2006/07/31 by Mina Aganagic, Vincent Bouchard, Albrecht Klemm · 2 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Geometry and complex manifolds #hep-th #math.AG #math.NT
paper · pdf · doi:10.1007/s00220-007-0383-3
published as Commun.Math.Phys.277:771-819,2008 · 62 pages, 1 figure; v2: minor corrections
arxiv created 2007/05/04 · openalex publication_date 2007/11/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 H3(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 P2 and P1 x P1. 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 C3/Z3.