Six-Dimensional Superconformal Theories and their Compactifications from Type IIA Supergravity

We describe three analytic classes of infinitely many AdSd supersymmetric solutions of massive IIA supergravity, for $d=7,5,4$. The three classes are related by simple universal maps. For example, the AdS₇×M₃ solutions (where M₃ is topologically S³) are mapped to AdS₅×Σ₂×M₃^', where Σ₂ is a Riemann surface of genus g≥2 and the metric on M₃^' is obtained by distorting M₃ in a certain way. The solutions can have localized D6 or O6 sources, as well as an arbitrary number of D8-branes. The AdS₇ case (previously known only numerically) is conjecturally dual to an NS5-D6-D8 system. The field theories in three and four dimensions are not known, but their number of degrees of freedom can be computed in the supergravity approximation. The AdS₄ solutions have numerical ``attractor'' generalizations that might be useful for flux compactification purposes.