Equivalence between the Lovelock–Cartan action and a constrained gauge theory
Eur. Phys. J. C
Equivalence between the LovelockCartan action and a constrained gauge theory
O. C. Junqueira 2
A. D. Pereira 1
G. Sadovski 2
T. R. S. Santos 2
R. F. Sobreiro 2
A. A. Tomaz 0 2
0 CBPFCentro Brasileiro de Pesquisas Físicas , Rua Dr. Xavier Sigaud 150, Rio de Janeiro, RJ 22290180 , Brazil
1 Departamento de Física Teórica, UERJUniversidade Estadual do Rio de Janeiro , Rua São Francisco Xavier 524, Maracanã, Rio de Janeiro, RJ 20550013 , Brazil
2 Instituto de Física, UFFUniversidade Federal Fluminense, Campus da Praia Vermelha , Av. General Milton Tavares de Souza s/n, Niterói, RJ 24210346 , Brazil
We show that the fourdimensional LovelockCartan action can be derived from a massless gauge theory for the S O(1, 3) group with an additional BRST trivial part. The model is originally composed of a topological sector and a BRST exact piece and has no explicit dependence on the metric, the vierbein or a mass parameter. The vierbein is introduced together with a mass parameter through some BRST trivial constraints. The effect of the constraints is to identify the vierbein with some of the additional fields, transforming the original action into the LovelockCartan one. In this scenario, the mass parameter is identified with Newton's constant, while the gauge field is identified with the spin connection. The symmetries of the model are also explored. Moreover, the extension of the model to a quantum version is qualitatively discussed.

In [1], Mardones and Zanelli proposed the most general
gravity action depending on the curvature and torsion without the
use of the Hodge dual operation for any spacetime
dimension. This result generalizes the Lovelock theorem [2] which
states, for any dimension, the most general gravity action
depending only on the curvature. The Zanelli–Mardones
result was baptized as Lovelock–Cartan theory of gravity.
The main motivations of this result, in spite of the fact that
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torsion degrees of freedom have never been observed in
gravity, is that torsion might be relevant at the quantum level and
the fact that curvature and torsion are at the same level from
the geometry point of view [1,3–9].
The fact that curvature and torsion are independent
quantities in Lovelock–Cartan (LC) theories enables the use of
the Einstein–Cartan formalism of gravity [3,10–12], which
is based on the vierbein and spin connection as
fundamental and independent variables. In this approach, gravity can
be interpreted as a kind of gauge theory where the gauge
symmetry is identified with the spacetime local isometries.
This equivalence opens the possibility of the application of
wellknown quantization techniques of gauge theories.
In spite of how similar such formulations of gravity are
with respect to gauge theories, their quantization as
fundamental theories still lacks. Essentially, these theories share
the same problems as pure metric and Palatini theories of
gravity [13–15]. In particular, the perturbative
renormalizability or unitarity problems remain [16], as well as
background independence [17,18] and so on.
In the present work we provide the construction of a gauge
theory which encodes the Lovelock–Cartan dynamics. The
gauge theory is constructed for the gauge group S O(1, 3)
over a fourdimensional manifold. In contrast to the gravity
theories in the Einstein–Cartan formalism, the gauge degrees
of freedom and the spacetime are independent, by
construction. The original action is massless and is formed by a
topological term and a BRST exact one and the fundamental fields
are the gauge connection, the ghost field and a quartet system
formed by two BRST doublets. Moreover, the gauge theory
is metric independent and also independent of the vierbein
field. On the side of the manifold, we provide no
dynamics to it. It is just a generic manifold where the gauge
theory lives on. Hence, with the help of extra BRST doublets,
we introduce an algebraic quadratic coupling with the
vierbein of the manifold. The extra doublets can be visualized
as Lagrange multipliers for extra constraints. The effect of
such constraints is to transform the gauge theory coupled to
the vierbein into the fourdimensional LC action. Essentially,
the constraints identify the gauge theory degrees of freedom
with spacetime, providing the LC dynamics to it.
The model enjoys a rich set of symmetries that can be
written as consistent Ward identities. This feature would be
important in a quantum version of the model. A possibility is
to quantize the gauge theory coupled to the classical vierbein.
The classical limit of such model would be the LC action. In
this scenario, the dynamics of spacetime would be ruled by a
quantum gauge theory composed of a topological piece and a
BRST exact one. Nevertheless, in this work, we remain at the
classical level. The formalization of the quantum version of
the model is left for future investigation due to the intricacies
of renormalizability and gauge fixing of metric free theories.
The article is organized as follows: In Sect. 2 we
provide a small overview of the Lovelock–Cartan action in four
dimensions. In Sect. 3 we construct the massless gauge
theory composed of a topological term and a BRST exact one.
We also provide a complete discussion of the symmetries of
the model in terms of Ward identities. In Sect. 4 we
introduce the massive constraint carrying the vierbein classical
field and discuss how the constraint leads to the LC action.
In addition, we generalize all Ward identities of the
previous section. In Sect. 5 we provide an extra discussion of the
BRST symmetry and a detailed, yet qualitative, discussion
of the quantum version of the model. Finally, in Sect. 6 we
display our conclusions.
2 Overview of the Lovelock–Cartan action in four dimensions The Lovelock–Cartan action [1] in four dimensions is given by
S0 =
(z1 abcd Rab Rcd + z2 Rab Rab)
+ z5 Rabea eb + z6T a Ta ],
over a fourdimensional manifold M . The quantities Rab =
dωab + ωacωcb and T a = Dea = dea + ωabeb are,
respectively, the curvature and torsion 2forms. The basic fields are
the vierbein 1form ea and the connection 1form ωab. All
parameters zi are dimensionless, while μ carries mass
dimension 1. Moreover, the action (2.1) also contains the invariant
tensors abcd and ηab.
The first term in (2.2) is recognized as the Gauss–Bonnet
topological term, while the second term is the Pontryagin
topological term. For z6 = −z5, the last two terms in (2.3) are
also reduced to a topological term, i.e., the Nieh–Yan term:
T a Ta − Rabea eb = d(ea T a ). On the other hand, because
DT = Re, these terms are actually the same up to
surface terms. Thus, generically, S0 is topological, while Sμ
is dynamical. Obviously, the first term in the action Sμ is
the Einstein–Hilbert action while the second term in Sμ is
the a cosmological constant term. Hence, μ2z3 is identified
with Newton’s constant while μ2z4/z3 with the cosmological
constant.
The action (2.1) is invariant under gauge transformations
for the group S O(1, 3) whose infinitesimal version are
δωab = Dξ ab = dξ ab + ωacξ cb − ωbcξc a ,
which describe the local spacetime isometries associated
with the strong equivalence principle. The quantity ξ ab is the
infinitesimal gauge parameter and D the exterior covariant
derivative in the adjoint representation. These
transformations correspond to transformations in the cotangent space at
a point x ∈ M . Moreover, since these transformations leave
the manifold coordinates unchanged, they can be interpreted
as gauge transformations. Thus, gravity can be interpreted as
a special type of gauge theories for which the fields have a
geometrical meaning.1 These transformation laws establish
that, from the gauge theory point of view, the gauge field of
the model is the spin connection ω, while the vierbein e is a
matter field.
3 A massless gauge theory
As discussed at the Introduction, the aim of the paper is to
show that the LC action (2.1) can be obtained from a trivial
theory (in the sense of containing just a topological and BRST
exact terms) by the introduction of a suitable algebraic linear
constraint. This section is devoted to the construction of such
trivial action.
3.1 Fundamental ingredients and action
We consider a massless S O(1, 3) gauge theory in a
fourdimensional manifold. The natural ingredients are the
fundamental fields of the theory, namely the gauge field ωab
and the ghost field cab, and the invariant tensors abcd and
1 The fields directly determine the spacetime dynamics.
ηab. The most general action with vanishing ghost number,
dimension four, polynomial on the fields and their derivatives
and not explicitly dependent on the metric or the vierbein is
the topological part of the LC action, namely S0. Here, for
consistency, we define it as
S0 =
(z1 abcd Rab Rcd + z2 Rab Rab),
where zi are dimensionless parameters which eventually will
be identified with the original parameters zi of the
topological action (2.2). Because the model is massless by
construction, there is no room for Sμlike terms. Hence, the vierbein
independence is ensured at this point. The fundamental fields
transform under BRST symmetry as
where s is the nilpotent BRST operator.
We also define a BRST quartet system of dimensionless
1forms, namely,
scab = −cacccb,
The 1forms η¯a and ηa have fermionic statistics while σ¯ a and
σ a have a bosonic one. Moreover, it is clear from the
transformations (3.3) that the quartet is a double BRST doublet.
This means that they are nonphysical fields, belonging to the
trivial sector of the BRST cohomology [19]. The existence
of the quartet system allows the introduction of an extra term
to the action,
Striv = s
+ z5 Rabη¯a σ b + z6 Dη¯a Dσa
+ z7[η¯a σa (σ¯ bσb − η¯bηb) + η¯a σbη¯a ηb]}
+ z4(σ¯ a σ b + η¯a ηb)(σ¯ cσ d + η¯cηd )]
+ z5 Rab(σ¯ a σ b + η¯a ηb) − z6( Dσ¯ a Dσa − Dη¯a Dηa )
+ z7[(σ¯ a σa + η¯a ηa )(σ¯ bσb − η¯bηb)
+ 2η¯a ηb(σ¯ a σb − σ¯bσ a ) + η¯a ηbη¯a ηb]}, (3.4)
which is trivial with respect to the BRST cohomology. The
parameters zi are dimensionless parameters.2
The action
ST = S0 + Striv
2 Eventually, they will be associated with the Lovelock–Cartan param
eters zi appearing in (2.1).
is dynamically empty because the topological term does not
contribute to the field equations and the BRST exact
sector ensures that Striv is dynamically trivial. Moreover, the
action (3.5) is highly nonperturbative3 due to the absence
of quadratic terms. We also remark that this action is
independent of the metric, the vierbein and the ghost fields. As
a consequence the independence of the vierbein field in the
action can be taken as an independence between the gauge
symmetry and the manifold isometries. This property ensures
that the dynamics of spacetime is not related to any of the
fields in (3.5).
3.2 Symmetries and Ward identities
All continuous symmetries of the action ST can be
characterized in a functional way through suitable Ward identities. It
is useful to define a set of BRST invariant sources in order to
control the nonlinear character of the BRST transformations
of the fields through the external action
Sext = s
( ab ωba +L ab cba − Xa η¯a − X¯ a ηa +Ya σ¯ a +Y¯a σ a )
+ Xa (σ¯ a − cba η¯b) − X¯ a cba ηb − Ya cba σ¯ b
+Y¯a (ηa − cba σ b)],
The full action is then
0 = ST + Sext.
We now can list all Ward identities.
• Slavnov–Taylor identity:
S( 0) = 0,
S( 0) =
δ 0 δ 0 δ 0 δ 0 δ 0 δ 0
δ ab δωba + δ L ab δcba + δ Xa δη¯a
+ δ X¯ a δηa + δYa δσ¯ a + δδY¯a0 δδσ a0 .
δ 0 δ 0 δ 0 δ 0
3 It is not difficult to check that the action (3.5), as it stands, has no
quadratic terms in the fields. As a consequence, there is no free theory
to be defined (and no treelevel propagators). Hence, a perturbative
expansion around a free theory is not at our disposal. In fact, all
nonvanishing terms in (3.5) are interacting terms. A theory of this type is
said to be highly nonperturbative. Of course, one can always define
background configurations and enforce a perturbative regime around
these configurations.
• Ghost equation:
c b b c
(−La cc + Lc ca −
cbωac + Xa η¯b + X¯ a ηb − Ya σ¯ b − Y¯a σ b)
is a linear breaking.
• Vierbein equation:
R(i) 0 =
• Rigid supersymmetries:
where i ∈ {1, 2, 3, 4}. The rigid supersymmetric
operators are
R(1) = σ a δηδa − η¯a δσδ¯ a − Y a δ a δ
δ X a − X¯ δY¯ a
= σ¯ a δηδ¯a + ηa δσδ a − X a δYδ a + Y¯ a δ ,
δ X¯ a
= σ¯ a δηδa − η¯a δσδ a − Y¯ a δ Xδ a − X¯ a δYδ a ,
while the only nonvanishing
which are linear in the fields.
• Rigid fermionic equations:
Q(i) 0 =
= −η¯a δηδa + X¯ a δ Xδ a ,
Q(2) = ηa δηδ¯a − X a δ Xδ¯ a
and the only nonvanishing breaking is
which is linear in the fields.
• First U 4(1) charge equation:
Q0 0 = 0,
Q0 = σ a δσδ a − σ¯ a δσδ¯ a + ηa δηδa − η¯a δηδ¯a
+ X a δ Xδ a − X¯ a δ Xδ¯ a + Y a δYδ a − Y¯ a δ .
δY¯ a
Equation (3.20) expresses the existence of a quantum
number associated with a U 4(1) symmetry among the
quartet fields.
• Second U 4(1) charge equation:
Q¯ 0 0 = −2(Xa σ¯ a + Y¯a ηa ),
Q¯ 0 = σ a δσδ a − σ¯ a δσδ¯ a − ηa δηδa + η¯a δηδ¯a
− X a δ Xδ a + X¯ a δ Xδ¯ a + Y a δYδ a − Y¯ a δ .
δY¯ a
Equation (3.22) expresses the existence of a second
quantum number associated with the other U 4(1)
symmetry among the quartet fields. Combination of (3.20) and
(3.22) results in
Q(eif)f 0 = (−1)i−1(Xa σ¯ a + Y¯a ηa ),
Q(e2ff) = Q(e1ff) + Q¯ 0
= ηa δηδa − η¯a δηδ¯a + X a δ Xδ a − X¯ a δ Xδ¯ a
= σ a δσδ a − σ¯ a δσδ¯ a + Y a δYδ a − Y¯ a δYδ¯ a . (3.25)
For completeness, we display the quantum numbers of the
fields (including the vierbein) in Table 1 and the quantum
numbers of the sources in Table 2.
c
¯
σ
¯
−1
1
0
3
3
η
¯
Table 1 Quantum numbers of the fundamental fields and the quartet
system
S(F ) =
Table 2 Quantum numbers of the sources
The commutation relations between the Ward operators
can also be obtained by a straightforward computation.
Starting with the Slavnov–Taylor operator, for instance, let F be a
general functional of even ghost and form number; we define
the Slavnov–Taylor operator action on F as
δF δF δF δF δF δF
δ ab δωba + δ L ab δcba + δ Xa δη¯a
δF δF δF δF δF δF
+ δ X¯ a δηa + δYa δσ¯ a + δY¯a δσ a
Its linearized version reads
SF =
δF δ δF δ δF δ
δ ab δωba + δωba δ ab + δ L ab δcba
δF δ δF δ δF δ
+ δcba δ L ab + δ Xa δη¯a + δη¯a δ Xa +
δF δ δF δ δF δ
+ δ X¯ a δηa + δηa δ X¯ a + δYa δσ¯ a
+ δδσ¯Fa δ Yδa + δδYF¯a δσδ a + δδσFa δ Yδ¯a .
From (3.15), (3.26) and (3.27) we get
R(i)S(F ) + SF R(i)(F ) = 0.
We also have the following commutation relations:
( R(i))2 = 0, { R(1), R(2)} = Q0,
{ R(1), R(3)} = 2 Q(1), { R(1), R(4)} = 0, { R(2),
R(3)} = 0, { R(2), R(4)} = 2 Q(2), { R(3), R(4)} = −Q¯ 0.
(3.29)
The rich set of Ward identities ensures that 0 is the most
general local classical action, polynomial in the fields and
their derivatives, with vanishing ghost number and
independent on the metric and the vierbein. Hence, the Ward
identities ensure the triviality of the model as well as the fact that
the model has no relation with spacetime dynamics.
3.3 A remark about the BRST triviality
The quartet system (3.3) is composed of BRST doublets, and
thus, these fields live at the trivial sector of the BRST
cohomology. Hence, Striv does not affect the physical dynamical
content of the topological action S0. Nevertheless, we can
decompose s as
s = so + δ, so2 = δ2 = {so, δ} = 0,
Moreover, it is easy to check that
Striv = so (something),
Thus, although s and δ define the quartet system as trivial,
they are not trivial with respect to so. This means that these
fields can be interpreted as physical under the so cohomology.
The LC action can be recovered from 0 with the
introduction of a set of suitable constraints. The only demand is that
the Ward identities could be broken only by linear terms in
the fields. Such a requirement is essential for the extension of
such Ward identities to the quantum level. In fact, it is easy
to see that these constraints are given by
where m is a mass parameter4 and ea the vierbein field, which
is taken to be a classical field.5 Within this construction, the
vierbein must be a BRST invariant quantity,
It is clear that these constraints introduce not only the
vierbein, but also a mass scale.
To employ the constraint without spoiling the BRST
triviality of 0, we introduce two extra BRST quartet systems,
namely,
sea = 0.
which will work as Lagrange multipliers for the constraints
(4.1). The quantum numbers of the new quartet systems are
displayed at Table 3.
Then the constraint action is given by
Sc = s
[θ¯a (σ a − mea ) + θa (σ¯ a − mea ) + γ¯ a ηa + γa η¯a ]
[ λ¯a (σ a − mea ) + λa (σ¯ a − mea ) + θ¯a (ηa − cba σ b)
− γa (σ¯ a −cb η¯ )+ρ¯a ηa +ρa η¯a −θa cb σ¯ +γ¯a cba ηb].
a b a b
4 This parameter will, eventually, be identified with μ appearing in
(2.1).
5 Although we are at classical level, we mean that the vierbein would
remain classical in a possible quantum scenario.
4.1 Constraint action
4 Introducing a massive constraint and the vierbein
Table 3 Quantum numbers of the constraint fields
γ
¯
ρ
¯
The action of interest is then
which is totally equivalent to the LC action (2.1) if a proper
relation between {zi , m} and {zi , μ} is obeyed. The first step
to check this is to set all external sources in (4.6) to zero.
Hence, we perform the elimination of the auxiliary fields
defined in (4.3) and (4.4) by the implementation of their
field equations (see equations from (4.21) to (4.28) in the
next subsection). This will lead to an action that is totally
equivalent to the Lovelock–Cartan action (2.1) if the gauge
parameters {zi , m} and the LC parameters {zi , μ} are related
accordingly to
z 1 = z1, z 2 = z2, z 3m2 = z3μ2,
z 4m4 = z4μ4, z 5m2 = z5μ2,
On the other hand, at quantum level the set {zi , m} might
need renormalization before being identified with the LC
parameters. A discussion of the quantum scenario can be
found in Sect. 5.1.
4.2 Generalized Ward identities The set of Ward identities enjoyed by the action (4.6) are listed here:
• Slavnov–Taylor identity:
S( ) = 0,
S( ) =
δ δ δ δ δ δ
δ ab δωba + δ Lab δcba + δ Xa δη¯a
δ δ δ δ δ δ
+ δ X¯ a δηa + δYa δσ¯ a + δY¯a δσ a
• Ghost equation:
a =
ab − m
remains a linear breaking.
• Vierbein equation:
which is linearly broken.
• Rigid supersymmetries:
where i ∈ {1, 2, 3, 4}. The rigid supersymmetric
operators are6
R(3) = R(3)
a δ a δ a δ
δθ a − θ¯ δγ¯ a + λ¯ δρ¯a − ρ
which are linear in the fields.
6 The symmetry R(4) is quadratically broken and, thus, it is not an inter
esting identity for the model. As a consequence, generalized versions
of Q(2) and Q¯ 0 are not at our disposal in the full model. Obviously,
since there is no generalization of the Q¯ 0 symmetry, there is no place
for generalizing Q(eif)f as well.
We can conclude at this point that the effect of the
introduction of the constraints is that most of the Ward identities
are linearly broken. Moreover, Eqs. (3.28), (3.29) and (3.30)
Q0 = σ a δσδ a − σ¯ a δσδ¯ a + ηa δηδa − η¯a δηδ¯a
+ X a δ Xδ a − X¯ a δ Xδ¯ a + Y a δYδ a − Y¯ a δ
δY a
+ ρa δρδa − ρ¯a δρδ¯a + γ
+ λa δλδa − λ¯a δλδ¯a + θ a δθδa − θ¯a δθδ¯a
a δ a δ
δγ a − γ¯ δγ a . (4.20)
Equation (4.19) still expresses the existence of a
quantum number associated with a U 4(1) symmetry among
the quartet fields, even though the symmetry is linearly
broken.
• Field equations:
• Rigid fermionic equation:
Q(1) = Q(1) + θ¯a + ρ¯a
but it is also linear in the fields.
• U 4(1) charge equation:
are easily generalized by rejecting all relations of R(4), Q(2)
and Q¯ 0.
5 Discussion
5.1 Quantization attempts
In light of gauge theories, let us take a closer look at the action
(2.1). First of all, we see that there are no quadratic terms7
in (2.1), only interacting terms in the fields ω and e. This is a
problem if one plans to quantize the LC action because this
property ruins the wellestablished perturbative program of
QFT, unless a background is previously chosen. Background
independence though requires that such a choice is arbitrary.
Another problem to be faced is the gauge fixing. A typical
gauge fixing is obtained by fixing the divergence of the gauge
field. However, to define the divergence of a field, the Hodge
dual operator is required. Hence, an explicit dependence on
the metric should be introduced.
The advantage in working with the action instead is
that the model can be interpreted as a typical gauge theory
for the gauge field ω and the fields defined in (3.3). Hence,
the theory is composed of a topological piece and a BRST
trivial sector. The addition of the constraint (3.5) introduces
a coupling with the vierbein in such a way that (some of) the
BRST trivial fields are identified with the vierbein. Thus, the
interpretation of the field ω as the spin connection is natural.
A quantum version of such model would also be highly
nonperturbative since there are no quadratic terms of ω in
. Moreover, the terms in the constraint action Sc are
algebraic, i.e., there are no kinetic terms. To face this problem one
should, perhaps, employ the strategies developed in [20]. The
authors in [20] claim that a BRST exact gauge fixing can be
added to the action, even though it depends explicitly on the
spacetime metric. The reason is that, since the gauge fixing
is BRST exact, physical observables do not depend on the
metric. In addition, the gauge fixing term provides quadratic
terms for ω, making a perturbative analysis possible.
Another important property of the model is the existence
of a rich set of Ward identities, which are broken linearly, at
most. This is a very welcome property, which ensures their
validity at the quantum level [19]. In particular, the vierbein
equation (4.12) ensures that the vierbein should not appear at
the counterterm. This last feature is quite strong and ensures
that the constraint could be employed at quantum level while
maintaining e classical.
To understand what a quantum version of the model would
mean, let us consider a gauge fixed action = + s g f ,
enjoying the above discussed properties. The partition
function can be written as
7 This is a problem that also appears in the case of the pure Einstein–
Hilbert action.
Z =
where D ≡ Dω Dc Dσ¯ Dσ Dη¯ Dη Dθ¯ Dθ Dρ¯ Dρ Dγ¯ Dγ
Dc¯ Db, while c¯ is the Faddeev–Popov antighost field and
b the Lautrup–Nakanishi field enforcing the referred gauge
fixing. The partition function defines the quantum version
of the model coupled to the vierbein classical field. Since
there is no previous dynamics for the spacetime, it is the
model itself that defines the spacetime dynamics. Moreover,
the classical limit of such a model would provide a classical
gravity limit, which is exactly the Lovelock–Cartan action
(2.1). Hence, we have constructed a topological and BRST
exact quantum model that assigns dynamics to spacetime
and has the Lovelock–Cartan case as its classical limit. In
this limit, the identification of ω with the spin connection
and μ with Newton’s constant is natural.
5.2 Further symmetry aspects
Let us assume the existence of consistent BRST exact gauge
fixed action which allows the construction of a suitable
partition function Z and the usual perturbative tools [20].
Moreover, it is also reasonable to assume that the gauge fixing
would not spoil any of the Ward identities.8 In addition, since
the field equations (4.21) are exact or linearly broken, the
validity of constraints (4.1) at quantum level is ensured.
One effect of the constraint (4.6) is that, besides the BRST
symmetry being preserved, its decomposition (3.31) is not.
In fact, it is easy to check that
which is consistent with Eq. (3.31) and s
quence, we have the relations
= 0. As a
conse= so Sc =
s Z = 0,
so Z = 0,
= mea ,
= mea ,
= 0,
The field equations (4.21) can be written in the form of
expectation values9 as
8 See the previous section.
9 The expectation values are taken with respect to the functional mea
sure D as defined in Sect. 5.1.
= 0,
= 0.
= −mea ,
= 0,
= 0,
= 0,
= mea ,
= 0,
= 0,
= 0.
Now, combining (5.4), (5.5) and (5.6), we get
= =
(3.32) and (3.33), we have
Due to (5.3), the BRST operator s commutes with the
expectation values in (5.4). Hence, due to the first four
relations in (5.4) and the s invariance of the vierbein we have
sηa sσ a
= 0. Moreover, from
From (5.7) and (5.8) we understand that the breaking of the
symmetries δ and so compensate each other, as in (5.2), while
the s symmetry remains a symmetry of the model. In
addition, it is clear that these breaks are directly related with the
vierbein and the mass parameter since so and δ exact terms
attain a nonvanishing vacuum expectation value equal to
mea .
6 Conclusions
We have constructed a massless gauge theory coupled with
the vierbein field through algebraic constraints quadratic in
the fields. Essentially, the action is composed of a
topological and a BRST exact term. The constraints also carry a mass
parameter which, eventually, is identified with Newton’s
constant. The interpretation of the model is that the gauge
theory induces a dynamics for the spacetime, resulting in the
Lovelock–Cartan action [1].
The constraints, being quadratic in the fields, ensure the
validity of a rich set of symmetries. These symmetries, in
the form of Ward identities, motivates the construction of a
quantum version of the model. However, due to intricacies
such as the gauge fixing problem and quantum stability, the
formal analysis of the quantization of the model is left for a
future paper.
Another possibility to be investigated is the generalization
of the model to other dimensions, at least at classical level.
Finally, an extra remark is that, at the classical level, the
model can be simplified to consider only Lovelock gravity [2]
or even general relativity. However, in a quantum version of
such simplified models, it seems that the Ward identities are
not strong enough to block the other terms of the Lovelock–
Cartan. So, they would probably appear in the counterterm,
requiring their introduction in the bare action.
Acknowledgements The Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPqBrazil) and the Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) are acknowledge
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