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

We extend the construction of bubbling 1/2 BPS solutions of Lin, Lunin and Maldacena (H. Lin, O. Lunin, J. Maldacena, JHEP 0410 (2004) 025, 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 S1×S1S1×S1 bosonic reduction of (1,0)(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=0F∧F=0). This is to be contrasted to the case of the S3×S3S3×S3 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)(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 AdS3×S3AdS3×S3 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.