Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

Abstract

This paper focuses on supergravity duals of BPS states in N = 4 super YangMills. 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 × S 1 , and finally on S 3 × S 1 × 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 Kähler metric on the base. Similarly, the 1/4 BPS configurations have an S 3 × S 1 isometry and a four-dimensional base, whose Kähler metric obeys another non-linear, Monge-Ampère 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 × 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. We discuss a large class of examples to show the emergence of a universal bubbling AdS picture for all 1/2, 1/4 and 1/8 BPS geometries.

(Non)-integrability of geodesics in D-brane backgrounds

“…From the point of view of AdS/CFT correspondence, it is equally interesting to look at field theories on R × S 3 , which are dual to geometries produced by spherical D3 branes. The most symmetric geometry of this type is a direct product of global AdS 5 and a five dimensional sphere, but less symmetric examples are also known [28,[69][70][71][72]. In this section we will apply the techniques developed in section 3 to identify the most general geometries of [28] with separable geodesics.…”

Brane webs and 1/4-BPS geometries

“…the authors of [23] to propose a similar picture for the 1/4-BPS droplets: it was argued that, by introducing droplets with Z = ± 1 2 in 2D Kahler space, one produces a regular geometry. It turns out that the 1/4-BPS case is more subtle.…”

“…19 Solution (6.2) is written in the notation introduced in [23]. More general family of 1/4-BPS geometries with SO(4) × U(1) isometries was constructed in [32], but due to the mixing between U(1) direction and other coordinates, these solutions break Z2 symmetry, so they are not relevant for describing the states (6.1), and we will not discuss these metrics further.…”

“…The appropriate map was derived in [23], and here we just write down the answer which will be used later on.…”

“…Notice that the 1/2-BPS geometries (6.22) can also be embedded into different class of 1/4-BPS solutions constructed in [32] by treatingφ + ψ rather than ψ as a Killing vector preserved by the 1/4-BPS geometries. The corresponding embedding was discussed in [32,23], but it appears to be irrelevant for viewing states i TrX n i as a subset of objects constructed out of (6.1).…”