The giant graviton expansion in AdS₅ × S⁵

The superconformal index of 12 12 -BPS states of N=4 𝒩=4 U(N) U(N) super Yang-Mills theory has a known infinite q q -series expression with successive terms suppressed by q^N qN . We derive a holographic bulk interpretation of this series by evaluating the corresponding functional integral in the dual AdS _5 × S^5 5×S5 . 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 S3 of maximal size inside the S^5 S5 . 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.