(Quadratically) Refined Discrete Anomaly Cancellation

In this work we study the cancellation of non-perturbative anomalies of gravitational theories with gauge group $\mathbb{Z}_k$ in six dimensions. These subtle anomalies require a classification of deformation classes of manifolds with discrete gauge bundles known as bordism groups. The consistency of the theory demands a cancellation of the fermion anomalies, which can be done by the transformation properties of 2-form fields in the theory. Since the 2-forms in six dimensions are themselves chiral, their formulation needs subtle topological information encoded in a so-called quadratic refinement. A matching between the fermionic anomalies and the defining properties of the quadratic refinement, lead to strong consistency constraints on the charged fermion spectrum. We explicitly determine these consistency conditions for the case of a single chiral 2-form and various discrete gauge groups. Since we provide a model-independent formulation, these restrictions hold universally for theories of this type.