Parity violating gravitational response and anomalous constitutive relations

TL;DR

This work computes the parity-violating part of the time-dependent gravitational response for an ideal gas of Weyl fermions up to third order in the derivative expansion, revealing two invariant functions $c_{\mathbb{V}}(q^0,q)$ and $c_{\mathbb{T}}(q^0,q)$ that encode the parity-odd energy-momentum response to vector and tensor metric perturbations. The authors derive explicit leading and third-order expressions for these functions, connect the static limit to anomalous constitutive relations and to the full parity-odd anomalous partition function, and show the necessity of an invariant, gauge- and diffeomorphism-compatible term $W_{\mathrm{inv}}$ (with $c_1=c_m$) to reconcile gauge and gravitational invariances at third order. They demonstrate that the linearized results are compatible with the structure proposed for the anomalous partition function in Jensen 2012kj, including a subtle sign interplay between vector and tensor channels, and they illuminate how the full time-dependent gravitational response can inform anomaly-driven, time-dependent phenomena in hydrodynamics. The findings provide a framework for connecting microscopic anomaly-induced transport to macroscopic, time-dependent gravitational responses and suggest avenues toward kinetic descriptions that incorporate metric perturbations.

Abstract

We compute the parity violating part of the time-dependent gravitational response function of an ideal gas of Weyl fermions up to third order in the derivative expansion and give its full tensorial structure. Our main results are two functions that parametrize the energy-momentum tensor in terms of gauge-invariant combinations of vector and tensor metric perturbations. The zero frequency limit of these functions is related with the anomalous constitutive relations and with the full anomalous partition function in the presence of gauge and mixed anomalies. In particular, our results imply the existence of a previously unknown invariant contribution to the parity-odd partition function at third derivative order that we explicitly construct. Beyond the static limit, the gravitational response function may provide valuable insights into time-dependent phenomena driven by anomalies.