Minisuperspace quantization of ``Bubbling AdS" and free fermion droplets

We quantize the space of 1/2 BPS configurations of type-IIB SUGRA found by Lin, Lunin and Maldacena (hep-th/0409174), directly in supergravity. We use the Crnkovic-Witten-Zuckerman covariant quantization method to write down the expression for the symplectic structure on this entire space of solutions. We find the symplectic form explicitly around AdS₅ × S⁵ and obtain a U(1) Kac-Moody algebra, in precise agreement with the quantization of a system of N free fermions in a harmonic oscillator potential, as expected from AdS/CFT. As a cross check, we also perform the quantization around AdS₅ × S⁵ by another method, using the known spectrum of physical perturbations around this background and find precise agreement with our previous calculation.