Geometry Quantization from Supergravity: the case of "Bubbling AdS"

TL;DR

The work develops and applies on-shell quantization, via the covariant CWZ symplectic current, to the LLM bubbling AdS geometries with fixed fluxes. By restricting the full gravitational plus 5-form symplectic structure to the LLM moduli space and carefully treating regular gauge perturbations, the authors derive the symplectic form for arbitrary droplet shapes and show exact agreement with the dual free-fermion phase-space description. This resolves previous discrepancies and demonstrates that the gravity side reproduces the correct Hilbert space and commutation relations for 1/2 BPS states in the large N limit, effectively providing a gravity-based derivation of the AdS/CFT correspondence in this sector. The methodology offers a general recipe to quantize moduli spaces of supergravity solutions and has potential applications to counting microstates, including 2-charge and 3-charge black hole geometries, and to exploring 1/N corrections in SUSY-protected regimes.

Abstract

We consider the moduli space of 1/2 BPS configurations of type IIB SUGRA found by Lin, Lunin and Maldacena (hep-th/0409174), and quantize it directly from the supergravity action, around any point in the moduli space. This quantization is done using the Crnkovic-Witten-Zuckerman covariant method. We make some remarks on the applicability and validity of this general on-shell quantization method. We then obtain an expression for the symplectic form on the moduli space of LLM configurations, and show that it exactly coincides with the one expected from the dual fermion picture. This equivalence is shown for any shape and topology of the droplets and for any number of droplets. This work therefore generalizes the previous work (hep-th/0505079) and resolves the puzzle encountered there.

Figures (4)

• Figure 1: An example of a 3-dimensional moduli space $\mathcal{M}$ foliated by 2-dimensional symplectic sheets corresponding to a fixed value of a gauge field flux $\mathcal{F}$. On-shell quantization can determine symplectic structure on the sheets, but cannot be used to quantize the flux.
• Figure 2: The surface $\Sigma_{2}$ caps the droplet in the $x_{1},x_{2}$ plane by extending into $y>0$. Fibering the $\tilde{S}^{3}$ over $\Sigma$, we get a closed 5-manifold supporting an $F_{5}$ flux proportional to the area of the droplet LLM.
• Figure 3: For (a collection of) droplets of characteristic size $\ell$, we split the plane into three regions: the near-infinity region $R_{1}=\{x:\text{dist}(x,\mathcal{D})\gg\ell\},$ the near-$\partial\mathcal{D}$ region $R_{2}=\{x:\text{dist}(x,\partial\mathcal{D})\ll\ell\},$ and the intermediate region $R_{3}$ covering the rest of the plane. To show that integral (\ref{['toshow']}) vanishes, we use large-$x$ asymptotics in $R_{1}$, wavy line approximation in $R_{2}$, and uniform convergence in $R_{3}$.
• Figure 4: The vector field $\delta U$ has vanishing integrals over topologically trivial ($\Gamma_{1,2}$) as well as nontrivial ($\Gamma_{3}$) contours in the complement of $\partial\mathcal{D}$. This is enough to show that (\ref{['tosolve1']}) has a global solution.