  Index theorems for the Dirac operator allow one to study spinors on manifolds
with boundary and torsion. We analyse the modifications of the boundary
Chern-Simons correction and APS eta invariant in the presence of torsion. The
bulk contribution must also be modified and is computed using a supersymmetric
quantum mechanics representation. Here we find agreement with existing results
which employed heat kernel and Pauli-Villars techniques. Nonetheless, this
computation also provides a stringent check of the Feynman rules of de Boer et
al. for the computation of quantum mechanical path integrals. Our results can
be verified via a duality relation between manifolds admitting a Killing-Yano
tensor and manifolds with torsion. As an explicit example, we compute the
indices of Taub-NUT and its dual constructed using this method and find
agreement for any finite radius to the boundary. We also suggest a resolution
to the problematic appearance of the Nieh-Yan invariant multiplied by the
regulator mass^2 in computations of the chiral gravitational anomaly coupled to
torsion.
