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Worldline N=1 and N=2 supersymmetric sigma models in curved background areuseful to describe spin one-half and spin one particles coupled to externalgravity, respectively. It is well known that worldline path integrals in curvedspace require regularization: we present here the mode-regularization for thesemodels, finding in particular the corresponding counterterms, both in the caseof flat and curved indices for worldline fermions. For N=1, using curvedindices we find a contribution to the counterterm from the fermions thatcancels the contribution of the bosons, leading to a vanishing totalcounterterm and thus preserving the covariance and supersymmetry of theclassical action. Conversely in the case of N=2 supersymmetries we obtain anon-covariant counterterm with both curved and flat indices. This workcompletes the analysis of the known regularization schemes for N=1,2 nonlinearsigma models in one dimension.

Mode Regularization for N=1,2 SUSY Sigma Model