| We study supersymmetric partition functions in several versions of the AdS/CFT correspondence.;We present an Index for superconformal field theories in d = 3, 4, 5, 6. This captures all information about the spectrum that is protected, under continuous deformations of the theory, purely by group theory. We compute our Index in N = 4 SYM at weak coupling using gauge theory and at strong coupling using supergravity and find perfect agreement at large N. We also compute this Index for supergravity on AdS4 x S7 and AdS7 x S4 and for the recently constructed Chern Simons matter theories.;We count 1/16 BPS states in the free gauge theory and find qualitative agreement with the entropy of big black holes in AdS5. We note that the near horizon geometry of some small supersymmetric black holes is an extremal BTZ black holes fibered on a compact base and propose a possible explanation for this, based on giant gravitons. We also find the partition function of the chiral ring of the N = 4 SYM theory at finite coupling and finite N.;Turning to AdS3, we study the low energy 1/4 and 1/2 BPS partition functions by finding all classical supersymmetric probe brane solutions of string theory on this background. If the background BNS field and theta angle vanish, AdS3 x S 3 x T4/K3 supports supersymmetric probes: D1 branes, D5 branes and bound states of D5 and D1 branes. In global AdS, upon quantization, these solutions give rise to states in discrete representations of the SL(2,R) WZW model on AdS 3.;We conclude that (a) the 1/4 BPS partition function jumps if we turn on a theta angle or NS-NS field (b) generic 1/2 BPS states are protected. We successfully compare our 1/2 BPS partition function with that of the symmetric product. We also discuss puzzles, and their possible resolutions, in reproducing the elliptic genus of the symmetric product.;Finally, we comment on the spectrum of particles in the theory of gravity dual to non-supersymmetric Yang Mills theory on S3 x time. |
