  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.
