Giant Graviton Expansion from Bubbling Geometry: Discreteness from Quantized Geometry

The superconformal index of half-BPS states in N=4 supersymmetric Yang-Mills with gauge group U(N) admits an expansion in terms of giant gravitons, I_{N}(q)=I_{∞}(q)∑_{m=0}^{∞}q^{mN}I[over ^]_{m}(q), where m is the number of giant gravitons and I_{∞}(q) is the graviton index. The expansion can be viewed as the implementation of trace relations for finite N. We derive this expansion directly in supergravity from the class of half-BPS solutions due to Lin, Lunin, and Maldacena in type IIB supergravity. The moduli space of these configurations can be quantized using covariant quantization methods. We show how this quantization leads to the precise expression for the expansion in terms of giant gravitons. Our proposal provides a derivation of the giant graviton expansion directly in terms of quantized supergravity degrees of freedom, and it recovers discrete data via quantum geometries that are classically nonsmooth.