### 7.2 M2-branes

In this section, I would like to briefly mention the main results involving the amount of progress recently achieved in the description of parallel M2-branes, referring to the relevant literature when appropriate. This will be done taking the different available perspectives on the subject: a purely kinematic approach, based on supersymmetry and leading to 3-algebras, a purely field theory approach leading to three dimensional CFTs involving Chern–Simons terms, a brane construction approach, in which one infers the low energy effective description in terms of an intersection of branes and the connection between all these different approaches.

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 .

##### Supersymmetry approach:
Inspection of the SYM supersymmetry transformations and the geometrical intuition coming from M2-branes suggest that one look for a supersymmetric field theory with field content involving eight scalar fields and their fermionic partners , and being invariant under a set of supersymmetry transformations whose most general form is

This was the original approach followed in [26], based on a real vector space with basis , , endowed with a triple product
where the set of are real, fully antisymmetric in and satisfy the fundamental identity
Closure of the supersymmetry algebra requires Eq. (501), but also shows the appearance of an extra gauge symmetry [26]. To deal properly with the latter, one must introduce an additional (non-dynamical) gauge field requiring one to consider a more general set of supersymmetry transformations [27, 274]
Here is a covariant derivative, whereas and define triple products on the algebra. Closure of the supersymmetry algebra determines a set of equations of motion that can be derived, which form a Lagrangian. It was soon realised that under the assumptions of a real vector space, essentially the only 3-algebra is the one defined by , with defining an inner product, and satisfying  [399, 412, 226]. Interestingly, it was pointed out in [488] that such supersymmetric field theory could be rewritten as a Chern–Simons theory. The latter provided a link between a purely kinematic approach, based on supersymmetry considerations, and purely field theoric results that had independently been developed.

##### Field theory considerations:
Conformal field theories have many applications. In the particular context of Chern–Simons matter theories in , they can describe interesting IR fixed points in condensed matter systems. Here I am interested in their supersymmetric versions to explore the AdS4/CFT3 conjecture. Let me start this overview with theories. Chern–Simons theories coupled to matter include a vector multiplet , the dimensional reduction of the four dimensional vector multiplet, in the adjoint representation of the gauge group , and chiral multiplets in representations of the latter. Integrating out the D-term equation and the gaugino, one is left with the action

where and are the bosonic and fermionic components of the chiral superfield and the gauge field is non-dynamical.

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.

##### Brane construction:
Following the seminal work of [282], one can associate low energy effective field theories with the dynamics of brane configurations stretching between branes. Consider a set of D3-branes wrapping the direction and ending on different NS5-branes according to the array

This gives rise to an gauge theory in dimensions, along the directions, whose field content includes a vector multiplet in the adjoint representation and 2 complex bifundamental hypermultiplets, describing the transverse excitations to both D3-branes and NS5-branes [282]. Adding D5-branes, as illustrated in the array below,
breaks supersymmetry to and adds massless chiral multiplets in the and representation of each of the gauge group factors. Field theoretically, this construction allows a set of mass deformations that can be mapped to different geometrical notions [282, 72, 12]:
1. Moving the D5-branes along the 78-directions generates a complex mass parameter.
2. 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.
3. 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 AdS4 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].

##### Matching field theory, branes and 3-algebra constructions:
The brane derivation of the supersymmetric field theory relevant to describe multiple M2-branes raised the natural question for what the connection was, if any, with the 3-algebra formulation that stimulated all these investigations. The answer was found in [28]. The main idea was to consider a 3-algebra based on a complex vector space endowed with a triple product

and an inner product
The change in the notation points out antisymmetry only occurs in the first two indices. Furthermore, the constants satisfy the following fundamental identity,
It was proven in [28] that this set-up manages to close the algebra on the different fields giving rise to some set of equations of motion. In particular, the conformal field theories described in [12] could be rederived for the particular choices
Thus, the 3-algebra approach based on complex vector spaces is also suitable to describe these string theory models. Furthermore, it provides us with a mathematical formalism capable of describing more general set-ups.