Pressure and Compressibility of Conformal Field Theories from the AdS/CFT Correspondence

The equation of state associated with 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 grav-ity 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 com-pressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to 3 and 6 dimensional con-formal field theories again yields mean field exponents associated with the compressibility at the critical point. [1