"Gauge/gravity in three dimensions"

Today's Theoretical Particle Physics seminar was given by Diego Rodriguez-Gomez (Queen Mary, U. of London), based mainly on 0809.3237 and 0903.3231.

Given the broad audience, he started by a small review of the AdS/CFT correspondence as a specially clean incarnation of 't Hooft's and Susskind's holographic principle. Whereas the 4d case has been relatively well understood for some time, the 3d case began to reveal itself only recently. In fact, until roughly two years ago, it was thought that the IR fixed point at the end of the RG flow of the maximally supersymmetric $\mathcal{N}=8$ SYM in 3d had no Lagrangian description. The reason for this belief was that the dual description has a non-constant dilaton blowing up at small radius: $e^{\phi} =(R/r)^{5/4}$. The BLG theory (anticipated by Schwarz) was an $\mathcal{N}=8$ Chern-Simons-like theory but it had an $SU(2)$ gauge group instead of the large $N$ needed in the AdS/CFT context. The resolution came from relaxing the requirement of maximal susy down to $\mathcal{N}=6$, which allowed arbitrary $N$ [ABJM].

Diego was interested in reducing the supersymmetry to $\mathcal{N}=2$ by putting the M2-branes at the tip of a Calabi-Yau four-fold. He focused on a specific example: the cone over $Q^{1,1,1}$, which is similar to the conifold in 6d and has the advantage to have been extensively studied (the metric is explicitly known). The simplest proposal (inspired by crystal models) is that the dual field theory is a quiver with four gauge groups, six fields, and a sextic superpotential $W$ with two terms. As in the ABJM case, all the global symmetries are not manifest, but appear by studying how scalar fields get identified. This results in a mesonic moduli space that is an $\mathcal{N}=2$ orbifold $Q^{1,1,1}/\mathbb{Z}_2$. Here are the toric diagram and the associated quiver:
Diego and his collaborators were able to show that the chiral operator spectrum is matching the supergravity harmonics, at least at large CS level $k$. This is a non-trivial check of the non-Abelian superpotential $W= C_1 A_1 B_1 C_2 A_2B_2 - C_1 A_1 B_2 C_2 A_2B_1$.

Remaining mysteries include the question of whether theories with $\mathcal{N}<3$ susy are conformal, which would require to have an equivalent in 3d of a-maximization; the inverse algorithm (from the CY4 to the quiver) and the connection to type IIA string theory; the small $k$ limit and monopoles operators, which is the genuinely M-theoretic limit, since at large $k$ the theory reduces to IIA.