Semi-universality of CFT${}_d$ entropy at large spin

Abstract

The thermal partition function, $Z$, of a $CF{T}_d$ on $S^{d-1}$ is parameterized by the inverse temperature $\beta$ along with $\lfloor d/2\rfloor$ angular velocities ${\omega }_i$. In this paper, we investigate the behaviour of this partition function when $n$ of the ${\omega }_i$ are scaled to unity (the largest allowed value) at fixed values of the other $(\lfloor d/2\rfloor -n)$ angular velocities. We argue that $\ln Z$ develops a simple pole in $(1-{\omega }_i)$ for each ${\omega }_i$ that is scaled to unity. The residue of this product of poles is a theory dependent (so non-universal) function of $\beta$ and the fixed angular velocities. The inverse Laplace transformation of this partition function constrains the functional form of the field theory entropy as a function of charges in a limit in which angular momenta and the twist are scaled as follows. While $n$ special angular momenta $J_1\dots J_n$ are scaled to infinity, the twist and the other angular momenta - collectively denoted $x_i$ - are also taken to infinity but at the slower rate that ensures that the scaled charges $x_i/(J_1{J}_2\dots J_n{)}^{\frac{1}{n+1}}$ are held fixed. In this limit, we demonstrate that the scaled entropy $S/(J_1{J}_2\dots J_n{)}^{\frac{1}{n+1}}$ depends only on the $\lfloor d/2\rfloor -n+1$ scaled charges defined above (the precise form of this dependence is non-universal). We verify our predictions (and compute all non-universal functions) in the case of free scalar theories (which show surprisingly rich behaviour) as well as large $N$, strongly coupled $\mathcal{N}=4$ Yang Mills theory. The last theory is analyzed in the bulk via the AdS/CFT correspondence. In the scaling limit described above, its phase diagram displays sharp phase transitions between black hole, grey galaxy, and thermal gas phases.

83 pages+appendices, 1 figure; v2: minor typos fixed