Pressure and compressibility of conformal field theories from the AdS/CFT correspondence

The equation of state associated with ${\cal N}=4$ supersymmetric Yang-Mills in 4 dimensions, for $SU(N)$ in the large $N$ limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang-Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure and the $P-V$ diagram is studied. It is known that there is a critical point where the heat capacity diverges and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large $N$ limit. The same analysis applied to 3 and 6 dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point.