  We employ the covariant background formalism to derive generic expressions
for the one-loop effective action in field theoretic orbifold
compactifications. The contribution of each orbifold sector is given by the
effective action of its fixed torus with a shifted mass matrix. We thus study
in detail the computation of the heat kernel on tori. Our formalism manifestly
separates UV sensitive (local) from UV-insensitive (nonlocal) renormalization.
To exemplify our methods, we study the effective potential of 6d gauge theory
as well as kinetic terms for gravitational moduli in 11d supergravity.
