### Abstract

This paper focuses on supergravity duals of BPS states in = 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} × 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.

Original language | English |
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Article number | 003 |

Journal | Journal of High Energy Physics |

Volume | 2007 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2007 Oct 1 |

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### Keywords

- AdS-CFT correspondence
- Gauge-gravity correspondence
- M-theory
- Superstring vacua

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Journal of High Energy Physics*,

*2007*(10), [003]. https://doi.org/10.1088/1126-6708/2007/10/003