Global gravitational anomalies and transport

TL;DR

This paper explores how global gravitational anomalies constrain parity odd transport coefficients in even-dimensional theories with chiral fields, confirming consistency between anomaly matching and direct calculations.

Contribution

It demonstrates the constraints of global gravitational anomalies on transport coefficients and verifies these through both anomaly matching and explicit perturbative calculations in six dimensions.

Findings

Global anomalies impose mod 2 constraints on transport coefficients.

Perturbative calculations agree with anomaly matching results.

The second Pontryagin class does not affect the eta-invariant in 6D.

Abstract

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.