Bubbling 1/4 BPS solutions in type IIB and supergravity reductions on

Abstract

We extend the construction of bubbling 1/2 BPS solutions of Lin, Lunin and Maldacena (hep-th/0409174) in two directions. First we enquire whether bubbling 1/2 BPS solutions can be constructed in minimal 6d supergravity and second we construct solutions that are 1/4 BPS in type IIB. We find that the S 1 × S 1 bosonic reduction of (1,0) 6d supergravity to 4d gravity coupled to 2 scalars and a gauge field is consistent only provided that the gauge field obeys a constraint (F ∧ F = 0). This is to be contrasted to the case of the S 3 × S 3 bosonic reduction of type IIB supergravity to 4d gravity, 2 scalars and a gauge field, where consistency is achieved without imposing any such constraints. Therefore, in the case of (1,0) 6d supergravity we are able to construct 1/2 BPS solutions, similar to those derived in type IIB, provided that this additional constraint is satisfied. This ultimately prohibits the construction of a family of 1/2 BPS solutions corresponding to a bubbling AdS 3 × S 3 geometry. Returning to type IIB solutions, by turning on an axion-dilaton field we construct a family of bubbling 1/4 BPS solutions. This corresponds to the inclusion of back-reacted D7 branes to the solutions of Lin, Lunin and Maldacena.

Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

“…In addition, solutions preserving an S 3 isometry (corresponding to the 1/8 BPS case) may be obtained by double analytic continuation of the AdS 3 solutions investigated in [20], as it was later done in [21]. (Note that 1/4 BPS and 1/8 BPS solutions of a different nature were also investigated in [4] and [22], respectively.) In both cases of S 3 isometry [20] and S 3 × S 1 isometry [18,19], the invariant tensor analysis and resulting description of the backgrounds are essentially complete.…”

“…In order to properly reduce the covariant derivative 7 ∇ µ appearing in the gravitino variation (2.12), we must keep in mind that Killing spinors ǫ may in fact be charged along the U(1) fiber [4]. We thus take 25) where n ∈ Z, and the sign is chosen for later convenience.…”

“…Note that here we have explicitly set η = −1. 4 We first want to consider the flux that is orthogonal to the (dt + ω)∧ dy ∧ Ω 3 directions. This flux component is easy to identify using (6.99) and (3.45).…”

“…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.…”

“…The four-dimensional metric, two scalars H and G, and the 2-form field strength F µν are related to their ten-dimensional counterparts according to [1,4,5] ds 2 10 = g µν dx µ dx ν + e H (e G dΩ 2 3 + e −G d Ω 2 3 ), F (5) = (1 + * 10 )F (2) ∧ Ω 3 .…”