  If there exists Lorentz and CPT violation in nature, then it is crucial to
discover and understand the underlying mechanism. In this contribution, we
discuss one such mechanism which relies on four-dimensional chiral gauge
theories defined over a spacetime manifold with topology $\mathbb{R}^3 \times
S^1$ and periodic spin structure for the compact dimension. It can be shown
that the effective gauge-field action contains a local Chern-Simons-like term
which violates Lorentz and CPT invariance. For arbitrary Abelian $U(1)$ gauge
fields with trivial holonomies in the compact direction, this anomalous Lorentz
and CPT violation has recently been established perturbatively with a
Pauli-Villars-type regularization and nonperturbatively with a lattice
regularization based on Ginsparg-Wilson fermions.
