3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3

Abstract

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 M^7 that were classified long ago. In particular we focus on the two cases M^7=Q^{1,1,1} and M^7=M^{1,1,1}, for the latter the Kaluza Klein spectrum being completely known. We show how the toric description of M^7 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 C^p. The ring of chiral primary fields is defined as the coordinate ring of C^p 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.

Figures (4)

• Figure 1: Gauge group $SU(N)_1 \times SU(N)_2 \times SU(N)_3$ and color representation assignments of the supersingleton fields $A_i, \, B_j,\, C_\ell$ in the $Q^{1,1,1}$ world volume gauge theory.
• Figure 2: Gauge group $U(N)_1 \times U(N)_2$ and color representation assignments of the supersingleton fields $V^A$ and $U^i$ in the $M^{1,1,1}$ world volume gauge theory.
• Figure 3: Schematic representation of the atlas on ${\mathbb P}^{2*}$. The three patches $W_\alpha$ cover the open ball and part of the boundary circle, which constitutes the set of coordinate singularities. This latter is made of three $S^2$'s: $\{\theta=0\}$, $\{\theta=\pi\}$ and $\{\mu=\pi/2\}$, which touch each other at the three points marked with a dot. Each $W_\alpha$ covers the whole ${\mathbb P}^{2*}$ except for one of the spheres (for example, $W_3$ does not cover $\{\mu=\pi/2\}$). The three most singular points are covered by only one patch (for example, $\{\mu=0\}$ is covered by the only $W_3$).
• Figure 4: Two coordinate patches for the sphere. They constitute the base for a local trivialization of a fibre bundle on $S^2$. Each patch covers only one of the poles, where the coordinates $(\theta,\phi)$ are singular.