Anomaly Cancellation in Six Dimensions

TL;DR

The work addresses how six-dimensional anomaly cancellation constrains Calabi–Yau compactifications, specifically the K3 surface, in heterotic string theory. It develops the six-dimensional Green–Schwarz mechanism, derives factorization conditions for gravitational, gauge, and mixed anomalies, and deduces topological data such as $h_{1,1}=20$ and ${\rm dim} H^1({\rm End}\,T)=90$, along with the one-Higgs-like relation $s - y=244$; these results are then used to classify Abelian $Z_N$ orbifolds of K3, including asymmetric ones, under anomaly-consistency checks. The paper provides explicit Green–Schwarz counterterms, charge-sum rules (e.g., $\sum_i Q_i^2=42$, $\sum_i Q_i^4=9$, $\alpha^{(1,2)}_{U(1)}=1,6$), and demonstrates that the K3 geometry underpins the spectra of orbifold models, with $Q^2=\frac{1}{2}$ for $U(1)$ charges and consistent embeddings yielding groups such as $E_8\times E_7\times U(1)$. This establishes a concrete link between anomaly cancellation and CY/K3 topology, guiding six-dimensional model-building and informing extensions to four dimensions via Wilson lines and non-standard embeddings.

Abstract

I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for Calabi-Yau compactification to six dimensions. This reflects the fact that K3 is the only non-trivial CY manifold in two complex dimensions. I explicitly construct the Green-Schwarz counterterms and derive sum rules for charges of additional enhanced U(1) factors and compare the results with all possible Abelian orbifold constructions of K3. This includes asymmetric orbifolds as well, showing that it is possible to regain a geometrical interpretation for this class of models. Finally, I discuss some models with a broken $E_7$ gauge group which will be useful for more phenomenological applications.