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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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