that wrap an $S^3$ in the $S^5$ of the AdS$_5\times$S$^5$ background. Since the restricted Schur polynomials furnish a suitable basis for the two giant system, we know that these operators capture all excitations (BPS and non supersymmetric) of the two giant system. For a study of excitations of the single giant system using restricted Schur polynomials see [@Koch:2010gp]. For a spacetime study of excitations of the single giant system using the Born-Infeld action see [@Das:2000st]. The most important result from [@Das:2000st; @Koch:2010gp] for us, is that all the deformations of the single threebrane giant graviton that we consider, are supersymmetric.

The mixing of these operators with restricted Schur polynomials that have three columns (or more) is suppressed by a factor of order ${1\over\sqrt{N}}$. This factor arises from the normalization of the restricted Schur polynomials: the three column restricted Schur polynomials (with one short column) have a two point function which is smaller than the two column restricted Schur polynomials by a factor of order ${1\over N}$[@Koch:2010gp]. Thus, at large $N$ we can focus on the two column restricted Schur polynomials, which is a huge simplification. The analog of the statement that for operators with a dimension of $O(1)$, different trace 