  The superconformal index of $\frac12$-BPS states of $N=4$ U(N) super
Yang-Mills theory has a known infinite $q$-series expression with successive
terms suppressed by $q^N$. We derive a holographic bulk interpretation of this
series by evaluating the corresponding functional integral in the dual $AdS_5
\times S^5$. The integral localizes to a product of small fluctuations of the
vacuum and of the collective modes of an arbitrary number of giant-gravitons
wrapping an $S^3$ of maximal size inside the $S^5$. The quantum mechanics of
the small fluctuations of one maximal giant is described by a supersymmetric
version of the Landau problem. The quadratic fluctuation determinant reduces to
a sum over the supersymmetric ground states, and precisely reproduces the first
non-trivial term in the infinite series. Further, we show that the terms
corresponding to multiple giants are obtained precisely by the matrix versions
of the above super-quantum-mechanics.
