M- theory solutions invariant under D(2,1;.). D(2,1;.)

We simplify and extend the construction of half-BPS solutions to 11-dimensional supergravity, with isometry superalgebra D(2,1;) circle plus D(2,1;). Their space-time has the form AdS(3)x S(3)x S-3 warped over a Riemann surface sigma. It describes near-horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data (, h, G), where is the real parameter of the isometry superalgebra, and h and G are functions on sigma whose differential equations and regularity conditions depend only on the sign of . The magnitude of enters only through the map of h,G onto the supergravity fields, thereby promoting all solutions into families parametrized by ||. By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2-dimensional CFT dual is AdS(3)x S(3)x S(3)x (2), modulo discrete identifications of the flat (2), and (ii) that solutions with < 0 cannot have more than one asymptotic higher-dimensional AdS region. We classify the allowed singularities of h and G near the boundary of sigma, and identify four local solutions: asymptotic AdS(4)/Z(2) or AdS(7) regions; highly-curved M5-branes; and a coordinate singularity called the cap. By putting these Lego pieces together we recover all known global regular solutions with the above symmetry, including the self-dual strings on M5 for <0, and the Janus solution for > 0, but now promoted to families parametrized by ||. We also construct exactly new regular solutions which are asymptotic to AdS(4)/Z(2) for < 0, and conjecture that they are a different superconformal limit of the self-dual string. Finally, we construct exactly > 0 solutions with highly curved M5-brane regions, which are the formal continuation of the self-dual string solutions across the decompactification point at = 0.