| We show that for in order for a type IIB supergravity half-BPS solution to be asymptotic to AdS5 x S5, it must have an invariance superalgebra which is a subalgebra of PSU(2, 2|4) and has 16 Killing spinors. Similarly, for an M-theory half-BPS solution to be asymptotic to AdS7 x S4 or AdS4 x S7, it must have an invariance superalgebra which is a subalgebra of OSp(8*|4) or OSp(8|4, R) respectively and has 16 Killing spinors. We find all relevant subalgebras of PSU(2, 2|4), OSp(8*|4), and OSp(8|4, R) and based on these are able to support the existence of certain solutions suspected based on probe analysis, as well conjecture the existence of unexpected solutions. We obtain one example, the solution invariant under OSp(4*|2) x OSp(4*|2) and asymptotic to AdS 7 x S4 and OSp(4|2, R) x OSp(4|2, R) and asymptotic to AdS4 x S7. Both solutions have the same spacetime isometry SO(2, 2) x SO(4) x SO(4), implying the metric form AdS3 x S 3 x S3 warped over a two-dimensional Riemann surface with boundary, Sigma. The BPS equations are reduced on this Ansatz, mapped to an integrable system of the sine-Gordon/Liouville type, and finally into a linear equation. The general local solutions are constructed explicitly in terms of one harmonic function on Sigma, and an integral transform of two further harmonic functions on Sigma. The solutions to the BPS equations are shown to automatically solve the Bianchi identities and field equations for the 4-form field, as well as Einstein's equations. The solutions have non-vanishing 4-form field strength on each of the three factors of AdS3 x S3 x S 3. The solutions asymptotic to AdS7 x S4 correspond to backreacted M2-branes and intersecting M5 branes, in the near horizon background of M5 branes. We impose regularity conditions on these local solutions and explicitly construct globally well-defined solutions. The solutions are interpreted as gravity duals to 1+1 dimensional defects in the 6-dimensional maximally symmetric CFT. |
