Topological Strings and (Almost) Modular Forms

Abstract

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Gamma, 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 Gamma. 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 P_2 and P_1 x P_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 C^3/Z_3.

62 pages, 1 figure; v2: minor corrections

References in corpus (19)

• Monopole Condensation, And Confinement In N=2 Supersymmetric Yang-Mills Theory
• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Geometric Engineering of Quantum Field Theories
• The Topological Vertex
• Topological Strings and Integrable Hierarchies
• Holomorphic Anomalies in Topological Field Theories
• Nonperturbative Results on the Point Particle Limit of N=2 Heterotic String Compactifications
• Matrix Model as a Mirror of Chern-Simons Theory
• Nonperturbative Effective Actions of N=2 Supersymmetric Gauge Theories
• On Orbifolds with Discrete Torsion
• Gravitational corrections in supersymmetric gauge theory and matrix models
• All Loop Topological String Amplitudes From Chern-Simons Theory
• N=2 Electric-magnetic duality in a chiral background
• The Donaldson-Witten function for gauge groups of rank larger than one
• N=2 symplectic reparametrizations in a chiral background
• Towards Integrability of Topological Strings I: Three-forms on Calabi-Yau manifolds
• The uses of Whitham hierarchies
• Higher-Order Gravitational Couplings and Modular Forms in $N=2,D=4$ Heterotic String Compactifications

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