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

The thermal partition function,

, of a

CFT_d C F T d

on

S^{d-1} S d − 1

is parameterized by the inverse temperature

\beta β

along with

\lfloor d/2\rfloor ⌊ d / 2 ⌋

angular velocities

\omega_i ω i

. In this paper, we investigate the behaviour of this partition function when

n n

of the

\omega_i ω i

are scaled to unity (the largest allowed value) at fixed values of the other

(\lfloor d/2\rfloor-n) ( ⌊ d / 2 ⌋ − n )

angular velocities. We argue that

\ln Z ln ⁡ Z

develops a simple pole in

(1-\omega_i) ( 1 − ω i )

for each

\omega_i ω i

that is scaled to unity. The residue of this product of poles is a theory-dependent (so non-universal) function of

\beta 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 n

special angular momenta

J_1\ldots J_n J 1 … J n

are scaled to infinity, the twist and the other angular momenta – collectively denoted

x_i 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}} x i / ( J 1 J 2 … J n ) 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}} S / ( J 1 J 2 … J n ) 1 n + 1

depends only on the

\lfloor d/2\rfloor-n+1 ⌊ d / 2 ⌋ − 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 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.