Mode Regularization for N=1,2 SUSY Sigma Model

TL;DR

This paper develops a mode-regularization scheme for N=1 and N=2 supersymmetric sigma models in curved backgrounds, identifying counterterms that preserve covariance and supersymmetry in N=1 but not in N=2 cases.

Contribution

It introduces a detailed mode-regularization approach for N=1 and N=2 supersymmetric sigma models, clarifying counterterms needed for covariance and supersymmetry preservation.

Findings

N=1 models have a vanishing total counterterm, preserving covariance and supersymmetry.

N=2 models require a non-covariant counterterm, affecting symmetry properties.

The work completes the analysis of regularization schemes for these models.

Abstract

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.