Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

TL;DR

<3-5 sentence high-level summary> The paper extends the Lin–Lunin–Maldacena (LLM) bubbling AdS construction beyond the 1/2 BPS sector to 1/4 and 1/8 BPS states in IIB supergravity, by performing a sequence of breathing-mode reductions on S^3, S^3×S^1, and CP^1. It shows that regularity is encoded in boundary data on Kaehler bases: a linear harmonic function governs the 1/2 BPS case, while nonlinear Monge–Ampère and curvature conditions control the 1/4 and 1/8 sectors, respectively, with bubbling described as droplets in higher-dimensional phase spaces. The work provides explicit examples (AdS$_5$×S^5, AdS$_3$×S^3×T^4, three-charge bubbles, LLM embeddings) and develops a unified droplet picture linking gravity to dual matrix models and eigenvalue distributions, highlighting topology changes and flux quantization as key features. These results pave the way for understanding nontrivial topology and interactions among droplets in reduced-supersymmetry sectors and their gauge-theory interpretations.

Abstract

This paper focuses on supergravity duals of BPS states in N=4 super Yang-Mills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S^3, then S^3 x S^1, and finally on S^3 x S^1 x CP^1. We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S^3). The 1/8 BPS geometries, which have an S^3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kahler metric on the base. Similarly, the 1/4 BPS configurations have an S^3 x S^1 isometry and a four-dimensional base, whose Kahler metric obeys another non-linear, Monge-Ampere type equation. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S^3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS_5 x S^5 ground state corresponds to excising the interior of an S^5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S^5 and/or the excision of additional droplets from the base. In the case of 1/4 BPS configurations, by enforcing regularity conditions, we identify three-dimensional surfaces inside the four-dimensional base which separate the regions where the S^3 shrinks to zero size from those where the S^1 shrinks.

Figures (6)

• Figure 1: Profile of $r$ versus $|z_1|$ for the configuration corresponding to a single hole of radius $0.1$ centered at the origin of the AdS disk (of unit radius). This picture corresponds to a maximal giant graviton expanding on $S^5$.
• Figure 2: An LLM configuration with three droplets and two holes.
• Figure 3: The LLM configuration of Fig. \ref{['fig:llmbub']} shown as droplets in the six-dimensional base given by (\ref{['eq:6dhbase']}). Here $r^2=|z_2|^2 +|z_3|^2$, and the additional $S^3$ directions are suppressed. Note that the physical space is comprised of the region outside of the droplets only.
• Figure 4: Schematic picture of a 1/2 BPS configuration corresponding to four dual giant gravitons excited on top of the AdS vacuum (central sphere). Giant gravitons expanding on $S^5$ are not pictured, but would correspond to giving the AdS sphere a non-trivial topology. These 1/2 BPS configurations always preserve an $\tilde{S}^3$ invariance corresponding to rotations in the $z_2$-$z_3$ planes.
• Figure 5: Picture of a 1/4 BPS configuration with five dual giant gravitons. The configuration is symmetric under $S^1$ rotations in the $z_3$ plane (which, however, cannot be directly visualized since the imaginary components of the axes are suppressed).