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Polynomial Structure of Topological String Partition Functions

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Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Part of the book series: Fields Institute Monographs ((FIM,volume 34))

Abstract

We review the polynomial structure of the topological string partition functions as solutions to the holomorphic anomaly equations. We also explain the connection between the ring of propagators defined from special Kähler geometry and the ring of almost-holomorphic modular forms defined on modular curves.

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Notes

  1. 1.

    Throughout the note, we shall simply call it Kähler structure by abuse of language.

  2. 2.

    See also [38, 29, 30, 33, 45, 31, 27, 1, 26, 4, 19, 6, 22, 21, 42, 5, 32, 7] for related works.

  3. 3.

    The quantity \(\mathcal{F}^{(g)}\) is really a section rather than a function, but in the literature it is termed topological string partition function which we shall follow in this note.

  4. 4.

    In this note, we shall use \(\bar{\partial }_{\bar{\imath }}\) and \(\partial _{\bar{\imath }}\) interchangeably to denote \(\frac{\partial } {\partial \bar{z}^{\bar{\imath }}}\) for some local complex coordinates \(z =\{ z^{i}\}_{i=1}^{\mathrm{dim}\mathcal{M}}\) chosen on the moduli space \(\mathcal{M}\).

  5. 5.

    This assumption is reasonable since these quantities have different singular behaviors when written in the canonical coordinates at the large complex structure.

  6. 6.

    See also [38, 29, 30, 33, 45, 31, 27, 1, 26, 4, 19, 6, 22, 21, 42, 5, 32, 7] for related works.

  7. 7.

    This is due to properties of special Kähler geometry and the particular form for the Picard-Fuchs equation, see [10] for details.

  8. 8.

    This is related to the ψ coordinate in [11] by \(z = (5\psi )^{-5}\).

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Acknowledgements

The author would like to thank Murad Alim, Emanuel Scheidegger and Shing-Tung Yau for valuable collaborations and inspiring discussions on related projects. Thanks also goes to Murad Alim, Emanuel Scheidegger, Teng Fei and Atsushi Kanazawa for carefully reading the draft and giving very helpful comments. He also wants to thank Professor Noriko Yui and the other organizers for inviting him to the thematic program Calabi-Yau Varieties: Arithmetic, Geometry and Physics at the Fields Institute, and the Fields Institute for providing excellent research atmosphere and partial financial support during his visiting.

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  1. Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, Canada, N2L 2Y5

    Jie Zhou

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  1. Jie Zhou
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Correspondence to Jie Zhou .

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Editors and Affiliations

  1. Mathematics Department, Stony Brook University, Stony Brook, New York, USA

    Radu Laza

  2. Leibniz Universität Hannover, Hannover, Germany

    Matthias Schütt

  3. Department of Math & Stats, Queen's University, Kingston, Ontario, Canada

    Noriko Yui

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Zhou, J. (2015). Polynomial Structure of Topological String Partition Functions. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2830-9_14

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