Topological Strings on Elliptic Fibrations

TL;DR

This work develops a mirror-symmetric framework for topological strings on elliptically fibered Calabi–Yau threefolds, expressing higher-genus amplitudes as polynomials in generators tied to the special geometry of the moduli space. A refined structure emerges when expanding in the base moduli: the fiber dependence is governed by SL$(2,Z)$ modular forms, enabling a recursion in terms of $E_2,E_4,E_6$ that parallels BCOV-type anomalies. The authors methodically construct the generator set, fix boundary conditions at various loci, and demonstrate the approach on several elliptic fibrations, connecting to Gopakumar–Vafa invariants and modular phenomena. This work bridges holomorphic anomaly methods, modular form technology, and elliptic fibration geometry, offering a systematic route to compute and understand higher-genus amplitudes and their arithmetic structure in this rich setting.

Abstract

We study topological string theory on elliptically fibered Calabi-Yau threefolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the moduli space. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at the boundary in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which governs this finer structure and which can be related to the anomaly equations for correlation functions.

Figures (1)

• Figure 1: The blown--up moduli space