From $Z$ to $a$: High-temperature relations, subleading semi-universality, and conformal anomalies

The free energy of any CFT, $ \ln Z(β; ω_i)$, admits two expansions: high temperature ($β\rightarrow 0$) and fast rotation ($ω_i \rightarrow 1$). We demonstrate that locality of the thermal effective action forces $\ln Z$ to take a simple analytic form at all orders in the high temperature expansion, and further imposes an infinite number of sharp relations on the coefficients in this expansion. All are homogeneous, except at order $β^1$ due to the Weyl anomaly. From this, the $a$-anomaly can be extracted from the partition function. The relations resum in the fast-spinning expansion into differential equations in $β$ obeyed by the semi-universal limit and its corrections. We verify the relations in a variety of CFTs. We generalize to any even $d$, but find no similar relations at odd $d$.