We present a trace formula for a Witten type Index for superconformal field
theories in d=3,5 and 6 dimensions, generalizing a similar recent construction
in d=4. We perform a detailed study of the decomposition of long
representations into sums of short representations at the unitarity bound to
demonstrate that our trace formula yields the most general index (i.e. quantity
that is guaranteed to be protected by superconformal symmetry alone) for the
corresponding superalgebras. Using the dual gravitational description, we
compute our index for the theory on the world volume of N M2 and M5 branes in
the large N limit. We also compute our index for recently constructed Chern
Simons theories in three dimensions in the large N limit, and find that, in
certain cases, this index undergoes a large N phase transition as a function of
chemical potentials.Comment: a small typo corrected, 46 page
