From Z to all: cyclic sum rules for thermal CFT partition functions

Abstract

The high-temperature expansion of twisted $S^{d-1}\times S^1$ partition functions of CFT${}_d$, $d=2\kappa$, has recently been shown to obey an infinite tower of universal constraints. We determine these constraints explicitly and demonstrate that they can be packaged into a single equation involving a large conformal diffeomorphism isomorphic to ${\mathbb{Z}}_{\kappa +1}$, which cyclically permutes the thermal circle and the rotation angles. Our equation asserts that the ${\mathbb{Z}}_{\kappa +1}$-invariant part of $\ln Z/\beta$ is a constant fixed by the a-anomaly. Our derivation assumes CP-invariance and locality of the thermal effective action, whose structure we constrain using an algebraic formalism starting from embedding space.

arXiv comment40 pages: 8 pages main Letter (including references), 2 pages appendices, and 30 pages Supplemental Material; 5 figures