Merging Heterotic Orbifolds and K3 Compactifications with Line Bundles

TL;DR

The work constructs a comprehensive correspondence between perturbative heterotic orbifolds on $T^4/\mathbb{Z}_N$ and smooth heterotic compactifications on $K3$ with line bundles for $SO(32)$ and $E_8\times E_8$. Using anomaly polynomials and detailed spectrum analyses, it demonstrates how orbifold data map to smooth line-bundle embeddings, particularly showing that many $N=2,3$ models admit clean one-line-bundle matches, while higher-twist cases ($N=4,6$) typically require multiple line bundles and blow-ups to achieve complete spectral agreement. A field-theoretic blow-up analysis of twisted flat directions explains how apparent mismatches are resolved and clarifies the role of $U(1)$ factors in the massless spectrum at the orbifold point versus the smooth geometry. The results provide a concrete framework to realize explicit line bundles on $K3$ that reproduce the non-Abelian sectors of orbifold spectra and illuminate the geometric interpretation of orbifold-to-smooth transitions, with implications for extending these ideas to higher-dimensional compactifications and Calabi–Yau contexts.

Abstract

We clarify the relation between six-dimensional Abelian orbifold compactifications of the heterotic string and smooth heterotic K3 compactifications with line bundles for both SO(32) and E_8 x E_8 gauge groups. The T^4/Z_N cases for N=2,3,4 are treated exhaustively, and for N=6 some examples are given. While all T^4/Z_2 and nearly all T^4/Z_3 models have a simple smooth match involving one line bundle only, this is only true for some T^4/Z_4 and T^4/Z_6 cases. We comment on possible matchings with more than one line bundle for the remaining cases. The matching is provided by comparisons of the massless spectra and their anomalies as well as a field theoretic analysis of the blow-ups.