  Worldline N=1 and N=2 supersymmetric sigma models in curved background are
useful to describe spin one-half and spin one particles coupled to external
gravity, respectively. It is well known that worldline path integrals in curved
space require regularization: we present here the mode-regularization for these
models, finding in particular the corresponding counterterms, both in the case
of flat and curved indices for worldline fermions. For N=1, using curved
indices we find a contribution to the counterterm from the fermions that
cancels the contribution of the bosons, leading to a vanishing total
counterterm and thus preserving the covariance and supersymmetry of the
classical action. Conversely in the case of N=2 supersymmetries we obtain a
non-covariant counterterm with both curved and flat indices. This work
completes the analysis of the known regularization schemes for N=1,2 nonlinear
sigma models in one dimension.
