Bubbling AdS space and 1/2 BPS geometries

TL;DR

The paper shows that all 1/2 BPS excitations of AdS×S configurations in type IIB string theory and M-theory can be described by smooth, horizonless geometries determined by simple boundary data on a two-dimensional plane, with the boundary data encoding droplets of free fermions in phase space.In IIB, the geometries are governed by a linear Laplace-type equation for a master function z on a plane, with regularity realized by boundary conditions z=±1/2 on y=0, yielding a direct correspondence between droplet topology and spacetime topology and flux quantization.In M-theory, the corresponding half-BPS sectors reduce to a Toda-like equation in the four-dimensional base; analytic continuations and U-dualities connect these to AdS5 compactifications with N=2 supersymmetry and to mass-deformed M2/M5 vacua, including a non-singular gauged SUGRA construction that uplifts to 11D.Across both theories, the work makes precise the dictionary between fermionic phase-space configurations, brane/flux transitions, and fully non-singular geometries, suggesting a robust pathway to quantizing these geometries via droplet data and highlighting topological transitions as a central feature of the AdS/CFT correspondence in the 1/2 BPS sector.

Abstract

We consider all 1/2 BPS excitations of $AdS \times S$ configurations in both type IIB string theory and M-theory. In the dual field theories these excitations are described by free fermions. Configurations which are dual to arbitrary droplets of free fermions in phase space correspond to smooth geometries with no horizons. In fact, the ten dimensional geometry contains a special two dimensional plane which can be identified with the phase space of the free fermion system. The topology of the resulting geometries depends only on the topology of the collection of droplets on this plane. These solutions also give a very explicit realization of the geometric transitions between branes and fluxes. We also describe all 1/2 BPS excitations of plane wave geometries. The problem of finding the explicit geometries is reduced to solving a Laplace (or Toda) equation with simple boundary conditions. We present a large class of explicit solutions. In addition, we are led to a rather general class of $AdS_5$ compactifications of M-theory preserving ${\cal N} =2$ superconformal symmetry. We also find smooth geometries that correspond to various vacua of the maximally supersymmetric mass-deformed M2 brane theory. Finally, we present a smooth 1/2 BPS solution of seven dimensional gauged supergravity corresponding to a condensate of one of the charged scalars.

Figures (10)

• Figure 1: Droplets representing chiral primary states. In the field theory description these are droplets in phase space occupied by the fermions. In the gravity picture this is a particular two-plane in ten dimensions which specifies the solution uniquely. In (a) we see the droplet corresponding to the $AdS \times S$ ground state. In (b) we see ripples on the surface corresponding to gravitons in $AdS \times S$. The separated black region is a giant graviton brane which wraps an $S^3$ in $AdS_5$ and the hole at the center is a giant graviton brane wrapping an $S^3$ in $S^5$. In (c) we see a more general state.
• Figure 2: Plane wave configurations correspond to filling the lower half plane. This can be understood from the fact that the plane wave solution is a limit of the $AdS \times S$ solution.
• Figure 3: We see various configurations whose solutions can be easily constructed as superpositions of the $AdS_5 \times S^5$ solution and the plane wave solution. In (a) we see an example of the type of configurations that can be obtained by superimposing the circular solution (\ref{['ads1']}). In (b) we see generic configurations that lead to solutions which have two Killing vectors and lead to static configurations in $AdS$. In (c) we see the solution corresponding to a superposition of D3 branes wrapping the $\tilde{S}^3$ in $S^5$. In (d) we see the configuration resulting from many such branes, which can be thought of as a superposition of branes on the $S^3$ of $AdS_5$ uniformly distributed along the angular coordinate $\tilde{\phi}$ of $S^5$. In (e) we see a configuration that can be viewed as an excitation of a plane wave with constant energy density. In (f) we see a plane wave excitation with finite energy.
• Figure 4: We can construct a five-manifold by adding the sphere $\tilde{S}^3$ fibered over the surface $\tilde{\Sigma}_2$. This is a smooth manifold since at the boundary of $\tilde{\Sigma}_2$ on the $y=0$ plane the sphere $\tilde{S}^3$ is shrinking to zero. The flux of $F_5$ is proportional to the area of the black region inside $\tilde{\Sigma}_2$. Another five manifold can be constructed by taking $\Sigma_2$ and adding the other three--sphere $S^3$. The flux is proportional to the area of the white region contained inside $\Sigma_2$.
• Figure 5: We see here an example of a two dimensional surface, $\Sigma_2$, that is surrounding a ring. If we add the three--sphere $\tilde{S}^3$ fibered over $\Sigma_2$ we get a five manifold with the topology of $S^4 \times S^1$.