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

TL;DR

This work identifies and explores a semi-universal structure in the twisted partition function Z of CFT_d on S^{d-1}×S^1, focusing on limits where multiple angular velocities approach unity. The authors show that ln Z develops simple poles as these ω_i→1 with residues that are theory-dependent, leading to a semi-universal form for the entropy in a large angular-momentum limit S ∝ (J_1…J_n)^{1/(n+1)} S^{int}(...). They derive and check these structures via derivative-expansion methods, exact free-scalar results, and holographic large-N N=4 SYM in AdS/CFT, where the bulk phase diagram features black-hole, grey-galaxy, and thermal-gas phases. The results imply that, despite non-universal microscopic details, the macroscopic thermodynamics in these regimes obey universal pole structures and a universal entropy scaling form, constraining the UV data of CFTs and relating to lightcone bootstrap insights. The work opens avenues for understanding semi-universal aspects across dimensions, massive deformations, and other manifolds, with potential implications for OPE coefficients and stress-tensor universality in extreme spin limits.

Abstract

The thermal partition function, $Z$, of a $CFT_d$ on $S^{d-1}$ is parameterized by the inverse temperature $β$ along with $\lfloor d/2\rfloor$ angular velocities $ω_i$. In this paper, we investigate the behaviour of this partition function when $n$ of the $ω_i$ 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 $(1-ω_i)$ for each $ω_i$ that is scaled to unity. The residue of this product of poles is a theory dependent (so non-universal) function of $β$ 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 $J_1\ldots J_n$ 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 $x_i/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}}$ are held fixed. In this limit, we demonstrate that the scaled entropy $S/(J_1 J_2 \ldots J_n)^{\frac{1}{n+1}}$ 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 ${\cal N}=4$ 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.

Figures (1)

• Figure 1: The integrand in \ref{['eq:int']} for $\omega \in \{1/2,2/3,0.9,0.95\}$ is plotted. As $\omega \to 1$, the denominator $(1-\omega^2\sin^2\theta)$ becomes small only in the vicinity of the equator $\theta=\pi/2$. Thus the integrand is sharply peaked there, and the dominant contribution to the integral comes from a narrow neighborhood around $\theta=\pi/2$. We make this precise below.