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 N 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 N M2-branes is a d = 3, U (N ) × U (N ) gauge theory with four complex scalar fields C I (I = 1, 2,3,4) in the (N, ¯N ) representation, their complex conjugate fields in the ¯ (N, N ) representation and their fermionic partners [12Jump To The Next Citation Point]. The theory includes non dynamical gauge fields with a Chern–Simons action with levels k and − k for the two gauge groups. This gauge theory is weakly coupled in the large k limit (k ≫ N) and strongly coupled in the opposite regime (k β‰ͺ N ), for which a weakly coupled gravitational description will be available if N ≫ 1.

Supersymmetry approach:
Inspection of the d = 3 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 XI = XIaT a48 and their fermionic partners Ψ = Ψ T a a, and being invariant under a set of supersymmetry transformations whose most general form is

δXI = i¯πœ–Γ IΨ , d d δΨ = ∂ XI Γ μΓ Iπœ– − 1-XI XJ XK f abc Γ IJK πœ– + 1XJ XJ XI gabc Γ Iπœ–. d μ d 6 a b c d 2 a b c d
This was the original approach followed in [26Jump To The Next Citation Point], based on a real vector space with basis T a, a = 1, ...N, endowed with a triple product
[Ta,T b,Tc] = fabcd Td, (500 )
where the set of abc f d are real, fully antisymmetric in a,b,c and satisfy the fundamental identity
f [abcef d]efg = 0. (501 )
Closure of the supersymmetry algebra requires Eq. (501View Equation), 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 A&tidle;μcd requiring one to consider a more general set of supersymmetry transformations [27, 274]
I I δX d = i¯πœ–Γ Ψd I μ I 1- I J K abc IJK 1- J J I abc I ,δΨd = D μX dΓ Γ πœ– − 6 X aX b X c f dΓ πœ– + 2X aX b X cg dΓ πœ– &tidle; c I abc ,δA μ d = i¯πœ–Γ μΓ IX aΨbh d . (502 )
Here Dμ is a covariant derivative, whereas gabcd and habcd 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 fabcd = f abcehed, with ( ) hab = Tr Ta,T b defining an inner product, and satisfying fabcd ∝ πœ€abcd [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 d = 3, 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 𝒩 = 2 theories. 𝒩 = 2 Chern–Simons theories coupled to matter49 include a vector multiplet A, the dimensional reduction of the four dimensional 𝒩 = 1 vector multiplet, in the adjoint representation of the gauge group G, and chiral multiplets Φ i in representations R i of the latter. Integrating out the D-term equation and the gaugino, one is left with the action

𝒩 =2 ∫ -k- 2- 3 ¯ μ ¯ μ S = 4π Tr (A ∧ dA + 3A ) + D μΟ•iD Ο•i + iψiγ D μψi 2 − 16π--(¯Ο•iTaR Ο•i)(Ο•¯jT bR Ο•j)(¯Ο•kT aR TbR Ο•k ) − 4-π(¯Ο•iTRaΟ•i)(¯ψjT aR ψj) (503 ) k2 i j k k k i j 8π- ¯ a ¯ a − k (ψiTRiΟ•i)(Ο•jTRjψj),
where Ο•i and ψi are the bosonic and fermionic components of the chiral superfield Φi and the gauge field A is non-dynamical.

There are 𝒩 = 3 generalisations, but since their construction is more easily argued for starting with the field content of an 𝒩 = 4 theory, let me review the latter first. The field content of the 𝒩 = 4 theories adds an auxiliary (non-dynamical) chiral multiplet φ in the adjoint representation of G and pairs chiral multiplets &tidle; Φi,Φi 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 W = Φ&tidle;i φΦi for each pair. 𝒩 = 3 theories are constructed by the addition of Chern–Simons terms, as in Eq. (503View Equation), and the extra superpotential W = − -kTr (φ2 ) 8π. Integrating out φ leads to a superpotential

W = 4π-(&tidle;Φ T aΦ )(Φ&tidle; Ta Φ ). (504 ) k i Ri i j Rj j
The resulting 𝒩 = 3 theory has the same action as Eq. (503View Equation) with the addition of the above superpotential.

In [12Jump To The Next Citation Point], an 𝒩 = 6 theory based on the gauge group U(N ) × U (N ) 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 A ,A 1 2 and their anti-bifundamental by B ,B 1 2, the superpotential then equals

k W = --Tr (φ2(2) − φ2(1)) + Tr (Biφ (1)Ai) + Tr (Aiφ (2)Bi). (505 ) 8π
After integrating out the auxiliary fields φ (i),
2π 4π W = --Tr (AiBiAjBj − BiAiBjAj ) = --Tr (A1B1A2B2 − A1B2A2B1 ). (506 ) k k
As discussed in [12Jump To The Next Citation Point], the four bosonic fields CI ≡ (A1,A2, B ∗1,B2∗) transform in the 4 of SU (4), matching the generic SO (𝒩 ) R-symmetry in d = 3 super-CFTs. For a more thorough discussion of global symmetries and gauge invariant observables, see [12Jump To The Next Citation Point].

It was argued in [12Jump To The Next Citation Point] that the 𝒩 = 6 theory constructed above was dual to N M2-branes on β„‚4 βˆ•β„€k for k ≥ 3. 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 [282Jump To The Next Citation Point], one can associate low energy effective field theories with the dynamics of brane configurations stretching between branes. Consider a set of N D3-branes wrapping the x6 direction and ending on different NS5-branes according to the array

NS5 : 1 2 3 4 5 x x x x NS5 : 1 2 3 4 x x x x x (507 ) D3 : 1 2 x x x 6 x x x .
This gives rise to an 𝒩 = 4 U (N ) × U (N ) gauge theory in d = 1 + 2 dimensions, along the {x1, x2} 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 [282Jump To The Next Citation Point]. Adding k D5-branes, as illustrated in the array below,
NS5 : 1 2 3 4 5 x x x x NS5 : 1 2 3 4 5 x x x x D5 : 1 2 3 4 x x x x 9 (508 ) D3 : 1 2 x x x 6 x x x ,
breaks supersymmetry to 𝒩 = 2 and adds k massless chiral multiplets in the N and ¯N representation of each of the U (N ) gauge group factors. Field theoretically, this 𝒩 = 2 construction allows a set of mass deformations that can be mapped to different geometrical notions [282, 72Jump To The Next Citation Point, 12Jump To The Next Citation Point]:
  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 k D5-branes and NS5-branes along the 01234 directions and merging them into an intermediate (1,k) 5-brane bound state generates a real mass of the same sign for both N and ¯N representations. This mechanism is a web deformation [72]. The merging is characterised by the angle πœƒ 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 (1,k ) 5-brane in the 01234 [5,9]πœƒ, where [5,9]πœƒ stands for the x5cos πœƒ + x4sinπœƒ direction. πœƒ is fixed by supersymmetry [14].

After the web deformation and at low energies, one is left with an 𝒩 = 2 U (N )k × U(N )−k 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 𝒩 = 3 theory described in the purely field theoretical context is realised in the brane construction by rotating the (1, k) 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 (1,k) 5-brane along 012 [3,7 ]πœƒ[4,8]πœƒ[5,9]πœƒ.This particular mass deformation ensures all massive adjoint fields acquire the same mass, enhancing the symmetry to 𝒩 = 3. Equivalently, there must exist an SO (3)R R-symmetry corresponding to the possibility of having the same SO (3) rotations in the 345 and 789 subspaces. Thus, the d = 3 supersymmetric field theory must be 𝒩 = 3.

The connection to 𝒩 = 6 is obtained by flowing the 𝒩 = 3 theory to the IR [12Jump To The Next Citation Point]. Indeed, by integrating out all the massive fields, we recover the field content and interactions described in the field theoretical 𝒩 = 6 construction. The enhancement to 𝒩 = 8 for k = 1,2 was properly discussed in [276].

It was realised in [12Jump To The Next Citation Point] that under T-duality in the 6 x direction and uplifting the configuration to M-theory, the brane construction gets mapped to N 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 1βˆ•k coupling constant. Thus, large k has a weakly coupled description. At large N, it is natural to consider the ’t Hooft limit: λ = N βˆ•k fixed. The gauge theory is weakly coupled for k ≫ N and strongly coupled for k β‰ͺ N. In the latter situation, the supergravity description becomes reliable and weakly coupled for N ≫ 1 [12Jump To The Next Citation Point].

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 [28Jump To The Next Citation Point]. The main idea was to consider a 3-algebra based on a complex vector space endowed with a triple product

a b -¯c abc¯ d [T ,T ;T ] = f d T , (509 )
and an inner product
a¯b (-a b) h = Tr T T . (510 )
The change in the notation points out antisymmetry only occurs in the first two indices. Furthermore, the constants ab¯c f d satisfy the following fundamental identity,
fef¯g fcb¯a + ffe¯a fcb¯g + f∗¯g¯af¯fce¯b + f∗¯a¯ge¯f cf¯b = 0 . (511 ) b d b d b d b d
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 𝒩 = 6 conformal field theories described in [12] could be rederived for the particular choices
- - - - f abcd = − fbacd , and f abcd = f∗cdab. (512 )
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.
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