The large $N$ vector model with angular velocity

Abstract: We study the free energy of the critical $O(N)$ vector model at large $N$ on $S^{1}\times S<sup>{2}$ with an angular velocity $\hatμ$ without the singlet constraint. The leading high-temperature behaviour is determined analytically both as an expansion about $\hatμr=0$ and $\hatμ<sup>{2}r<sup>{2}=1$ where $r$ is the radius of the sphere. We supplement the analytic results with a numerical analysis that agrees with both the analytical expansions in their respective regimes and smoothly interpolates between them. The leading high-temperature contribution to the free energy develops a pole at $\hatμ<sup>{2}r<sup>{2}=1$, in agreement with expectations from the thermal effective field theory. Its residue coincides with that of the massless free theory. Sub-leading terms, however, exhibit non-analytic dependence on the angular velocity and distinguish the critical fixed-point result from the free theory answer. The residue at the pole can also be obtained by placing the $O(N)$ model on the pp-wave geometry. We show that the residue agrees with that obtained from the direct computation.