A Class of N=1 Dual String Pairs and its Modular Superpotential

TL;DR

The paper probes N=1 dual string pairs by comparing F-theory compactifications with modular superpotentials to dual heterotic models on CY^{19,19}, deriving the spectrum from fourfold cohomology and relating heterotic bundle moduli and fivebranes to F-theory data. It shows that F-theory superpotentials are generated by worldvolume instantons wrapping rational curves, and that a modular correction factor, captured by η(τ)^{-12}, is necessary to preserve modular weight and duality consistency, with curve-counting arguments from mirror symmetry supporting this correction. On the heterotic side, the superpotential arises from worldsheet instantons producing a product of E8 theta-functions, and the modular structure matches the F-theory computation under a diagonal T-duality limit. Together, these results provide a concrete check of N=1 duality between F-theory and heterotic vacua and illuminate how modularity and instanton effects underpin the effective superpotential in these compactifications.

Abstract

We compare the N=1 F-theory compactification of Donagi, Grassi and Witten with modular superpotential - and some closely related models - to dual heterotic models. We read of the F-theory spectrum from the cohomology of the fourfold and discuss on the heterotic side the gauge bundle moduli sector (including the spectral surface) and the necessary fivebranes. Then we consider the N=1 superpotential and show how a heterotic superpotential matching the F-theory computation is built up by worldsheet instantons. Finally we discuss how the original modular superpotential should be corrected by an additional modular correction factor, which on the F-theory side matches nicely with a `curve counting function' for the del Pezzo surface. On the heterotic side we derive the same factor demanding correct T-duality transformation properties of the superpotential.