Topological Strings and (Almost) Modular Forms

TL;DR

This paper explores the modular properties of topological string amplitudes on Calabi-Yau threefolds, revealing their quasi-modular or almost holomorphic modular nature depending on polarization, with applications to gauge theories and Gromov-Witten invariants.

Contribution

It demonstrates how topological string amplitudes can be characterized as (almost) modular forms and provides a method to relate amplitudes at different moduli space points.

Findings

Genus g amplitudes are quasi-modular or almost holomorphic modular forms.

Certain combinations of amplitudes are both modular and holomorphic.

Predictions for Gromov-Witten invariants of orbifold C^3/Z_3.

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.