Anomaly-Free Supersymmetric Models in Six Dimensions

TL;DR

Schwarz analyzes the consistency of six-dimensional $N=1$ supersymmetric theories by enforcing anomaly cancellation through the eight-form polynomial $I$. He develops and applies the factorization condition $I = (tr R^2 + \sum u_\alpha tr F_\alpha^2)(tr R^2 + \sum v_\alpha tr F_\alpha^2)$, explores perturbative heterotic compactifications on K3, and examines non-perturbative sectors arising from small instantons. The paper identifies two infinite anomaly-free families, $SO(2n+8)\times Sp(n)$ and $SU(n)\times SU(n)$, and shows their factorized anomalies can persist at arbitrarily large rank, with the latter linked to 4D $N=2$ superconformal theories and Lie superalgebras $OSp(2n+8|n)$ and $SU(n|n)$. These results broaden the landscape of consistent 6D vacua, highlight non-perturbative avenues beyond conventional string constructions, and suggest deeper connections between anomaly structures, Higgsing, and potential string-theoretic realizations of high-rank gauge groups.

Abstract

The conditions for the cancellation of all gauge, gravitational, and mixed anomalies of $N=1$ supersymmetric models in six dimensions are reviewed and illustrated by a number of examples. Of particular interest are models that cannot be realized perturbatively in string theory. An example of this type, which we verify satisfies the anomaly cancellation conditions, is the K3 compactification of the $SO(32)$ theory with small instantons recently proposed by Witten. When the instantons coincide it has gauge group $SO(32) \times Sp(24)$. Two new classes of models, for which non-perturbative string constructions are not yet known, are also presented. They have gauge groups $SO(2n+8)\times Sp(n)$ and $SU(n)\times SU(n)$, where $n$ is an arbitrary positive integer.