Anomalies and the Helicity of the Thermal State

TL;DR

This work shows that the thermal helicity, defined as $\langle \mathfrak{L}_{12}\mathfrak{L}_{34}\dots \mathcal{P}_{2n-1} \rangle$, is a finite, parity-odd observable that, per unit volume, forms a homogeneous polynomial in the temperature $T$ and chemical potentials $\mu$. The authors establish a concrete link between this polynomial and the theory’s anomalies: in theories without chiral gravitino, the polynomial can be derived from the anomaly polynomial $\mathcal{P}_{anom}$ via the replacement $F \mapsto \mu$, $p_1(\mathfrak{R}) \mapsto -T^2$, $p_{k>1}(\mathfrak{R}) \mapsto 0$, enabling a direct computation of anomaly-induced transport from thermodynamic data. This connection is demonstrated through sphere partition function methods in free theories (fermions and chiral bosons) and corroborated by AdS/CFT expectations, yielding a universal Cardy-like relation in higher dimensions. However, the case of a free chiral gravitino shows a breakdown of the replacement rule, with gravitino contributions enhancing the helicity by a factor $(2n-1)$ compared to a fermion, indicating a qualitative distinction for massless higher-spin fields. Overall, the paper provides a versatile framework connecting finite-temperature observables to anomalies, with implications for non-equilibrium dynamics and potential tests of modular invariance and holography in diverse dimensions.

Abstract

We study the thermal expectation value of the following observeable at finite temperature T and chemical potential μ: < L_{12} L_{34} ... L_{d-3,d-2} P_{d-1} > where L_{ij} denote the angular momenta, and P_i denotes the spatial momentum in d spacetime dimensions with d even. We call this observeable the thermal helicity. Using a variety of arguments, we motivate the surprising assertion that thermal helicity per unit volume is a polynomial in T and μ. Further, in field theories without chiral gravitino, we conjecture that this polynomial can be derived from the anomaly polynomial of the theory. We show that this conjecture is related to the recent conjecture on gravitational anomaly induced transport made in arXiv:1201.2812 . We support these statements by various sphere partition function computations in free theories.