Perturbative Couplings and Modular Forms in N=2 String Models with a Wilson Line

TL;DR

This work develops a comprehensive framework to compute perturbative couplings in four-parameter $D=4$, $N=2$ string models with a Wilson line, using generalized modular forms (Siegel and Jacobi) to obtain the one-loop prepotential $F_0^{\rm het}$ and the Wilsonian gravitational coupling $F_1$. It extends heterotic/Type II duality tests beyond the previously studied $N_V=3$–$4$ cases to $N_V=5$ by incorporating the Wilson line modulus $V$ through a hatting procedure that maps Jacobi forms to Siegel forms. The authors provide explicit expressions for both STU and STU-V models, relate the heterotic data to Calabi–Yau instanton numbers, and verify a nontrivial duality check for a concrete CY model ($X_2$). The results establish a robust link between perturbative automorphic data and Calabi–Yau topological invariants, and they indicate a path to generalize to more Wilson lines and to connect with broader nonperturbative indices in related theories.

Abstract

We consider a class of four parameter D=4, N=2 string models, namely heterotic strings compactified on K3 times T2 together with their dual type II partners on Calabi-Yau three-folds. With the help of generalized modular forms (such as Siegel and Jacobi forms), we compute the perturbative prepotential and the perturbative Wilsonian gravitational coupling F1 for each of the models in this class. We check heterotic/type II duality for one of the models by relating the modular forms in the heterotic description to the known instanton numbers in the type II description. We comment on the relation of our results to recent proposals for closely related models.