Topological Strings and (Almost) Modular Forms |
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Authors: | Mina Aganagic Vincent Bouchard Albrecht Klemm |
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Institution: | (1) Department of Mathematics, University of California, Berkeley, CA 94720, USA;(2) Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720, USA;(3) Department of Physics, University of Wisconsin, Madison, WI 53706, USA |
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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 . |
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Keywords: | |
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