On discrete Twist and Four-Flux in N=1 heterotic/F-theory compactifications

TL;DR

This work establishes and tests a precise correspondence between four-form flux data on elliptically fibered Calabi–Yau fourfolds in F-theory and discrete twist data on the heterotic side, encapsulated in the proposed relation $G^2=-\pi_*\gamma^2$. Through a detailed computation of the Euler characteristic $e(X^4)$ across smooth and singular fibrations, the authors derive Plücker-type corrections for cusp and tacnode curves, develop a comprehensive treatment of codimension-3 loci, and validate the duality by matching $e(X^4)/24$ with the heterotic three-brane count $n_3$ (and, in the smooth case, with the five-brane data $n_5$). They further connect the spectral-cover description of heterotic bundles to del Pezzo fibrations in F-theory, illuminating the discrete moduli and the proposed $(1,1)$-Hodge shift between the two pictures. The results give robust, case-by-case confirmations across $E$- and $D$-series singularities and provide a general framework for computing Euler characteristics in the presence of complex codimension-two and -three singularities, strengthening the heterotic/F-theory duality in four dimensions. The work has broad implications for constructing and checking consistent 4D $N=1$ vacua with controlled flux and discrete data in string compactifications.

Abstract

We give an indirect argument for the matching $G^2=-π_* γ^2$ of four-flux and discrete twist in the duality between N=1 heterotic string and $F$-theory. This treats in detail the Euler number computation for the physically relevant case of a Calabi-Yau fourfold with singularities.