Supersymmetric particle in a spacetime with torsion and the index theorem

The supersymmetric Lagrangian compatible with the presence of torsion in the background spacetime requires, in addition to the minimal coupling, an interaction between the spin and the torsion of the form $1/2{N}_{{μ}{ν}{λ}{ρ}}{{ψ}}^{{μ}}{{ψ}}^{{ν}}{{ψ}}^{{λ}}{{ψ}}^{{ρ}},$ where $N$ is the Nieh-Yan 4-form. This gives rise to a coupling between helicity $(h)$ and the Nieh-Yan density of the form $hN.$ The classical Lagrangian allows computing the index for the Dirac operator on a four-dimensional compact manifold with curvature and torsion using the path integral representation for the index. This calculation provides an independent check for the recent result of the chiral anomaly in spaces with torsion.