Global gravitational anomalies and transport.

We investigate the constraints imposed by global gravitational anomalies on parity odd
induced transport coefficients in even dimensions for theories with chiral fermions, gravitinos and self dual tensors.
The $\eta$- invariant for the large diffeomorphism corresponding to the $T$ transformation on a torus constraints
the coefficients in the thermal effective action up to mod 2. We show that the result obtained for the parity odd
transport for gravitinos using global anomaly matching is consistent with the direct perturbative calculation.
In $d=6$ we see that the second Pontryagin class in the anomaly polynomial does not contribute to the $\eta$-invariant
which provides a topological explanation of this observation in the `replacement rule'. We then perform a direct
perturbative calculation for the contribution of the self dual tensor in $d=6$ to the parity odd transport coefficient
using the Feynman rules proposed by Gaum\'{e} and Witten. The result for the transport coefficient agrees with that
obtained using matching of global anomalies.