  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 ${1/2}
N_{\mu\nu\lambda\rho}\psi^\mu\psi^\nu\psi^\lambda\psi^\rho$, 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.
