Symmetric mass generation for K\"{a}hler-Dirac fermions

We show that theories of massless K\"{a}hler-Dirac fermions suffer from an anomaly which is topological in origin and breaks a $U(1)$ symmetry, unique to K\"{a}hler-Dirac theories, down to a discrete subgroup depending on the Euler number of the background space and the number of flavors. Furthermore, in the presence of interactions, cancellation of this anomaly places constraints on the number of massless fermions. We write down Yukawa interactions respecting this discrete symmetry which we argue are capable of generating fermion masses without breaking symmetries and, if carefully chosen, can yield chiral low energy theories. We show that the simplest example of such a model exhibits a ${\rm Spin}(7)\times {\rm Spin}(4)$ flavor symmetry. If the ${\rm Spin}(7)$ symmetry is subsequently Higgsed to ${\rm Spin}(6)$ the resultant low energy theory possesses the symmetries and fermion representations of the Pati-Salam model.