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

The high-temperature expansion of twisted $S^{d-1}\times S^1$ partition functions of CFT$_d$, $d = 2κ$, 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 $\mathbb{Z}_{κ+1}$, which cyclically permutes the thermal circle and the rotation angles. Our equation asserts that the $\mathbb{Z}_{κ+1}$-invariant part of $\ln Z/β$ 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.