The M2/M5 BPS partition functions from supergravity

In the framework of the AdS/CFT duality, we calculate the supersymmetric partition function of the superconformal field theories living in the world volume of either N M2-branes or N M5-branes. We use the dual supergravity partition function in a saddle point approximation over supersymmetric Black Holes. Since our BHs are written in asymptotically global AdSd+1 co-ordinates, the dual SCFTs are in R x Sd for d = 2, 5. The resulting partition function shows phase transitions, constraints on the phase space and allowed us to identify unstable BPS Black holes in the AdS phase. These configurations should correspond to unstable configurations in the dual theory. We also report an intriguing relation between the most general Witten Index, computed in the above theories, and our BPS partition functions.