Perturbative and non-perturbative infrared behavior of ...

5.2. Generalizations
representation of the gauge group and assume that the chiral multiplets come in pairs Φi, ˜ Φi
in conjugate representations of the gauge group. Note that this procedure closely resembles the
construction of the N = 2 theory in 3 + 1 dimensions, where the combination of Φ and ˜ Φ forms
a hypermultiplet. We add the superpotential terms ˜ ΦiφΦ i and − k
8π Trφ2 . The latter breaks the
supersymmetry to N = 3. Since φ is an auxiliary field and has not any kinetic term we can
integrate it out. The superpotential now reads
W = 4π
k (˜ ΦiT a Ri Φi)( ˜ ΦjT a Rj Φj) (5.1.3)
The full theory is the sum of (5.1.2) (with the addition of the same terms for the conjugate chiral
multiplets) and (5.1.2). The relative coefficients are fixed by supersymmetry.
Let us now specialize to the case of U(N)k × U(N)−k gauge group and two hypermultiplets
(A1,B1) and (A2,B2) in the bifundamental representation of the gauge group. The pedices of
the two U(N) factors of the gauge group indicate that we are taking the two Chern-Simons levels
equal but opposite in sign. The above construction leads us to the following superpotential
W = 2π
k ǫabǫ ˙a˙
b
Tr AaB˙aAbB˙ b
(5.1.4)
which exhibits an SU(2) × SU(2) symmetry acting separately on the A’s and on the B’s. Moreover,
theories with N superconformal symmetries in 2+1 dimensions have an SO(N) R-symmetry
which in this case is SO(3) ≃ SU(2)R. It is realized on the component fields in the following way.
The vector multiplet fermions form a triplet and a singlet of SU(2)R, and the three scalar fields
transform as a doublet, as
a triplet. The lowest component of the chiral superfields A1 and B ∗ 1
A2 and B ∗ 2
do. The R-symmetry does not commute with the global SU(2) × SU(2) symmetry,
and they together form an SU(4)R ≃ SO(6)R R-symmetry. It is a R-symmetry because the
supercharges cannot be singlets under it. Thus, the theory (5.1.2) with the superpotential (5.1.4)
enjoys N = 6 supersymmetry.
The coupling constant in (5.1.4) is 1/k and for large k it is weakly coupled. In the large N
limit with N/k fixed one can expand in 1/N 2 and the effective coupling constant in the leading
order, planar diagrams is the ’t Hooft coupling λ ≡ N/k. Thus, the theory is weakly coupled for
k ≫ N and strongly coupled for k ≪ N.
5.2 Generalizations
In this section we review some generalizations of the ABJM model to different gauge groups
and to less supersymmetric field theories. We focus on unitary gauge groups, but many other
possibilities have been analyzed [88].
The first generalization we consider is to U(N)k × U(M)−k Chern-Simons matter theories,
with M = N but the same matter content and interactions as in the ABJM model [88]. In
complete analogy with the previous section, the theory is constructed from the N = 3 Chern-
Simons matter theory and the special form of the superpotential (5.1.4) leads to the conclusion
that it has an enhanced N = 6 superconformal symmetry. The field theory has been named the
ABJ model, after the authors who explained its gravity dual [70].
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