ap-3-10.dvi


Acta Polytechnica Vol. 50 No. 3/2010

M2-Branes in N =3 Harmonic Superspace

E. Ivanov

Invited Talk at the Conference “Selected Topics in Mathematical and Particle Physics”,
In Honor of the 70-th Birthday of Jiri Niederle, Prague, 5–7 May 2009

Abstract

We give a brief account of the recently proposed N = 3 superfield formulation of the N = 6,3D superconformal theory of
Aharony et al (ABJM) describing a low-energy limit of the system of multipleM2-branes on theAdS4 × S7/Zk background.
This formulation is given in harmonic N = 3 superspace and reveals a number of surprising new features. In particular,
the sextic scalar potential of ABJM arises at the on-shell component level as the result of eliminating appropriate auxiliary
fields, while there is no explicit superpotential at the off-shell superfield level.

1 Preliminaries: AdS/CFT

1.1 AdS/CFT in IIB superstring

As the starting point, I recall the essentials of the orig-
inal AdS/CFT correspondence (for details see [1] and
references therein).
It is the conjecture that the IIB superstring on

AdS5× S5 is in some sense dual to maximally super-
symmetricN =4,4D superYang-Mills (SYM) theory.
This hypothesis is to a large extent based upon the
coincidence of the symmetry groups of both theories.
Indeed,

AdS5×S5 ∼
SO(2,4)
SO(1,4)

×
SO(6)
SO(5)

⊂
SU(2,2|4)

SO(1,4)× SO(5)
,

so the superisometries of this background constitute
the supergroup SU(2,2|4). On the other hand, the
supergroup SU(2,2|4) defines superconformal invari-
ance of N = 4 SYM, with SO(2,4) and SO(6) ∼
SU(4) being, respectively, 4D conformal group and
R-symmetry group.
Some related salient features of the AdS/CFT cor-

respondence are as follows.
• AdS5×S5 (plus a constant closed 5-formon S5) is
the bosonic “body” of the maximally supersym-

metric curved solution
SU(2,2|4)

SO(1,4)× SO(5)
of IIB,

10D supergravity. It preserves 32 supersymme-
tries.

• N = 4 SYM action with the gauge group U(N)
is the low-energy limit of a gauge-fixed action of
a stack of N coincident D3-branes on AdS5×S5:
4 worldvolume co-ordinates of the latter system
become the Minkowski space-time co-ordinates,
while 6 transverse (u(N) algebra-valued) D3-
brane co-ordinates yield just 6 scalar fields of the
nonabelian N =4,4D gauge multiplet.

• This system has the following on-shell content: 6
bosons and 16/2 = 8 fermions (all u(N) algebra

valued); 2 “missing” bosonic degrees of freedom
which are required by world-volume N = 4 su-
persymmetry come from a gauge field. This is a
“heuristic” explanationwhy just D3-branes, with
the gauge fields contributing non-trivial degrees
of freedom on shell, matter in the case of the
AdS5/CFT4 correspondence.

1.2 AdS/CFT in M-theory

Recently, there has been a surge of interest in another
example of AdS/CFT duality, this time related to M-
theory and the IIA superstring.
The fundamental (thoughnot explicitly formulated

as yet) M-theory can be defined as a strong-coupling
limit of the IIA, 10D superstringwith 11D supergrav-
ity as the low-energy limit. It has the following maxi-
mally supersymmetric classical curved solution:

AdS4 × S7 ∼
SO(2,3)
SO(1,3)

×
SO(8)
SO(7)

⊂
OSp(8|4)

SO(1,3)× SO(7)

(plus a constant closed 7-formon S7), which preserves
32 supersymmetries.
When trying to treat this option within the gen-

eralAdS/CFTcorrespondence (like thepreviouslydis-
cussed AdS5× S5 example), there arise the following
natural questions.
• What is the CFT dual to this geometry?
1. It should be some 3D analog of N =4 SYM
and should arise as a low-energy limit of
multipleM2-branes (membranesofM-theory,
analogs of D3-branes of the IIB superstring).

2. Hence it should contain8 (gauge algebraval-
ued) scalar fields which originate from the
transverse co-ordinates of M2-branes.

3. It should contain off-shell 16 physical
fermions (16 other fermionic modes can be
gauged away by the relevant κ symmetry).

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Acta Polytechnica Vol. 50 No. 3/2010

4. Finally, it should be superconformal, with
OSp(8|4) realized as N = 8,3D supercon-
formal group.

• On shell there should be 8+16/2=8+8 de-
grees of freedom. Hence the gauge fields should
not contribute any degree of freedom on shell in
this special case (in a drastic contrast with the
“type IIB /N =4 SYM” correspondence).

The unique possibility which meets all these demands
is that the dual theory is some supersymmetric exten-
sion of Chern-Simons gauge theory [2].

2 Chern-Simons theories
The standard bosonic Chern-Simons (CS) action is as
follows

Scs =
k

4π
Tr

∫
d3x�mns ·(

Am∂nAs +
2i
3

AmAnAs

)
(2.1)

⇒ Fmn = ∂mAn − ∂nAm +[Am, An] = 0,

i.e. the YM field An is pure gauge on shell.
The N = 1 superextension of the CS action is ob-

tainedbyextending An toN =1gaugesupermultiplet

An ⇒ (An, χα), α =1,2;
Lcs(A) ⇒ Lcs(A)−Tr(χ̄χ) .

(2.2)

The fermionic field χ is auxiliary, and no dynami-
cal (Dirac) equation for it appears. The same phe-
nomenon takes place in the case of N =2 and N =3
superextensions of the pure CS action. The physical
fermionic fields (having standard kinetic terms) can
appear only from the matter supermultiplets coupled
to the CS one.
Keeping inmind these general properties of super-

symmetric Chern-Simons theories, Schwarz assumed
[2] that the theory dual to AdS4×S7 must be N = 8
superextension of the 3D CS theory, i.e. one should
deal with the on-shell supermultiplet (Am, φ

I , ψBα ),
I =1, . . . ,8, B =1, . . . ,8.
How to gain physical kinetic terms for 16 (u(N)

algebra-valued) fermions? The recipe: place the latter
into matter multiplets of the manifest N = 1, N = 2
or N =3 supersymmetries, consider the relevant com-
bined “CS+matter” actions and realize extra super-
symmetries as the hidden ones mixing the CS super-
multiplet with the matter multiplets.

3 BLG and ABJM models

3.1 Attempts toward N=8 CS theory

Thefirst attempt to formulate the appropriateCS the-
orywas undertakenby J. Schwarz in 2004 [2]. He used
N =2,3D superfield formalismand tried to construct

N = 8 superconformal CS theory as N = 2 CS the-
ory plus 4 complexmatter chiral superfields (with the
off-shell content consisting of 8 physical bosons, 16
fermions and 8 auxiliary fields). However, these at-
tempts failed. As became clear later, the reason for
this failure is that the standard assumption that both
matter and gauge fields are in the adjoint of the gauge
group prove to be wrong in this specific case.
Sucha theorywas constructedbyBaggerandLam-

bert [3] and Gustavsson [4]. The basic assumption of
BLG was that the scalar fields and fermions take val-
ues in an unusual “three-algebra”

[Ta, Tb, Tc] = f
d

abc Td . (3.1)

The gauge group acts as automorphisms of this alge-
bra, gauge fields being still in the adjoint. The totally
antisymmetric “structure” constants of the 3-algebra
should satisfy a fundamental Jacobi-type identity

f dabc f
egh

d +some permutations of indices= 0 . (3.2)

BLG managed to define N = 8 (on-shell) super-
symmetry in such a system and to construct the in-
variant Lagrangian

LN=8=L̃cs(A)+covariantized kin.terms of φI , ψA +
6-th order potential of φI + . . . ,

where L̃cs(A) is some generalization of the Lagrangian
in (2.1). All terms involve the constants f dabc and
contain only one free parameter, the CS level k.

3.2 Problems with the BLG
construction

Assuming that the 3-algebra is finite-dimensional and
no ghosts are present among the scalar fields, the only
solution of the fundamental identity (3.2) proved to be
f abcd = �abcd, a, b =1,2,3,4.
Thus the only admissible gauge group is SO(4) ∼

SU(2)L × SU(2)R and φI , ψA are in the “bi-
fundamental” representation of this gauge group (in
fact these are just SO(4) vectors). No generalization
to the higher-dimensional gauge groups with the fi-
nite number of generators and positive-definedKilling
metric is possible.
The SU(2)×SU(2) gaugegroupcase canbe shown

to correspond just to twoM2-branes. How to describe
the system of N M2-branes?

3.3 Way out: ABJM construction

Aharony, Bergman, Jafferis, Maldacena in 2008 [5]
proposed a way to evade this restriction on the gauge
group. Their main observation was that there is no
need in exotic 3-algebras to achieve this at all! The
fields φI , ψA should be always in the bi-fundamental

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Acta Polytechnica Vol. 50 No. 3/2010

of the gauge group U(N)×U(N), while the double set
of gauge fields should be in the adjoint.
The ABJM theory is in fact dual to M-theory on

AdS4 × S7/Zk, and in general it respects only N = 6
supersymmetry and SO(6) R-symmetry. The invari-
ant action is a low-energy limit of the worldvolume
action of N coincident M2-branes on this manifold.
For the gauge group SU(2) × SU(2), the ABJM

theory is equivalent to the BLG theory.
The full on-shell symmetry of the ABJM action is

the N = 6,3D superconformal symmetry OSp(6|4).
Characteristic features of this action are the presence
of sextic scalar potential of special form and the ab-
sence of any free parameter except for the CS level k.
This k is common for both U(N) CS actions which
should appear with the relative sign minus (only in
this case there is an invariance under N = 6 super-
symmetry).

3.4 Superfield formulations

Off-shell superfield formulations make manifest un-
derlying supersymmetries and frequently reveal un-
usual geometricproperties of supersymmetric theories.
Thus it was advantageous to find a superfield formu-
lation of the ABJM model with the maximal number
of supersymmetries being manifest and off-shell.
N =1 and N =2 off-shell superfield formulations

were given in refs. [6, 7, 8]. They allowedone to partly
clarify the origin of the interaction of scalar and spinor
component fields. On-shell N = 6 and N = 8 formu-
lations were also constructed for both the ABJM and
BLG models (see e.g. [9, 10, 11]).
The maximally possible off-shell supersymmetry

for the CS theory coupled to matter is N =3,3D su-
persymmetry [12, 13]. Thus it was an urgent problem
to reformulate the generalABJMmodels inN =3,3D
superspace. This was recently done in [14].
This formulation uses the N = 3,3D version [12]

of the N =2,4D harmonic superspace [15, 16].

4 N =3 superfield formulation
of the ABJM model

4.1 N =3,3D harmonic superspace

N = 3,3D harmonic superspace (HSS) is an ex-
tension of the standard real N = 3,3D superspace
by the harmonic variables parametrizing the sphere
S2 ∼ SU(2)R/U(1)R:

(xm, θ(ik)α ) ⇒ (x
m, θ(ik)α , u

±
j ) , (4.1)

u±i ∈ SU(2)R/U(1)R , u
+iu−i =1 ,

m, n = 0,1,2; i, k, j =1,2; α =1,2 .

The most important feature of the N = 3,3D HSS is
the presence of an analytic subspace in it, with a lesser

number ofGrassmannvariables (two3D spinors as op-
posed to three such spinor coordinates of the standard
superspace)

(ζM) ≡ (xmA , θ
++
α , θ

0
α, u

±
k ) , (4.2)

θ++α = θ
(ik)
α u

+
i u
+
k , θ

0
α = θ

(ik)
α u

+
i u

−
k .

It is closed under both the N =3,3D Poincaré super-
symmetry and its superconformal extension OSp(3|4).
All the basic objects of the N =3 superspace for-

mulation live as unconstrained superfields on this sub-
space:
1. Gauge superfields

V ++(ζ), δV ++ = −D++Λ(ζ)− [V ++,Λ] , (4.3)
Λ=Λ(ζ) .

2. Matter superfields (hypermultiplets)

(q+(ζ), q̄+(ζ)), (4.4)

q+ = u+i f
i +(θ++αu−k − θ

0αu+k )ψ
k
α +∞

of aux. fields.

In eq. (4.3), D++ is the analyticity-preserving deriva-
tive on the harmonic sphere S2.

4.2 N =3 action
The N = 3 superspace formulation of the U(N) ×
U(N) ABJM model [14] involves:
1. The gauge superfields V ++L and V

++
R for the left

and right gauge U(N) groups. Both of them have the
following field contents in the Wess-Zumino gauge:

V ++ ∼
(
Am, φ

(kl), λα, χ
(kl)
α , X

(kl)
)

, (4.5)

i.e. (8+8) fields.
2. The hypermultiplets (q+a)

B
A, (q̄

+a)AB, a = 1,2,
in the bi-fundamental of U(N)×U(N): A =1, . . . , N;
B = 1, . . . , N. Each hyper q+a contributes (8+16)
physical fields off shell ((8+8) on shell).
The full superfield action is as follows:

SN3 = SCS(V
++
L )− SCS(V

++
R )+∫

dζ(−4) q̄+a ∇
++q+a , (4.6)

∇++q+a = D++q+a + V ++L q
+a − q+aV ++R .

4.3 Some salient features of the N =3
formulation

• Though the gauge superfieldCS actions are given
by integrals over the harmonic superspace, their
variations with respect to V ++L , V

++
R are repre-

sented by integrals over the analytic subspace

δSCS = −
ik

4π
Tr

∫
dζ(−4)δV ++W++ , (4.7)

W++ = W++(ζ), ∇++W++ =0 .

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Acta Polytechnica Vol. 50 No. 3/2010

As a result, the equations of motion are written
solely in termsof analytic superfields in the simple
form:

W++L = −i
4π
k

q+aq̄+a , W
++
R = −i

4π
k

q̄+a q
+a ,

∇++q+a = ∇++q̄+a =0 . (4.8)

• The N = 3 superfield action, in contrast to the
N = 0, N = 1 and N = 2 superfield ABJM
actions, does not involve any explicit superfield
potential, only minimal couplings to the gauge
superfields. The correct 6-th order scalar poten-
tial emergeson-shell after eliminatingappropriate
auxiliary fields from both the CS and hypermul-
tiplet sectors.

• Three hidden supersymmetries completing the
manifest N = 3 supersymmetry to N = 6 are
realized by simple transformations

δV ++L =
8π
k

�α(ab)θ0αq
+
a q̄
+
b ,

δV ++R =
8π
k

�α(ab)θ0αq̄
+
a q̄
+
b , (4.9)

δq+a = i�α(ab)∇0αq
+
b ,

where ∇0α is the properly covariantized derivative
with respect to θ0α.

• The hidden R-symmetry transformations extend-
ing theR-symmetry of the N =3 supersymmetry
to SO(6) also have a very transparent represen-
tation in terms of the basic analytic superfields.

• The N = 3 harmonic superspace formulation
makes manifest that the hidden N = 6 super-
symmetry is compatiblewith other product gauge
groups, e.g. with U(N) × U(M), N �= M, and
with other types of bi-fundamental representation
for the hypermultiplets. The hidden supersym-
metry transformations have the universal form
in all cases and suggest a simple criterion as to
which gauge groups admit this hidden supersym-
metry. In this way one can e.g. reproduce, at the
N =3 superfield level, the classification of admis-
sible gauge groups worked out at the component
level by Schnabl and Tachikawa in [17].

• The enhancement of the hidden N =6 supersym-
metry toN =8andR-symmetry SO(6) to SO(8)
in the case of the gauge group SU(2)k ×SU(2)−k
is also very easily seen in the N = 3 superfield
formulation. Actually, this enhancementarisesal-
ready in the case of the gauge group U(1)× U(1)
with a doubled set of hypermultiplets (with 16
physical bosons as compared to 8 such bosons in
the “minimal” U(1)× U(1) case [18]).

5 Outlook
In conclusion, letme list some further problemswhich
can be studied within the N = 3 superfield formula-
tion sketched above.

• Construction and study of the quantum effective
action of the ABJM-type models in the N = 3
superfield formulation. The fact that the super-
field equations of motion are given solely in the
analytic subspace hopefully implies some power-
ful non-renormalizability theorems [19].

• Computing the correlation functions of compos-
ite operators directly in the N =3 superfield ap-
proach as comprehensive checks of the considered
version of the AdS4/CF T3 correspondence.

• A study of interrelations between the low-energy
actions of M2- and D2-branes using the Higgs
mechanism [20], in which the second system is
interpreted as a Higgs phase of the first one.

• Constructing the full effective actions of M2-
branes in terms of the N = 3 superfields (with
a Nambu-Goto action for scalar fields in the case
of oneM2-braneand its nonabeliangeneralization
for N branes).

• ETC . . .

Acknowledgement

I thank the Organizers of Jiri Niederle’s Fest for invit-
ing me to present this talk and my co-authors in
refs. [14] and [19] for our fruitful collaboration. I
acknowledge a support from grants of the Votruba-
Blokhintsev and the Heisenberg-LandauPrograms, as
well as from RFBR grants 08-02-90490, 09-02-01209
and 09-01-93107.

References

[1] Aharony, O., Gubser, S. S., Maldacena, J. M.,
Ooguri, H., Oz, Y.: Large N field theories, string
theory and gravity, Phys. Rept. 323 (2000) 183,
arXiv:hep-th/9905111.

[2] Schwarz, J. H.: Superconformal Chern-Simons
theories, JHEP 0411 (2004) 078,
arXiv:hep-th/0411077.

[3] Bagger, J., Lambert, N.: Modeling multi-
ple M2’s, Phys. Rev. D75 (2007) 045020,
arXiv:hep-th/0611108;
Gauge symmetry and supersymmetry of multi-
ple M2-branes, Phys. Rev. D77 (2008) 065008,
arXiv:0711.0955 [hep-th];
Comments on multiple M2-branes, JHEP 0802
(2008) 105, arXiv:0712.3738 [hep-th];
Three-algebras and N = 6 Chern-Simons
gauge theories, Phys. Rev. D79 (2009) 025002,
arXiv:0807.0163 [hep-th].

[4] Gustavsson, A.: Algebraic structures on par-
allel M2-branes, Nucl. Phys. B811 (2009) 66,
arXiv:0709.1260 [hep-th];
Selfdual strings and loop space Nahm equa-

93



Acta Polytechnica Vol. 50 No. 3/2010

tions, JHEP 0804 (2008) 083, arXiv:0802.3456
[hep-th].

[5] Aharony, O., Bergman, O., Jafferis, D. L., Mal-
dacena, J.: N =6 superconformalChern-Simons-
matter theories, M2-branes and their gravity du-
als, JHEP 0810 (2008) 091, arXiv:0806.1218
[hep-th].

[6] Benna, M., Klebanov, I., Klose, T., Smed-
back, M.: Superconformal Chern-Simons The-
ories and AdS4/CF T3 Correspondence, JHEP
0809 (2008) 072, arXiv:0806.1519 [hep-th].

[7] Mauri,A.,Petkou,A.C.: An N =1superfieldac-
tion for M2 branes, Phys. Lett. B666 (2008) 527,
arXiv:0806.2270 [hep-th].

[8] Cherkis, S., Sämann,C.: MultipleM2-branes and
generalized 3-Lie algebras,Phys. Rev.D78 (2008)
066019, arXiv:0807.0808 [hep-th].

[9] Cederwall, M.: N = 8 superfield formulation
of the Bagger-Lambert-Gustavssonmodel, JHEP
0809 (2008) 116, arXiv:0808.3242 [hep-th];
Superfieldactions for N =8and N =6conformal
theories in three dimensions, JHEP 0810 (2008)
070, arXiv:0809.0318 [hep-th].

[10] Bandos, I. A.: NB BLG model in N =
8 superfields, Phys. Lett. B669 (2008) 193,
arXiv:0808.3568 [hep-th].

[11] Samtleben, H., Wimmer, R.: N = 8 Superspace
Constraints for Three-dimensional Gauge Theo-
ries, JHEP 1002 (2010) 070, arXiv:0912.1358
[hep-th].

[12] Zupnik, B. M., Khetselius, D. V.: Three-
dimensional extended supersymmetry in har-
monic superspace, Sov. J. Nucl. Phys. 47 (1988)
730.

[13] Kao, H.-C., Lee, K.: Self-dual Chern-Simons
Higgs systems with N = 3 extended supersym-
metry, Phys. Rev. D 46 (1992) 4691.

[14] Buchbinder, I. L., Ivanov, E. A., Lechtenfeld, O.,
Pletnev, N. G., Samsonov, I. B., Zupnik, B. M.:
ABJM models in N = 3 harmonic super-
space, JHEP 0903 (2009) 096, arXiv:0811.4774
[hep-th].

[15] Galperin, A., Ivanov, E., Kalitzin, S., Ogievet-
sky, V., Sokatchev, E.: Unconstrained N =
2 Matter, Yang-Mills and supergravity theories
in harmonic superspace, Class. Quant. Grav. 1
(1984) 469.

[16] Galperin, A. S., Ivanov, E. A., Ogievetsky, V. I.,
Sokatchev, E. S.: Harmonic Superspace, Cam-
bridge University Press, 2001, 306 p.

[17] Schnabl, M., Tachikawa, Y.: Classification of
N=6 superconformal theories of ABJM type,
arXiv:0807.1102 [hep-th].

[18] Bandres, M. A., Lipstein, A. E., Schwarz, J. H.:
Studies of the ABJM Theory in a Formulation
with Manifest SU(4) R-Symmetry, JHEP 0809
(2008) 027, arXiv:0807.0880 [hep-th].

[19] Buchbinder, I. L., Ivanov, E. A., Lechtenfeld, O.,
Pletnev, N. G., Samsonov, I. B., Zupnik, B. M.:
Quantum N = 3, d = 3 Chern-Simons Matter
Theories in Harmonic Superspace, JHEP 0910
(2009) 075, arXiv:0909.2970 [hep-th].

[20] Mukhi, S., Papageorgakis, C.: M2 to D2, JHEP
0805 (2008) 085, arXiv:0803.3218 [hep-th].

Evgeny Ivanov
E-mail: eivanov@theor.jinr.ru
Bogoliubov Laboratory of Theoretical Physics, JINR
141980, Dubna, Moscow Region, Russia

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