Abstract:
The thermal partition function, Z, of a CFT_4 on S^{3} is parameterized by the inverse temperature $\beta$ along with 2 angular velocities $\omega_i$. In this project we investigate the behaviour of this partition function when the $\omega_i$ are scaled to unity (the largest allowed value). We argue that \ln Z develops a simple pole in (1-\omega_i). The residue of this product of poles is a theory dependent (so non universal) function of $\beta$ and the fixed angular velocities. The Legendre 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 the angular momenta are scaled to infinity, the twist T is also taken to infinity but at the slower rate such that the scaled twist $T/(J1J2)^{\frac{1}{3}}$ is held fixed. In this limit we demonstrate that the scaled entropy $S/(J_1 J_2 )^{\frac{1}{3}}$ depends only on the scaled twist (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, 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.}.