Anomaly-free sets of fermions

We present new techniques for finding anomaly-free sets of fermions. Although the anomaly cancellation conditions typically include cubic equations with integer variables that cannot be solved in general, we prove by construction that any chiral set of fermions can be embedded in a larger set of fermions which is chiral and anomaly-free. Applying these techniques to extensions of the standard model, we find anomaly-free models that have arbitrary quark and lepton charges under an additional U(1) gauge group.