Page 1
arXiv:1108.4060v1 [hep-th] 19 Aug 2011
MFA-11-36
(1,0) superconformal models
in six dimensions
Henning Samtlebena, Ergin Sezginb, Robert Wimmera
aUniversit´ e de Lyon, Laboratoire de Physique, UMR 5672, CNRS et ENS de Lyon,
46 all´ ee d’Italie, F-69364 Lyon CEDEX 07, France
bGeorge P. and Cynthia W. Mitchell Institute
for Fundamental Physics and Astronomy
Texas A&M University, College Station, TX 77843-4242, USA
Abstract
We construct six-dimensional (1,0) superconformal models with non-abelian
gauge couplings for multiple tensor multiplets. A crucial ingredient in the
construction is the introduction of three-form gauge potentials which com-
municate degrees of freedom between the tensor multiplets and the Yang-
Mills multiplet, but do not introduce additional degrees of freedom. Gener-
ically these models provide only equations of motions. For a subclass also
a Lagrangian formulation exists, however it appears to exhibit indefinite
metrics in the kinetic sector. We discuss several examples and analyze the
excitation spectra in their supersymmetric vacua. In general, the models
are perturbatively defined only in the spontaneously broken phase with the
vev of the tensor multiplet scalars serving as the inverse coupling constants
of the Yang-Mills multiplet. We briefly discuss the inclusion of hypermul-
tiplets which complete the field content to that of superconformal (2,0)
theories.
Page 2
Contents
1 Introduction
2
2Non-abelian tensor fields in six dimensions
2.1Minimal tensor hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2Extended tensor hierarchy . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
7
3 Superconformal field equations
3.1Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2Minimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3Extended model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Adding hypermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Supersymmetric vacua and excitation spectrum . . . . . . . . . . . . .
3.6A model with adjoint tensor multiplets . . . . . . . . . . . . . . . . . .
8
9
10
12
13
13
15
4Action
4.1
4.2
4.3
4.4
4.5
16
16
17
18
20
22
Conditions for existence of an action . . . . . . . . . . . . . . . . . . .
The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Multiplet structure of excitations . . . . . . . . . . . . . . . . . . . . .
Example: SO(5) gauge group . . . . . . . . . . . . . . . . . . . . . . .
Example: Nilpotent gauge group . . . . . . . . . . . . . . . . . . . . .
5Conclusions23
A Conventions 25
1
Page 3
1 Introduction
One of the discoveries of the seminal analysis in [1] is the existence of interacting quan-
tum field theories in five and six dimensions. Of particular interest are six-dimensional
(2,0) superconformal theories which are supposed to describe the low energy limit of
multiple coincident M5 branes.
However, no Lagrangian description for these theories is known and it is in general
believed that no such formulation exists: The M/string theory origin implies that these
theories have no free (dimensionless) parameter, which would allow a parametrization
to weak coupling and thus make the existence of a Lagrangian description plausible.
This conclusion was also drawn from symmetry properties which imply that tree level
amplitudes have to vanish [2]. In addition, these (2,0) theories consist of chiral tensor
multiplets and so far it has often been considered as impossible to define non-abelian
gauge couplings for such multiplets.
Regarding the first aspect the situation is similar to that of multiple M2 branes, as
it was before the recent developments that were triggered by the discovery of the three
dimensional N = 8 superconformal BLG model [3, 4]. The meaning of this N = 8
model in the M/string theory context is rather unclear, but subsequently a N = 6
superconformal theory (ABJM model) was formulated for an arbitrary number of M2
branes [5]. The decisive observation in [5] is that an orbifold compactification of the M
theory/supergravity background provides a dimensionless, though discrete, parameter
k which allows a parametrization to weak coupling and thus also a Lagrangian formu-
lation. The orbifold compactification breaks N = 8 supersymmetry down to N = 6
except for k = 1,2, where the theory is strongly coupled. The N = 6 ABJM model has
the same field content as the N = 8 multiplet and it has been argued that monopole
operators enhance the supersymmetry to N = 8 for k = 1,2 [6, 7] (for U(2) gauge
group see [8, 9]).
We take here an analogous route. Instead of focusing on (2,0) supersymmetry we
construct (1,0) superconformal models for interacting multiple tensor multiplets. One
major obstacle, the nonabelian gauging of the (self dual) tensor fields, is resolved by the
introduction of a tensor hierarchy [10, 11, 12] which besides the Yang-Mills gauge field
and the two-form gauge potentials of the tensor multiplets contains also three-form
gauge potentials. We therefore have an extended tensor gauge freedom with p = 0,1,2
p-form gauge parameters.
We then formulate essentially unique supersymmetry transformations for the vari-
ous fields, where we find a suitable extension of the structures introduced in [13]. While
in [13] the 2-form potential is a singlet, here it carries a representation of the local gauge
group, which is facilitated by the introduction of a 3-form potential that mediates cou-
plings between the tensor and vector multiplets. While the brane interpretation of our
models requires further investigation, it is worth mentioning that the field content of
the model in [13] is known to arise in the worldvolume description of D6 branes stretch
between NS fivebranes [14, 15, 16, 17, 18]. The closure of the supersymmetry algebra
2
Page 4
into translations and extended tensor gauge transformations puts the system on-shell
with a particular set of e.o.m. For example the tensor multiplet field strength has to
satisfy its self-duality condition and the Yang-Mills field strength is related to the field
strength of the three-form potentials by a first-order duality equation. Consequently,
the three-form gauge potentials do not introduce additional degrees of freedom. They
communicate degrees of freedom between the tensor multiplets and the Yang-Mills mul-
tiplet. We also describe the extension of the tensor hierarchy to higher p-form gauge
potentials and briefly discuss the inclusion of hypermultiplets which complete the field
content to that of superconformal (2,0) theories.
Consistency of the tensor hierarchy imposes a number of conditions on the possible
gauge groups and representations. We discuss several solutions of these conditions.
Generically these models provide only equations of motions, but for a subclass also a
Lagrangian formulation exists. In particular we find a Lagrangian model with SO(5)
gauge symmetry. However, the existence of a Lagrangian description necessarily im-
plies indefinite metrics for the kinetic terms. It is at the moment not clear if the
resulting ghost states can be decoupled with the help of the large extended tensor
gauge symmetry. This and other questions regarding the quantization of the theory
we have to leave for a further investigation. A general feature of all considered cases
is that the models are perturbatively defined only in the spontaneously broken phase
with the vev of the tensor multiplet scalars serving as inverse coupling constants of the
Yang-Mills multiplets.
To write down a Lagrangian for a self dual field strength is in general a formidable
task. For a single M5 brane, in which case the e.o.m. are known [19], this has been
done in [20, 21].
We consider these difficulties to be of a different category than
finding a superconformal non-abelian theory. When we formulate a Lagrangian we
understand that the first order duality equations are consistently imposed in addition
to the second order Lagrangian e.o.m., just as in the democratic formulation of ten-
dimensional supergravity [22].
Finally we want to comment on some recent attempts and proposals for the descrip-
tion of the (2,0) theory. The low energy description of the theory when compactified
on a small circle is expected to be given by the maximal supersymmetric Yang-Mills
theory in five dimensions. Recent attempts tried to basically rewrite five-dimensional
Yang-Mills theory in six dimensions [23, 24] or introduced non-abelian gaugings at the
cost of locality [25]. Furthermore, it was recently proposed that the (2,0) theory is
identical to five-dimensional super Yang-Mills theory for arbitrary coupling or compact-
ification radius [26, 27]. It is not clear yet how one could obtain Yang-Mills theories in
five dimensions from the models presented here (even when including hypermultiplets).
Clearly a mechanism more complicated than a trivial dimensional reduction has to be
considered.
The paper is organized as follows: in section 2 we present the general non-abelian
hierarchy of p-forms in six dimensions. We show that all couplings are parametrized
in terms of a set of dimensionless tensors that need to satisfy a number of algebraic
3
Page 5
consistency constraints. In particular, we find that non-abelian charged tensor fields
require the introduction of St¨ uckelberg-type couplings among the p-forms of different
degree. In section 3, we extend the non-abelian vector/tensor system to a supersym-
metric system. Closure of the supersymmetry algebra puts the system on-shell and we
derive the modified field equations for the vector and tensor multiplets. In particular,
we obtain the first-order duality equation relating vector fields and three-form gauge
potentials. In section 3.4 we sketch the extension of the model upon inclusion of hyper-
multiplets and gauging of their triholomorphic isometries. In section 3.5 we derive the
general conditions for maximally supersymmetric vacua and compute the fluctuation
equations by linearizing the equations of motion around such a vacuum. Finally, we
give in section 3.6 an explicit example with an arbitrary compact gauge group and
tensor fields transforming in the adjoint representation.
Section 4 presents the additional conditions on the couplings in order to allow for
a Lagrangian formulation. We give the full action in section 4.2. In section 4.3 we
calculate the fluctuation equations induced by the action and show that the degrees
of freedom arrange in the free vector and self-dual tensor multiplet as well as in cer-
tain ‘non-decomposable’ combinations of the two. We illustrate the general analysis
in sections 4.4 and 4.5 with two explicit models that provide solutions to the con-
sistency constraints with compact gauge group SO(5) and nilpotent gauge group N8,
respectively. Finally, we summarize our findings in section 5.
2 Non-abelian tensor fields in six dimensions
In this section, we present the general (non-abelian) couplings of vectors and anti-
symmetric p-form fields in six dimensions. While the standard field content of the
ungauged theories falls into vector and tensor multiplets, it is a general feature of
these theories that the introduction of gauge charges generically requires the introduc-
tion of and couplings to three-form potentials. The specific couplings can be derived
successively and in a systematic way by building up the non-abelian p-form tensor hi-
erarchy, as worked out in [10, 11, 12], see also [28, 29, 30] for some applications to the
specific 6D context. Rather than going again step by step through the derivation of
the general couplings, we directly present the final result as parametrized by a set of
constant tensors (generalized structure constants) that need to satisfy a system of al-
gebraic consistency equations (generalized Jacobi identities). In section 2.1 we present
the couplings for the minimal field content required to introduce non-abelian couplings
between vector and tensor fields. In section 2.1, we extend the system to include also
four-form gauge potentials.
2.1Minimal tensor hierarchy
The basic p-form field content of the theories to be discussed is a set of vector fields Ar
and two-form gauge potentials BI
µ,
µν, that we label by indices r and I, respectively. In
4
Page 6
addition, we will have to introduce three-form gauge potentials that we denote by Cµνρr.
The fact that three-form potentials are labeled by an index r dual to the vector fields
is in anticipation of their dynamics: in six dimensions, these fields will be the on-shell
duals to the vector fields. For the purpose of this section however, the dynamics of
these fields is not yet constrained, the construction of the tensor hierarchy remains
entirely off-shell, and the indices ‘r’ and ‘r’ might be taken as unrelated. Similarly,
throughout this section, the self-duality of the field strength of the two-form gauge
potentials, which is a key feature of the later six-dimensional dynamics, is not yet an
issue.
The full non-abelian field strengths of vector and two-form gauge potentials are
given as
Fr
µν
≡ 2∂[µAr
ν]− fstrAs
µAt
ν+ hr
IBI
µν,
HI
µνρ
≡ 3D[µBI
νρ]+ 6dI
rsAr
[µ∂νAs
ρ]− 2fpqsdI
rsAr
[µAp
νAq
ρ]+ gIrCµνρr, (2.1)
in terms of the antisymmetric structure constants fstr= f[st]r, a symmetric d-symbol
dI
of different degree.1Covariant derivatives are defined as Dµ ≡ ∂µ− Ar
action of the gauge generators Xron the different fields given by Xr·Λs≡ −(Xr)tsΛt,
Xr· ΛI≡ −(Xr)JIΛJ, etc. The field strengths are defined such that they transform
covariantly under the set of non-abelian gauge transformations
rs= dI
(rs), and the tensors gIr, hr
Iinducing St¨ uckelberg-type couplings among forms
µXr with an
δAr
µ
= DµΛr− hr
IΛI
µ,
∆BI
µν
= 2D[µΛI
ν]− 2dI
rsΛrFs
µν− gIrΛµν r,
∆Cµνρr
= 3D[µΛνρ]r+ 3bIrsFs
[µνΛI
ρ]+ bIrsHI
µνρΛs+ ... ,(2.2)
where we have introduced the compact notation
∆BI
µν
≡ δBI
≡ δCµνρr− 3bIrsBI
µν− 2dI
rsAr
[µδAs
ν],
∆Cµνρr
[µνδAs
ρ]− 2bIrsdI
pqAs
[µAp
νδAq
ρ].(2.3)
The ellipsis in the last line of (2.2) represent possible terms that vanish under projection
with gIr. This system is completely defined by the choice of the invariant tensors gIr,
hr
p-form gauge fields can be used to gauge away some of the p-forms, giving mass to
others by the St¨ uckelberg mechanism. However, for the general analysis of couplings,
we find it the most convenient to work with the uniform system (2.2) and to postpone
possible gauge fixing to the analysis of particular models.
Consistency of the tensor hierarchy requires that the gauge group generators in the
various representations are parametrized as
I, bIrs, dI
rs, and frst. It is obvious from (2.2) that the shift symmetry action on the
(Xr)st
(Xr)IJ
= −frst+ dI
= 2hs
rsht
I,
IdJ
rs− gJsbIsr,(2.4)
1We use canonical dimensions such that a p-form has mass dimension p and as a result all constant
tensors fstr, dI
I, are dimensionless.
rs, gIr, hr
5
Page 7
in terms of the constant tensors appearing in the system. The second relation exposes
an important feature of the tensor hierarchy: tensor fields can be charged under the
gauge group only if either hr
vanishing St¨ uckelberg-type couplings in the field strengths (2.1). This corresponds to
the known results [31, 32] that in absence of such couplings and the inclusion of addi-
tional three-form gauge potentials, the free system of self-dual tensor multiplets does
not admit any non-abelian deformations. On the other hand, the first relation of (2.4)
shows that in presence of hr
tion’ (Xr)stare not just given by the structure constants but acquire a modification,
symmetric in its indices (rs).
Furthermore, consistency of the system, i.e. covariant transformation behavior of
the field strengths (2.1) under the gauge transformations (2.2) requires the constant
tensors to satisfy a number of algebraic consistency constraints. A first set of con-
straints, linear in f, g, h, is given by
Ior gIrare non-vanishing, i.e. they require some non-
I, the gauge group generators in the ‘adjoint representa-
2?dJ
rtbIsu+ 2dK
r(udI
v)s− dI
rsdJ
uv
?hs
?hu
J
= 2fr(usdI
v)s− bJsrdJ
uvgIs,
?dJ
rsbIut+ dJ
rubKstδJ
IJ
= frsubIut+ frtubIsu+ gJubIurbJst, (2.5)
and ensures the invariance of the d- and the b-symbol under gauge transformations.
The remaining constraints are bilinear in f, g, h and take the form
f[pqufr]us−1
3hs
IdI
u[pfqr]u
hr
= 0 ,
IgIs
= 0 ,
frsthr
I− dJ
rsht
Jhr
I
= 0 ,
gJshr
KbIsr− 2hs
Ihr
KdJ
rs
= 0 ,
− frtsgIt+ dJ
rths
JgIt− gItgJsbJtr
= 0 .(2.6)
They may be understood as generalized Jacobi identities of the system: together with
(2.5) they ensure the closure of the gauge algebra according to
[Xr,Xs] = −(Xr)stXt,(2.7)
for the generators (2.4), as well as gauge invariance of the tensors f, g and h. The
first equation of (2.6) shows that the standard Jacobi identity is modified in presence
of a non-vanishing hI
restrictive, it admits rather non-trivial solutions and we will discuss explicit examples
of solutions in sections 3.6, 4.4, and 4.5, below. The system admits different abelian
limits with frst= 0 = gIrand either hr
(2.5), (2.6) are trivially satisfied. A slightly more general solution is given by vanishing
hr
particular choice dI
the Yang-Mills multiplet to an uncharged self-dual tensor multiplet as described in [13].
r. Even though the set of constraints (2.5), (2.6) looks highly
Ior dI
rsvanishing, in which the constraints
I= 0 = gIrwith frstrepresenting the structure constants of a Lie algebra. With the
rs= dIδrs, the vector-tensor system then reduces to the coupling of
6
Page 8
The covariant field strengths (2.1) satisfy the modified Bianchi identities
D[µFr
D[µHI
νρ]
=
1
3hr
IHI
µνρ,
νρσ]
=
3
2dI
rsFr
[µνFs
ρσ]+1
4gIrH(4)
µνρσ r,(2.8)
where the non-abelian field strength H(4)
the second equation. In turn, its Bianchi identity is obtained from (2.8) as
µνρσ r of the three-form potential is defined by
D[µH(4)
νρστ]r
= −2bIrsFs
[µνHI
ρστ]+ ... ,(2.9)
where the ellipsis represents possible terms that vanish under projection with gIr. We
finally note that the general variation of the field-strengths is given by
δFr
µν
= 2D[µδAr
ν]+ hr
I∆BI
µν,
δHI
µνρ
= 3D[µ∆BI
νρ]+ 6dI
rsFr
[µνδAs
ρ]+ gIr∆Cµνρr,
δH(4)
µνρσ r
= 4D[µ∆Cνρσ]r− 6bIrsFs
[µν∆BI
ρσ]+ 4bIrsHI
[µνρδAs
σ]+ ... , (2.10)
again with the ellipsis representing possible terms that vanish under projection with gIr.
2.2Extended tensor hierarchy
The field content introduced in the last section were the p-forms Ar
which in particular we have defined their non-abelian field strengths. Strictly speak-
ing, in the entire system, only a subset of the three-form potentials have appeared,
defined by projection with the tensor gIras gIrCµνρr. As it turns out, this trunca-
tion is precisely the ‘minimal field content’ required in order to write down an action
and/or define a consistent set of equations of motion. Off-shell on the other hand, the
tensor hierarchy may be extended to the full set of three-form potentials, which then
necessitates the introduction of four-form gauge potentials, etc.
For later use, we present in this section the results of the general tensor hierarchy for
the four-form gauge potentials which we denote by C(4)
H(5)
µ, BI
µν, Cµνρr, for
µνρλαwith covariant field strength
α . The full version of the Bianchi identity (2.9) then reads
D[µH(4)
νρστ]r
= −2bIrsFs
[µνHI
ρστ]+1
5krαH(5)
µνρστ α, (2.11)
where now the field strength H(5)
α itself satisfies the Bianchi identity
D[µH(5)
νρλστ] α
=
10
3cαIJHI
[µνρHJ
λστ]−5
2ct
αsFs
[µνH(4)
ρλστ]t+ ··· ,(2.12)
up to terms vanishing under projection with the tensor krα. The new constant tensors
krα, cαIJ, and ct
αsare constrained by the relations
krαcαIJ
= hs
[IbJ]rs,krαct
αs= frst− bIrsgIt+ dI
rsht
I,gKrkrα= 0 ,(2.13)
7
Page 9
which extend the constraints (2.5), (2.6). As a consistency check, we note that equa-
tions (2.5), (2.6) imply the orthogonality relations
gKrhs
[IbJ]rs
= 0 ,
gKr?frst− gItbIrs+ ht
IdI
rs
?
= 0 ,(2.14)
showing that (2.13) does not imply new constraints among the previous tensors. Fur-
thermore, consistency of the extended system requires an additional relation among b-
and d-symbol to be satisfied
bJr(sdJ
uv)
= 0 ,(2.15)
as also noted in [28]. The new tensor gauge transformations take the form
∆Cµνρr
= 3D[µΛνρ]r+ 3bIrsFs
[µνΛI
ρ]+ bIrsHI
µνρΛs− krαΛµνρα,
∆C(4)
µνρσ α
= 4D[µΛνρσ]α− 8cαIJH[I
[µνρΛJ]
σ]+ 6ct
αsFs
[µνΛρσ]t
+ ct
αsH(4)
µνρσ tΛs+ ... ,(2.16)
where the first equation completes the corresponding transformation law of (2.2) and
the second transformation is given up to terms that vanish under projection with the
tensor krα. Accordingly, the general variation of the non-abelian field strengths from
(2.11), (2.12) is given by
δH(4)
µνρσ r
= 4D[µ∆Cνρσ]r− 6bIrsFs
[µν∆BI
ρσ]+ 4bIrsHI
[µνρδAs
σ]+ krα∆C(4)
µνρσ α,
δH(5)
µνρστ α
= 5D[µ∆C(4)
νρστ] α− 10ct
αsFs
µν∆Cρστ] t− 20cαIJH[I
[µνρ∆BJ]
στ]
− 5ct
αsδAs
[µH(4)
νρστ] t+ ... . (2.17)
Continuing along the same line, the tensor hierarchy can be continued by introducing
five-form and six-form potentials together with their field strengths and non-abelian
gauge transformations.For the purpose of this paper we will only need the vec-
tor/tensor system up to the four-form gauge potentials given above.
3 Superconformal field equations
In the previous section we have introduced the tensor hierarchy for p-form gauge po-
tentials (p = 1,2,3) with the associated generalized field strengths (2.1) and Bianchi
identities (2.8). Gauge covariance w.r.t. the extended tensor gauge symmetry (2.2)
implies a number of conditions on the (dimensionless) invariant tensors and generators
of the gauge group (2.4)–(2.6), but otherwise does not contain any information about
the dynamics of theses fields.
The aim of this section is to complete the bosonic fields of the tensor hierarchy
into supersymmetry multiplets in order to obtain a non-abelian superconformal model
8
Page 10
for the (1,0) vector and tensor multiplets. With the given (bosonic) field content of
the tensor hierarchy (2.1), a supersymmetric tensor hierarchy will contain Yang-Mills
multiplets (Ar
I, respectively.The index i = 1,2 indicates the Sp(1) R-symmetry, the field Yij
denotes the auxiliary field of the off-shell vector multiplets. In addition one has to
accommodate within this structure the three-form potential Cµνρrwhose presence was
crucial in the last section in order to describe non-abelian charged tensor fields.
µ,λir,Yij r), and tensor multiplets (φI,χiI,BI
µν), labeled by indices r and
3.1 Supersymmetry
The coupling of a single (1,0) self-dual tensor multiplet to a Yang-Mills multiplet was
introduced in [13] and as a first step we give the necessary generalization for a non-
abelian coupling of an arbitrary number of these tensor multiplets. To this end, we
introduce supersymmetry transformations such that they close into translations and
the extended tensor gauge symmetry (2.2) according to
[δǫ1,δǫ2] = ξµ∂µ+ δΛ+ δΛµ+ δΛµν, (3.1)
with field dependent transformation parameters for the respective transformations.
These parameters are given by
ξµ
Λr
ΛI
≡
= −ξµAr
= −ξνBI
= −ξρCρµν r− bIrsΛsBI
1
2¯ ǫ2γµǫ1,
µ,
νµ+ dI
µ rsΛrAs
µ+ ξµφI,
µν−2
Λµν r
3bIrpdI
qsΛsAp
[µAq
ν],(3.2)
as will be shown shortly. With dI
ing algebra of [13].2The supersymmetry transformations for the Yang-Mills multiplet
are given by
rs= α′dIδrs, bIrs= 0, this reproduces the correspond-
δAr
µ
= −¯ ǫγµλr,
δλir
=
1
8γµνFr
µνǫi−1
2Yij rǫj+1
4hr
IφIǫi,
δYij r
= −¯ ǫ(iγµDµλj)r+ 2hr
I¯ ǫ(iχj)I. (3.3)
Here the generalization w.r.t. the transformations for the off-shell pure Yang-Mills
multiplet is parametrized by the constant tensor hr
of the tensor multiplets on the r.h.s. of the transformations. These additional terms
are necessary for the supersymmetry algebra to close to the generalized tensor gauge
symmetry (3.1), (3.2). E.g. the last term in δλiris required to produce the proper δΛµ
action in the commutator of supersymmetries on the vector field Ar
last term in δYij rensures the proper closure of the supersymmetry algebra on λir. It
Iand brings in the fields (φI,χiI,BI
µν)
µ. Likewise, the
2Note that in canonical dimensions, the tensor dI
rsis dimensionless.
9
Page 11
then comes as a non-trivial consistency check, that the variation of this last term is
precisely what is needed for closure of the algebra on Yij r. Even though, fields from
the tensor multiplets appear in these transformation rules, the Yang-Mills multiplet by
itself, using the necessary tensor multiplet transformations, still closes off-shell.
Next we give the supersymmetry transformations of the tensor multiplet
δφI
δχiI
∆BI
= ¯ ǫχI,
=
= −¯ ǫγµνχI,
1
48γµνρHI
µνρǫi+1
4γµDµφIǫi−1
2dI
rsγµλir¯ ǫγµλs,
µν
∆Cµνρr
= −bIrs¯ ǫγµνρλsφI,(3.4)
where we have used the same notation (2.3) for general variation introduced in the
tensor hierarchy. We also note that γµνρǫiacts as a self-duality projector such that
only HI +
r.h.s. of these transformations has been generalized by the introduction of the general
d-symbol, and the inclusion of covariant field strengths and derivatives on the now
charged fields of the tensor multiplets. In particular, the important new ingredient in
these transformation rules is the three-form potential Cµνρrwhich is contained in the
definition of HI
(2.10). Its presence has been vital in establishing the non-abelian bosonic vector-tensor
system in the last section, and similarly, its presence turns out to be indispensable for
closure of the supersymmetry algebra here. To group it with the tensor multiplet in
(3.4) is a mere matter of convenience; with the same right it might be considered as a
member of the gauge multiplet (indeed, as mentioned before by its dynamics the three-
form potential will be the dual of the vector fields Ar
transformation (3.4), mixing Yang-Mills and tensor multiplet fields, displays its dual
role as a messenger between these two multiplets. Note that we have given in (3.4)
the supersymmetry transformation for the uncontracted three-form Cµνρr, although all
the explicit couplings only contain the contracted expression gK rCµνρr. We will come
back to this difference in the following.
Closure of the supersymmetry algebra on the tensor multiplet according to (3.1) is
now rather non-trivial and heavily relies on the extra terms arising from variation of
the three-form potential. In particular, the algebra closes only on-shell on the tensor
multiplets. In the search for new model or theory such a property may be considered
as feature that provides a certain uniqueness. We will discuss these equations and their
individual origin now in detail.
µνρ, see (A.1), is actually alive in δχiI. W.r.t. the couplings discussed in [13], the
µνρand contributing to its supersymmetry transformation according to
µ). The form of its supersymmetry
3.2Minimal model
We first investigate the equations of motion resulting from supersymmetrization of the
bosonic field content of the minimal tensor hierarchy of section 2.1. In particular, this
model includes only the projected subset gK rCµνρrof three-form gauge potentials. The
10
Page 12
resulting tensor multiplet field equations are given by
HI −
µνρ
= −dI
rs¯λrγµνρλs,
γσDσχiI
=
1
2dI
rsFr
στγστλis+ 2dI
rsYij rλs
j+?dI
rshs
J− 2bJsrgIs?φJλir,
DµDµφI
= −1
2dI
rs
?Fr
µνFµν s− 4Yr
ijYij s+ 8¯λrγµDµλs?
?¯λrχJ− 3dI
− 2?bJsrgIs− 8dI
rshs
J
rshr
Jhs
KφJφK.(3.5)
The first equation, which imposes a self duality condition on the three-from field
strength, originates in the closure of supersymmetry on the associated two-form poten-
tial BI
field equation is obtained by the supersymmetry transformation of the χiI- equation.
The fact that the tensor fields are charged under the gauge group has rather non-
trivial consequences, namely supersymmetry variation of the field equations (3.5) in
turn implies the following first-order equations of motion for the Yang-Mills multiplets
µν. The closure on δχiIgives the fermionic equations of motion while the scalar
gKrbIrs
?Ys
µνφI− 2¯λsγµνχI?
?φIγµDµλs
ijφI− 2¯λs
(iχI
j)
?
= 0 ,
gKrbIrs
?Fs
=
1
4!εµνλρστgKrH(4)λρστ
r
,
gKrbIrs
i+1
2γµλs
iDµφI?
= gKrbIrs
?1
4Fs
+3
µνγµνχI
i+
1
24HI
µνργµνρλs
i− Ys
ijχj I
2hs
JφIχJ
i+1
3dI
uvγµλu
i¯λsγµλv?
.
(3.6)
The first equation is the algebraic equation for the auxiliary field Yij r, while the second
equation provides the anticipated duality of vector fields and three-form potentials by
relating their respective field strengths. In particular, derivation of this equation and
use of the Bianchi identity (2.9) gives rise to a standard second-order equation of Yang-
Mills type for the vector fields Ar
inferred from closure of the supersymmetry algebra on the three-form gauge potentials
gKrCµνρr. The appearance of the Yang-Mills dynamics (3.6) from supersymmetry of
the tensor field equations (3.5) is in strong contrast to the model of [13] (in which
effectively gKr= 0, and the tensor field are not charged) where the vector fields remain
entirely off-shell or can alternatively be set on-shell with field equations that do not
contain the tensor multiplet fields. Moreover, in the model of [13], an algebraic equation
analogous to the first equation of (3.6) is excluded by the appearance of an anomaly
in its supersymmetry variation (see also [33]). We should stress that in the present
model, such anomalies are actually absent due to the particular Fierz identities (A.6),
(A.7) in combination with the identity (2.15). I.e. the quartic fermion terms in the
supersymmetry variation of (3.6) cancel precisely, which yields a strong consistency
check of the construction.
To summarize, the system of equations of motion (3.5), (3.6) consistently trans-
forms into itself under supersymmetry. It describes a novel system of supersymmetric
µ. Equivalently, the first two equations of (3.6) can be
11
Page 13
non-abelian couplings for multiple (1,0) tensor multiplets in six dimensions. The equa-
tions of motion contain no dimensionful parameter and hence the system is at least
classically (super)-conformal. A crucial ingredient to the model are the three-form
gauge potentials Cµνρrwhich are related by first-order duality equations to the vector
fields of the theory and thus do not constitute new dynamical degrees of freedom. This
is similar to the situation of Chern-Simons matter theories in the context of multiple
M2 branes [5], [3]. The actual model depends on the explicit choice of the gauge group
and representations and the associated invariant tensors of the gauge group which have
to satisfy the conditions (2.4)–(2.6). The task that remains is to find explicit solutions
for these constraints. We will discuss different examples in sections 3.6, 4.4 and 4.5
below.
3.3Extended model
The above described model represents the minimal field content and equations of mo-
tion, required for closure of the supersymmetry algebra and the supersymmetry of the
equations of motions. In particular, it relies on the projected subset gK rCµνρrof three-
form gauge potentials. Just as for the bosonic tensor hierarchy in section 2.2, one may
seek to extend the above supersymmetric system to the full set of three-form gauge
potentials. With the supersymmetry transformation of general Cµνρrgiven by (3.4),
closure of the supersymmetry algebra leads to the following uncontracted equations
bIrs
?Ys
µνφI− 2¯λsγµνχI?
?φIγµDµλs
ijφI− 2¯λs
(iχI
j)
?
= 0 ,
bIrs
?Fs
=
1
4!εµνλρστH(4)λρστ
r
,
bIrs
i+1
2γµλs
iDµφI?
= bIrs
?1
+ hs
4Fs
µνγµνχI
i+
1
24HI
µνργµνρλs
i− Ys
ijχj I+
J
?2φIχJ
i−1
2φJχI
i
?+1
3dI
uvγµλu
i¯λsγµλv?
,
(3.7)
In order to have this system close under supersymmetry it is necessary to introduce also
a four-form gauge potential. Consequently the tensor hierarchy has to be continued one
step further as described in section 2.2. The resulting supersymmetry transformation
of the four-form potential is
∆C(4)
µνρσ α
= 2cαIJφ[I¯ ǫγµνρσχJ], (3.8)
Furthermore, supersymmetry of the field equations (3.7) induces the first-order field
equations
1
5!εµνρλστkrαH(5)µνρλσ
α
= 2krα?cαIJ
?φIDµφJ− 2¯ χIγµχJ?− ct
µνρσ α is given by a first-order duality equations,
αubJtv¯λuγµλv?
. (3.9)
This shows that the dynamics of C(4)
which relates these four-form potentials to the Noether current of some underlying
global symmetry. In particular, this first-order equation ensures that the four-form
gauge potentials do not constitute new dynamical degrees of freedom.
12
Page 14
3.4 Adding hypermultiplets
Another possible extension of the supersymmetric model presented above is the inclu-
sion of hypermultiplets. As is well known, global supersymmetry requires the hyper-
scalars to parametrize a hyper-K¨ ahler manifold Mh, more precisely superconformal
symmetry requires Mhto be a hyper-K¨ ahler cone. The above presented non-abelian
theories can be extended to include gaugings of isometries on the hyper-K¨ ahler cone
along the lines of [34, 35, 36], from which the additional couplings and in particular the
resulting scalar potential can be inferred. While we defer the details of this extension
to another publication, here we only sketch a few relevant elements of the construc-
tion. Within in the above construction, gauging of triholomorphic isometries on the
hyper-K¨ ahler cone is achieved by introducing an embedding tensor ϑrαthat encodes
the coupling of vector fields Ar
algebraic conditions
µto hyper-K¨ ahler isometries Kαand is subject to the
fprsϑsα
= fβγαϑpβϑrγ,hr
Iϑrα= 0 ,(3.10)
with the structure constants fαβγof the algebra of hyper-K¨ ahler isometries. On the
other hand, in the presence of hypermultiplets, the vector multiplet equations of motion
(3.7) allow for a consistent modification, in particular in the Y -field equation as
bIrs
?Ys
ijφI− 2¯λs
(iχI
j)
?
= krαPij
α, (3.11)
with the constant tensor krαfrom (2.13), and the moment maps Pij
triholomorphic hyper-K¨ ahler isometries. It is only by means of this algebraic equation
for Ys
the existence of an action eventually leads to the identification
αassociated with the
ijthat the hyperscalars enter the tensor multiplet field equations. Further requiring
krα
= ϑrα, (3.12)
i.e. relates the gauging of hyper-K¨ ahler isometries to a modification of the vector and
tensor multiplet field equations.
3.5Supersymmetric vacua and excitation spectrum
We study now supersymmetric vacua for the minimal model of section 3.2 and the
excitation spectrum in such vacua, i.e. the linearized field equations. The algebraic
equation for the vector field strength, the second equation in (3.6), indicates that
the expectation value of the tensor multiplet scalar φIserves as an (inverse) coupling
constant. This notion will become more evident in the subsequent sections where we
discuss models which provide a Lagrangian. Consequently, the perturbative analysis is
limited to the spontaneously broken phase where φIhas a (large) expectation value.
The Killing spinor equations of the theory (4.3) are obtained from (3.3), (3.4)
0
!≡ δλir=
1
8γµνFr
µνǫi−1
2Yij rǫj+1
4hr
IφIǫi,
0
!≡ δχiI=
1
48γµνρHI +
µνρǫi+1
4γµDµφIǫi, (3.13)
13
Page 15
and characterize solutions that preserve some fraction of supersymmetry. These equa-
tions show that a Lorentz-invariant solution preserving all supersymmetries corre-
sponds to setting the scalar fields to constant values φI
0satisfying
φI
0hr
I
= 0 ,(3.14)
and setting all other fields to zero. Expanding the scalar fluctuations as φI≡ φI
ϕIand imposing the condition (3.14) one obtains at the linearized level for the field
equations (3.5), (3.6) the system:
0+
(dBI+ gIrCr)−= 0 ,NI
rYr
ij= 0 ,
/ ∂ χiI+ 2NI
rλir= 0 ,NI
rdAr− gIr ∗dCr= 0 ,
✷ϕI− NI
r∂ · Ar= 0 ,NI
r/ ∂λir= 0 ,(3.15)
where we have defined the matrices
Krs ≡ φI
0bIrs,NI
r ≡ gIsKsr. (3.16)
and used that NI
tion (3.14).
rhr
J= 0, by the first identity in (2.14) and the susy vacuum condi-
Unbroken gauge symmetry.
(3.14) the vector gauge transformations ΛrXr are broken down to the subgroup of
transformations ΛrXrwhich satisfy
For a generic supersymmetric vacuum which satisfies
XrJIφJ
0= − NI
r
!= 0, (3.17)
where the index r labels the subset of unbroken generators (2.4). The rest of the
extended tensor gauge symmetry (2.2) remains intact. Consequently, in the case that
the gauge group is not completely broken, the matrix NI
the matrix Krs, always has some null-directions. The fluctuation equations (3.15)
show that for these null-directions the fields of the corresponding vector multiplets
drop out of this perturbative analysis. This is nothing else than the above mentioned
observation that the perturbative analysis is valid only in the spontaneously broken
phase and that the unbroken sector of the Yang-Mills multiplet is (infinitely) strongly
coupled and perturbatively not visible. This part of the spectrum decouples and should
be integrated out for a proper treatment.
In general it is rather difficult to break the gauge symmetry completely with a
single scalar field. The addition of hypermultiplets as sketched in section 3.4 may
offer additional possibilities in this directions. This is for example comparable with the
situation of N = 2 SQCD, for which mixed Coulomb-Higgs phases with vev’s for vector
multiplet and hypermultiplet scalars exist where the theory is completely higgsed. In
such a case there would be regions in the moduli space of vacua where the complete
spectrum of the models discussed here is perturbatively accessible. For the extended
models of section 3.3 on the other hand, the coupling of the Yang-Mills multiplet is
given by the matrix Krswhich may have less null directions than the matrix NI
r, and for invertible gIralso
r.
14
Page 16
3.6 A model with adjoint tensor multiplets
A particular solution to the constraints (2.5), (2.6) is given by choosing some semi-
simple compact gauge group G with Lie-algebra g, identifying both I and r with
the adjoint representation of G, and the tensor grswith the Cartan-Killing metric.
Moreover we set
hs
r
≡ 0 ,dp
rs≡ drstgpt,bprs ≡ fprs, (3.18)
with the totally symmetric d-symbol drstand the totally antisymmetric structure con-
stants frst. As will be discuss in detail in the next section, for a solution of this form
the resulting theory does not admit an action and is described by the set of equations
of motion (3.5), (3.6) only.
With grsbeing the (invertible) Cartan-Killing metric, the matrices N and K intro-
duced in (3.16) are essentially the same,
Nr
s= grtKts=: Krs= −φt
0ftsr, (3.19)
and the matrix Krsdefines the adjoint action of the vev φ0. By a gauge rotation the
φ0can always be chosen to lie in the Cartan subalgebra t, and we decompose g into
the orthogonal sum g = t ⊕ ˜ g. In that case, the unbroken sector of the Yang-Mills
multiplets, which drops out of the fluctuation equations, spans the Cartan subalgebra
t on which the action of Krs vanishes.On the orthogonal complement ˜ g and for
generic choice of φ0, the matrix Krsis invertible, and using the Cartan-Weyl basis we
introduce the notation K
no summation over repeated indices in this case).
Before giving the explicit excitation equations for this specific model we discuss the
gauge fixing of the vector field gauge symmetry, which for hs
symmetry, see (2.2). A convenient gauge, which disentangles the scalar field and gauge
field fluctuations is given by the Lorenz gauge condition
˜ r
red˜ s= k˜ rδ˜ r
˜ sfor the reduced matrix on this subspace (there is
r= 0 is an ordinary gauge
∂ · Ar= 0. (3.20)
Since the gauge fields are determined by first-order equations the Lorenz gauge, and not
a ’t Hooft Rξ-gauge, decouples the scalar and gauge field kinetic terms. The fluctuation
equations (3.15) thus take the form
/ ∂ χir= 0 ,dCr= 0 ,
(dBr+ Cr)−= 0 ,
✷ϕr= 0 ,
/ ∂ χi˜ r+ 2k˜ rλi˜ r= 0 , dA˜ r−1
k˜ r
∗dC˜ r= 0 ,
/ ∂λi˜ r= 0 ,Y˜ r
ij= 0 ,(3.21)
where we have split the gauge indices as r = (r, ˜ r) according to the decomposition
g = t ⊕ ˜ g. For the unbroken sector t, the first line of (3.21) together with the second
15
Page 17
line for r = r thus describe a free tensor multiplet coupled to the three-form potential
Crwhich has vanishing field strength and may be gauged away. Alternatively, one
may employ the two-form shift symmetry in (2.2) with gauge parameter Λr
Br= 0. Then the linearized equations describe a self-dual closed field Crwhich gives
an equivalent description of the free tensor multiplet.
The broken sector ˜ g is described by the second line of (3.21) for r = ˜ r together with
the last two lines. Here also the Yang-Mills multiplet is present but the structure is
somewhat unusual. The multiplet structure is not the direct sum of a free tensor and
Yang-Mills multiplet, but forms a multiplet that we call henceforth non-decomposable,
as can be seen in particular from the fermionic field equations. This seems to be a
general feature of the models considered here and will be discussed in section 4.3.
The third equation in the right column again demonstrates the dual role of the three-
form potential C˜ r: Acting with d∗on this equation implies the second-order free field
equation ✷A˜ r= 0. The original equation then fixes C˜ rin terms of A˜ rup to an two-
form b˜ rwhose field strength has to be self dual in the Br= 0 gauge, see the second
line of (3.21). The three-form potential C˜ rtherefore shifts or communicates degrees of
freedom between the gauge and tensor multiplet.
µνto set
4 Action
So far, we have found a set of field equations that consistently transform into each
other under (1,0) supersymmetry. The full system is entirely determined by the choice
of the constant tensors gIr, hr
straints (2.5), (2.6). In this section we present the additional conditions, which these
tensors have to satisfy in order for the field equations to be integrated to an action. We
give the full supersymmetric action and discuss the general structure of supersymmet-
ric vacua and the fluctuation equations around such vacua. The non-unitarity of the
action manifests itself in the generic appearance of some unusual ‘non-decomposable’
multiplet couplings. Finally, we illustrate the general analysis by two concrete mod-
els, with compact gauge group SO(5) and a nilpotent eight-dimensional gauge group,
respectively.
I, bIrs, dI
rs, and frstsubject to the set of algebraic con-
4.1Conditions for existence of an action
The existence of an action first of all requires the existence of a constant non-degenerate
metric ηIJ by which tensor multiplet indices can be raised and lowered, in order to
provide a kinetic term for the scalar fields and the other fields of the tensor multiplets.
Further inspection of the field equations (3.5)–(3.7) then shows that their integrability
to an action requires the identifications
hr
I= ηIJgJr,dI
rs=1
2ηIJbJrs, (4.1)
16
Page 18
i.e. in particular a b-symbol that is symmetric in its indices (rs). Moreover, in the
process of computing the action, one finds that the identity (2.15) needs to be imposed
in order to ensure the existence of a proper topological term. From (4.1) it is obvious
that the models we have discussed in section 3.6 indeed do not admit an action.
To summarize, with these identifications, the algebraic consistency conditions (2.5),
(2.6), (2.15) reduce to
bI r(ubI
vs)
= 0 ,
?bJ
r(ubI
v)s− bJ
uvbI
6f[pqufr]us− gs
rs+ bK rsbK
uvηIJ?gs
IbI
J
= 2fr(usbI
v)s,
u[pfqr]u
= 0 ,
2frstgr
I− bJ
gr
rsgt
Jgr
I
= 0 ,
Kgs
[IbJ]sr
gr
= 0 ,
IgIs
= 0 .(4.2)
Finding non-trivial solutions to these constraints is a formidable task. We will give in
sections 4.4, 4.5 below some explicit solutions that are inspired from similar construc-
tions in gauged supergravity theories.
4.2The action
In case the constant tensors satisfy all algebraic conditions (4.2), the equations of
motion (3.5), (3.6) can be lifted to an action. In fact, one may verify a somewhat
stronger conclusion: the identifications (4.1) and thus the set of constraints (4.2) appear
already to be necessary in order to construct a conserved supercurrent underlying the
equations of motion (3.5), (3.6) from a canonical structure for the fermions [37].
In order to write an action, we ignore for the moment the subtleties of writing an
action for a self-dual three-form field strength, but give a standard second-order action,
keeping in mind that the corresponding first-order equation of (3.5) is supposed to be
imposed after having derived the second-order equations of motion, just as in the
democratic formulation of ten-dimensional supergravities [22].3The full action then
reads
L = −1
8DµφIDµφI−1
µνρHµνρ
+1
2¯ χIγµDµχI+
1
16bIrsφI?Fr
µνFµν s− 4Yr
ijYij s+ 8¯λrγµDµλs?
µν¯λsγµνχI+ bIrsYr
−
1
96HI
I
−
1
48bIrsHI
J)φI¯λrχJ+1
µνρ¯λrγµνρλs−1
4bIrsFr
ij¯λisχj I
2(bJsrgs
I− 4bIsrgs
8bIrsgr
Jgs
KφIφJφK−
1
48Ltop
−
1
24bIrsbI
uv¯λrγµλu¯λsγµλv,(4.3)
which shows explicitly the role of the scalar fields φIas inverse coupling constants for
the Yang-Mills multiplet. Like the equations of motion, this action contains no dimen-
sionful parameter such that the system is (super)-conformal at least at the classical
3Alternatively, this self-duality can be implemented by using a non-abelian version [38] of the
Henneaux-Teitelboim action [39] that breaks manifest space-time covariance.
17
Page 19
level. The topological term is given by integrating
dV δLtop= 6?bIrsδAr∧Fs∧HI− ∆BI∧?gr
and has the explicit form
IH(4)−1
2bIrsFr∧Fs?− gr
I∆Cr∧HI?,(4.4)
dV Ltop= −6gr
ICr∧HI+ bIrsBI∧Fr∧Fs− bIrshr
Jhs
KBI∧BJ∧BK
+ BI∧?hs
IbJ
subJvrAu∧Av∧dAr+3
4(bIrsfpqr+ 4bJqsXpIJ)fuvsAp∧Aq∧Au∧Av?
qsbJvrAp∧Aq∧Au∧Av∧dAr.
−
1
10fupsbJ
(4.5)
It can be understood in compact form as the boundary contribution of a manifestly
gauge-invariant seven-dimensional term
?
∂M7
Ltop ∝
?
M7
?bIrsFr∧Fs∧HI− HI∧DHI
?
.(4.6)
As usual, gauge invariance of the topological term may lead to quantization conditions
for the various coupling constants. For the tensor multiplet, it is straightforward to
verify that the action (4.3) induces the field equations (3.5) from above. For the fields
of the vector multiplet, we obtain the first and the last of the uncontracted equations
(3.7), while the duality equation relating Fr
form (3.6). In addition, variation w.r.t. the vector field gives rise to the Yang-Mills
equation
µνand H(4)
µνρσ r only appears in its contracted
bIrsDν?φIFs
µν− 2¯λsγµνχI?
=
?φIDµφJ− 2¯ χIγµχJ?XrIJ− 2φIbIpqXrsq¯λpγµλs
−
1
12bIrsεµνρλστFνρsHλστ I,(4.7)
that can alternatively obtained as a derivative of the uncontracted duality equation
(3.7) upon use of the first-order equation (3.9).
We note that the last constraint equation of (4.2) shows that non-trivial solutions
to these constraints (i.e. solutions in which the tensor fields are charged) exist only
if the metric ηIJ is indefinite, which in turn implies that some of the scalars (and
some of the two-forms) in (4.3) have a negative kinetic term. This somewhat reminds
the situation for the three-dimensional BLG theories [3, 4] with Lorentzian three-
algebra [40, 41, 42, 43, 44], and certainly requires further investigation. We also note
that similar structures as encountered in this section have appeared in generic 6d
supergravity theories [45, 46, 47]
4.3Multiplet structure of excitations
The supersymmetry transformations of the model (4.3) are still given by equations (3.3),
(3.4), such that the Killing spinor equations remain of the form (3.13). In particular,
the existence of a maximally supersymmetric vacuum is still encoded in the condition
(3.14) on the scalar expectation values. In this vacuum, the linearized field equations
18
Page 20
obtained from (4.3) extend the fluctuation equations (3.21) by the linearization of the
second-order equation for the vector fields (4.7), which takes the form
Krs
?✷As
µ− ∂µ∂νAs
ν
?
= NIr
?∂µϕI− ∂νBI
IϕI, the equation turn into the free Klein-
µ.4With this gauge fixing, the full
νµ
?
. (4.8)
With the gauge fixing ∂νBI
Gordon equation for the vector field components Ar
set of linearized field equations obtained from (4.3) is given as
νµ= 0 = ∂µAr
µ+gr
(dBI+ gIrCr)−= 0 ,KrsYs
ij= 0 ,
/ ∂ χiI+ 2NI
rλir= 0 ,NI
rdAr− gIr ∗dCr= 0 ,
✷ϕI= 0 ,Krs/ ∂λis− 2NrIχiI= 0 ,
Krs✷As
µ= 0 , (4.9)
with the matrices Krsand NI
a proper choice of basis such that Krsis diagonal, the lowest order dynamics contains
rK= rank(K) vector multiplets. The fluctuation equations (4.9) decouple into various
multiplets which we denote as follows, and whose multiplicities are given in Table 1:
rfrom (3.16). We note that NI
rgrJ= 0 = grINs
I. With
(V) : ✷Aµ= 0,
/ ∂λ = 0,
(T) : ✷ϕ = 0,
/ ∂χ = 0,(dB)−= 0,
(T′) : ✷ϕ = 0,
/ ∂χ = 0,(dB)−= −gC−, dC = 0,
(TV) : ✷ϕ = 0,κdA =
∗dC ,(dB)−= −gC−,
/ ∂λ = 0,
/ ∂χ = −2gκλ,
(VT) : ✷ϕ = 0,
✷Aµ= 0,(dB)−= 0,
/ ∂χ = 0,
/ ∂λ = 2g χ. (4.10)
We have kept the coupling constants g and κ to keep track of the scales of gIrand φI
respectively. The first two multiplets (V) and (T) are the free vector and self-dual tensor
multiplet, respectively, the third one (T’) is the self-dual tensor multiplet enhanced by
a non-propagating three-form potential.
decomposable’ combination of a free vector multiplet and a self-dual tensor multiplet
for which the vector multiplet acts as a source. It is obvious from the fermionic field
equations that these two multiplets cannot be separated. This is the type of coupling
we have encountered in the broken sector ˜ g of the model described in section 3.6. The
last line (VT) describes the dual version of such a ‘non-decomposable’ coupling, here a
free self-dual tensor multiplet acts as the source for a vector multiplet. This situation is
similar to the observation made in [33] regarding the BSS model [13]. Diagonalizing for
example the χ-equation and using the relations for NI
rN = rank(N) TV-multiplets. Is straightforward to verify that only the combination
0,
The fourth line (TV) describes the ‘non-
rgiven above shows that there are
4Alternatively, this can be achieved by choosing Lorenz gauge for the vector fields and fixing
the tensor gauge freedom by ∂νBI
equation in this gauge turns into the massless Klein-Gordon equation.
νµ≡ ∂µϕI. This is a consistent gauge choice since the scalar field
19
Page 21
multiplet
#
(V) (T)(T’)
rg− rN
(TV)
rN
(VT)
rN
rK− 2rN
nT− rN− rg
Table 1: Multiplicities of the different structures (4.10) appearing in the lowest order
fluctuations (4.9) expressed in terms of the number of tensor multiplets nTand the
ranks rg, rN, rKof the matrices gIr, NI
rand Krs, respectively.
of (TV) and (VT) can be derived from an action, which implies that they appear with
equal multiplicity and thus the λ-equation in (4.9) describes rK−2rNvector multiplet
(VT) fermions. In a similar fashion one finds the multiplicities of the other couplings
in (4.10) as obtained from the equations (4.9) and which are entirely encoded in the
rank of the matrices gIr, NI
following, we will illustrate these general structures in some explicit examples.
rand Krs. We collect the explicit result in table 1. In the
4.4 Example: SO(5) gauge group
The constraints (4.2) constitute a rather non-trivial system of consistency conditions for
the undetermined constant tensors and structure constants. Fortunately, a number of
solutions can be inferred from analogous construction in gauged supergravity theories.
In this section, we discuss a solution to (4.2) that is inspired by gaugings of the maximal
six-dimensional supergravity theory [29].
Let the indices I and r parametrize the vector and spinor representations of the
group SO(5,5), respectively, let ηIJto be corresponding invariant metric, and set
bI
rs
≡ γI
rs,frst≡ − 4γIJK
rs
γIJ ptgp
K, (4.11)
where we have chosen a real representation of gamma matrices. With this choice, the
first equation of (4.2) is the well known magic identity for SO(5,5) gamma-matrices.
The second equation reduces to a non-trivial SO(5,5) gamma-matrix identity if in
addition one imposes the tensor gIrto be gamma-traceless according to
gIrγIrs = 0 , (4.12)
i.e. to parametrize the real 144crepresentation. Some further calculation shows that
the remaining equations of (4.2) which are quadratic in gIrthen reduce to the last two
equations which transform in the 10 + 126c+ 320 of SO(5,5). A particular solution
to these equations can be found by choosing gIrto live within the 15 ⊂ 144cupon
breaking to the maximal subgroup GL(5) ⊂ SO(5,5). This simply follows from the
fact that the symmetric tensor product (15 ⊗ 15)symdoes not contain any represen-
tation that lies in the 10 + 126c+ 320 in which the bilinear constraints transform.
Representing the 15 parameters as a symmetric 5×5 matrix, the resulting gauge group
is CSO(p,q,r) with p + q + r = 5 according to the signature of the matrix, cf. [29]
for details. In particular, these gaugings include the theory with compact gauge group
20
Page 22
SO(5). It is instructive, to give the bosonic field content in representations of this
gauge group:
Ar
µ
−→ 1+5+ 5−3+ 10+1,
(φI,BI
µν) −→ 5+2+ 5−2,
Cµνρr −→ 1−5+ 5+3+ 10−1, (4.13)
where the subscripts refer to GL(1) charges under the embedding GL(1) × SO(5) ⊂
GL(5) ⊂ SO(5,5), under which the tensor gIrhas charge −1. In particular, the tensor
multiplets transform in two copies of the fundamental representation of the gauge
group. The scalar field content shows that the gauge invariant cubic potential of (4.3)
for this theory vanishes identically.
In order to elucidate the structure of the SO(5) theory, we will calculate the fluc-
tuations around a maximally supersymmetric solution according to general analysis of
section 4.3. It follows from (3.14) and the particular form of gIrthat maximal super-
symmetry of the vacuum amounts to restricting φI
For the supersymmetric SO(5) invariant vacuum φI
(3.16) vanish identically. As a result, the linearized field equations (4.9) simply describe
ten copies of the self-dual tensor multiplet whereas as discussed above in the unbroken
phase, the vector multiplets are invisible in this perturbative analysis. In the notation
of section 4.3 we find five copies of (T) and of (T’), respectively.
Let us instead consider a non-vanishing value of φI
gauge symmetry at the vacuum down to SO(4) but preserves all supersymmetries.
Accordingly, the bosonic fields break into
0to values within the 5+2of (4.13).
0= 0, both matrices Krs, NIrfrom
0in the 5+2which breaks the
Ar
µ
−→ 1+5+ 1−3+ 4−3+ 4+1+ 6+1,
(φI,BI
µν) −→ 1+2+ 4+2+ 1−2+ 4−2,
Cµνρr
−→ 1−5+ 1+3+ 4+3+ 4−1+ 6−1, (4.14)
under SO(4) × GL(1). In this case, the only non-vanishing entries in the kinetic vec-
tor matrix Krs are the off-diagonal entries in its 4+3× 4−1 and 4−1× 4+3 blocks,
corresponding to eight non-vanishing eigenvalues, of which four are negative.
cordingly, the vector fields from the 1+5+ 1−3+ 6+1(which include the fields in the
adjoint representation of the unbroken gauge group) do not appear in the lowest order
fluctuations (4.9). On the other hand, the matrix gIras chosen above has its only
non-vanishing entries in the (1+2+4+2)×(1−3+4−3) block. This shows in particular,
that from the three-form fields Cµνρr, only the components in the 1+3+ 4+3appear
in the action (4.3). Evaluating the linearized field equations (4.9) for these fields, one
verifies that these indeed fall into the structures identified in (4.10). The explicit re-
sult for the representation content of the various multiplets is displayed in table 2. In
order to correctly keep track of the GL(1) charges, it is worth to keep in mind that
the gauge coupling constant g and the scalar vacuum expectation value κ appearing in
these equations are of charge −1 and +2, respectively.
Ac-
21
Page 23
(T) (T’) (TV)
4+3: Cm
(VT)
1−2: (ϕ−,χ−,B−)1+3: C+
1+2: (ϕ+,χ+,B+) 4+2: (ϕm,χm,Bm) 4−2(˜ ϕm, ˜ χm,˜Bm)
4+1: (Am,λm)4−3: (˜Am,˜λm)
Table 2: Lowest order fluctuations around the SO(4) invariant vacuum.
4.5 Example: Nilpotent gauge group
Another solution to the constraints (4.2) may be obtained from the gauged supergrav-
ities of [30]. In this case, vector and tensor multiplets are supposed to come in the
spinor and vector representation, respectively, of the group SO(9,1). Since real gamma
matrices exist, and their algebra is the same as in the previous example, with the choice
(4.11), the first two equations of (4.2) again reduce to gamma-tracelessness (4.12) of
the tensor gIr. However, in this case, the remaining constraint equations turn out to
admit a unique solution, which is given by
gIr
≡ g ζrζsζtγI
st, (4.15)
with gauge coupling constant g and an arbitrary constant SO(9,1) spinor ζr. This
choice corresponds to a nilpotent gauge group whose algebra N+
so(9,1) according to the three-grading
8 is embedded into
so(9,1) −→ N−
8⊕ (so(8) ⊕ so(1,1)) ⊕ N+
8,(4.16)
see [30] for further details. Under the little group SO(7) of the spinor defining (4.15),
the multiplets decompose as
Ar
µ
−→ 1−1+ 7−1+ 8+1,
(φI,BI
µν) −→ 1+2+ 1−2+ 80,
Cµνρr
−→ 1+1+ 7+1+ 8−1, (4.17)
where again we keep the charges under the GL(1) under which the gauge coupling
constant carries charge −3. A distinctive feature of this model as compared to the
previous one, is a nonvanishing cubic scalar potential. More precisely, the scalar La-
grangian takes the form
L = −1
8DµφiDµφi−1
8∂µφ+Dµφ−+ g3(φ+)3,(4.18)
where (φ+,φ−,φi) represent the 1+2+ 1−2+ 80 scalars in the 80 according to the
decomposition (4.17). A maximally supersymmetric vacuum is found by choosing a
non-vanishing φi
little group down from SO(7) to G2. In this case, the matrix Krsin (4.9) remains
invertible, such that all fields contribute to the linearized fluctuation equations. Eval-
uating the fluctuation equations, one confirms that all fluctuations again fall into the
structures identified in (4.10). The final result for the representation content of the
various multiplets is displayed in table 3.
0, which breaks one generator of the nilpotent gauge group, and the
22
Page 24
(V)(T)(TV)(VT)
7 + 77 + 111
Table 3: Lowest order fluctuations around the G2invariant vacuum.
5Conclusions
In this paper, we have constructed a general class of six-dimensional (1,0) superconfor-
mal models with non-abelian vector and tensor multiplets. The construction is based
on the non-abelian hierarchy of p-form fields and strongly relies on the introduction of
further three-form gauge potentials. These are related to the vector fields by a first-
order duality equation and do not constitute new degrees of freedom, however they
play a crucial role in communicating the degrees of freedom between the vector and
tensor multiplets. The models are parametrized by a set of dimensionless constant
tensors, which are constrained to satisfy a number of algebraic identities (2.5)–(2.6).
Generically these models provide only equations of motions which we have derived
from closure of the supersymmetry algebra. For particular choice of the parameters,
the equations of motion may be integrated to an action. However, the kinetic metrics
in the vector and the tensor sector appear with indefinite signature. It will require
further work to understand the fate of the resulting ghost states and if one can for
example decoupled them with the help of the large extended tensor gauge symmetry.
For the M2-brane theories, a similar structure has appeared in the theories based on
Lorentzian signature 3-algebras. In these models, the ghost states have been elimi-
nated at the cost of breaking conformal symmetry by further gauging of particular
shift symmetries [44], which are however absent in the models constructed here. The
cubic potential, if non-vanishing, will generically be unbounded from below. However,
since the indefinite metric brings in negative norm states, the relation E = ||Q||2≥ 0
is no longer valid in such cases (E = energy, Q = supercharge) and a non-vanishing
cubic potential is in principle possible and not forbidden by supersymmetry.
We have discussed several explicit examples which satisfy all algebraic consistency
conditions. An arbitrary compact gauge group with tensor fields in the adjoint repre-
sentation can be realized on the level of equations of motion. Lagrangian models have
been given for the compact gauge group SO(5) and for a particular eight-dimensional
nilpotent gauge group embedded in SO(9,1). All these models share some peculiar
features. The fluctuation spectrum of excitations around a supersymmetric vacuum
contain not only free vector and tensor multiplets, but also certain ‘non-decomposable’
combinations of couplings between the two, which we have collected in (4.10). More-
over, null-directions in the kinetic vector matrix may appear for unbroken gauge sym-
metries and cause that the fields of the corresponding vector multiplets drop out of
this perturbative analysis. In general this analysis is valid only in the spontaneously
broken phase, however, the unbroken sector of the Yang-Mills multiplet is still (in-
23
Page 25
finitely) strongly coupled and perturbatively not visible. The corresponding part of
the spectrum decouples and should be integrated out for a proper treatment.
Let us note that although we have used in our explicit examples the algebraic struc-
ture underlying gauged supergravity theories in order to find solutions to the algebraic
consistency constraints (4.2), none of these theories can be obtained as a suitable flat-
space limit of the supergravities of [48, 29, 30]. E.g. globally supersymmetric theories
derived as a flat space limit of these supergravities (if they exist) would not have ghosts
in the scalar sector, which seems to be an inevitable feature of the theories presented
here.
An obvious direction of further investigation is the study of the constraints (4.2)
and (2.5)–(2.6) for models with and without action, respectively. Especially for the
case with an action, it would be highly interesting to understand, if the model with
compact SO(5) gauge group that we have presented in section 4.4 corresponds to
very particular solution of these constraints or if it may be generalized to other gauge
groups. In this context, it may be interesting to pursue the comparison to the five-
dimensional superconformal models classified and studied in [49], which may elucidate
the geometric role of the set of algebraic consistency constraints (4.2) that underlie
our construction. Another interesting research direction is the generalization of the
analysis our maximally supersymmetric vacua of these models to such states which
only preserve a fraction of supersymmetry.
An intriguing question about the models is, how much of the presented structures
can be carried over to (2,0) theories. Although there is no propagating-(2,0) vector
multiplet, the present construction has illustrated the possible relevance of the inclusion
of non-propagating degrees of freedom. As a first step in this direction, we have briefly
sketched in section 3.4 the inclusion of hypermultiplets to the gauged models. Adding
nThypermultiplets with flat target space completes the present field content from (1,0)
to the (2,0) theories. A different extension of our models within the (1,0) framework
could be obtained by studying the possibilities of coupling linear multiplets as sketched
in [50]. A pending question is of course the quantization of the models, in particular the
decoupling of the ostensible ghost states and if the conformal symmetry is preserved at
the quantum level. Last but not least, the study of anomalies for the presented models
with their new gauge symmetries and non-abelian couplings raises an entirely new set
of questions.
It seems clear from our discussion and the many open questions that we are still far
from a profound understanding of the models we have presented in this paper. On the
other hand, given the hitherto lack of non-abelian models in six dimensions the very
existence of these models is rather fascinating. They provide new and very intriguing
structures that deserve more study and may yet reserve further surprises. We look
forward to further analysis.
24
Page 26
Acknowledgement
This work is supported in part by the Agence Nationale de la Recherche (ANR). We
wish to thank F. Delduc for helpful discussions. R.W. thanks Yu-tin Huang for useful
discussions in the early stages of the project and E.S. thanks University of Lyon for
hospitality. The research of E. S. is supported in part by NSF grants PHY-0555575
and PHY-0906222.
Appendix
A Conventions
In this appendix, we summarize our space-time and spinor conventions. We work with
a flat space-time metric of signature (−+++++) and Levi-Civita tensor ε012345= 1.
(Anti-)selfdual three-forms are defined such as to satisfy
H±
µνρ
= ±1
3!ελστµνρHλστ ±, (A.1)
with Hµνρ= H+
µνρ+H−
µνρ. Six-dimensional gamma-matrices satisfy the basic relations
{γµ,γν} = 2gµν,
γ7 ≡ γ012345,γ2
7= 1 ,
γa1···an
=
sn
(6 − n)!εa1···anb1···b6−nγb1···b6−nγ7,sn=
?−1 : n = 0,1,4,5
+1 : n = 2,3,6
,
(A.2)
as well as the particular identities γλγµνργλ= 0, γµνργλγµνρ= 0, γµνργλστγµνρ= 0.
The spinor chiralities are given by Spinor chiralities
γ7ǫ = ǫ ,γ7λr= λr,γ7χX= −χX.(A.3)
In addition, the fermions carry Sp(1) indices for which we use standard northwest-
southeast conventions according to λi= εijλj, etc. Accordingly, their bilinear products
satisfy the symmetry properties
¯λiγ(n)χj
= tn¯ χjγ(n)λi,tn=
?−1 : n = 0,3,4
+1 : n = 1,2,5,6
.(A.4)
The Fierz identities are of the form
ǫj
2¯ ǫi
1
= −1
4
?
ξµγµεij+1
6ξij
µνργµνρ
?1 − γ7
2
,
withξµ≡1
2¯ ǫ2γµǫ1,ξij
µνρ≡1
2¯ ǫ2iγµνρǫj
1.(A.5)
25
Page 27
In addition, we will employ some particular Fierz identities, cubic in a spinor λr
0 = trs,uv
?3¯ ǫγρλu¯λsγµνρλv+ 4¯ ǫγ[µλu¯λsγν]λv− ¯ ǫγµνρλu¯λsγρλv?
?¯ ǫ(iγµλu
with an arbitrary tensor trs,uv= t[rs],(uv)satisfying tr(s,uv)= 0. These identities can be
derived by making use of the following well known γ-matrix identity
,(A.6)
0 = trs,uv
j)¯λsγµλv− 3¯ ǫγµλu¯λs
(iγµλv
j)
?
,(A.7)
ηµνγµ
δℓ,(αiγν
βj,γk)= 0(A.8)
as follows. Multiplication of this identity by
trs,uv(γρσ)ηm
αiǫδℓλβj,uλγk,sληm,v
(A.9)
and using the conventions
γµ
αi,βj= γαβεij,λαiγµ
αβλβ
j=¯λiγµλj
(A.10)
yields the identity (A.6). Similarly, multiplication of (A.8) with
trs,uvλδn,sλαu
j λβk,vǫγℓ
(A.11)
produces the identity (A.7).
References
[1] E. Witten, Some comments on string dynamics, hep-th/9507121.
[2] Y.-t. Huang and A. E. Lipstein, Amplitudes of 3D and 6D maximal
superconformal theories in supertwistor space, JHEP 1010 (2010) 007,
[1004.4735].
[3] J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple
M2-branes, Phys. Rev. D77 (2008) 065008, [0711.0955].
[4] A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B811
(2009) 66–76, [0709.1260].
[5] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, N = 6
superconformal Chern-Simons-matter theories, M2-branes and their gravity
duals, JHEP 10 (2008) 091, [0806.1218].
[6] D. Bashkirov and A. Kapustin, Supersymmetry enhancement by monopole
operators, JHEP 1105 (2011) 015, [1007.4861].
[7] H. Samtleben and R. Wimmer, N = 6 superspace constraints, SUSY
enhancement and monopole operators, JHEP 1010 (2010) 080, [1008.2739].
26
Page 28
[8] A. Gustavsson and S.-J. Rey, Enhanced N = 8 supersymmetry of ABJM theory
on R8and R8/Z2, 0906.3568.
[9] O.-K. Kwon, P. Oh, and J. Sohn, Notes on supersymmetry enhancement of
ABJM theory, JHEP 08 (2009) 093, [0906.4333].
[10] B. de Wit and H. Samtleben, Gauged maximal supergravities and hierarchies of
nonabelian vector-tensor systems, Fortschr. Phys. 53 (2005) 442–449,
[hep-th/0501243].
[11] B. de Wit, H. Nicolai, and H. Samtleben, Gauged supergravities, tensor
hierarchies, and M-theory, JHEP 02 (2008) 044, [arXiv:0801.1294].
[12] E. A. Bergshoeff, J. Hartong, O. Hohm, M. H¨ ubscher, and T. Ortin, Gauge
theories, duality relations and the tensor hierarchy, JHEP 04 (2009) 123,
[arXiv:0901.2054].
[13] E. Bergshoeff, E. Sezgin, and E. Sokatchev, Couplings of self-dual tensor
multiplet in six dimensions, Class. Quant. Grav. 13 (1996) 2875–2886,
[hep-th/9605087].
[14] I. Brunner and A. Karch, Branes and six-dimensional fixed points, Phys.Lett.
B409 (1997) 109–116, [hep-th/9705022].
[15] I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in
six-dimensions, JHEP 9803 (1998) 003, [hep-th/9712143].
[16] A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric
theories, Nucl.Phys. B529 (1998) 180–206, [hep-th/9712145].
[17] I. Brunner and A. Karch, Six-dimensional fixed points from branes,
hep-th/9801156.
[18] S. Ferrara, A. Kehagias, H. Partouche, and A. Zaffaroni, Membranes and
five-branes with lower supersymmetry and their AdS supergravity duals,
Phys.Lett. B431 (1998) 42–48, [hep-th/9803109].
[19] P. S. Howe, E. Sezgin, and P. C. West, Covariant field equations of the
M-theory five-brane, Phys. Lett. B399 (1997) 49–59, [hep-th/9702008].
[20] P. Pasti, D. P. Sorokin, and M. Tonin, Covariant action for a D = 11 five-brane
with the chiral field, Phys. Lett. B398 (1997) 41–46, [hep-th/9701037].
[21] I. A. Bandos et. al., Covariant action for the super-five-brane of M-theory,
Phys. Rev. Lett. 78 (1997) 4332–4334, [hep-th/9701149].
27
Page 29
[22] E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest, and A. Van Proeyen, New
formulations of D = 10 supersymmetry and D8-O8 domain walls, Class. Quant.
Grav. 18 (2001) 3359–3382, [hep-th/0103233].
[23] N. Lambert and C. Papageorgakis, Nonabelian (2,0) tensor multiplets and
3-algebras, JHEP 1008 (2010) 083, [arXiv:1007.2982].
[24] H. Singh, Super-Yang-Mills and M5-branes, 1107.3408.
[25] P.-M. Ho, K.-W. Huang, and Y. Matsuo, A non-abelian self-dual gauge theory
in 5+1 dimensions, JHEP 07 (2011) 021, [1104.4040].
[26] M. R. Douglas, On D = 5 super Yang-Mills theory and (2,0) theory, JHEP
1102 (2011) 011, [1012.2880].
[27] N. Lambert, C. Papageorgakis, and M. Schmidt-Sommerfeld, M5-branes,
D4-branes and quantum 5D super-Yang-Mills, JHEP 1101 (2011) 083,
[1012.2882].
[28] J. Hartong and T. Ortin, Tensor hierarchies of 5- and 6-dimensional field
theories, JHEP 09 (2009) 039, [0906.4043].
[29] E. Bergshoeff, H. Samtleben, and E. Sezgin, The gaugings of maximal D = 6
supergravity, JHEP 03 (2008) 068, [0712.4277].
[30] M. G¨ unaydin, H. Samtleben, and E. Sezgin, On the magical supergravities in six
dimensions, Nucl. Phys. B848 (2011) 62–89, [1012.1818].
[31] X. Bekaert, M. Henneaux, and A. Sevrin, Deformations of chiral two forms in
six-dimensions, Phys.Lett. B468 (1999) 228–232, [hep-th/9909094].
[32] X. Bekaert, M. Henneaux, and A. Sevrin, Chiral forms and their deformations,
Commun.Math.Phys. 224 (2001) 683–703, [hep-th/0004049].
[33] P. S. Howe and E. Sezgin, Anomaly-free tensor-Yang-Mills system and its dual
formulation, Phys. Lett. B440 (1998) 50–58, [hep-th/9806050].
[34] G. Sierra and P. K. Townsend, The gauge invariant N = 2 supersymmetric
sigma model with general scalar potential, Nucl. Phys. B233 (1984) 289.
[35] B. de Wit, M. Rocek, and S. Vandoren, Gauging isometries on hyperk¨ ahler
cones and quaternion-K¨ ahler manifolds, Phys. Lett. B511 (2001) 302–310,
[hep-th/0104215].
[36] B. de Wit, B. Kleijn, and S. Vandoren, Superconformal hypermultiplets, Nucl.
Phys. B568 (2000) 475–502, [hep-th/9909228].
28
Page 30
[37] D. Burke and R. Wimmer, Quantum energies and tensorial central charges of
confined monopoles, 1107.3568.
[38] H. Samtleben, Actions for non-abelian twisted self-duality, Nucl. Phys. B851
(2011) 298–313, [1105.3216].
[39] M. Henneaux and C. Teitelboim, Dynamics of chiral (self-dual) p-forms, Phys.
Lett. B206 (1988) 650.
[40] U. Gran, B. E. W. Nilsson, and C. Petersson, On relating multiple M2 and
D2-branes, JHEP 10 (2008) 067, [0804.1784].
[41] J. Gomis, G. Milanesi, and J. G. Russo, Bagger-Lambert theory for general Lie
algebras, JHEP 06 (2008) 075, [0805.1012].
[42] S. Benvenuti, D. Rodriguez-Gomez, E. Tonni, and H. Verlinde, N = 8
superconformal gauge theories and M2 branes, JHEP 01 (2009) 078,
[0805.1087].
[43] P.-M. Ho, Y. Imamura, and Y. Matsuo, M2 to D2 revisited, JHEP 07 (2008)
003, [0805.1202].
[44] M. A. Bandres, A. E. Lipstein, and J. H. Schwarz, Ghost-free superconformal
action for multiple M2-branes, JHEP 07 (2008) 117, [0806.0054].
[45] H. Nishino and E. Sezgin, New couplings of six-dimensional supergravity, Nucl.
Phys. B505 (1997) 497–516, [hep-th/9703075].
[46] H. Nishino and E. Sezgin, The complete N = 2,d = 6 supergravity with matter
and Yang-Mills couplings, Nucl. Phys. B278 (1986) 353–379.
[47] F. Riccioni, All couplings of minimal six-dimensional supergravity, Nucl. Phys.
B605 (2001) 245–265, [hep-th/0101074].
[48] M. Duff, J. T. Liu, H. Lu, and C. Pope, Gauge dyonic strings and their global
limit, Nucl.Phys. B529 (1998) 137–156, [hep-th/9711089].
[49] E. Bergshoeff, S. Cucu, T. D. Wit, J. Gheerardyn, R. Halbersma, S. Vandoren,
and A. V. Proeyen, Superconformal N = 2, D = 5 matter with and without
actions, JHEP 10 (2002) 045, [hep-th/0205230].
[50] E. Sezgin and Y. Tanii, Superconformal sigma models in higher than two
dimensions, Nucl. Phys. B443 (1995) 70–84, [hep-th/9412163].
29
Download full-text