Mode Regularization for \(N=1,2\) SUSY Sigma Model

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 regu- larization: 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 world- line 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.