Quantum states to brane geometries via fuzzy moduli spaces of giant gravitons

Eighth-BPS local operators in 𝒩 = 4 SYM are dual to quantum states arising from the quantization of a moduli space of giant gravitons in AdS 5 × S 5 . Earlier results on the quantization of this moduli space give a Hilbert space of multiple harmonic oscillators in 3 dimensions. We use these results, along with techniques from fuzzy geometry, to develop a map between quantum states and brane geometries. In particular there is a map between the oscillator states and points in a discretization of the base space in the toric fibration of the moduli space. We obtain a geometrical decomposition of the space of BPS states with labels consisting of U(3) representations along with U( N ) Young diagrams and associated group theoretic multiplicities. Factorization properties in the counting of BPS states lead to predictions for BPS world-volume excitations of specific brane geometries. Some of our results suggest an intriguing complementarity between localisation in the moduli space of branes and localisation in space-time.