this moment. The B-model Yukawa couplings of the mirror pair are also 3-points correlation functions.

Partily supported by Grant-in Aid for Scientific Research (B-09440015), the Ministry of Education, Scuence and Culture, Japan

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kobe University Rokko, Kobe, 657-8501, Japan

    Masa-Hiko Saito

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  1. Masa-Hiko Saito
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Editor information

Editors and Affiliations

  1. University of Oklahoma, Norman, Oklahoma, USA

    B. Brent Gordon

  2. University of Alberta, Edmonton, Alberta, Canada

    James D. Lewis

  3. Universität-GH-Essen, Essen, Germany

    Stefan Müller-Stach

  4. Tokyo Institute of Technology, Tokyo, Japan

    Shuji Saito

  5. Queen’s University, Kingston, Ontario, Canada

    Noriko Yui

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Saito, MH. (2000). Prepotentials of Yukawa Couplings of Certain Calabi-Yau 3-Folds and Mirror Symmetry. In: Gordon, B.B., Lewis, J.D., Müller-Stach, S., Saito, S., Yui, N. (eds) The Arithmetic and Geometry of Algebraic Cycles. NATO Science Series, vol 548. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4098-0_14

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  • DOI: https://doi.org/10.1007/978-94-011-4098-0_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-6194-7

  • Online ISBN: 978-94-011-4098-0

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