# (1,0) superconformal models in six dimensions

Article (PDF Available)inJournal of High Energy Physics 2011(12) · August 2011with25 Reads
DOI: 10.1007/JHEP12(2011)062 · Source: arXiv
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 communicate degrees of freedom between the tensor multiplets and the Yang-Mills multiplet, but do not introduce additional degrees of freedom. Generically 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 hypermultiplets which complete the field content to that of superconformal (2,0) theories.
arXiv:1108.4060v1 [hep-th] 19 Aug 2011
MFA-11-36
(1,0) superconformal models
in six dimensions
Henning Samtleben
a
, Ergin Sezgin
b
, Robert Wimmer
a
a
Universit´e de Lyon, Laboratoire de Physique, UMR 5672, C NRS et ENS de Lyon,
46 alee d’Italie, F-69364 Lyon CEDEX 07, France
b
George 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 additio nal 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 cont ent to that of superconformal (2,0)
theories.
Conte nts
1 Intr oduction
2
2 Non-ab elian tensor ﬁelds in six dimensions 4
2.1 Minimal tensor hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Extended tensor hierar chy . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Superconformal ﬁeld equations 8
3.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Minimal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Extended model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Adding hypermultiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.5 Supersymmetric vacua and excitation spectrum . . . . . . . . . . . . . 13
3.6 A model with adjoint t ensor multiplets . . . . . . . . . . . . . . . . . . 15
4 Action 16
4.1 Conditions for existence of an action . . . . . . . . . . . . . . . . . . . 16
4.2 The action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Multiplet structure of excitations . . . . . . . . . . . . . . . . . . . . . 18
4.4 Example: SO(5) gauge group . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Example: Nilpotent gauge group . . . . . . . . . . . . . . . . . . . . . 22
5 Conclusions 23
A Conventions 25
1
1 Introduction
One of the discoveries of the seminal analysis in [
1] is the existence of intera cting quan-
tum ﬁeld theories in ﬁve and six dimensions. O f particular interest are six-dimensional
(2, 0) superconformal theories which are supposed to describe the low energy limit o f
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 pla usible.
This conclusion was also drawn from symmetry pro perties which imply that tr ee level
amplitudes have to vanish [
2]. In addition, these (2, 0) theories consist of chiral tensor
multiplets and so far it ha s often been considered as impossible to deﬁne non-abelian
gauge couplings for such multiplet s.
Regarding the ﬁrst aspect the situation is similar to that of multiple M2 branes, as
it was befor e 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 cont ext 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 t hus 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 ha s 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 ro ute. 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 hierar chy [
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 potent ials. We therefore have a n extended tensor gauge freedom with p = 0, 1, 2
p-form gauge parameters.
We then formulate essentially unique supersymmetry transformations for the var i-
ous ﬁelds, where we ﬁnd a suitable extension of the structures intr oduced 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 int r oduction of a 3-form potential that mediates cou-
plings between the tensor and vector multiplets. While the brane interpretation of our
models requires further investigatio n, it is worth mentioning that the ﬁeld cont ent 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
into translations and extended tensor g auge 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 intr oduce additional degrees of freedom. They
communicate degrees of freedom between the tensor multiplets a nd the Yang-Mills mul-
tiplet. We also describe the extension of the tensor hierarchy to higher p-for m gauge
potent ials 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 o f 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 t he 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 g eneral 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 L agrangian we
understand that the ﬁrst or der duality equations are consistently imposed in addition
to the second order L agrangian e.o.m., just as in the democratic formulation of ten-
dimensional supergr avity [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. R ecent attempts tried to basically rewrite ﬁve-dimensional
Yang-Mills theory in six dimensions [
23, 24] or introduced non-abelian gauging s 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 a s 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 tha t need to satisfy a number of algebraic
3
consistency constraints. In particular, we ﬁnd that non-abelian charged tensor ﬁelds
require the introduction of St¨uckelberg -type couplings among the p-f orms of diﬀerent
degree. In section 3, we extend the non-abelian vector/tensor system to a sup ersym-
metric system. Closure of the supersymmetry algebra puts the system on-shell and we
derive the modiﬁed ﬁeld equations for the vector and tensor mult iplets. In particular,
we obtain the ﬁrst-order duality equation relating vector ﬁelds and three-form gauge
potent ials. In section 3.4 we sketch the extension of the model upon inclusion of hyper-
multiplets a nd gauging of their triholomorphic isometries. In section
3.5 we derive the
general conditions for maximally sup ersymmetric 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 gaug e group and
tensor ﬁelds transforming in the adjoint representation.
Section 4 presents the additional conditions on t he couplings in order to allow for
a Lagrangian formula t ion. We give the full action in section
4.2. In section 4.3 we
calculate the ﬂuctuation equa t ions 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 constraint s with compact gauge group SO(5) and nilpo tent gauge group N
8
,
respectively. Finally, we summarize our ﬁndings in section 5.
2 Non-abelian tensor ﬁelds i n six dimension s
In this section, we present the general (no n-abelian) couplings of vectors and anti-
symmetric p-for m ﬁelds in six dimensions. While the standard ﬁeld content of the
ungauged theories falls into vector and tensor multiplets, it is a g eneral feature of
these theories that the introduction of gauge charges g enerically 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-f orm 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 ag ain step by step through the derivation of
the general couplings, we directly present the ﬁnal result as parametrized by a set of
constant t ensors (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.
2.1 Minimal tensor hierarchy
The basic p-form ﬁeld content of the theories to be discussed is a set of vector ﬁelds A
r
µ
,
and two-f orm gauge po t entials B
I
µν
, that we label by indices r and I, respectively. In
4
addition, we will have to introduce three-form gauge potentials that we denote by C
µνρ r
.
The fact that three-fo r m 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
potent ials, 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-fo r m gauge potentials are
given as
F
r
µν
2
[µ
A
r
ν]
f
st
r
A
s
µ
A
t
ν
+ h
r
I
B
I
µν
,
H
I
µνρ
3D
[µ
B
I
νρ]
+ 6 d
I
rs
A
r
[µ
ν
A
s
ρ]
2f
pq
s
d
I
rs
A
r
[µ
A
p
ν
A
q
ρ]
+ g
Ir
C
µνρ r
, (2.1)
in terms of the antisymmetric structure constants f
st
r
= f
[st]
r
, a symmetric d-symbol
d
I
rs
= d
I
(rs )
, and the tensors g
Ir
, h
r
I
inducing St¨uckelberg -type couplings among forms
of diﬀerent degree.
1
Covariant derivatives are deﬁned as D
µ
µ
A
r
µ
X
r
with an
action o f the gauge g enerato r s X
r
on the diﬀerent ﬁelds given by X
r
· Λ
s
(X
r
)
t
s
Λ
t
,
X
r
· Λ
I
(X
r
)
J
I
Λ
J
, etc. The ﬁeld strengths are deﬁned such that they transform
covariant ly under the set of non-abelian gauge transformations
δA
r
µ
= D
µ
Λ
r
h
r
I
Λ
I
µ
,
B
I
µν
= 2D
[µ
Λ
I
ν]
2 d
I
rs
Λ
r
F
s
µν
g
Ir
Λ
µν r
,
C
µνρ r
= 3D
[µ
Λ
νρ] r
+ 3 b
Ir s
F
s
[µν
Λ
I
ρ]
+ b
Ir s
H
I
µνρ
Λ
s
+ . . . , (2.2)
where we have introduced the compact notation
B
I
µν
δB
I
µν
2d
I
rs
A
r
[µ
δA
s
ν]
,
C
µνρ r
δC
µνρ r
3 b
Ir s
B
I
[µν
δA
s
ρ]
2 b
Ir s
d
I
pq
A
s
[µ
A
p
ν
δA
q
ρ]
. (2.3)
The ellipsis in the last line of (
2.2) represent possible terms that vanish under projection
with g
Ir
. This system is completely deﬁned by the choice of the invariant tensors g
Ir
,
h
r
I
, b
Ir s
, d
I
rs
, and f
rs
t
. It is obvio us from (2.2) that the shift symmetry action on the
p-form gauge ﬁelds can be used to gauge away some of the p-for ms, giving mass to
others by the St¨uckelberg mechanism. However, for the general analysis of couplings,
we ﬁnd it the most convenient t o work with the uniform system (
2.2) and to postp one
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
(X
r
)
s
t
= f
rs
t
+ d
I
rs
h
t
I
,
(X
r
)
I
J
= 2 h
s
I
d
J
rs
g
Js
b
Isr
, (2.4)
1
We use canonical dimensions such that a p-form has mass dimension p and as a result all constant
tensors f
st
r
, d
I
rs
, g
Ir
, h
r
I
, are dimensionless.
5
in terms of the constant tensors appearing in the system. The second relation expo ses
an importa nt feature of the tensor hierarchy: tensor ﬁelds can be charged under the
gauge group only if either h
r
I
or g
Ir
are non-vanishing, i.e. they require some non-
vanishing St¨uckelberg-type couplings in the ﬁeld strengths (
2.1). This corresponds to
the known results [
31, 32] t hat in absence of such couplings and the inclusion of addi-
tional three-for m 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 h
r
I
, the gauge group generators in the ‘adjoint representa-
tion’ (X
r
)
s
t
are no t just given by the structure constants but acquire a modiﬁcation,
symmetric in its indices (rs).
Furthermore, consistency of the system, i.e. covaria nt 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 o f con-
straints, linear in f, g, h, is given by
2
d
J
r(u
d
I
v)s
d
I
rs
d
J
uv
h
s
J
= 2f
r(u
s
d
I
v)s
b
Jsr
d
J
uv
g
Is
,
d
J
rs
b
Iut
+ d
J
rt
b
Isu
+ 2 d
K
ru
b
Kst
δ
J
I
h
u
J
= f
rs
u
b
Iut
+ f
rt
u
b
Isu
+ g
Ju
b
Iur
b
Jst
, (2.5)
and ensures the invar iance of the d- and the b-symbol under gauge transformations.
The remaining constraints are bilinear in f , g, h and take the form
f
[pq
u
f
r]u
s
1
3
h
s
I
d
I
u[p
f
qr]
u
= 0 ,
h
r
I
g
Is
= 0 ,
f
rs
t
h
r
I
d
J
rs
h
t
J
h
r
I
= 0 ,
g
Js
h
r
K
b
Isr
2h
s
I
h
r
K
d
J
rs
= 0 ,
f
rt
s
g
It
+ d
J
rt
h
s
J
g
It
g
It
g
Js
b
Jtr
= 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
[X
r
, X
s
] = (X
r
)
s
t
X
t
, (2.7)
for the generator s (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 h
I
r
. Even though the set of constraints (
2.5), (2.6) looks highly
restrictive, it admits rather non-t rivial solutions and we will discuss explicit examples
of solutions in sections 3.6, 4.4, and 4 .5, below. The system a dmits diﬀerent abelian
limits with f
rs
t
= 0 = g
Ir
and either h
r
I
or d
I
rs
vanishing, in which the constraints
(2.5), (2 .6) are trivially satisﬁed. A slightly more general solution is given by vanishing
h
r
I
= 0 = g
Ir
with f
rs
t
representing the structure constants of a Lie algebra. With the
particular choice d
I
rs
= d
I
δ
rs
, the vector-tensor system then reduces to the coupling of
the Yang-Mills multiplet to an uncharged self-dual tensor multiplet as describ ed in [
13].
6
The covariant ﬁeld strengths (2.1) satisfy the modiﬁed Bianchi identities
D
[µ
F
r
νρ]
=
1
3
h
r
I
H
I
µνρ
,
D
[µ
H
I
νρσ]
=
3
2
d
I
rs
F
r
[µν
F
s
ρσ]
+
1
4
g
Ir
H
(4)
µνρσ r
, (2.8)
where the non-abelian ﬁeld strength H
(4)
µνρσ r
of the three-fo r m potential is deﬁned by
the second equation. In turn, its Bianchi identity is obtained from (
2.8) as
D
[µ
H
(4)
νρστ ] r
= 2 b
Ir s
F
s
[µν
H
I
ρστ ]
+ . . . , (2.9)
where the ellipsis represents possible terms that vanish under projection with g
Ir
. We
ﬁnally note that the general variation of the ﬁeld-strengths is given by
δF
r
µν
= 2D
[µ
δA
r
ν]
+ h
r
I
B
I
µν
,
δH
I
µνρ
= 3D
[µ
B
I
νρ]
+ 6 d
I
rs
F
r
[µν
δA
s
ρ]
+ g
Ir
C
µνρ r
,
δH
(4)
µνρσ r
= 4D
[µ
C
νρσ]r
6 b
Ir s
F
s
[µν
B
I
ρσ]
+ 4 b
Ir s
H
I
[µνρ
δA
s
σ]
+ . . . , (2.10)
again with the ellipsis representing possible terms that vanish under projection with g
Ir
.
2.2 Extended tensor hierarchy
The ﬁeld content introduced in the last section were the p-forms A
r
µ
, B
I
µν
, C
µνρ r
, for
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 g
Ir
as g
Ir
C
µνρ r
. 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 ot her hand, the
tensor hierarchy may be extended to the full set of three-form potentials, which then
necessitates the introduction of four-form gauge potentia ls, etc.
For later use, we present in this section the results of the general t ensor hierarchy for
the four-form gauge potentials which we denote by C
(4)
µνρλ α
with covariant ﬁeld strength
H
(5)
α
. The full version o f the Bianchi identity (
D
[µ
H
(4)
νρστ ] r
= 2 b
Ir s
F
s
[µν
H
I
ρστ ]
+
1
5
k
r
α
H
(5)
µνρστ α
, (2.11)
where now the ﬁeld strength H
(5)
α
itself satisﬁes the Bianchi identity
D
[µ
H
(5)
νρλστ ] α
=
10
3
c
α IJ
H
I
[µνρ
H
J
λστ ]
5
2
c
t
α s
F
s
[µν
H
(4)
ρλστ ] t
+ · · · , (2.12)
up to terms vanishing under projection with the tensor k
r
α
. The new constant tensors
k
r
α
, c
α IJ
, and c
t
α s
are constrained by the relations
k
r
α
c
α IJ
= h
s
[I
b
J]rs
, k
r
α
c
t
α s
= f
rs
t
b
Ir s
g
It
+ d
I
rs
h
t
I
, g
Kr
k
r
α
= 0 ,(2.13)
7
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
g
Kr
h
s
[I
b
J]rs
= 0 ,
g
Kr
f
rs
t
g
It
b
Ir s
+ h
t
I
d
I
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 additiona l relation among b-
and d-symbol to be satisﬁed
b
Jr(s
d
J
uv)
= 0 , (2.15)
as also not ed in [
28]. The new tensor gauge transformations take the form
C
µνρ r
= 3D
[µ
Λ
νρ] r
+ 3 b
Ir s
F
s
[µν
Λ
I
ρ]
+ b
Ir s
H
I
µνρ
Λ
s
k
r
α
Λ
µνρ α
,
C
(4)
µνρσ α
= 4 D
[µ
Λ
νρσ] α
8 c
α IJ
H
[I
[µνρ
Λ
J]
σ]
+ 6 c
t
α s
F
s
[µν
Λ
ρσ] t
+ c
t
α s
H
(4)
µνρσ t
Λ
s
+ . . . , (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 k
r
α
. Accordingly, the general variation of the non-abelian ﬁeld strengths from
(
2.11), (2.1 2) is given by
δH
(4)
µνρσ r
= 4D
[µ
C
νρσ]r
6 b
Ir s
F
s
[µν
B
I
ρσ]
+ 4 b
Ir s
H
I
[µνρ
δA
s
σ]
+ k
r
α
C
(4)
µνρσ α
,
δH
(5)
µνρστ α
= 5 D
[µ
C
(4)
νρστ ] α
10 c
t
α s
F
s
µν
C
ρστ ] t
20c
α IJ
H
[I
[µνρ
B
J]
στ ]
5 c
t
α s
δA
s
[µ
H
(4)
νρστ ] t
+ . . . . (2.17)
Continuing along the same line, the tensor hierar chy can be continued by introducing
ﬁve-form and six-form potentials together with their ﬁeld strengths and non-abelian
gauge transfor mations. 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 eld 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 informat ion 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
for the (1, 0) vector and tensor multiplets. With the given (bosonic) ﬁeld content of
the tensor hierarchy (
2.1), a supersymmetric tensor hierar chy will contain Yang-Mills
multiplets (A
r
µ
, λ
i r
, Y
ij r
), and tensor mult iplets (φ
I
, χ
i I
, B
I
µν
), labeled by indices r and
I, respectively. The index i = 1, 2 indicates the Sp(1) R-symmetry, the ﬁeld Y
ij
denotes the auxiliary ﬁeld of the oﬀ-shell vector multiplets. In addition one ha s to
accommodate within this structure the three-form potential C
µνρ r
whose presence was
crucial in the last section in or der to describe non- abelian charged tensor ﬁelds.
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 a n 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
[δ
ǫ
1
, δ
ǫ
2
] = ξ
µ
µ
+ δ
Λ
+ δ
Λ
µ
+ δ
Λ
µν
, (3.1)
with ﬁeld dependent transformation parameters for the respective transformations.
These parameters are given by
ξ
µ
1
2
¯ǫ
2
γ
µ
ǫ
1
,
Λ
r
= ξ
µ
A
r
µ
,
Λ
I
µ
= ξ
ν
B
I
νµ
+ d
I
rs
Λ
r
A
s
µ
+ ξ
µ
φ
I
,
Λ
µν r
= ξ
ρ
C
ρµν r
b
Ir s
Λ
s
B
I
µν
2
3
b
Ir p
d
I
qs
Λ
s
A
p
[µ
A
q
ν]
, (3.2)
as will be shown shortly. With d
I
rs
= α
d
I
δ
rs
, b
Ir s
= 0, this reproduces the correspond-
ing algebra of [
13].
2
The supersymmetry transformations for the Yang-Mills multiplet
are given by
δA
r
µ
= ¯ǫγ
µ
λ
r
,
δλ
i r
=
1
8
γ
µν
F
r
µν
ǫ
i
1
2
Y
ij r
ǫ
j
+
1
4
h
r
I
φ
I
ǫ
i
,
δY
ij r
= ¯ǫ
(i
γ
µ
D
µ
λ
j)r
+ 2h
r
I
¯ǫ
(i
χ
j)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 h
r
I
and brings in the ﬁelds (φ
I
, χ
i I
, B
I
µν
)
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 δλ
i r
is required to produce the proper δ
Λ
µ
action in the commutator of supersymmetries on the vector ﬁeld A
r
µ
. Likewise, the
last term in δY
ij r
ensures the proper closure of t he supersymmetry algebra on λ
i r
. It
2
Note that in canonical dimensions, the tensor d
I
rs
is dimensionless.
9
then comes as a non-trivial consistency check, that the variation of this last term is
precisely what is needed for closure of the alg ebra on Y
ij r
. Even tho ugh, ﬁelds from
the tensor multiplets appear in these transformatio n 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
δφ
I
= ¯ǫχ
I
,
δχ
i I
=
1
48
γ
µνρ
H
I
µνρ
ǫ
i
+
1
4
γ
µ
D
µ
φ
I
ǫ
i
1
2
d
I
rs
γ
µ
λ
i r
¯ǫγ
µ
λ
s
,
B
I
µν
= ¯ǫγ
µν
χ
I
,
C
µνρ r
= b
Ir s
¯ǫγ
µνρ
λ
s
φ
I
, (3.4)
where we have used the same notation (
2.3) for general variat ion introduced in the
tensor hierarchy. We also note that γ
µνρ
ǫ
i
acts as a self-duality projector such that
only H
I +
µνρ
, see (A.1), is actually alive in δχ
i I
. W.r.t. the couplings discussed in [13], the
r.h.s. of these transformations has been generalized by the int r oduction of the general
d-symbo l, 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 C
µνρ r
which is cont ained in the
deﬁnition of H
I
µνρ
and contributing to its supersymmetry transformation according to
(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 A
r
µ
). The form of its supersymmetry
transformation (
3.4), mixing Yang-Mills and tensor multiplet ﬁelds, displays its dual
role as a messenger between these two multiplet s. No t e that we have given in (
3.4)
the supersymmetry tra nsfor ma t ion for the uncont racted three-for m C
µνρ r
, although all
the explicit couplings only contain the contracted expression g
K r
C
µνρ r
. 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.
3.2 Minimal 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 g
K r
C
µνρ r
of three-form gauge potentials. The
10
resulting t ensor multiplet ﬁeld equations are given by
H
I
µνρ
= d
I
rs
¯
λ
r
γ
µνρ
λ
s
,
γ
σ
D
σ
χ
iI
=
1
2
d
I
rs
F
r
στ
γ
στ
λ
is
+ 2d
I
rs
Y
ij r
λ
s
j
+
d
I
rs
h
s
J
2b
Jsr
g
Is
φ
J
λ
ir
,
D
µ
D
µ
φ
I
=
1
2
d
I
rs
F
r
µν
F
µν s
4 Y
r
ij
Y
ij s
+ 8
¯
λ
r
γ
µ
D
µ
λ
s
2
b
Jsr
g
Is
8d
I
rs
h
s
J
¯
λ
r
χ
J
3 d
I
rs
h
r
J
h
s
K
φ
J
φ
K
. (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 B
I
µν
. The closure on δχ
i I
gives the fermionic equations of motion while the scalar
ﬁeld equation is obtained by the supersymmetry transformat ion of the χ
iI
- equation.
The fa ct that t he tensor ﬁelds are charg ed under the gauge group has rather non-
trivial consequences, namely supersymmetry variation of t he ﬁeld equations (
3.5) in
turn implies the following ﬁrst-order equations of motion for the Yang-Mills multiplets
g
Kr
b
Ir s
Y
s
ij
φ
I
2
¯
λ
s
(i
χ
I
j)
= 0 ,
g
Kr
b
Ir s
F
s
µν
φ
I
2
¯
λ
s
γ
µν
χ
I
=
1
4!
ε
µνλρστ
g
Kr
H
(4) λρστ
r
,
g
Kr
b
Ir s
φ
I
γ
µ
D
µ
λ
s
i
+
1
2
γ
µ
λ
s
i
D
µ
φ
I
= g
Kr
b
Ir s
1
4
F
s
µν
γ
µν
χ
I
i
+
1
24
H
I
µνρ
γ
µνρ
λ
s
i
Y
s
ij
χ
j I
+
3
2
h
s
J
φ
I
χ
J
i
+
1
3
d
I
uv
γ
µ
λ
u
i
¯
λ
s
γ
µ
λ
v
.
(3.6)
The ﬁrst equation is the algebraic equation for the auxiliary ﬁeld Y
ij r
, while the second
equation provides the anticipated duality of vector ﬁelds and three-form potentials by
relating their respective ﬁeld strengths. In par t icular, derivatio n of this equation and
use of t he Bianchi identity (
2.9) gives rise to a standa r d second-order equation of Yang-
Mills type for the vector ﬁelds A
r
µ
. Equivalently, the ﬁrst two equations of (
3.6) can be
inferred from closure of the supersymmetry algebra o n the three-form gauge potentials
g
Kr
C
µνρ r
. The appeara nce 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 g
Kr
= 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 combinat ion 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 o f the construction.
To summarize, the system o f equations of motion (3.5), (3.6) consistently trans-
forms into itself under supersymmetry. It describes a novel system of supersymmetric
11
non-abelian couplings for multiple (1, 0) tensor multiplets in six dimensions. The equa-
tions of motio n contain no dimensionful parameter and hence the system is a t least
classically (super)-conformal. A crucial ingredient to the model are the three-form
gauge potentials C
µνρ r
which 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 represent ations a nd the associated invariant t ensors 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.
3.3 Extended 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 g
K r
C
µνρ r
of three-
form gauge potentials. Just as for t he bosonic tensor hierarchy in section
2.2, one may
seek to extend the above supersymmetric system to the full set of three-form gauge
potent ials. With the supersymmetry transformation of general C
µνρ r
given by (3 .4),
closure of the supersymmetry algebra leads to the following uncontracted equations
b
Ir s
Y
s
ij
φ
I
2
¯
λ
s
(i
χ
I
j)
= 0 ,
b
Ir s
F
s
µν
φ
I
2
¯
λ
s
γ
µν
χ
I
=
1
4!
ε
µνλρστ
H
(4) λρστ
r
,
b
Ir s
φ
I
γ
µ
D
µ
λ
s
i
+
1
2
γ
µ
λ
s
i
D
µ
φ
I
= b
Ir s
1
4
F
s
µν
γ
µν
χ
I
i
+
1
24
H
I
µνρ
γ
µνρ
λ
s
i
Y
s
ij
χ
j I
+
+ h
s
J
2φ
I
χ
J
i
1
2
φ
J
χ
I
i
+
1
3
d
I
uv
γ
µ
λ
u
i
¯
λ
s
γ
µ
λ
v
,
(3.7)
In order to have this system close under supersymmetry it is necessary to intr oduce also
a four-f orm gauge po tential. Consequently the tensor hierarchy has to be cont inued one
step further as described in section 2.2. The resulting supersymmetry transformation
of the four-form potential is
C
(4)
µνρσ α
= 2c
α IJ
φ
[I
¯ǫγ
µνρσ
χ
J]
, (3.8)
Furthermore, supersymmetry of the ﬁeld equations (
3.7) induces t he ﬁrst-order ﬁeld
equations
1
5!
ε
µνρλστ
k
r
α
H
(5) µνρλσ
α
= 2k
r
α
c
α IJ
φ
I
D
µ
φ
J
2¯χ
I
γ
µ
χ
J
c
t
α u
b
Jtv
¯
λ
u
γ
µ
λ
v
. (3.9)
This shows that the dynamics of C
(4)
µνρσ α
is given by a ﬁrst-order duality equations,
which relates these four-fo rm potentials to the Noether current of some underlying
global symmetry. In particular, this ﬁrst-order equation ensures that t he four-form
gauge potentials do not constitute new dynamical degrees of freedom.
12
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 M
h
, more precisely superconformal
symmetry requires M
h
to 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 ϑ
r
α
that encodes
the coupling of vector ﬁelds A
r
µ
to hyper-K¨ahler isomet r ies K
α
and is subject to the
algebraic conditions
f
pr
s
ϑ
s
α
= f
βγ
α
ϑ
p
β
ϑ
r
γ
, h
r
I
ϑ
r
α
= 0 , (3.10)
with the structure constants f
αβ
γ
of the algebra of hyper-ahler isometries. On the
other hand, in the presence of hypermultiplets, the vector multiplet equations of motion
(
3.7) allow f or a consistent modiﬁcation, in particular in the Y -ﬁeld equation as
b
Ir s
Y
s
ij
φ
I
2
¯
λ
s
(i
χ
I
j)
= k
r
α
P
ij
α
, (3.11)
with the constant tensor k
r
α
from (
2.13), and the moment maps P
ij
α
associated with the
triholomorphic hyper-K¨ahler isometries. It is only by means of this algebraic equation
for Y
s
ij
that the hyperscalars enter the tensor multiplet ﬁeld equations. Further requiring
the existence of an action eventua lly leads to the identiﬁcation
k
r
α
= ϑ
r
α
, (3.12)
i.e. relates the gauging of hyper-K¨ahler isometries to a modiﬁcatio n of the vector and
tensor multiplet ﬁeld equations.
3.5 Supersymmetric 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. t he 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 φ
I
serves as an (inverse) coupling
constant. This no t ion will become more evident in the subsequent sections where we
discuss models which provide a Lagrangian. Consequent ly, the perturbative analysis is
limited t o the spontaneously broken phase where φ
I
has a (large) expectation value.
The Killing spinor equations of t he theory (
4.3) are obtained from (3.3), (3.4)
0
!
δλ
i r
=
1
8
γ
µν
F
r
µν
ǫ
i
1
2
Y
ij r
ǫ
j
+
1
4
h
r
I
φ
I
ǫ
i
,
0
!
δχ
i I
=
1
48
γ
µνρ
H
I +
µνρ
ǫ
i
+
1
4
γ
µ
D
µ
φ
I
ǫ
i
, (3.13)
13
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 φ
I
0
satisfying
φ
I
0
h
r
I
= 0 , (3.14)
and setting all other ﬁelds to zero. Expanding the scalar ﬂuctuations as φ
I
φ
I
0
+
ϕ
I
and imposing the condition (
3.14) one obtains at the linearized level for the ﬁeld
equations (
3.5), (3.6) the system:
( dB
I
+ g
Ir
C
r
)
= 0 , N
I
r
Y
r
ij
= 0 ,
/
χ
iI
+ 2 N
I
r
λ
ir
= 0 , N
I
r
dA
r
g
Ir
dC
r
= 0 ,
ϕ
I
N
I
r
· A
r
= 0 , N
I
r
/
λ
ir
= 0 , (3.15)
where we have deﬁned the matrices
K
rs
φ
I
0
b
Ir s
, N
I
r
g
Is
K
sr
. (3.16)
and used that N
I
r
h
r
J
= 0, by the ﬁrst ident ity in (
2.14) and the susy vacuum condi-
tion (
3.14).
Unbroken gauge symmetry. For a generic supersymmetric vacuum which satisﬁes
(
3.14) the vector gauge transformations Λ
r
X
r
are broken down to the subgroup of
transformations Λ
r
X
r
which satisfy
X
r J
I
φ
J
0
= N
I
r
!
= 0 , (3.17)
where the index r
labels the subset of unbr oken generators (2.4). The rest of the
extended tensor gauge symmetry (
2.2) remains intact. Consequently, in the case tha t
the gauge group is not completely broken, the matrix N
I
r
, and for invertible g
Ir
also
the matrix K
rs
, 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 o f this perturbative analysis. This is nothing else than the above mentioned
observation that the perturbative a nalysis is valid only in the spontaneously broken
phase and that the unbro ken 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 t he
situation of N = 2 SQCD, for which mixed Coulomb-Hig gs 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 K
rs
which may have less null directions tha n the mat r ix N
I
r
.
14
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 g
rs
with the Cartan-Killing metric.
Moreover we set
h
s
r
0 , d
p
rs
d
rst
g
pt
, b
p rs
f
prs
, (3.18)
with the totally symmetric d-symbol d
rst
and the totally antisymmetric structure con-
stants f
rst
. 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 describ ed by the set of equations
of motion (3.5), (3.6) only.
With g
rs
being the (invertible) Carta n-Killing metric, the matrices N and K intro-
duced in (3.16) are essentially the same,
N
r
s
= g
rt
K
ts
=: K
r
s
= φ
t
0
f
ts
r
, (3.19)
and the matrix K
r
s
deﬁnes the adjoint action of the vev φ
0
. By a gauge rotation the
φ
0
can always be chosen to lie in the Cart an 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 K
r
s
vanishes. On the orthogonal complement
˜
g and for
generic choice of φ
0
, the matrix K
r
s
is invertible, and using the Cartan-Weyl basis we
introduce t he notation K
˜r
red
˜s
= k
˜r
δ
˜r
˜s
for the reduced matrix o n this subspace (there is
no summation over repeated indices in t his case).
Before giving the explicit excitation equations for this speciﬁc model we discuss the
gauge ﬁxing of the vector ﬁeld gauge symmetry, which for h
s
r
= 0 is an ordinary gauge
symmetry, see (2.2). A convenient gauge, which disent angles the scalar ﬁeld and gauge
ﬁeld ﬂuctuations is given by the Lorenz gauge condition
· A
r
= 0 . (3.20)
Since the gauge ﬁelds are determined by ﬁrst-order equations the Lorenz gauge, and not
a ’t Hooft R
ξ
-gauge, decouples the scalar and gauge ﬁeld kinetic terms. The ﬂuctuation
equations (
3.15) thus take the form
/
χ
ir
= 0 , dC
r
= 0 ,
( dB
r
+ C
r
)
= 0 , ϕ
r
= 0 ,
/
χ
i˜r
+ 2 k
˜r
λ
i˜r
= 0 , dA
˜r
1
k
˜r
dC
˜r
= 0 ,
/
λ
i˜r
= 0 , Y
˜r
ij
= 0 , (3.21)
where we have split the ga uge indices a s r = (r
, ˜r) a ccording to the decomposition
g = t
˜
g. For the unbroken sector t, the ﬁrst line o f (3.21) together with the second
15
line for r = r thus describe a free tensor multiplet coupled to the three-form potential
C
r
which has vanishing ﬁeld strength and may be gauged away. Alternatively, one
may employ the two-form shift symmetry in (
2.2) with gauge parameter Λ
r
µν
to set
B
r
= 0. Then the linearized equations describe a self-dual closed ﬁeld C
r
which gives
an equivalent description of the free tensor multiplet.
The broken sector
˜
g is described by the second line of (3.2 1) 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 t he right column again demonstrates the dua l 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
˜r
in terms of A
˜r
up to an two -
form b
˜r
whose ﬁeld strength has to be self dual in the B
r
= 0 gauge, see the second
line of (
3.21). The three-for m potential C
˜r
therefore shifts or communicates degrees of
freedom between the gauge and tensor multiplet.
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 g
Ir
, h
r
I
, b
Ir s
, d
I
rs
, and f
rs
t
subject to the set of algebraic con-
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