  On-shell Pauli-Villars regularization of the one-loop divergences of
supergravity theories is used to study the anomaly structure of supergravity
and the cancellation of field theory anomalies under a $U(1)$ gauge
transformation and under the T-duality group of modular transformations in
effective supergravity theories with three K\"ahler moduli $T^i$ obtained from
orbifold compactification of the weakly coupled heterotic string. This
procedure requires constraints on the chiral matter representations of the
gauge group that are consistent with known results from orbifold
compactifications. Pauli-Villars regulator fields allow for the cancellation of
all quadratic and logarithmic divergences, as well as most linear divergences.
If all linear divergences were canceled, the theory would be anomaly free, with
noninvariance of the action arising only from Pauli-Villars masses. However
there are linear divergences associated with nonrenormalizable
gravitino/gaugino interactions that cannot be canceled by PV fields. The
resulting chiral anomaly forms a supermultiplet with the corresponding
conformal anomaly, provided the ultraviolet cut-off has the appropriate field
dependence, in which case total derivative terms, such as Gauss-Bonnet, do not
drop out from the effective action. The anomalies can be partially canceled by
the four-dimensional version of the Green-Schwarz mechanism, but additional
counterterms, and/or a more elaborate set of Pauli-Villars fields and
couplings, are needed to cancel the full anomaly, including D-term
contributions to the conformal anomaly that are nonlinear in the parameters of
the anomalous transformations.
