Semi-Universality of $O(N)$ model on $S^1\times S^2$
We study the thermal partition function of the critical large-$N$ $O(N)$ model on rotating $S^1\times S^2$. Rotation makes the saddle latitude dependent, with its leading profile governed by the local temperature. Using a Weyl-covariant hydrostatic expansion, we determine the partition function through four-derivative order in high-temperature and semi-universal limits. In the near-light-speed limit, the geometry around the equator reduces to a pp-wave and captures the expected semi-universal behavior of the partition function. We compute the corresponding theory-dependent residue directly on this limiting geometry, both at high and low temperatures. The high-temperature result agrees with the near-light-speed limit of the sphere calculation, while the low-temperature expansion reveals interaction-dependent corrections arising from the response of the saddle to the thermal source.