't Hooft anomalies for staggered fermions

We show that the phase structure of certain staggered fermion theories can be understood on the basis of exact anomalies. These anomalies arise when staggered fermions are coupled to gravity which can be accomplished by replacing them by discrete K\"{a}hler-Dirac fermions. We first show the existence of a perturbative anomaly in even dimensions which breaks an exact $U(1)$ symmetry of the massless theory down to $Z_4$. If we attempt to gauge this $Z_4$ symmetry we find a 't Hooft anomaly which can only be cancelled for multiples of two K\"{a}hler-Dirac fields. This result is consistent with the cancellation of a further mixed non-perturbative 't Hooft anomaly between the global $Z_4$ and a reflection symmetry. In four dimensional flat space, theories of two staggered fields yield eight Dirac or sixteen Majorana fermions in the continuum limit and this critical number of fermions agrees with results in condensed matter theory literature on the fermion content required to gap boundary fermions in $4+1$ dimensional topological superconductors. It is also consistent with constraints stemming from the cancellation of spin-$Z_4$ anomalies of Weyl fermions. Indeed, cancellation of 't Hooft anomalies is a necessary requirement for symmetric mass generation and this result gives a theoretical explanation of recent numerical work on the phase diagram of interacting staggered fermions. As an application of these ideas we construct a lattice model whose low energy continuum limit is conjectured to yield the Pati-Salam GUT theory.