In M-theory, the only AdS 7 supersymmetric solutions are AdS 7 × S 4 and its orbifolds. In this paper, we find and classify new supersymmetric solutions of the type AdS 7 × M 3 in type II supergravity. While in IIB none exist, in IIA with Romans mass (which does not lift to M-theory) there are many new ones. We use a pure spinor approach reminiscent of generalized complex geometry. Without the need for any Ansatz, the system determines uniquely the form of the metric and fluxes, up to solving a system of ODEs. Namely, the metric on M 3 is that of an S 2 fibered over an interval; this is consistent with the Sp(1) R-symmetry of the holographically dual (1,0) theory. By including D8 brane sources, one can numerically obtain regular solutions, where topologically M 3 ∼ = S 3 .

Motivated by a recently found class of AdS 7 solutions, we classify AdS 5 solutions in massive IIA, finding infinitely many new analytical examples. We reduce the general problem to a set of PDEs, determining the local internal metric, which is a fibration over a surface. Under a certain simplifying assumption, we are then able to analytically solve the PDEs and give a complete list of all solutions. Among these, one class is new and regular. These spaces can be related to the AdS 7 solutions via a simple universal map for the metric, dilaton and fluxes. The natural interpretation of this map is that the dual CFT 6 and CFT 4 are related by twisted compactification on a Riemann surface Σ g . The ratio of their free energy coefficients is proportional to the Euler characteristic of Σ g . As a byproduct, we also find the analytic expression for the AdS 7 solutions, which were previously known only numerically. We determine the free energy for simple examples: it is a simple cubic function of the flux integers.

We describe three analytic classes of infinitely many AdS(d) supersymmetric solutions of massive IIA supergravity, for d=7,5,4. The three classes are related by simple universal maps. For example, the AdS(7)×M(3) solutions (where M(3) is topologically S(3)) are mapped to AdS(5)×Σ(2)×M(3)', where Σ(2) is a Riemann surface of genus g≥2 and the metric on M(3)' is obtained by distorting M(3) in a certain way. The solutions can have localized D6 or O6 sources, as well as an arbitrary number of D8-branes. The AdS(7) 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 AdS(4) solutions have numerical "attractor" generalizations that might be useful for flux compactification purposes.