Gapped Chiral Fermions

In principle, there is no obstacle to gapping fermions preserving any global symmetry that does not suffer a 't Hooft anomaly. In practice, preserving a symmetry that is realised on fermions in a chiral manner necessitates some dynamics beyond simple quadratic mass terms. We show how this can be achieved using familiar results about supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking. We present simple models that gap fermions while preserving a symmetry group under which they transform in chiral representations. For example, we show how to gap a collection of 4d fermions that carry the quantum numbers of one generation of the Standard Model, but without breaking electroweak symmetry. We further show how to gap fermions in groups of 16 while preserving certain discrete symmetries that exhibit a mod 16 anomaly.