The giant graviton expansion in $AdS_5 \times S^5$

Abstract

The superconformal index of \frac1212-BPS states of \mathcal{N}=4𝒩=4U(N)U(N) super Yang-Mills theory has a known infinite qq-series expression with successive terms suppressed by q^NqN. We derive a holographic bulk interpretation of this series by evaluating the corresponding functional integral in the dual AdS_5 × S^55×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^3S3 of maximal size inside the S^5S5. 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.