  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 used the dual
supergravity partition function in a saddle point approximation over
supersymmetric Black Holes. Since our BHs are written in asymptotically global
$AdS_{d+1}$ co-ordinates, the dual SCFTs are in $R x S^{d}$ for $d=2,5$. The
resulting partition function shows phase transitions, constraints on the phase
space and allowed us to identify unstable BPS Black hole 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.
