Anomalies and the helicity of the thermal state

We study the thermal expectation value of the following observeable at finite temperature T and chemical potential mu: < L12L34 ... Ld-3,d-2Pd-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 mu. 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.