content_image/340334.jpg content_image/340342.jpg
ignore the coupling to gauge field and concentrate on the metric only. 

<img src='content_image/340334.jpg'>

L is the radius of curvature for AdS$_{4}$ , and we have adopted the global coordinate above. Sasaki-Einstein metric of M 1 , 1 , 1 is here written as a nontrivial U (1) bundle over$_{CP}$ 2 ×$_{CP}$ $^{1}$. θ, φ parametrize$_{CP}$ $^{1}$, while µ, ˜ ψ, ˜ θ, ˜ φ are for$_{CP}$ $^{2}$. Their sizes are adjusted so that$_{CP}$$^{2}$×$_{CP}$1 as a whole becomes K¨ ahler-Einstein. The angles range as 0 ≤ θ, ˜ θ ≤ π, 0 ≤ φ, ˜ φ ≤ 2 π, 0 ≤ ψ ≤ 4 π and 0 ≤ µ ≤ π/ 2. 

The above metric for M 1 , 1 , 1 obviously has isometry group SU (3) × SU (2) × U (1). The U (1) part of the Killing vector, ∂$_{ψ}$ , is called the Reeb vector and accounts for the R-symmetry of the dual field theory. The rest should appear as additional global symmetry, like flavor symmetry. The N = 2 Chern-Simons theory we will consider in general is dual to orbifold M 1 , 1 , $^{1}$/$_{Z}$$_{k}$ , which is obtained by identifying φ ∼ φ + 2 π k . SU (2) part of the isometry is then broken to U (1). 

The Chern-Simons dual of AdS$_{4}$ × M 1 , 1 , 1 was proposed in [6], and can be summarised by a quiver diagram Fig. 1. We have U ( N ) × U ( N ) × U ( N ) gauge group which is denoted as three nodes in the quiver diagram. The lines connecting two nodes represent chiral multiplets which are in bifundamental representations of the two relevant gauge groups. It is essential to note that the CS levels are not given symmetrically, and take values ( k, k, − 2 k ). We also have superpotential, 

<img src='content_image/340342.jpg'>

The duality can be justified with the help of toric geometry. The mesonic vacuum moduli space, which is determined by the usual D-term and F-term flatness conditions, is a toric variety and equivalent to the cone over the orbifold M 1 , 1 , $^{1}$/$_{Z}$$_{k}$ [6, 7, 17]. Of course, for k = 1 the vacuum moduli space has an enhanced isometry group SU (3) × SU (2) × U (1). X i $_{12}$, X i $_{23}$, X i 31 provide three fundamental representations of SU (3). The SU (2) broken by the orbifold is manifest in the quiver diagram. 

