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Topological Strings and (Almost) Modular Forms

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Abstract

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 and IP 1  ×  IP 1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold \({{\mathbb {C}^3} / {\mathbb {Z}_3}}\).

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Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Mina Aganagic

  2. Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA, 94720, USA

    Vincent Bouchard

  3. Department of Physics, University of Wisconsin, Madison, WI, 53706, USA

    Albrecht Klemm

Authors
  1. Mina Aganagic
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  2. Vincent Bouchard
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  3. Albrecht Klemm
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Correspondence to Mina Aganagic.

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Communicated by N.A. Nekrasov

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Aganagic, M., Bouchard, V. & Klemm, A. Topological Strings and (Almost) Modular Forms. Commun. Math. Phys. 277, 771–819 (2008). https://doi.org/10.1007/s00220-007-0383-3

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  • DOI: https://doi.org/10.1007/s00220-007-0383-3

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