Chern-Simons terms from thermal circles and anomalies

TL;DR

The paper demonstrates how anomalies affect the thermodynamic partition function in equilibrium by introducing a Euclidean-vacuum consistency condition that fixes base-manifold Chern-Simons terms through a thermal anomaly polynomial $P_T$. It constructs a master form $V_T$ whose derivatives give the anomaly-induced flavor, heat, and spin currents, and shows that the Chern-Simons coefficients are entirely determined by the anomaly data via a replacement rule. This framework yields the complete set of thermodynamic response parameters fixed by anomalies and provides a constructive method to compute anomaly-induced transport in hydrodynamics across dimensions. The results unify the treatment of gauge and gravitational anomalies in the thermal setting and have implications for hydrodynamic constitutive relations and entropy-current consistency.

Abstract

We compute the full contribution of flavor and (or) Lorentz anomalies to the thermodynamic partition function. Apart from the Wess-Zumino consistency condition the Euclidean generating function must satisfy an extra requirement which we refer to as `consistency with the Euclidean vacuum.' The latter requirement fixes all Chern-Simons terms that arise in a particular Kaluza-Klein reduction of the theory. The solution to both conditions may be encoded in a `thermal anomaly polynomial' which we compute. Our construction fixes all the thermodynamic response parameters of a hydrodynamic theory associated with anomalies.