Complex Matrix Model and Fermion Phase Space for Bubbling AdS Geometries

TL;DR

The paper tackles how bubbling AdS half-BPS droplet geometries relate to a complex matrix model describing chiral primaries in ${ m N=4}$ SYM. It proves that the singlet holomorphic sector of the complex matrix model is exactly equivalent to the holomorphic sector of two-dimensional free fermions and, via second quantization, to one-dimensional free fermions, enabling a direct link to the Wigner phase-space distribution $u(p,q,t)$. Using AdS/CFT, the authors provide evidence that bubbles in bubbling AdS geometries correspond to droplets in the fermionic phase space, and they derive an explicit mapping between ${ m Tr}(Z^J)$ operators and the Wigner distribution. They further construct states corresponding to (dual) giant gravitons in the fermionic Hilbert space, reproducing the anticipated gravity-side droplet configurations, with results valid at finite $N$. Overall, the work furnishes a concrete, finite-$N$ bridge between matrix-model dynamics, fermionic phase-space, and bubbling AdS geometries, enriching the holographic understanding of half-BPS sectors.

Abstract

We study a relation between droplet configurations in the bubbling AdS geometries and a complex matrix model that describes the dynamics of a class of chiral primary operators in dual N=4 super Yang Mills (SYM). We show rigorously that a singlet holomorphic sector of the complex matrix model is equivalent to a holomorphic part of two-dimensional free fermions, and establish an exact correspondence between the singlet holomorphic sector of the complex matrix model and one-dimensional free fermions. Based on this correspondence, we find a relation of the singlet holomorphic operators of the complex matrix model to the Wigner phase space distribution. By using this relation and the AdS/CFT duality, we give a further evidence that the droplets in the bubbling AdS geometries are identified with those in the phase space of the one-dimensional fermions. We also show that the above correspondence actually maps the operators of N=4 SYM corresponding to the (dual) giant gravitons to the droplet configurations proposed in the literature.

Figures (6)

• Figure 1: Circular droplet corresponding to $AdS_5 \times S^5$.
• Figure 2: Fluctuation around the circular droplet.
• Figure 3: planar contribution.
• Figure 4: nonplanar contribution.
• Figure 5: A giant graviton in $AdS_5 \times S^5$.