Massive U(1)s and Heterotic Five-Branes on K3

TL;DR

This work analyzes six-dimensional heterotic string compactifications on K3 with arbitrary Abelian and non-Abelian bundles and H5-branes for both $SO(32)$ and $E_8\times E_8$. By performing a detailed dimensional reduction, the authors derive the perturbative Green-Schwarz counter-terms and show how Abelian $U(1)$ gauge factors acquire masses through couplings to two-form and four-form fields, with the masses controlled by the first Chern classes of the bundles and embedding data, while tadpole cancellation involves the second Chern characters. They explicitly compute the non-perturbative H5-brane contributions to the Green-Schwarz mechanism and demonstrate their role in anomaly cancellation, presenting concrete examples on toric K3 realize-ations with up to three (1,1) forms. The results connect the six-dimensional GS structure to four-dimensional expectations and provide a framework for understanding F-theory lifts of multiple heterotic $U(1)$ factors, as well as guidance for constructing explicit vacua. Overall, the paper systematizes the interplay between bundle data, H5-branes, and anomaly cancellation in 6D heterotic theories on K3, with practical implications for dualities and higher-dimensional compactifications.

Abstract

We systematically consider heterotic SO(32) and E8 x E8 compactifications on K3 with Abelian and non-Abelian backgrounds as well as an arbitrary number of five-branes. The masses of the U(1) factors depend on the first Chern classes of the bundles and some combinatorial factors specifying the embedding in SO(32) or E8. The form of the generalised Green-Schwarz counter-terms in six dimensions constrains the possible heterotic five-brane actions. Some supersymmetric examples on K3 realisations as toric complete intersection spaces with up to three explicit two-forms are given.

Figures (1)

• Figure 1: A six-dimensional anomalous diagram and its two possible types of Green-Schwarz counter diagrams.