On-Shell Action for Bubbling Geometries

We study the half-BPS sector of the AdS/CFT correspondence. In the fullsector of the correspondence, on-shell actions in type IIB supergravity havebeen played crucial roles, for example in the GKP-Witten relation. We derive,therefore, an on-shell action that reproduces a class of half-BPS solutionsfound by Lin, Lunin and Maldacena. These solutions describe the bubblinggeometries in the supergravity. We show that this on-shell action for the LLMsolutions is dual to the classical limit of an observable of the N freefermions in a harmonic oscillator potential in 1+1 dimensions. The observableconsists of three parts: an expectation value of the total energy, matrixelements of the momentum squared, and matrix elements of the position squared.Our results suggest that there exists a holographic correspondence on atwo-dimensional surface in the bubbling geometries.Comment: 12 pages, 3 figures, v2: references added, v3: section 3, 4 and 5 improved, a reference and a figure added, typos correcte