The main conclusion is that the effective theory describing M2-branes is a , gauge theory with four complex scalar fields () in the representation, their complex conjugate fields in the representation and their fermionic partners [12]. The theory includes non dynamical gauge fields with a Chern–Simons action with levels and for the two gauge groups. This gauge theory is weakly coupled in the large limit () and strongly coupled in the opposite regime , for which a weakly coupled gravitational description will be available if .

There are generalisations, but since their construction is more easily argued for starting with the field content of an theory, let me review the latter first. The field content of the theories adds an auxiliary (non-dynamical) chiral multiplet in the adjoint representation of and pairs chiral multiplets into a set of hypermultiplets by requiring them to transform in conjugate representations, as the notation suggests. The theory does not contain Chern–Simons terms, but a superpotential for each pair. theories are constructed by the addition of Chern–Simons terms, as in Eq. (503), and the extra superpotential . Integrating out leads to a superpotential

The resulting theory has the same action as Eq. (503) with the addition of the above superpotential.In [12], an theory based on the gauge group was constructed. Its field content includes two hypermultiplets in the bifundamental and the Chern–Simons levels of the two gauge groups were chosen to be equal but opposite in sign. Denoting the bifundamental chiral superfields by and their anti-bifundamental by , the superpotential then equals

After integrating out the auxiliary fields , As discussed in [12], the four bosonic fields transform in the of , matching the generic R-symmetry in super-CFTs. For a more thorough discussion of global symmetries and gauge invariant observables, see [12].It was argued in [12] that the theory constructed above was dual to M2-branes on for . Below, I briefly review the brane construction in which their argument is based. This will provide a nice example of the notion of geometrisation (or engineering) of supersymmetric field theories provided by brane configurations.

- Moving the D5-branes along the 78-directions generates a complex mass parameter.
- Moving the D5-branes along the 5-directions generates a real mass, of positive sign for the fields in the fundamental representation and of negative sign for the ones in the anti-fundamental.
- Breaking the D5-branes and NS5-branes along the 01234 directions and merging them into an intermediate 5-brane bound state generates a real mass of the same sign for both and representations. This mechanism is a web deformation [72]. The merging is characterised by the angle relative to the original NS5-brane subtended by the bound state in the 59-plane. The final brane configuration is made of a single NS5-brane in the 012345 directions and a 5-brane in the , where stands for the direction. is fixed by supersymmetry [14].

After the web deformation and at low energies, one is left with an Yang–Mills–Chern–Simons theory with four massless bi-fundamental matter multiplets (and their complex conjugates), and two massless adjoint matter multiplets corresponding to the motion of the D3-branes in the directions 34 common to the two 5-branes.

The enhancement to an theory described in the purely field theoretical context is realised in the brane construction by rotating the 5-brane in the 37 and 48-planes by the same amount as in the original deformation. Thus, one ends with a single NS5-brane in the 012345 and a 5-brane along .This particular mass deformation ensures all massive adjoint fields acquire the same mass, enhancing the symmetry to . Equivalently, there must exist an R-symmetry corresponding to the possibility of having the same rotations in the 345 and 789 subspaces. Thus, the supersymmetric field theory must be .

The connection to is obtained by flowing the theory to the IR [12]. Indeed, by integrating out all the massive fields, we recover the field content and interactions described in the field theoretical construction. The enhancement to for was properly discussed in [276].

It was realised in [12] that under T-duality in the direction and uplifting the configuration to
M-theory, the brane construction gets mapped to M2-branes probing some configuration of
KK-monopoles. These have a supergravity description in terms of hyper-Kähler geometries [224]. Flowing
to the IR in the dual gravitational picture is equivalent to probing the near horizon of these geometries,
which includes the expected AdS_{4} factor times a quotient of the 7-sphere.

The Chern–Simons theory has a coupling constant. Thus, large has a weakly coupled description. At large , it is natural to consider the ’t Hooft limit: fixed. The gauge theory is weakly coupled for and strongly coupled for . In the latter situation, the supergravity description becomes reliable and weakly coupled for [12].

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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