Master des Sciences de la Matière M2
STAGE 2006-2007
LE DIFFON, Arnaud
2ème année
École normale supérieure de Lyon
Spécialité Champs et particules
Université Claude Bernard Lyon 1
Hidden symmetries in Supergravity.
Laboratoire de Physique
ENS LYON
46 allée d'Italie. LYON
Responsable de stage : Henning SAMTLEBEN
Avril-Août 2007
TABLE DES MATIÈRES
2
Table des matières
1 Introduction.
3
2 Dimensional reduction and gauged supergravity theories.
2.1
2.2
Dimensional reduction and global symmetry group.
The gauging of supergravity theory. . . . . . . . . .
2.2.1 Denition of a gauged supergravity theory.
2.2.2 The embedding tensor. . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3 Dimensional reduction of D = 11 maximal supergravity to D = 9.
3.1
3.2
4
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Underlying symmetries and charges of compactied D=9 supergravity from D=11.
The gauging of the D=9 supergravity theory. . . . . . . . . . . . . . . . . . . . . .
3.2.1 The embedding tensor and gauge generators in D = 9 supergravity. . . . . .
3.2.2 Constraints on gauge generators for valid gauging. . . . . . . . . . . . . . .
R+ -symmetry coupling to gravity sector.
4.1 Modied tensorial quantities : Christoel symbol and Riemann tensor. .
4.1.1 The modied Christoel symbol. . . . . . . . . . . . . . . . . . .
4.1.2 The modied Riemann tensor. . . . . . . . . . . . . . . . . . . .
4.2 The modied spin connection. . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Generalized metric compatibility and denition of the Riemann tensor .
4.4 Generalized Christoel symbol with non-zero torsion. . . . . . . . . . . .
4.5 Ricci tensor and discussion. . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Variation of modied Christoel symbol and Ricci tensor under general
and gauge eld. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
. . . . . . . . . . . .
variation of metric
. . . . . . . . . . . .
5
5
6
6
6
8
8
11
11
12
15
15
15
16
17
18
19
19
21
A General relativity formulas.
22
B Modied Christoel symbol and spin connection.
24
C The R+ -modied Riemann tensor
25
References
29
B.1 Transformation law of Christoel symbol and spin connection. . . . . . . . . . . . . . . . . . .
B.2 Generally covariant and R+ -gauge invariant Christoel symbol and spin connection. . . . . .
C.1 Computation with the modied Christoel symbol. . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Computation with modied spin connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
24
25
26
3
1 Introduction.
The last century has seen the birth and the success of quantum mechanics and general relativity. More
precisely, Einstein's equations of general relativity have given accurate predictions about the large scale
structure of the universe, gravitational interactions and astrophysical observations. The quantum theory of
elds has achieved a great understanding of fundamental interactions at small length scale. The unication
of electroweak and strong interactions in the framework of the standard model led to many conclusive tests
in particle physics. Everything is ne as long as we describe things separately at either small or large scales.
But, when we want to describe situations such as black holes or the early universe, we need to unify the two
formalisms. String theory is a possible candidate that could oer a theory of quantum gravity and a unied
theory of all the known interactions [1], [2], [3].
Unication is intrinsic to physics and Maxwell's theory of electromagnetism is a successful example of the
unication of electricity and magnetism, in the second half of XIXth century. Putting together the concept
of a unied nature of particles and basic interactions led to the unication of weak and electromagnetic
interactions and the standard model. One of the great challenges in modern physics is the unication of
gravity with the three other interactions within a consistent quantum theory. In the 60's, it was believed that
unication could be achieved by merging Poincaré group with internal symmetries (isospin, SU (3),...). But
Coleman and Mandula showed that these attempts could not be realized with the mathematical framework
of Lie groups and Lie algebras [4]. As a consequence, supersymmetry was postulated to extend the Poincaré
algebra with a spacetime symmetry that relates fermions to bosons. Even if there is no evidence of its existence
at the present time, supersymmetry oers an attractive framework to unify spacetime with internal symmetry.
Moreover, UV divergences in perturbative expansion of quantum eld theory can be cancelled out just by
invoking supersymmetry. In addition, string theory needs supersymmetry to solve some inconsistencies. Before
that, the concept of gauge invariance had become the keystone of particle physics ; so supersymmetry was
proposed to be elevated at the level of gauge theory. Doing this, it was realized that local supersymmetry
contained gravity. In fact, two global supersymmetric transformations already led to spacetime translations.
Supergravity, obtained as the gauging of supersymmetry (local supersymmetry), was found independently of
string theory [5], [6], [7], [8]. Later, it was realized that supergravity was a quantum eld theory limit or low
energy limit of string theories [9], [10].
Another important feature to unify fundamental interactions are Kaluza-Klein theories. Initially, it was
an attempt to geometrize electromagnetism, like gravity in general relativity. It was the idea of Kaluza in
1919 to postulate the existence of a fth dimension. By splitting a theory of pure gravity in 5 dimensions
(Lagrangian and eld equations) into a 4-dimensional gravity plus a spatial dimension, one could recover
Einstein gravity and Maxwell theory [11]. The analysis remained at the classical level. It was Klein, in 1926,
who extended this idea to quantum level and suggested that the additional dimension was compactied on
a circle, that is to say that, at each point of the 4-dimensional spacetime lays a small circle [12], [13]. So,
a 5-dimensional world with one of its spatial dimension compactied on a circle is equivalent formally to a
4-dimensional world and U (1) gauge symmetry [14].
After the discovery of Yang-Mills gauge theories in 1954 [15], it was shown that it was possible to derive
Yang-Mills elds from higher-dimensional gravity via higher-dimensional Kaluza-Klein theories [16], [17], [18]
and spontaneous compactication [19]. The idea of this approach is to think about internal symmetries of
non-abelian gauge theories in four dimensions as spacetime symmetries of the extra dimensions. Thus, the
gauge group of non-abelian theories is a subgroup of the higher-dimensional general coordinate group. It was
E. Witten who the rst gave a convincing argument of using a higher-dimensional theory with D = 11 [20].
Trying to recover the gauge group of the standard model SU (3) × SU (2) × U (1), a dimensional reduction
is perfomed. By associating to each symmetry group, the corresponding manifold, we obtain the following
4
1 INTRODUCTION.
geometrical internal manifold CP2 × S2 × S1 . The dimension of this object is dimCP2 + dimS2 + dimS1 =
4 + 2 + 1 = 7. So, the internal space needs a minimal dimension of 7 and this is an argument in favour of a
higher-dimensional spacetime with 11 dimensions.
Kaluza-Klein theories found a new eld of application with the appearance of supergravity. For a review
of supergravity see [5], [6]. When supergravity was extended to higher dimension, dimensional reduction was
a way to obtain extended supersymmetry. It appeared that for mathematical consistency reasons, D ≤ 11
was needed for supergravity [21]. Besides, in the 70's E. Cremmer, B. Julia and J. Scherk [22] built the
N = 1 D = 11 supergravity lagrangian, describing the dynamics of the elfbein eMA , the majorana gravitino
ΨM and a three-form gauge eld CM N P and the extended N = 8 D = 4 theory was obtained. For a review
see [23], [14], [24], [19].
But, the gauge groups of reduced theories and eective theories depend upon the compactication of
internal space (ex : S 7 for eective theory in D = 4, etc). The question of the choice of the internal space and
the way we are performing the compactication is tricky. In the late 70's, it was proposed a reduction ansatz
by Scherk and Schwarz [24]. By choosing appropriate internal groups and parameters, it was possible to obtain
from an abelian theory with a global symmetry, a non-abelian gauge theory. This process of deformation is
known as the gauging of a theory and in Scherk-Schwarz reductions, mass parameters are used to parametrize
the deformations of the initial theory (for a use of SS reductions in D = 9 theory, see [25]).
In this framework, a systematic way to obtain the gaugings of maximal supergravities was developped in
[26], [27], [28]. Starting from a Kaluza-Klein like dimensional reduction on a torus, and using reduced elds
that do not depend on torus internal coordinates, it is possible to obtain all the deformations of the reduced
theory. So, the rst part of my work is dedicated to the presentation of this method. Then, I will apply
it to the reduction of D = 11 supergravity down to D = 9 supergravity and derive the possible gaugings
of the nine-dimensional theory. I will recover the solutions found in [25]. Finally, I will study a peculiar
R+ -symmetry of ungauged supergravities, the so-called "trombone" symmetry [29] [30], which is scaling and
on-shell symmetry, and construct the gauging of this symmetry.
5
2 Dimensional reduction and gauged supergravity theories.
In this part, we present dimensional reduction on a torus and a systematic approach to construct the
deformations of such a reduced theory [26], [27], [28]. We will see how the internal symmetries group arises
from compactication of additional space dimensions and how, starting from the internal global symmetry
group G0 , we obtain the gauge group G of the reduced theory. For the dimensional reduction of gravity and
supergravity, we refer to [31], [29].
2.1 Dimensional reduction and global symmetry group.
We consider pure Einstein gravity on a D-dimensional manifold MD with world coordinates xµ̂ , µ̂ =
0, ... , D − 1. The metric gµ̂ν̂ has a Lorentzian signature (−; +, ... , +) and the information about its dynamic
is contained in the Einstein-Hilbert action
Z
√
SEH = dD x LEH , LEH = −gR(D)
(1)
where g = det(gµ̂ν̂ ), R(D) is the scalar curvature of gµ̂ν̂ (see appendix A for details). The equations of
motion are given by Einstein equations in D dimensions
1
Rµ̂ν̂ − gµ̂ν̂ R = Gµ̂ν̂ = 0
2
(2)
We want to dimensionally reduce this theory of pure gravity on a torus down to d = D − q space-time
dimensions. In other words, we provide MD with a principal bundle structure, with structure group [U (1)]q
and base manifold Md , so that locally MD = Md × Tq with Md a d-dimensional space-time manifold and
T q a q -dimensional torus. We introduce world coordinates xµ , µ = 0, ... , d − 1 on Md and coordinates y a ,
a = 1, ... , q on T q . The ansatz for the metric on MD :
(3)
ds2 = gµ̂ν̂ dxµ̂ dxν̂
µ
ν
= g̃µν dx dx + ρ
2/q
a
Mab (dy +
Aaµ dxµ )(dy b
+
Abν dxν )
(4)
where g̃µν , Aaµ , ρ and Mab depend on xµ only. g̃µν is the metric on Md , Aaµ are the q Kaluza-Klein vector
elds. The metric on the internal space T q is composed of the dilaton ρ and the unimodular matrix Mab .
They appear as scalar elds from a d-dimensional point of view.
If we want to completely perform the reduction, we have to take the previous ansatz (3) and put it into
the Einstein-Hilbert action. Computing all the reduced quantities, we obtain an eective lagrangian for the
reduced theory and the equations of motion corresponding to every dynamical degrees of freedom that appear
in the reduction. But, we need not to do this in order to identify the dierent underlying symmetries. We
will see this in the following section with the D = 9 reduced supergravity theory.
Note that for the moment we only speak about the reduction of pure gravity. For supergravity theories
[32], there are other bosonic degrees of freedom represented by p-forms (3-form gauge eld for D = 11
supergravity, 3-form and 2-form gauge elds in D = 9 supergravity, etc). We will perform dimensional
reduction over a torus of these elds too, and in addition to the ansatz reduction for the metric, we demand
6
2 DIMENSIONAL REDUCTION AND GAUGED SUPERGRAVITY THEORIES.
that these bosonic elds are constant along the internal manifold coordinates (in this case the torus). For
instance, the condition for the 3-form gauge eld Cµ̂ν̂ ρ̂ contained in the bosonic sector by D = 11 supergravity
theory is given by :
∂
Cµ̂ν̂ ρ̂ = 0
∂y a
(5)
2.2 The gauging of supergravity theory.
2.2.1 Denition of a gauged supergravity theory.
Gauging a theory consists in turning a global symmetry or a part of it into a local one. Making this, the
corresponding symmetry parameters are allowed to have a space-time dependence in the gauged theory. In
a way, supergravity can be obtained from the gauging of global supersymmetry. But, what we consider will
be only ordinary bosonic symmetries.
Assuming that we have a supersymmetric theory with a global symmetry group G0 , we want to make a
part of or G0 itself local without breaking the supersymmetry and this puts constraints on possible gaoge
groups. In section 3, we will apply this method to D = 9 supergravity theory, dimensionally reduced from
D = 11 supergravity.
2.2.2 The embedding tensor.
Consider an ungauged theory with global symmetry group G0 . To this symmetry group corresponds an
Lie algebra G0 , generated by the symmetry generators ti , i = 1, ..., dim(G0 ). The algebra is dened by the
structure constants fij k of G0 :
(6)
[ti , tj ] = fij k tk
We want to gauge the theory but also we want to preserve gauge invariance. To do this, we need to
minimally couple vector gauge elds to symmetry generators, and then, we are going to replace partial
derivatives ∂µ by covariant derivatives Dµ .
We assume that the theory contains abelian vector elds AM
µ that transform in some representation V ,
labelled by an index M , of the global symmetry group G0 . Each vector eld is a U (1) gauge eld. So, they
transform not only under G0 -transformations Li , but also under local gauge transformations with space-time
dependent parameter ΛM (x) :
δ L AM
µ
δ Λ AM
µ
= −Li tiN
= ∂µ Λ
M
AN
µ
M
(7)
(8)
So, we have to construct the coupling of these vector elds AM
µ to the G0 symmetry generators tα in the
covariant derivative of the gauged theory :
i
Dµ = ∂µ − gAM
µ ΘM ti
(9)
2.2 The gauging of supergravity theory.
7
ΘM α is called the embedding tensor : it is a rectangular matrix that allows us to couple vector elds
and symmetry generators. g is the gauge coupling constant and can be absorbed into ΘM α . The embedding
tensor can be seen as a map Θ : V → G0 . The image of this map denes the gauge group G of local
symmetry and the gauge transformations are now parametrized by ΛM (x). For instance, a eld BM in the
dual representation V̄ of the vector gauge elds transforms under the gauge group G as
δBM = gΛN ΘN i tiM
P
(10)
BP = gΛN XN M P BP
Gauge eld transformations are also given by :
M
P
δAM
µ = ∂µ Λ (x) + gΛ (x)XN P
M
(11)
AN
µ
where we have introduced the gauge group generators XM = ΘM ß ti . In the vector eld representation,
they take the role of generalized structure constants for the gauge group XM N P = ΘM i tiN P .
Note that the embedding tensor is not invariant under the global symmetry group G0 , but to ensure the
closure of the gauge group and the gauge covariance of the construction, Θ has to be invariant under gauge
transformations δ = ΛM δM , ie
δM ΘN
i
= gΘM j (tjN
P
ΘP
i
(12)
− fjk i ΘN k ) = 0
It is equivalent to demand the XM N P to be gauge invariant, and the condition δM XN P
expressed like
[XM , XN ] = −XM N P XP
Q
= 0 can be
(13)
This equation gives us the closure of the gauge group. These relations are quadratic constraints on Θ.
The embedding tensor has to satisfy these constraints in order to describe a valid gauging.
In addition to this quadratic constraints, a linear constraint on Θ is needed. The consequence of this is a
suppression of some irreducible component of the embedding tensor. In terms of the gauge generators these
linear constraints are expressed as [27] :
X(M NK) = dI, M N Z I, K
(14)
where dI, M N is a G invariant tensor, the tensor Z I, K is implicitly dened as a linear function of the
embedding tensor and the index I is referring to the representation of two-forms.
I will not say much more about this. The consequence of the linear constraints on gauge generators are
presented in the following section.
Thus, the embedding tensor can be used to parametrize the gauging. Any Θ with the appropriate linear
constraint and the quadratic constraint will describe a valid gauging. The construction of the gauged theory
only requires these constraints for consistency. Note that Θ transforms under the global symmetry group
G0 , so we formally preserve the G0 symmetry in the gauged theory. This illustrates the fact that the set
of all possible gaugings is G0 invariant. But if we consider a particular gauging, the embedding tensor that
parametrizes this gauging breaks the G0 invariance down to the local gauge group G ⊂ G0 .
8
3 DIMENSIONAL REDUCTION OF D = 11 MAXIMAL SUPERGRAVITY TO D = 9.
3 Dimensional reduction of D = 11 maximal supergravity to D = 9.
The unique supergravity theory in D = 11 space-time dimensions contains in the bosonic sector the
metric and an antisymmetric 3-form gauge eld Cµ̂ν̂ ρ̂ [22]. Its eld strength is Gµ̂ν̂ ρ̂λ̂ = 4∂[µ̂ Cν̂ ρ̂λ̂] and we
have a gauge symmetry δCµ̂ν̂ ρ̂ = 3∂[µ̂ Λν̂ ρ̂] . The bosonic part of the D = 11 supergravity lagrangian is :
√
1
2 µ̂ν̂ ρ̂λ̂σ̂τ̂ κ̂τ̂ χ̂θ̂ξ̂
( D)
µ̂ν̂ ρ̂λ̂
LD=11 = −g R −
Gµ̂ν̂ ρ̂λ̂ Gσ̂τ̂ κ̂τ̂ Cχ̂θ̂ξ̂
(15)
G
G
+ 2
2 · 3! µ̂ν̂ ρ̂λ̂
72
By dimensional reduction every bosonic degree of freedom of D = 9 supergravity can be tracked back
to the D = 11 supergravity, so that we don't have to add bosonic eld at D = 9. We only consider in the
following the bosonic degrees of freedom of the theory.
3.1 Underlying symmetries and charges of compactied D=9 supergravity from D=11.
When we dimensionally reduce the 11-dimensional maximal supergravity on a torus T 2 = S1 × S1 , we
obtain residual symmetries in the 9-dimensional theory, which are characterized by corresponding charges.
From the reduction on the torus, we obtain a GL(2)-symmetry due to torus symmetry. The symmetry
GL(2) = SL(2) × GL(1) is characterized by two charges G , associated with SL(2)-symmetry, and B , with
GL(1)-symmetry. These two symmetries are symmetries of both lagrangian and the equations of motion,
i.e. o-shell symmetries. In addition, the 9-dimensionnal has a R+ -symmetry, known in the literature as a
trombone symmetry [29], [30]. It is an on-shell symmetry except for D = 2 theories in which R+ -symmetry
is o-shell. It is a symmetry for which the vielbein eld and metric are charged. This symmetry is a generic
feature in ungauged supergravities. For instance, we nd this symmetry in D = 11 supergravity theory. This
symmetry acts on the eld as follows
gµ̂ν̂ → λ2 gµ̂ν̂ , Cµ̂ν̂ ρ̂ → λ3 Cµ̂ν̂ ρ̂ , ψµ̂ → λ1/2 ψµ̂
(16)
where λ ∈ R+ . It is a symmetry of the eld equations of supergravity theories but the lagrangian
transforms as L → λ9 L. The weights of the elds are determined by the following rule : for the bosonic
elds the weights equal the number of Lorentz indices, and for fermions it is one-half. Note that instead of
choosing a charge λ ∈ R+ , we can use a GL(1) charge A and scales elds and Lagrangian with λ = e−A . It
is completely equivalent and using the previous scaling we obtain the charges A :
eµ a → e−A eµ a
gµν
Aµ1 ...µk
(17)
−2A
gµν
(18)
−kA
Aµ1 ...µk
(19)
→ e
→ e
Concerning the SL(2)-symmetry, the representations under SL(2) follow directly from elds' indices
structure (α gives doublet under SL(2)). We can reconstruct SL(2), by using an abelian subgroup of SL(2)
(SO(1, 1)), and the charges G are linked with this abelian subgroup. A more detailed explanation of this can
be found in [25].
The GL(1) charge B can be computed by using the formula given in [28],
3.1 Underlying symmetries and charges of compactied D=9 supergravity from D=11.
eφ → e(d−2)B/n eφ
Bµm
−[(d−2)/n+1]B
→ e
µ
Aµ1 ...µk
[(d−2)p/n+p−k]B
−kB
Aµ1 ...µk
→ e
Am1 ...mp µp+1 ...µk
→ e
Am1 ...mk
Bm
k(d−2)B/
→ e
n Am
9
(20)
(21)
(22)
Am1 ...mp µp+1 ...µk
1 ...mk
(23)
(24)
where d is the dimension of the reducd theory and n the dimension of the torus.
Let us make this a little more explicit. If we begin with the bosonic eld content of 11-dimensional
supergravity, we have the vielbein (metric of spacetime) and an antisymmetric three-form gauge potential :
D = 11 : {êµ̂ â , Ĉµ̂ν̂ ρ̂ }
(25)
avec µ̂, â = 0, ..., 10.
By xing two spatial coordinates on a torus T 2 , we obtain the dimensional reduction or compactication
of D = 11 supergravity. So, we have the reduced elds in nine dimensions from the 11-dimensional ones, with
the following indices µ = 0, ..., 8, a, b, c = 9, 10, α = 1, 2 :
ĝµ̂ν̂

 gµν
(α)
→
g = Aµ
 µa
gab = scalar elds : φ = g10,10 ; ϕ = g9,9 ; χ = g9,10

Cµνρ



(α)
Cµνa = Bµν
Ĉµ̂ν̂ ρ̂ →

C
= Aµ = Cµ,9,10 = −Cµ,10,9

 µab
Cabc = 0
(26)
(27)
The bosonic eld content in 9-dimensional supergravity is given by the vielbein eld (spacetime metric
in D=9), 1-form and 2-form gauge potential which SL(2) doublets, 1-form and 3-form gauge potentials, and
scalar elds as SL(2) singlets (one dilaton and two axions) :
(α)
D = 9 : {eaµ , φ, ϕ, χ, A(α)
µ , Aµ , Bµν , Cµνρ }
(28)
Given that we have achieved the dimensional reduction, we can now compute the dierent charges for
every elds. Using the formulas (17)-(19) for R+ charge A, (20)-(24) for GL(1) charge B with d = 9 and
n = 2, and the reference [25] for SL(2) charge G , we obtain the dierent results, gathered in TAB.1.
In the reference [25], dimensional reduction is not based upon a torus reduction directly but upon successive S1 reductions. A D = 9 supergravity theories can be obtained by dimensionally reducing D = 11
supergravity theory on a circle in D = 10 IIA supergravity, then reducing this one on a circle we obtain
D = 9 supergravity. Or, we can reduce D = 10 IIB supergravity on a circle and obtain D = 9 supergravity
too.
3 DIMENSIONAL REDUCTION OF D = 11 MAXIMAL SUPERGRAVITY TO D = 9.
10
Field content
A
B
G
eµ a
1
0
0
φ
0
−7/2
0
ϕ
0
0
2
χ
0
0
−2
(1)
Aµ
1
−6
0
(2)
Aµ
1
9/2
1
(1)
Aµ
1
9/2
−1
(2)
Bµν
2
−3/2
1
Bµν
2
−3/2
−1
Cµνρ
3
3
0
L
7
0
0
1 Symmetries of the 9-dimensional supergravity : a R+ -symmetry at 9D A, a GL(1)-symmetry B
and a SL(2)-symmetry G both from dimensional reduction.
Tab.
Tab.
[25].
Field content
α0
eµ a
φ
9
7
β0
γ0
δ0
0
0
√6
√7
7
4
8
7
0
− √47
ϕ
0
χ
0
Aµ
3
− 34
2
0
3
4
1
2
−2
0
0
0
Aµ
0
(1)
Aµ
0
(2)
Bµν
3
(1)
Bµν
3
(2)
Cµνρ
3
L
9
Origin
11D
− 34
1
2
0
−1
2
− 14
1
2
1
2
− 41
0
4
0
0
8
IIA
IIB
IIB
−1
2
2 Charges α0 , β 0 , γ 0 , δ 0 related to the four symmetries of the 9-dimensional supergravity described in
Concerning the symmetries :
• D = 10 IIA has two R+ -symmetries, one that scales the lagrangian with a D = 11 origin, and one
that leaves the lagrangian invariant ;
• D = 10 IIB has an SL(2)-symmetry and a R+ -symmetry that does not leave the lagrangian invariant.
The Kaluza-Klein reduction of the massless IIA or IIBgives the unique D = 9 massless supergravity theory
whose eld content has already been given above (28). The massless 9-dimensional theory inherits several
global symmetries from its parents : two R+ -symmetries with parameters α0 and β 0 from IIA supergravity
and one R+ -symmetry with parameter δ 0 plus an SL(2)-symmetry with parameter γ 0 from IIB. There is a
relation between the four parameters 49 α0 − 83 β 0 = γ 0 + 21 δ 0 . The dierent values are summarized in TAB.2.
We try to rewrite the four symmetries α0 , β 0 , γ 0 and δ 0 , that are found in the paper [25], as function of
our three symmetry parameters A, B and G .
We obtain the following relation between charges of the symmetries :
9
A−
7
1
= − B
12
= G
8
=
A+
7
D = 11 origin : α0 =
D = 10 IIA origin : β 0
γ0
D = 10 IIB origin : δ 0
2
B
7
3
− G
8
(29)
(30)
(31)
4
B
21
(32)
3.2 The gauging of the D=9 supergravity theory.
SL(2) avec ta , a = 1, 2, 3 : 300
GL(1) avec t0 : 100
R+ avec tR+ : 100
11
Aµ : 1−6
1
300 ⊗ 1−6
=
3−6
1
1
−6
100 ⊗ 1−6
=
1
1
1
−6
100 ⊗ 1−6
=
1
1
1
9/2
Aβµ : 21
9/2
9/2
9/2
300 ⊗ 21 = 21 ⊕ 41
9/2
9/2
100 ⊗ 21 = 21
9/2
9/2
100 ⊗ 21 = 21
3 Irreducible components of tensor product g0 ⊗ V . Il s'agit du tenseur Θi
M = (0, β), β = 1, 2
Tab.
M,
avec i = a, 0, R+ et
3.2 The gauging of the D=9 supergravity theory.
3.2.1 The embedding tensor and gauge generators in D = 9 supergravity.
Previously, we identied the global symmetry group G0 = SL(2)×GL(1)×R+ of the compactied theory.
We want to realize all the possible gaugings of the theory, and to do that we parametrize the deformations
by an embedding tensor Θ as explained in section 2.2. The gauging process breaks the global symmetry G0
down to a local gauge group G ⊂ G0 . We use the embedding tensor to build from global symmetry generators
the local gauge group generators XM .
The embedding tensor is given by its dierent irreducible components ΘM i (dual of the quantities collected
in TAB.3), with the following notation SL(2)−B
−A :
(33)
Θ0 a = 36−1
−9/2
Θβ a = 2−1
Θ0
0
=
Θβ
0
=
Θ0
R+
=
Θβ
R+
=
−9/2
(34)
+ 4−1
(35)
16−1
−9/2
2−1
16−1
−9/2
2−1
(36)
(37)
(38)
where the vector index M is expressed as M = (0, α).
Till the end, we will parametrize SL(2) triplet with SL(2) doublet indices α. In fact, the two notations
are equivalent for we have the same number of SL(2) generators : ta , a = 1, 2, 3 gives us 3 generators and
t(αβ) , α = 1, 2, β = 1, 2 gives 3 generators too.
The generators of the gauge group G will be given by this expression :
XM N P = ΘM i (ti )N P = ΘM
(αβ)
+
(t(αβ) )N P + ΘM 0 (t0 )N P + ΘM R (tR+ )N P
(39)
In order to have an explicit form for the generators XM N P , we give a name to every irreducible component
of the embedding tensor in agreement with the representation.
12
3 DIMENSIONAL REDUCTION OF D = 11 MAXIMAL SUPERGRAVITY TO D = 9.
Θ0
(αβ)
Θβ
(40)
= ξ (αβ)
(γδ)
=
δ)
µ(γ δβ
(41)
Θ0
0
= Ξ
(42)
Θβ
0
= ρβ
(43)
Θ0
R+
= Υ
(44)
Θβ
R+
= ζβ
(45)
with β, γ, δ = 1, 2 as SL(2) doublets. The 4 representation is absent in 41 because of the linear
constraints.
The action of the generators of the symmetry group G0 = SL(2) × GL(1) × R+ on the vector elds AM
µ
is given by :
(t0 )N
P
(tR+ )N P
(t(αβ) )N P
=
(t0 )0 0
(t0 )β γ
= −6
= 92 δβγ
(46)
(47)
P
= δN
(t(αβ) )0 0 = 0
=
δ (t(αβ) )γ δ = δ(α
β)γ
(48)
With the expression of the generators of G0 and the embedding tensor, we can build the generators of
the gauge group G of 9-dimensional supergravity theory, using the equation (39) :
(49)
X00 0 = −6 Ξ + Υ
X0ν
0
X0ν σ
(50)
ν
= X00 = 0
9
σ
Ξ + Υ δνσ + ξ (γδ) δ(γ
δ)ν
=
2
(51)
(52)
Xβ0 0 = −6ρβ + ζβ
(53)
Xβν 0 = Xβ0 ν = 0
1 γ σ
µ δγ βν + δβσ γν +
2
Xβν σ =
9
ρβ + ζβ
2
(54)
3.2.2 Constraints on gauge generators for valid gauging.
We also get the following linear constraints, demanding that the gauge generators be antisymmetric when
every indice of the gauge generator is referring to the same representation (for example X(β0) 0 6= 0) :
Xβν
0
= 0 → −6 Ξ + Υ = 0
σ
γ
γλ 9
ρβ + ζβ
X(βν) = 0 → µ = −2
2
σ
σ
σλ 9
= X[βν]
= −2
ρβ + ζβ βν
2
X00
(55)
(56)
(57)
3.2 The gauging of the D=9 supergravity theory.
13
So the linear constraints are :
(58)
Υ = 6Ξ
µγ
= −2γλ
9
ρβ + ζβ
2
(59)
The embedding tensor is thus parametrized in terms of Ξ, ρ, ζ and ξ . These parameters will be linked
thanks to the quadratic constraints. We obtain the quadratic constraints by evaluating in the vector eld
representation and expanding the following relation between the gauge group generators XM :
(60)
[XM , XN ] = −XM N P XP
Hence,
21
Ξ(−6ρβ + ζβ ) + ξ (δ) δβ (−6ρε + ζε )
2
9
ρκ + ζκ ξ (κδ) δβ δντ + δν δβτ
2
9
9
21
τ
Ξ
ρβ + ζβ δν −
ρν + ζν δβτ
2
2
2
9
21 δτ
(δτ )
βν
ρδ + ζδ
Ξ − ξ
2
2
9
(τ δ)
ξ
−
ρν + ζν δβ + (−6ρβ + ζβ ) δν
2
9
ρδ + ζδ
ξ (δτ ) βν − δτ βκ ξ (κλ) λν + ξ (δκ) κν δβν
2
21
α(−6ρβ + ζβ )δντ
2
ρε ζλ ελ
9
9
ρδ + ζδ δσ
ρσ + ζσ (−βν δατ + αν δβτ − αβ δντ )
2
2
= 0
(61)
+
+
= 0
(62)
+
+
= 0
(63)
= 0
(64)
= 0
(65)
After the reductions of constraints, we get the quadratic constraints in their simplest and equivalent form,
that gauge generators have to satisfy in order to give a consistent gauging :
ρδ ξ (δτ ) = 0
(12ρδ + 5ζδ ) ξ
(66)
= 0
(67)
Ξ ρκ = 0
(68)
ζδ ξ
(δτ )
Ξ ζκ
(τ σ ν)δ
λε
ρλ ζε = 0
(69)
= 0
(70)
= 0
(71)
14
3 DIMENSIONAL REDUCTION OF D = 11 MAXIMAL SUPERGRAVITY TO D = 9.
mass parameters
(m1 , m2 , m3 )
(m4 , m̃4 )
(m11 , mIIA )
mIIB
Tab.
SL(2, R)
triplet
doublet
doublet
singlet
4 This table indicates the dierent multiplets that the D = 9 mass parameters form under SL(2).
Every solution of (66) to (71) gives a consistent gauging. These equations give several set of solutions. A
good way to determine them all is to start from the possibilities given by the fth equation. This equation
has three dierent solutions :
(a) ξ = 0
(72)
(b) ρδ = ζδ = 0
(73)
5
ζδ
12
(74)
(c) ρδ = −
One set of solution consists in choosing (a) ξ = 0 and Ξ = 0. We can make a rotation on ρδ and ζδ . In
that way, ρ2 = ζ2 = 0 and ρ1 , ζ1 remains arbitrary. All the quadratic constraints are satised.
Then we consider the situation where ξ 6= 0 and Ξ 6= 0. To satisfy the fth equation, we choose the
solution (b) ρδ = ζδ = 0. (b) gives a solution of (66), (67), (68), (69), and (71). So, ξ and Ξ remain arbitrary.
5
ζδ ,
Finally, we keep ξ 6= 0 (at least one of its components) and Ξ = 0. With the last solution (c) ρδ = − 12
conditions (68), (69), and (70) are fullled. By using the same redenition as previously (ρ2 = ζ2 = 0 and
ρ1 , ζ1 arbitrary), we obtain that two components ξ (11) = ξ (12) = 0 and ξ (22) 6= 0.
These three dierent sets of solution are in good agreement with the three independent gaugings found in
the paper [25] about non-abelian gauged supergravities in nine dimensions. Their method is quite dierent.
To gauge massless D = 9 supergravity, they introduce mass parameters that gauge the global symmetries
of the theory. By performing Scherk-Schwarz reductions with several mass parameters, they obtain gauged
theories.
We can link our parameters ξ (triplet under SL(2)), ρ, ζ (doublet under SL(2)) and Ξ (singlet under
SL(2)) to their mass parameters collected in TAB.4, extracted from [25].
Their solutions are the following :
• Case 1 with { mIIA , m4 }, one of the two components of the two doublets. This corresponds to our
rst set of solutions with ρ1 and ζ1 dierent from zero and the others set to zero.
• Case 2 with { m
~ , mIIB }, the triplet and the singlet. This is related to our second set of solutions ξ 6= 0
and Ξ 6= 0.
• Case 3 with { m4 = − 12
5 mIIA , m2 = m3 }. In our last set of solution, we have a condition between
one of the two components of the two doublets ρ1 and ζ1 with the same numerical factor, and only one of
the component of the triplet is dierent from zero.
15
4
R+ -symmetry
coupling to gravity sector.
In the previous section, we have seen that the construction of the gauging involves this R+ -symmetry and
overall that the gauging of R+ -symmetry could not ben achieved. The solutions of quadratic constraints in
D = 9 supergravity are involving several gauged symmetries. Before going further, gauge covariant derivative
Dµ can be rewritten as :
+
R
Dµ = ∂µ − AM
tR + −
µ ΘM
|
{z
}
=Aµ
J
AM
µ ΘM tJ
|
{z
}
, J = 0, 1, 2, 3
(75)
No coupling to gravity sector
where Aµ is the gauge eld associated with R+ -symmetry. One consequence of the quadratic constraint
is that Aµ is uncharged under R+ -symmetry. The part of Dµ which does not couple to gravity plays a role
in keeping Aµ uncharged.
However, we have seen in (16) that metric is charged under R+ . When we try to gauge R+ -symmetry,
as gravity is charged, this will imply modications in the gravitational sector and to study them we have to
restrict to gravity sector. This section explain how we compute the relevant quantities to take into account
the modications induced by the coupling of Aµ to gravity.
4.1 Modied tensorial quantities : Christoel symbol and Riemann tensor.
4.1.1 The modied Christoel symbol.
The transformation law of Christoel symbols under the following rescaling of the metric, also known as
Weyl rescaling :
gµν → e2φ(x) gµν
(76)
Γσµν → Γσµν + ∂µ φ(x)δνσ + ∂ν φ(x)δµσ − gµν gστ ∂ρ φ(x)
(77)
It motivates the denition of the modied Christoel symbols
Γ̂σµν = Γσµν − Aµ δνσ − Aν δµσ + gµν gστ Aτ
(78)
which are invariant under simultaneous metric rescaling and gauge transformation of the abelian gauge
eld Aµ (x), associated with R+ -symmetry :
→ e2φ(x) gµν
(79)
Aµ → Aµ + ∂µ φ(x)
(80)
gµν
4 R+ -SYMMETRY COUPLING TO GRAVITY SECTOR.
16
4.1.2 The modied Riemann tensor.
The Riemann tensor dened by Christoel symbols is given in appendix (130). The modied Riemann
tensor is obtained by replacing Christoel symbols with the hatted ones (78) :
R̂ρ µσν (Γ̂) = ∂σ Γ̂ρνµ − ∂ν Γ̂ρσµ + Γ̂ρσλ Γ̂λνµ − Γ̂ρνλ Γ̂λσµ
(81)
Note that we did not use covariant derivatives since Γ̂ is invariant under R+ . Then, after some algebra
given in appendix C.1, we get this expression :
ρλ
R̂ρ µσν (Γ̂) = Rρ µσν + Fνσ δµρ + Mρλ
σµ ∇ν Aλ − Mνµ ∇σ Aλ
ρ
λδ
ρλ
ρλ
+
δσρ Mλδ
µν − δν Mµσ Aλ Aδ + gµν g Aσ Aλ − gσµ g Aν Aλ
(82)
λ δ
λδ and F
where the tensor Mλδ
νσ is the eld strength of the abelian vector eld related
µν = δµ δν − gµν g
+
to R -symmetry. The modied Riemann tensor R̂ρ µσν no longer obeys symmetries (131)-(135) of standard
Riemann tensor Rρ µσν . If we try to separate this tensor into a part which have the exact same symmetry as
usual Riemann tensor, and some other antisymmetric part, we obtain :
1
R̂ρµσν (Γ̂) = R̂oρµσν + gρµ Fνσ + (gρσ Fνµ + gρν Fµσ + gσµ Fρν + gµν Fσρ )
2
(83)
with the following expression for R̂oρµσν :
λδ
R̂oρµσν = Rρµσν + Qλδ
ρσµν ∇(λ Aδ) + Sρσµν Aλ Aδ
(84)
λδ
parametrized by tensors Qρσµν and Sρσµν
Qλδ
ρσµν
λδ
Sρσµν
= gρδ δµλ δνδ + gµν δρλ δσδ − gσµ δρλ δνδ − gνρ δµλ δσδ
λδ
= Qλδ
ρσµν + gµσ gρν − gρσ gµν g
(85)
(86)
R̂oρµσν is antisymmetric under exchange of two rst indices and two last indices. This tensor is also
symmetric under exchange of pair indices.
R̂oρµσν
= −R̂oµρσν
(87)
R̂oρµσν
R̂oρµσν
−R̂oρµνσ
R̂oσνρµ
(88)
=
=
(89)
4.2 The modied spin connection.
17
4.2 The modied spin connection.
Let us now compute the modied Riemann tensor using the spin connection. If we want to couple fermions,
spin connection is indispensable. So, we establish the modied spin connection ω̂µ a b replacing in the following
formula :
ωµ a b = eν a eλ b Γνµλ − eλ b ∂µ eλ a
(90)
Christoel symbols and partial derivative by modied Christoel symbols and gauge covariant derivative
w.r.t R+ -symmetry Dµ = ∂µ − qAµ . It gives us :
ω̂µ a b = eν a eλ b Γ̂νµλ − eλ b Dµ eλ a
(91)
where Dµ eλ a = (∂µ − qAµ )eλ a , with q = 1.
The explicit expression of the modied spin connection then is as follows :
ω̂µ a b = ωµ a b − eµ a eλ b Aλ + eµ b eδ a Aδ
(92)
The usual Riemann tensor can be dened as curvature of the spin connection :
Ra b = dω a b + ω a c ∧ ω c b = (Ra b )µν dx µ ∧ dx ν
(93)
Explicitly :
Ra bµν
= (dω a b )µν + (ω a c ∧ ω c b )µν
= ∂µ ων a b − ∂ν ωµ a b + ωµ a c ων c b − ων a c ωµ c b
(94)
(95)
The covariant curvature of the modied spin connection gives us :
R̂ρ µσν (ω̂) = eρ a eµ b Ra bσν = eρ a eµ b [∂σ ω̂ν a b − ∂ν ω̂σ a b + ω̂σ a c ω̂ν c b − ω̂ν a c ω̂σ c b ]
ρλ
= Rρ µσν + Mρλ
σµ ∇ν Aλ − Mνµ ∇σ Aλ
ρ
λδ
ρλ
ρλ
+
δσρ Mλδ
µν − δν Mµσ Aλ Aδ + gµν g Aσ Aλ − gσµ g Aν Aλ
(96)
(97)
For details about the calculus, see appendix C.2. We obtain the same modied Riemann tensor up to a
eld strength term δµρ Fσν , dierence that will be explained in the next subsection.
4 R+ -SYMMETRY COUPLING TO GRAVITY SECTOR.
18
4.3 Generalized metric compatibility and denition of the Riemann tensor
Note that the usual metric compatibility remains valid in the R+ -symmetry coupling to gravity :
∇µ gνσ = ∂µ gνσ − Γλµν gλσ − Γλµσ gνλ = 0
∇µ eν a = ∂µ eν a − Γλµν eλ a + ωµ a b eν b = 0
(98)
(99)
These equation are not covariant w. r. t. R+ . But also, we have a generalised metric compatibility
by constructing a dierential operator Dµ with the modied Christoel symbol Γ̂σµν and the abelian R+ symmetry gauge eld Aµ , dened by its action under an arbitrary vector eld Xν , with charge q under
R+ :
Dµ Xν
ˆ µ Xν
∇
ˆ µ Xν − qAµ Xν
= ∇
= ∂µ Xν −
Γ̂δµν Xδ
(100)
(101)
Applying this operator on both the vielbein and the metric, we get the generalized metric compatibility
conditions :
Dµ gνσ = ∂µ gνσ − 2Aµ δνλ gλσ − 2Aµ δσλ gνλ − Γ̂λµν gλσ − Γ̂λµσ gνλ = 0
Dµ eν a = ∂µ eν a − Aµ eν a − Γ̂λµν eλ a + ω̂µ a b eν b = 0
(102)
(103)
These two equations are covariant versions under metric rescaling and R+ gauge transformation of (98)
and (99). They are simultaneously valid. These generalized identities are gonna be useful in explaining the
eld strength term dierence between R̂ρ µσν (ω̂) and R̂ρ µσν (Γ̂).
The usual Riemann tensor is dened with the commutator of two generally covariant derivative ∇µ . We
have the following denitions, in a coordinate basis and a non-coordinate one, respectively :
[∇σ , ∇ν ] Xµ = Rρ µσν (Γ)Xρ
(104)
bσν (ω)Ya
(105)
[∇σ , ∇ν ] Yb = Ra
This two denitions of the Riemann tensor normally coïncide, by multiplication with the appropriate
vielbein Rρ µσν (Γ) = eρ a eµ b Ra bσν (ω). If we try to dene the modied Riemann tensor with spin connection
R̂ρ µσν (ω̂), it corresponds to the following approach. In a non-coordinate basis and using an R+ -symmetry
uncharged eld Yb , we get :
ˆ σ, ∇
ˆ ν ]Yb = R̂a (ω̂)Ya
[∇
bσν
(106)
Multiplying this by the appropriate vielbein and using generalized metric compatibility, we get the Riemann tensor previously established with the modied spin connection :
[Dσ , Dν ] (eµ b Yb ) = R̂ρ µσν (ω̂)(eρ b Yb )
(107)
4.4 Generalized Christoel symbol with non-zero torsion.
19
where eµ b Yb is now charged under R+ .
Computing the same commutator applied on a charged eld Xµ = eµ b Yb in a coordinate basis, we obtain :
(108)
[Dσ , Dν ] Xµ = R̂ρ µσν (Γ̂) + Fνσ δµρ Xρ = R̂ρ µσν (ω̂)Xρ
Thus, the dierence between the two modied Riemann tensor R̂ρµσν (Γ̂) and R̂ρµσν (ω̂) computed above
is perfectly consistent.
4.4 Generalized Christoel symbol with non-zero torsion.
The action of the previous dierential operator Dµ (100) can be expressed in terms of a partial derivative
ˆ
and a generalized Christoel symbol Γ̂ρµν = Γ̂ρµν + qAµ δνρ . For q = 1, the modication corresponds to a
non-trivial torsion.
The torsion is dened as the antisymmetric part of the generalized Christoel symbol :
ˆδ
ˆδ
ˆδ
T δµν = 2Γ̂[µν] = Γ̂µν − Γ̂νµ
(109)
So we obtain the following expression for the torsion :
T δµν = q Aµ δνδ − Aν δµδ
(110)
The generalized Christoel symbol depends upon the torsion in a non trivial way and for a given value
of R+ -symmetry charge q = 1. Dening the contorsion as
1 δ
T µν + Tµ δν + Tν δµ = gµν gλδ Aλ − Aν δµδ
2
we can express the generalized Christoel symbol like :
Kδ µν =
1 δ
ˆ
Γ̂δµν = Γδµν + Kδ µν = Γδµν +
T µν + Tµ δν + Tν δµ
2
(111)
(112)
The non-zero torsion is consistent with the simultaneous validity of metric compatibility and generalized
metric compatibility equations, respectively (98)-(99) and (102)-(103). The modications induced by R+ symmetry consists in the addition of a non-zero torsion.
4.5 Ricci tensor and discussion.
If we want to establish modied Einstein equations, we have to compute the modied Ricci tensor. At
that point, we are going to discuss dierent approaches.
Applying the same strategy as for the Christoel symbol, we can compute the transformation law of Ricci
tensor under metric rescaling :
λρ
Rµν → Rµν + Nµν
∇λ ∂ρ φ(x) + (D − 2)Mλρ
µν (∂λ φ(x)) (∂ρ φ(x))
with
(113)
4 R+ -SYMMETRY COUPLING TO GRAVITY SECTOR.
20
(
λρ
Nµν
Mλρ
µν
= (2 − D)δµλ δµλ − gµν g λρ
= δµλ δνρ − gµν g λρ
(114)
By adding gauge eld terms, in the same way that we did with the modied Christoel symbol, we obtain
the following modied Ricci tensor R̃µν , generally covariant and R+ -gauge invariant :
λρ ˜
R̃µν = Rµν − Nµν
∇λ Aρ (x) + (D − 2)Mλρ
µν Aλ (x)Aρ (x)
(115)
˜ λ Aρ (x) = ∂λ Aρ (x) − Γ̂δ Aρ (x)
∇
λρ
(116)
with
Note however that this generally covariant and R+ -gauge invariant Ricci tensor is not unique. Any Ricci
tensor that diers from R̃µν up to a R+ -gauge invariant quantity, say a eld strength in this precise situation,
will do the job.
E.g. for R̂(Γ̂) we obtain :
R̂ρµσν (Γ̂) → R̂µν (Γ̂) = gρσ R̂ρµσν (Γ̂)
= R̂(µν) −
D
2
Fµν
where
λδ
λδ σ
R̂(µν) = Rµν − Nµν
∂(λ Aδ) + Nµν
Γλδ Aσ + (D − 2)Mλρ
(µν) Aλ Aρ
(117)
Alternatively, from R̂(ω̂) we nd :
R̂ρµσν (ω̂) → R̂µν (ω̂) = gρσ R̂ρµσν (ω̂)
2−D
= R̂(µν) +
Fµν
2
And,
R̂µν (ω̂) = R̂(µν) +
2−D
Fµν
2
(118)
If we take the modied Riemann tensor obtained with spin connection, we will have the same symmetric
part in the modied Ricci tensor. The only dierence, as expected, lies in the coecient before the eld
strength term.
The dierent approaches are totally equivalent : we obtain the same symmetric part (R̃µν , R̂ρµσν (Γ̂) and
R̂ρµσν (ω̂)) and the dierence comes from the same gauge invariant quantity up to a numerical coecient.
To obtain the modied Einstein equations, we use the symmetric part of modied Ricci tensor R̂(µν) ,
with scalar curvature as R̂ = gµν R̂(µν) :
1
R̂(µν) − gµν R̂ = 0
2
(119)
Taking into account the modications of R+ -symmetry, we obtain a deformed gravity and the extension
to matter sector seems straightforward.
4.6 Variation of modied Christoel symbol and Ricci tensor under general variation of metric and gauge eld.21
4.6 Variation of modied Christoel symbol and Ricci tensor under general variation
of metric and gauge eld.
Let us nally compute the variation to rst order of the modied Christoel symbol and modied Ricci
tensor (the symmetric part) under general variation of the gauge eld Aµ and the metric gµν . These transformation law will be important for supersymmetry calculations later on. So with this transformation of abelian
gauge eld,
Aµ (x) → Aµ (x) + δAµ
(120)
the variation of the symmetric part of the modied Ricci tensor is given by the following covariant
expression :
1 λδ ˆ
δ R̂(µν) = − Nµν
∇(λ δAδ)
2
(121)
Under a general transformation of the metric,
gµν → gµν + δgµν
(122)
we have the variation of the modied Christoel symbol :
1
δ Γ̂δµν = g δλ Dµ δ gνλ + Dν δ gµλ − Dλ δ gµν
2
(123)
We can express the variation of the modied Ricci tensor with the modied Christoel symbol and
covariant derivative. It gives us a generalized Palatini identity for Ricci tensor (139) :
ˆ σ (δ Γ̂σνµ ) − ∇
ˆ ν (δ Γ̂σσµ )
δ R̂(µν) = ∇
(124)
Conclusion.
Thus, we applied the algebraic procedure of gauging to D = 9 supergravity theory and obtained the same
solutions as in [25]. The next step will be to apply the same process to bigger internal groups like exceptional
Lie groups for D ≤ 5 supergravity theories. In addition, interesting features appeared when we took into
account the coupling of R+ -symmetry to gravity. It would be interesting to check the consistency of this
coupling with the transformation rules of supersymmetry, in reduced supergravity of arbitrary dimension.
Concerning the internship, it gave me the possibility to learn basic methods of dimensional reduction and
gauging of supergravity theories.
A GENERAL RELATIVITY FORMULAS.
22
Appendix.
A General relativity formulas.
The denition of spacetime covariant derivative is :
∇µ V ν
= ∂µ V ν + Γνµλ V λ
∇µ Vν
= ∂µ Vν −
Γλµν Vλ
(125)
(126)
The connection or Christoel symbol, that appears there, obeys two properties : torsion free Γλµν = Γλ(µν)
and metric compatibility ∇ρ gµν = 0. Expanding the metric compatibility and making permutations of the
three indices, we obtain the expression of the connection in function of the rst derivative of the metric :
∇ρ gµν
Γσµν
= ∂ρ gµν − Γλρµ gλν − Γλρν gµλ
1 σρ
=
g (∂µ gνρ + ∂ν gµρ − ∂ρ gµν )
2
(127)
(128)
The Riemann tensor is dened as the curvature of the connection and is computed with the commutator
of two covariant derivatives :
[∇σ , ∇ν ]V ρ = Rρµσν V µ
(129)
Rρ µσν = ∂σ Γρνµ − ∂ν Γρσµ + Γρσλ Γλνµ − Γρνλ Γλσµ
(130)
Expanding the commutator we nd :
The Riemann tensor has several properties :
Rρµσν
= −Rµρσν
(131)
Rρµσν
= −Rρµνσ
(132)
Rρµσν
= Rσνρµ
(133)
Rρ[µσν] = 0
(134)
R[ρµσν] = 0
(135)
The Riemann tensor satises the Bianchi identity :
∇[λ Rρµ]σν = 0
(136)
The contraction of two indices of the Riemann gives us the Ricci tensor, which is a symmetric tensor :
Rµν = Rρ µρν = Rνµ
The Ricci scalar ( or scalar curvature) is the trace of the Ricci tensor :
(137)
23
(138)
R = g µν Rµν
We have the Palatini identity to rst order in variation :
δRρ µσν = ∇σ (δΓρνµ ) − ∇ν (δΓρσµ )
(139)
Instead of using a set of basis vectors derived from a coordinate system ê(µ) = ∂µ , we can use sets of
basis vectors which are not related to coordinate system ê(a) . This new basis is orthonormal and known as
the tetrad or vielbein. The relation between the two basis is given by :
ê(µ) = eµ a ê(a) ê(a) = eµ a ê(µ)
(140)
The vielbein and inverse vielbein satisfy the relation :
eµ a eν a = δνµ eµ a eµ b = δba
(141)
The vielbein allows us to link the metric in world coordinates gµν and in canonical form ηab :
(142)
gµν eµ a eν b = ηab
The covariant derivative applied on quantity with Latin index is given by the spin connection ωµ a b :
∇µ X ab = ∂µ X ab + ωµ a c X c b − ωµ c b X ac
(143)
The usual Christoel symbol and spin connection are linked together with the following formula :
Γνµλ = eν a ∂µ eλ a + eν a eλ b ωµ a b
(144)
a
(145)
ωµ
a λ
a
b
= eν e
ν
b Γµλ
λ
− e b ∂µ eλ
Besides, we have the equivalent of mertric compatibility with the vielbein :
∇µ eν a = ∂µ eν a − Γλµν eλ a + ωµ a b eν b = 0
(146)
Finally, the Riemann tensor is dened as the curvature of the spin connection :
Ra b = dω a b + ω a c ∧ ω c b
For pure gravity, we have the following lagrangian formulation :
Z
√
−gR(D) dD x
SEH =
(147)
(148)
The Einstein equations are obtained by minimizing the action w.r.t the metric (Euler-Lagrange equation
for gravity) :
1
Rµν − gµν R = 0
2
(149)
B MODIFIED CHRISTOFFEL SYMBOL AND SPIN CONNECTION.
24
B Modied Christoel symbol and spin connection.
B.1 Transformation law of Christoel symbol and spin connection.
If we perform a Weyl rescaling of the metric, with a space-time dependent parameter φ(x) :
0
(150)
gµν → gµν = e2φ(x) gµν
the Christoel symbols transform in the following non trivial way :
1 0 σρ
0
0
0
g (∂µ gνρ + ∂ν gµρ − ∂ρ gµν )
2
= Γσµν + ∂µ φ(x)δνσ + ∂ν φ(x)δµσ − gµν g σρ ∂ρ φ(x)
0
Γσµν → Γµνσ
=
(151)
(152)
In order to compute the transformation law of the spin connection, we use the formula given in ?? annexe
A and transform the metric as above and the vielbein as follows :
0
(153)
eµ a → eµ a = eφ(x) eµ a
0µ
µ
e
a
→e
a
0
0
−φ(x) µ
=e
e
(154)
a
So the spin connection transforms as :
0
ωµ a b → ωµ
= ωµ
B.2 Generally covariant and
tion.
0
0
0
ν
− e λb ∂µ eλ
= eν a e λb Γµλ
a
b
a λ
a
b
R+ -gauge
(155)
a
ρa
+ eµ e b ∂λ φ(x) − eµb e
∂ρ φ(x)
(156)
invariant Christoel symbol and spin connec-
In order to have Christoel symbol and spin connection which are generally covariant and invariant w.r.t
metric rescaling and gauge transformation of abelian R+ -symmetry gauge eld,
→ e2φ(x) gµν
(157)
Aµ → Aµ + ∂µ φ(x)
(158)
gµν
we can replace partial derivative by covariant one w.r.t to R+ -gauge symmetry (Dµ = ∂µ − qAµ ) in
the usual expression of the connections ; or we make connections covariant by hands, compensating every
additional term in the transformation law by gauge eld. Both methods are equivalent and we obtain the
modied Christoel symbol and spin connection :
Γ̂σµν
1 σρ
g (Dµ gνρ + Dν gµρ − Dρ gµν )
2
σ
= Γµν − Aµ δνσ − Aν δµσ + gµν gσρ Aρ
=
(159)
(160)
25
ω̂µ a b = eν a eλ b Γ̂νµλ − eλ b Dµ eλ a
= ωµ b − eµ e b Aλ + eµ b e a Aδ
a
a λ
δ
(161)
(162)
Obviously, these two modied connections are invariant under metric scaling and gauge transformation.
C The R+-modied Riemann tensor
C.1 Computation with the modied Christoel symbol.
The modied Riemann tensor is obtained by replacing in the classical formula (130) Christoel symbols
with the hatted one (160) :
R̂ρ µσν = ∂σ Γ̂ρνµ − ∂ν Γ̂ρσµ + Γ̂ρσλ Γ̂λνµ − Γ̂ρνλ Γ̂λσµ
(163)
We want to express the derivative term of the modied Christoel symbol in function of gauge eld and
derivatives of the gauge eld :
∂σ Γ̂ρνµ − ∂ν Γ̂ρσµ = ∂σ Γρνµ − ∂σ Aν δµρ − ∂σ Aµ δνρ + (∂σ gνµ )gρδ Aδ + gνµ (∂σ gρδ )Aδ + gνµ gρδ (∂σ Aδ ) (164)
− ∂ν Γρσµ − ∂ν Aσ δµρ − ∂ν Aµ δσρ + (∂ν gσµ )gρδ Aδ + gσµ (∂ν gρδ )Aδ + gσµ gρδ (∂ν Aδ )
To realize this, we want derivative terms of the metric to disappear ; so using these formulas :
∂σ g ρδ = −g τ δ g λρ ∂σ gτ λ
∂ν g ρδ = −g τ δ g λρ ∂ν gτ λ
(165)
we get :
∂σ Γ̂ρνµ − ∂ν Γ̂ρσµ = ∂σ Γρνµ − ∂ν Γρσµ + (∂ν Aσ − ∂σ Aν )δµρ + ∂ν Aµ δσρ − ∂σ Aµ δνρ + gνµ gρδ ∂σ Aδ − gσµ gρδ ∂ν Aδ
+ (gεν g ρδ Γεσµ + gσµ g τ δ Γρντ + gσµ g λρ Γδνλ )Aδ − (gεσ gρδ Γενµ + gνµ gτ δ Γρστ + gνµ gλρ Γδσλ )Aδ
Then, we have to expand the multiplication of the modied Christoel symbols :
Γ̂ρσλ Γ̂λνµ − Γ̂ρνλ Γ̂λσµ = (Γρσλ − Aσ δλρ − Aλ δσρ + gσλ gρδ Aδ )(Γλνµ − Aν δµλ − Aµ δνλ + gνµ gλτ Aτ )
− (Γρνλ − Aν δλρ − Aλ δνρ + gνλ gρδ Aδ )(Γλσµ − Aσ δµλ − Aµ δσλ + gσµ gλτ Aτ )
Putting everything together, we obtain :
R̂ρ µσν
= Rρ µσν + (∂ν Aσ − ∂σ Aν )δµρ + ∂ν Aµ δσρ − ∂σ Aµ δνρ + gνµ gρδ ∂σ Aδ − gσµ gρδ ∂ν Aδ
+ gσµ gλρ Γδνλ Aδ − gνµ gλρ Γδσλ Aδ + Γλσµ δνρ Aλ − Γλνµ δσρ Aλ
+ Aµ Aν δσρ − Aµ Aσ δνρ + gνµ gρδ Aσ Aδ − gσν gρδ Aν Aδ − gνµ δσρ Aλ gλτ Aτ + gσµ δνρ Aλ gλτ Aτ
If we parametrize the modied Riemann tensor by only one tensorial object M, we have the following
expression :
C THE R+ -MODIFIED RIEMANN TENSOR
26
ρλ
= Rρ µσν + Fνσ δµρ + Mρλ
σµ ∇ν Aλ − Mνµ ∇σ Aλ
ρ
λδ
ρλ
ρλ
+
δσρ Mλδ
µν − δν Mµσ Aλ Aδ + gµν g Aσ Aλ − gσµ g Aν Aλ
(166)
= δµλ δνδ − gµν gλδ
(167)
Fνσ = ∂ν Aσ − ∂σ Aν
(168)
R̂ρ µσν
Mλδ
µν
Another way to express the modied Riemann tensor is to exhibit a part which has the same symmetry
as the usual Riemann tensor. We choose to name this term R̂oρµσν . Then, what we nally get is :
R̂ρµσν
= R̂oρµσν + gρµ Fνσ +
1
gρσ Fνµ + gρν Fµσ + gσµ Fρν + gµν Fσρ
2
(169)
with the following expression for R̂oρµσν :
(170)
λδ
R̂oρµσν = Rρµσν + Qλδ
ρσµν ∇(λ Aδ) + Sρσµν Aλ Aδ
where
Qλδ
ρσµν
λδ
Sρσµν
= gρδ δµλ δνδ + gµν δρλ δσδ − gσµ δρλ δνδ − gνρ δµλ δσδ
λδ
= Qλδ
ρσµν + gµσ gρν − gρσ gµν g
(171)
(172)
C.2 Computation with modied spin connection.
The computation of the modied Riemann tensor based upon the modied spin connection is analoguous
to the previous one. We replace in the usual expression of the Riemann tensor the spin connection by the
modied one.
R̂ρ µσν
= eρ a eµ b R̂a bσν = eρ a eµ b [∂σ ω̂ν a b − ∂ν ω̂σ a b + ω̂σ a c ω̂ν c b − ω̂ν a c ω̂σ c b ]
h
= eρ a eµ b ∂σ (ων a b − eν a eλ b Aλ + eν b eλ a Aλ ) − ∂ν (ωσ a b − eσ a eλ b Aλ + eσ b eλ a Aλ )
(173)
+ (ωσ a c − eσ a eλ c Aλ + eσ c eλ a Aλ )(ων c b − eν c eδ b Aδ + eν b eδ c Aδ )
−
i
(ων a c − eν a eλ c Aλ + eν c eλ a Aλ )(ωσ c b − eσ c eδ b Aδ + eσ b eδ c Aδ )
We use the same kind of formulas as previously ; but this time we have to replace derivative term of the
vielbein :
∂σ eν a = Γλσν eλ a − ωσ a b eν b
λ
∂σ e
a
=
Γλσδ eδ a
− ωσ
b
λ
ae b
∂σ eνa = Γλσν eλa − ωσa b eνb
λa
∂σ e
=
Γλσδ eδa
a λb
− ωσb e
(174)
(175)
After some some calculus and simplications, and using the property of antisymmetry :
ωµa b = −ωµ b a
(176)
C.2 Computation with modied spin connection.
27
we nally obtained the expression of the modied Riemann tensor in the case of the spin connection :
R̂ρ µσν
ρλ
= Rρ µσν + Mρλ
σµ ∇ν Aλ − Mνµ ∇σ Aλ
ρ
λδ
ρλ
ρλ
+
δσρ Mλδ
µν − δν Mµσ Aλ Aδ + gµν g Aσ Aλ − gσµ g Aν Aλ
(177)
RÉFÉRENCES
28
Références
[1] M. B. Green and J. H. Schwarz. Supersymmetrical string theories. Phys. Lett., B 109(1982) :444448.
[2] M. B. Green, J. H. Schwarz, and E. Witten. Superstring Theory. Vol. 1, 2. Cambridge, UK. Univ. Pr.
Cambridge Monographs On Mathematical Physics, 1987.
[3] J. Polchinski. String Theory. Vol. 1, 2. Cambridge, UK. Univ. Pr., 1998.
[4] M. B. Green and J. H. Schwarz. All possible symmetries of the S matrix. Phys. Rev., 159(1967) :1251.
[5] D. Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara. Progress toward a theory of supergravity. Phys.
Rev. D, 13(1976) :32143218.
[6] S. Deser and B. Zumino. Consistent supergravity. Phys. Lett., B 62(1976) :335.
[7] D. Z. Freedman and P. van Nieuwenhuizen.
14(1976) :912.
Phys. Rev. D,
Properties of supergravity theory.
[8] A. H. Chamseddine and P. C. West. Supergravity as a gauge theory of supersymmetry. Nucl. Phys.,
B129(1977) :39.
[9] J. H. Schwarz. Superstring theory. Phys. Rept., 89(1982) :223332.
[10] M. B. Green, J. H. Schwarz, and L. Brink. N=4 Yang-Mills and N=8 supergravity as limits of string
theories. Nucl. Phys., B198(1982) :474492.
[11] Th. Kaluza. Zum Unitätsproblem in der Physik. Sitzungsber. Preuss. Akad. Wiss. Phys. Math., Klasse
1 996(1982).
[12] O. Klein. Quantentheorie und fünfdimensionale Relativitätstheorie. Z. F. Physik, 37(1926) :895.
[13] O. Klein. Nature, 118(1926) :516.
[14] J. Strathdee A. Salam. On Kaluza-Klein theory. Ann. Phys., 141(1982) :316.
[15] P. G. O. Freund and M. A. Rubin. Conservation of isotopic spin and isotopic gauge invariance. Phys.
Rev., 96(1954) :191.
[16] Y. M. Cho. Higher dimensional unications of gravitation and gauge theories.
16(1975) :2029.
J. Math. Phys.,
[17] Y. M. Cho and P. G. O. Freund. Unied geometry of internal space with spacetime. Phys. Rev. D,
12(1975) :3789.
[18] Y. M. Cho and P. G. O. Freund. Non-abelian gauge elds as Nambu-Goldstone elds. Phys. Rev. D,
12(1975) :1711.
[19] E. Cremmer and J. Scherk. Spontaneous compactication of extra space dimensions. Nucl. Phys.
[20] E. Witten. Search for a realistic Kaluza-Klein theory. Nucl. Phys.
[21] W. Nahm. Supersymmetries and their representations. Nucl. Phys.
[22] E. Cremmer, B. Julia, and J. Scherk. Supergravity theory in 11 dimensions. Phys. Lett.
[23] P. G. O. Freund and M. A. Rubin. Dynamics of dimensional reduction. Phys. Lett., B 97(1980) :233.
[24] J. Scherk and J. Schwarz. How to get masses from extra dimensions. Nucl. Phys.
[25] E. Bergshoe, Tim de Wit, Ulf Gran, Roman Linares, and Diederik Roest. Non-abelian gauged supergravities in nine dimensions. JHEP, 10(2002) :061.
[26] B. de Wit, H. Samtleben, and M. Trigiante. Gauging maximal supergravities. Fortschritte der Physik,
52(2004) :489.
[27] B. de Wit and H. Samtleben. Gauged maximal supergravities and hierarchies of nonabelian vector-tensor
systems. Fortschritte der Physik, 52(2005) :442.
RÉFÉRENCES
29
[28] B. de Wit, H. Samtleben, and M. Trigiante. On lagrangians and gaugings of maximal supergravities.
Nucl. Phys., B 655(2003) :93.
[29] D. Roest. M-theory and gauge supergravities. PhD thesis, 2005, Groningen.
[30] M. Blau, J. Figueroa-O'Farrill, and C. Hull. A new maximally symmetric background of IIB superstring
theory. JHEP, 01(2002) :047.
[31] R. Coquereaux and A. Jadczyk. Riemannian geometry, ber bundles, Kaluza-Klein theories and all that.
World Sci. Lect. Notes Phys Vol. 16, 1988.
[32] B. de Wit. Supergravity, in Unity from Duality : Gravity, Gauge Theory and Strings. Springer, 2003.