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Eighth-BPS local operators in N=4 SYM are dual to quantum states arising fromthe quantization of a moduli space of giant gravitons in AdS5xS5. Earlierresults on the quantization of this moduli space give a Hilbert space ofmultiple harmonic oscillators in 3 dimensions. We use these results, along withtechniques from fuzzy geometry, to develop a map between quantum states andbrane geometries. In particular there is a map between the oscillator statesand points in a discretization of the base space in the toric fibration of themoduli space. We obtain a geometrical decomposition of the space of BPS stateswith labels consisting of U(3) representations along with U(N) Young diagramsand associated group theoretic multiplicities. Factorization properties in thecounting of BPS states lead to predictions for BPS world-volume excitations ofspecific brane geometries. Some of our results suggest an intriguingcomplementarity between localisation in the moduli space of branes andlocalisation in space-time.

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