3-D superconformal theories from Sasakian seven manifolds: New nontrivial evidences for AdS(4) / CFT(3)

In this paper we discuss candidate superconformal N=2 gauge theories that realize the AdS/CFT correspondence with M-theory compactified on the homogeneous Sasakian 7-manifolds M7 that were classified long ago. In particular we focus on the two cases M7=Q1,1,1 and M7=M1,1,1, for the latter the Kaluza-Klein spectrum being completely known. We show how the toric description of M7 suggests the gauge group and the supersingleton fields. The conformal dimensions of the latter can be independently calculated by comparison with the mass of baryonic operators that correspond to 5-branes wrapped on supersymmetric 5-cycles and are charged with respect to the Betti multiplets. The entire Kaluza-Klein spectrum of short multiplets agrees with these dimensions. Furthermore, the metric cone over the Sasakian manifold is a conifold algebraically embedded in some Cp. The ring of chiral primary fields is defined as the coordinate ring of Cp modded by the ideal generated by the embedding equations; this ideal has a nice characterization by means of representation theory. The entire Kaluza-Klein spectrum is explained in terms of these vanishing relations. We give the superfield interpretation of all short multiplets and we point out the existence of many long multiplets with rational protected dimensions, whose presence and pattern seem to be universal in all compactifications. © 2000 Elsevier Science B.V.