JHEP11(2004)078
Published by Institute of Physics Publishing for SISSA/ISAS
Received: November 23, 2004
Accepted: November 26, 2004
Superconformal Chern-Simons theories
John H. Schwarz
California Institute of Technology
Pasadena, CA 91125, U.S.A.
E-mail: jhs@theory.caltech.edu
Abstract: We explore the possibilities for constructing lagrangian descriptions of three-
dimensional superconformal classical gauge theories that contain a Chern-Simons term, but
no kinetic term, for the gauge fields. Classes of such theories with N = 1 and N = 2 super-
symmetry are found. However, interacting theories of this type with N = 8 supersymmetry
do not exist.
Keywords: Chern-Simons Theories, AdS-CFT and dS-CFT Correspondence,
Supersymmetry and Duality.
c
° SISSA/ISAS 2005 http://jhep.sissa.it/archive/papers/jhep112004078 /jhep112004078 .pdf
JHEP11(2004)078
Contents
1. Introduction 1
2. Supersymmetry of Chern-Simons theories 3
3. N = 1 models 4
3.1 The gauge multiplet 4
3.2 The matter theory 5
3.3 The gauged theory 6
4. N = 2 models 6
4.1 The gauge multiplet 6
4.2 The matter theory 7
4.3 The gauged theory 8
5. The N = 8 theory? 8
6. Discussion 10
1. Introduction
Many examples of conformal field theories are known in two dimensions and in four dimen-
sions. However, much less is known in three dimensions. From the perspective of AdS/CFT
one is particularly interested in conformally invariant gauge theories, where the rank of the
gauge group is related to the amount of flux in the dual AdS description. M theory admits
compactifications involving AdS
4
, the most symmetrical choice being AdS
4
× S
7
[1]. Ac-
cording to the AdS/CFT conjecture [2] this should be dual to a three-dimensional gauge
theory with the superconformal symmetry OSp(8|4). This gauge theory should have gauge
group U(N) if the dual M theory background has N units of flux through the seven-sphere.
The situation ought to be rather analogous to the case of type IIB superstring theory com-
pactified on AdS
5
× S
5
, with N units of flux, for which the dual gauge theory is N = 4
supersymmetric Yang-Mills theory with a U(N) gauge group and the superconformal sym-
metry is PSU(4|2, 2). There are some significant differences, however. For one thing the
type IIB superstring background contains a constant dilaton field, whose value corresponds
to the Yang-Mills coupling constant. There is no analogous scalar field in the M theory
case. Therefore the dual three-dimensional CFT should not have an adjustable coupling,
and therefore it is expected to be strongly coupled. This makes it a logical possibility that
there is no explicit lagrangian description of this theory, but it does not imply that this
must be the case.
– 1 –
JHEP11(2004)078
The usual viewpoint, which surely is correct, is the following: The low-energy effective
world-volume theory on a collection of N coincident D2-branes of type IIA superstring
theory is a maximally supersymmetric U(N) Yang-Mills theory in three dimensions. This
theory, which is not conformal because the Yang-Mills coupling in three dimensions is
dimensionful, has an SO(7) R symmetry corresponding to rotations of the transverse di-
rections. In the flow to the infrared the gauge coupling increases, which corresponds to the
string coupling (the vev of the dilaton) increasing. This in turn corresponds to the radius of
the circular 11th dimension increasing. In the limit that the coupling becomes infinite, one
reaches the conformally-invariant fixed point theory that describes a collection of coincident
M2-branes in eleven dimensions. This theory should have an enhanced SO(8) R symmetry
corresponding to rotations of the eight transverse dimensions. One question that we wish
to explore in this paper is whether it is possible to find an alternative characterization of
this fixed-point theory with an explicit classical lagrangian.
One can anticipate the field content of these theories from the relation to M2-branes
(in the M theory case) and D3-branes (in the type IIB superstring theory case). The world-
volume field content of a single D3-brane contains a vector, six scalars, and four Majorana
spinors. To describe N coincident D3-branes (at low energy) it is just a matter of promoting
these to N × N hermitean matrices and constructing an interacting superconformal field
theory with U(N) gauge symmetry. This is achieved by N = 4 SYM theory, of course. In
the case of an M2-brane, the physical world-volume field content consists of eight scalars
and eight (two-component) Majorana spinors. So a natural guess is that these should be
made into N ×N matrices and the U(N) global symmetry should be gauged. However, this
is not entirely obvious, because unlike the case of D-branes, there is no simple interpretation
in terms of strings stretched between the branes. When viewed in terms of the maximally
supersymmetric SYM theory that flows to the desired fixed-point theory one sees this field
content except that one of the matrix scalars is replaced by a propagating gauge field. In
the abelian case these can be related by a duality transformation, but in the nonabelian
case there is no simple way of doing that. Rather than trying to carry out such a duality
transformation, we will start with the postulated field content, which is clearly required
for exhibiting the desired Spin(8) R symmetry.
The U(N) gauge theory should have N = 8 super-Poincar´e symmetry and scale invari-
ance, which together ought to imply the full OSp(8|4) superconformal symmetry.
1
If one
succeeds in constructing such a theory, then it would be reasonable to expect that quantum
corrections do not destroy the scale invariance, like in the case of N = 4 SYM theory.
The scalars and spinors in the proposed three-dimensional CFT give an equal number
of physical bosonic and fermionic degrees of freedom. Therefore, to maintain supersymme-
try when U(N) gauge fields are added, the number of bosonic degrees of freedom should
not change. This should be contrasted with the case of N = 4 SYM theory, where the
transverse polarizations of the gauge fields are required to achieve an equal number of
bosonic and fermionic degrees of freedom. Starting from the free theory with global U(N)
1
Usually, but not always, Poincar´e invariance together with scale invariance implies conformal invariance.
(See [3] and references therein.)
– 2 –
JHEP11(2004)078
symmetry in three dimensions there are three alternative ways to introduce the gauge fields
that one might consider: (1) Add gauge field couplings to make the global U(N) symmetry
local, but do not introduce kinetic terms for the gauge fields. (2) Add gauge field cou-
plings to make the global U(N) symmetry local and add F
2
kinetic terms for the gauge
fields. (3) Add gauge field couplings to make the global U(N ) symmetry local and add a
Chern-Simons term for the gauge fields. We claim that choice number (1) is inconsistent
with supersymmetry, because the gauge fields would give rise to constraints that would
effectively subtract bosonic degrees of freedom. Similarly, choice number (2) is unaccept-
able, because the gauge fields would add bosonic degrees of freedom. Also F
2
is dimension
4, and scale invariance of the classical theory only allows dimension 3 terms. This leaves
choice (3), which I claim is exactly right. The Chern-Simons term is dimension 3 and its
inclusion does not lead to either an increase or a decrease in the number of propagating
bosonic degrees of freedom, so it is conceivable that supersymmetry can be achieved.
To be honest, it is quite mysterious how a Chern-Simons term could be generated in
the IR flow of the SYM theory discussed earlier. This is especially a concern since the
SYM theory that flows to the fixed point in question is parity conserving. So how could
the theory be parity violating at the fixed point? In the end, we will not find such an
N = 8 theory, and maybe this is one of the reasons why.
As we have said, the problem that we would most like to solve is the explicit construc-
tion of a lagrangian for the three-dimensional CFT that has maximal supersymmetry and
is dual to M theory on AdS
4
× S
7
. However, most of this paper will address more modest
goals: the construction of three-dimensional gauge theories with N = 1 and N = 2 super-
symmetry and classical scale invariance. This will provide a framework for explaining why
an N = 8 super Chern-Simons theory cannot be constructed. However, it is conceivable
that one could construct a lagrangian description of the desired OSp(8|4) superconformal
theory by modifying one or more of our assumptions.
2. Supersymmetry of Chern-Simons theories
Pure Chern-Simons theory has a lagrangian that is proportional to
L
CS
= tr
·
²
µνρ
µ
A
µ
∂
ν
A
ρ
+
2i
3
A
µ
A
ν
A
ρ
¶¸
. (2.1)
It gives the classical field equation
F
µν
= ∂
µ
A
ν
− ∂
ν
A
µ
+ i[A
µ
, A
ν
] = 0 . (2.2)
A curious fact about this theory is that it has any desired amount of supersymmetry, if one
simply decrees that A
µ
is invariant under each of the supersymmetry transformations. The
reason this is possible is that this theory has no propagating on-shell degrees of freedom.
To prove this assertion one needs to verify the super-Poincar´e algebra, especially that the
commutator of two supersymmetry transformations is a translation. Since the supersym-
metry transformation is trivial, this means that the translation symmetry transformation
should also be trivial.
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JHEP11(2004)078
Since A
µ
by itself is certainly not a complete off-shell supermultiplet, the supersym-
metry algebra should only hold on-shell. This means that in verifying the closure of the
algebra, one is allowed to use the field equation F
µν
= 0. This is a familiar situation; many
of the nicest supersymmetric theories, such as N = 4 super Yang-Mills theory, do not have
a straightforward formulation in terms of off-shell supermultiplets of the full supersymme-
try algebra. So the proof of the assertion that we are making is simply to show that an
infinitesimal translation by a constant amount a
ρ
is trivial modulo a gauge transformation
and the equations of motion. This is the case because an infinitesimal translation shifts
A
µ
by a
ρ
∂
ρ
A
µ
, which differs from a
ρ
F
ρµ
by an infinitesimal gauge transformation
δA
µ
= ∇
µ
Λ = ∂
µ
Λ + i[A
µ
, Λ] (2.3)
for the choice Λ = a
ρ
A
ρ
. This then vanishes by the equations of motion. Of course, this
triviality of translation invariance is not a big surprise since Chern-Simons is a topological
theory.
We will be interested in coupling the Chern-Simons gauge field to other fields. For
this purpose it is convenient to have complete off-shell supermultiplets. This enables one
to combine supersymmetric expressions without substantial modification of the supersym-
metry transformation formulas, as we will see. Pure Chern-Simons theories with off-shell
supersymmetry were constructed in [5] for N = 1, 2, 4. That work did not discuss coupling
these supermultiplets to other matter supermultiplets.
3. N = 1 models
3.1 The gauge multiplet
One of the nice things about the N = 1 theories in three dimensions that we want to con-
struct is that it is easy to implement supersymmetry by using superfields. The Grassmann
coordinates of N = 1 superspace consist just of a two-component Majorana spinor. There
are two kinds of multiplets that we will be interested in: gauge multiplets and scalar multi-
plets. In this section we discuss the gauge multiplet. This superfield is a spinor. However,
in this case we find it convenient to work with the component fields that survive in the
three-dimensional analog of Wess–Zumino gauge. These are the gauge field A
µ
and a Ma-
jorana two-component spinor χ. Both of these are in the adjoint representation of the Lie
algebra and can be represented as hermitean matrices in some convenient representation,
which will be specified later when they are coupled to scalar supermultiplets.
Since we are mainly interested in classical considerations in this paper, we will not
specify the overall normalization of the action at this time. This would need to be con-
sidered carefully in defining the quantum theory, of course. With this understanding we
choose the N = 1 Chern-Simons lagrangian to be
L
CS
= tr
·
²
µνρ
µ
A
µ
∂
ν
A
ρ
+
2i
3
A
µ
A
ν
A
ρ
¶
− ¯χχ
¸
. (3.1)
This theory differs from the pure Chern-Simons theory discussed in the preceding section
only by the addition of the auxiliary fermi field χ. Note that the lagrangian has dimension
three for the choices dim A = 1 and dim χ = 3/2, and then the action is scale invariant.
– 4 –