In this dissertation, the effective gauge field action of the chiral gauge theory is calculated, which
is defined over a four-dimensional spacetime manifold (M = R³ x S¹) with one compactified coordinate.
The gauge fields and fermionic fields have periodic boundary conditions over the compact
dimension. The Abelian gauge-field configurations Aμ(x) can depend upon the compactified coordinate
x⁴ and have a vanishing component A₄. The existence of CPT (and Lorentz) violation
is derived perturbatively with a generalized Pauli-Villars regularization. For more justification,
the CPT anomaly is established nonperturbatively with a lattice regularization scheme based on