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In the framework of the AdS/CFT duality, we calculate the supersymmetricpartition function of the superconformal field theories living in the worldvolume of either $N$ $M2$-branes or $N$ $M5$-branes. We used the dualsupergravity partition function in a saddle point approximation oversupersymmetric 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 $d=2,5$. Theresulting partition function shows phase transitions, constraints on the phasespace and allowed us to identify unstable BPS Black hole in the $AdS$ phase.These configurations should correspond to unstable configurations in the dualtheory. We also report an intriguing relation between the most general WittenIndex, computed in the above theories, and our BPS partition functions.

The M2/M5 BPS Partition Functions from Supergravity