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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.
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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
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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
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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
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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
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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
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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
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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)
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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
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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.
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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)
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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
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