Equivalence between the Lovelock–Cartan action and a constrained gauge theory
Abstract
We show that the four-dimensional Lovelock–Cartan action can be derived from a massless gauge theory for the SO(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 Lovelock–Cartan 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.
1 Introduction
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 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, 4, 5, 6, 7, 8, 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, 11, 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 well-known 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, 14, 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 SO(1, 3) over a four-dimensional 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 four-dimensional 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 first term in (2.2) is recognized as the Gauss–Bonnet topological term, while the second term is the Pontryagin topological term. For \(z_6=-z_5\), the last two terms in (2.3) are also reduced to a topological term, i.e., the Nieh–Yan term: \(T^aT_a-R_{ab}e^ae^b=d(e_aT^a)\). On the other hand, because \(DT=Re\), these terms are actually the same up to surface terms. Thus, generically, \(S_0\) is topological, while \(S_\mu \) is dynamical. Obviously, the first term in the action \(S_\mu \) is the Einstein–Hilbert action while the second term in \(S_\mu \) is the a cosmological constant term. Hence, \(\mu ^2z_3\) is identified with Newton’s constant while \(\mu ^2z_4/z_3\) with the cosmological constant.
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
3.2 Symmetries and Ward identities
- Slavnov–Taylor identity:where$$\begin{aligned} {\mathcal {S}}(\Sigma _0)=0, \end{aligned}$$(3.9)$$\begin{aligned} {\mathcal {S}}(\Sigma _0)= & {} \int \left( \frac{\delta \Sigma _0}{\delta \Omega _a^{b}} \frac{\delta \Sigma _0}{\delta \omega ^a_{b}}+\frac{\delta \Sigma _0}{\delta L_a^{b}}\frac{\delta \Sigma _0}{\delta c^a_{b}}+\frac{\delta \Sigma _0}{\delta X_a}\frac{\delta \Sigma _0}{\delta \bar{\eta }^a}\right. \nonumber \\&\left. +\,\frac{\delta \Sigma _0}{\delta \bar{X}_a}\frac{\delta \Sigma _0}{\delta \eta ^a}{+}\frac{\delta \Sigma _0}{\delta Y_a}\frac{\delta \Sigma _0}{\delta \bar{\sigma }^a} {+}\frac{\delta \Sigma _0}{\delta \bar{Y}_a}\frac{\delta \Sigma _0}{\delta \sigma ^a}\right) . \end{aligned}$$(3.10)
- Ghost equation:where$$\begin{aligned} \int \frac{\delta \Sigma _0}{\delta c^a_{\phantom {a}b}}=\Delta _a^{\phantom {a}b}, \end{aligned}$$(3.11)is a linear breaking.$$\begin{aligned} \Delta _a^{b}= & {} \int (-L_a^{c}c^b_{c}+L_c^{b}c^c_{a}- \Omega _a^{c}\omega ^b_{c}\nonumber \\&+\,\Omega _c^{b}\omega ^c_{a}+X_a\bar{\eta }^b+ \bar{X}_a\eta ^b-Y_a\bar{\sigma }^b-\bar{Y}_a\sigma ^b)\nonumber \\ \end{aligned}$$(3.12)
- Vierbein equation:$$\begin{aligned} \frac{\delta \Sigma _0}{\delta e^a}=0. \end{aligned}$$(3.13)
- Rigid supersymmetries:where \(i\in \{1,2,3,4\}\). The rigid supersymmetric operators are$$\begin{aligned} R^{(i)}\Sigma _0=\Delta ^{(i)}, \end{aligned}$$(3.14)while the only non-vanishing \(\Delta ^{(i)}\) are$$\begin{aligned} R^{(1)}= & {} \sigma ^a\frac{\delta }{\delta \eta ^a}-\bar{\eta }^a\frac{\delta }{\delta \bar{\sigma }^a}-Y^a\frac{\delta }{\delta X^a}-\bar{X}^a\frac{\delta }{\delta \bar{Y}^a},\nonumber \\ R^{(2)}= & {} \bar{\sigma }^a\frac{\delta }{\delta \bar{\eta }^a}+\eta ^a\frac{\delta }{\delta \sigma ^a}-X^a\frac{\delta }{\delta Y^a}+\bar{Y}^a\frac{\delta }{\delta \bar{X}^a},\nonumber \\ R^{(3)}= & {} \bar{\sigma }^a\frac{\delta }{\delta \eta ^a}-\bar{\eta }^a\frac{\delta }{\delta \sigma ^a}-\bar{Y}^a\frac{\delta }{\delta X^a}-\bar{X}^a\frac{\delta }{\delta Y^a},\nonumber \\ R^{(4)}= & {} \sigma ^a\frac{\delta }{\delta \bar{\eta }^a}+\eta ^a\frac{\delta }{\delta \bar{\sigma }^a}+Y^a\frac{\delta }{\delta \bar{X}^a}-X^a\frac{\delta }{\delta \bar{Y}^a}, \end{aligned}$$(3.15)which are linear in the fields.$$\begin{aligned} \Delta ^{(1)}= & {} X_a\bar{\eta }^a-\bar{X}_a\eta ^a-Y_a\bar{\sigma }^a+\bar{Y}_a\sigma ^a,\nonumber \\ \Delta ^{(4)}= & {} -2X_a\eta ^a, \end{aligned}$$(3.16)
- Rigid fermionic equations:where$$\begin{aligned} Q^{(i)}\Sigma _0=\Lambda ^{(i)}, \end{aligned}$$(3.17)and the only non-vanishing breaking is$$\begin{aligned} Q^{(1)}= & {} -\bar{\eta }^a\frac{\delta }{\delta \eta ^a}+\bar{X}^a\frac{\delta }{\delta X^a},\nonumber \\ Q^{(2)}= & {} \eta ^a\frac{\delta }{\delta \bar{\eta }^a}-X^a\frac{\delta }{\delta \bar{X}^a}, \end{aligned}$$(3.18)which is linear in the fields.$$\begin{aligned} \Lambda ^{(1)}=\bar{X}_a\bar{\sigma }^a-\bar{Y}_a\bar{\eta }^a, \end{aligned}$$(3.19)
- First \(U^{4}(1)\) charge equation:where$$\begin{aligned} Q_0\Sigma _0=0, \end{aligned}$$(3.20)Equation (3.20) expresses the existence of a quantum number associated with a \(U^4(1)\) symmetry among the quartet fields.$$\begin{aligned} Q_0= & {} \sigma ^a\frac{\delta }{\delta \sigma ^a}-\bar{\sigma }^a \frac{\delta }{\delta \bar{\sigma }^a}+\eta ^a\frac{\delta }{\delta \eta ^a}- \bar{\eta }^a\frac{\delta }{\delta \bar{\eta }^a}\nonumber \\&+\,X^a\frac{\delta }{\delta X^a}-\bar{X}^a\frac{\delta }{\delta \bar{X}^a}+ Y^a\frac{\delta }{\delta Y^a}-\bar{Y}^a\frac{\delta }{\delta \bar{Y}^a}.\nonumber \\ \end{aligned}$$(3.21)
- Second \(U^{4}(1)\) charge equation:where$$\begin{aligned} \bar{Q}_0\Sigma _0=-2(X_a\bar{\sigma }^a+\bar{Y}_a\eta ^a), \end{aligned}$$(3.22)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$$\begin{aligned} \bar{Q}_0= & {} \sigma ^a\frac{\delta }{\delta \sigma ^a}- \bar{\sigma }^a\frac{\delta }{\delta \bar{\sigma }^a}- \eta ^a\frac{\delta }{\delta \eta ^a}+\bar{\eta }^a\frac{\delta }{\delta \bar{\eta }^a}\nonumber \\&-\,X^a\frac{\delta }{\delta X^a}+\bar{X}^a\frac{\delta }{\delta \bar{X}^a}+ Y^a\frac{\delta }{\delta Y^a}-\bar{Y}^a\frac{\delta }{\delta \bar{Y}^a}.\nonumber \\ \end{aligned}$$(3.23)where$$\begin{aligned} Q_{\mathrm{{eff}}}^{(i)}\Sigma _0=(-1)^{i-1}(X_a\bar{\sigma }^a+\bar{Y}_a\eta ^a), \end{aligned}$$(3.24)$$\begin{aligned} Q_{\mathrm{{eff}}}^{(1)}= & {} \frac{1}{2}(Q_0-\bar{Q}_0)\nonumber \\= & {} \eta ^a\frac{\delta }{\delta \eta ^a}-\bar{\eta }^a \frac{\delta }{\delta \bar{\eta }^a}+X^a\frac{\delta }{\delta X^a}-\bar{X}^a\frac{\delta }{\delta \bar{X}^a},\nonumber \\ Q_{\mathrm{{eff}}}^{(2)}= & {} Q_{\mathrm{{eff}}}^{(1)}+\bar{Q}_0\nonumber \\= & {} \sigma ^a\frac{\delta }{\delta \sigma ^a}-\bar{\sigma }^a \frac{\delta }{\delta \bar{\sigma }^a}+Y^a\frac{\delta }{\delta Y^a}-\bar{Y}^a\frac{\delta }{\delta \bar{Y}^a}. \end{aligned}$$(3.25)
Quantum numbers of the fundamental fields and the quartet system
Field | \(\omega \) | \(\bar{c}\) | e | \(\bar{\sigma }\) | \(\sigma \) | \(\bar{\eta }\) | \(\eta \) |
---|---|---|---|---|---|---|---|
Q-charge | 0 | 0 | 0 | \(-\)1 | 1 | \(-\)1 | 1 |
\(\bar{Q}\)-charge | 0 | 0 | 0 | \(-\)1 | 1 | 1 | \(-\)1 |
Ghost n\(^ o\) | 0 | 1 | 0 | 0 | 0 | \(-\)1 | 1 |
Form rank | 1 | 0 | 1 | 1 | 1 | 1 | 1 |
Statistics | 1 | 1 | 1 | 1 | 1 | 0 | 2 |
Quantum numbers of the sources
Source | \(\Omega \) | L | \(\bar{X}\) | X | \(\bar{Y}\) | Y |
---|---|---|---|---|---|---|
Q\(-\)charge | 0 | 0 | \(-\)1 | 1 | \(-\)1 | 1 |
\(\bar{Q}\)\(-\)charge | 0 | 0 | 1 | \(-\)1 | \(-\)1 | 1 |
Ghost n\(^ o\) | \(-\)1 | \(-\)2 | 0 | \(-\)2 | \(-\)1 | \(-\)1 |
Form rank | 3 | 4 | 3 | 3 | 3 | 3 |
Statistics | 2 | 2 | 3 | 3 | 2 | 2 |
3.3 A remark about the BRST triviality
4 Introducing a massive constraint and the vierbein
4.1 Constraint action
Quantum numbers of the constraint fields
Field | \(\bar{\theta }\) | \(\theta \) | \(\bar{\lambda }\) | \(\lambda \) | \(\bar{\gamma }\) | \(\gamma \) | \(\bar{\rho }\) | \(\rho \) |
---|---|---|---|---|---|---|---|---|
Q-charge | \(-\)1 | 1 | \(-\)1 | 1 | \(-\)1 | 1 | \(-\)1 | 1 |
\(\bar{Q}\)-charge | 1 | \(-\)1 | \(-\)1 | 1 | \(-\)1 | 1 | 1 | \(-\)1 |
Ghost n\(^ o\) | \(-\)1 | \(-\)1 | 0 | 0 | \(-\)2 | 0 | \(-\)1 | 1 |
Form rank | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
Statistics | 2 | 2 | 3 | 3 | 1 | 3 | 2 | 4 |
4.2 Generalized Ward identities
- Slavnov–Taylor identity:where$$\begin{aligned} {\mathcal {S}}(\Sigma )=0, \end{aligned}$$(4.8)$$\begin{aligned} {\mathcal {S}}(\Sigma )= & {} \int \left( \frac{\delta \Sigma }{\delta \Omega _a^{b}} \frac{\delta \Sigma }{\delta \omega ^a_{b}}+\frac{\delta \Sigma }{\delta L_a^{b}}\frac{\delta \Sigma }{\delta c^a_{b}}+\frac{\delta \Sigma }{\delta X_a}\frac{\delta \Sigma }{\delta \bar{\eta }^a} \right. \nonumber \\&\left. +\,\frac{\delta \Sigma }{\delta \bar{X}_a}\frac{\delta \Sigma }{\delta \eta ^a} +\frac{\delta \Sigma }{\delta Y_a}\frac{\delta \Sigma }{\delta \bar{\sigma }^a}+ \frac{\delta \Sigma }{\delta \bar{Y}_a}\frac{\delta \Sigma }{\delta \sigma ^a} \right. \nonumber \\&\left. +\,\bar{\lambda }^a\frac{\delta \Sigma }{\delta \bar{\theta }^a}+ \lambda ^a\frac{\delta \Sigma }{\delta \theta ^a}+\bar{\rho }^a \frac{\delta \Sigma }{\delta \bar{\gamma }^a}+\rho ^a \frac{\delta \Sigma }{\delta \gamma ^a}\right) .\nonumber \\ \end{aligned}$$(4.9)
- Ghost equation:where$$\begin{aligned} \int \left( \frac{\delta \Sigma }{\delta c^a_{\phantom {a}b}}{+}\bar{\theta }_a\frac{\delta \Sigma }{\delta \bar{\lambda }_b}{+}\theta _a\frac{\delta \Sigma }{\delta \lambda _b}{+}\bar{\gamma }_a\frac{\delta \Sigma }{\delta \bar{\rho }_b}{+}\gamma _a\frac{\delta \Sigma }{\delta \rho _b}\right) =\widetilde{\Delta }_a^{\phantom {a}b},\nonumber \\ \end{aligned}$$(4.10)remains a linear breaking.$$\begin{aligned} \widetilde{\Delta }_a^{\phantom {a}b}=\Delta _a^{\phantom {a}b}-m\int \left( \bar{\theta }_a+\theta _a\right) e^b \end{aligned}$$(4.11)
- Vierbein equation:which is linearly broken.$$\begin{aligned} \frac{\delta \Sigma }{\delta e^a}=m\left( \bar{\lambda }^a+\lambda ^a\right) , \end{aligned}$$(4.12)
- Rigid supersymmetries:where \(i\in \{1,2,3,4\}\). The rigid supersymmetric operators are^{6}$$\begin{aligned} \widetilde{R}^{(i)}\Sigma =\widetilde{\Delta }^{(i)}, \end{aligned}$$(4.13)while$$\begin{aligned} \widetilde{R}^{(1)}= & {} R^{(1)}+\theta ^a\left( \frac{\delta }{\delta \gamma ^a}+\frac{\delta }{\delta \lambda ^a}\right) +\bar{\gamma }^a\left( \frac{\delta }{\delta \bar{\theta }^a}-\frac{\delta }{\delta \bar{\rho }^a}\right) \nonumber \\&-\left( \bar{\theta }^a+\bar{\rho }^a\right) \frac{\delta }{\delta \bar{\lambda }^a}-\left( \lambda ^a-\gamma ^a\right) \frac{\delta }{\delta \rho ^a},\nonumber \\ \widetilde{R}^{(2)}= & {} R^{(2)}+\gamma ^a\frac{\delta }{\delta \theta ^a}-\bar{\theta }^a\frac{\delta }{\delta \bar{\gamma }^a}+\bar{\lambda }^a\frac{\delta }{\delta \bar{\rho }^a}-\rho ^a\frac{\delta }{\delta \lambda ^a},\nonumber \\ \widetilde{R}^{(3)}= & {} R^{(3)}+\bar{\gamma }^a\frac{\delta }{\delta \theta ^a}+\bar{\theta }^a\frac{\delta }{\delta \gamma ^a}-\bar{\rho }^a\frac{\delta }{\delta \lambda ^a}-\bar{\lambda }^a\frac{\delta }{\delta \rho ^a},\nonumber \\ \end{aligned}$$(4.14)with$$\begin{aligned} \widetilde{\Delta }^{(i)}=\Delta ^{(i)}+\Upsilon ^{(i)}, \end{aligned}$$(4.15)which are linear in the fields.$$\begin{aligned} \Upsilon ^{(1)}= & {} m\left( \bar{\theta }_a-\theta _a+\bar{\rho }_a\right) e^a,\nonumber \\ \Upsilon ^{(2)}= & {} m{\rho }_ae^a,\nonumber \\ \Upsilon ^{(3)}= & {} m\bar{\rho }_ae^a, \end{aligned}$$(4.16)
- Rigid fermionic equation:where$$\begin{aligned} \widetilde{Q}^{(1)}\Sigma =\widetilde{\Lambda }^{(1)}, \end{aligned}$$(4.17)but it is also linear in the fields.$$\begin{aligned} \widetilde{Q}^{(1)}= & {} Q^{(1)}+\left( \bar{\theta }^a+\bar{\rho }^a\right) \frac{\delta }{\delta \rho ^a}+\bar{\gamma }^a\left( \frac{\delta }{\delta \gamma ^a}+\frac{\delta }{\delta \lambda ^a}\right) ,\nonumber \\ \widetilde{\Lambda }^{(1)}= & {} \Lambda ^{(1)}-m\bar{\gamma }_ae^a, \end{aligned}$$(4.18)
- \(U^{4}(1)\) charge equation:where$$\begin{aligned} \widetilde{Q}_0\Sigma = m\left( \bar{\lambda }_a-\lambda _a\right) e^a, \end{aligned}$$(4.19)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.$$\begin{aligned} \widetilde{Q}_0= & {} \sigma ^a\frac{\delta }{\delta \sigma ^a}- \bar{\sigma }^a\frac{\delta }{\delta \bar{\sigma }^a}+\eta ^a \frac{\delta }{\delta \eta ^a}-\bar{\eta }^a\frac{\delta }{\delta \bar{\eta }^a}\nonumber \\&+\,X^a\frac{\delta }{\delta X^a}-\bar{X}^a\frac{\delta }{\delta \bar{X}^a}+Y^a\frac{\delta }{\delta Y^a}-\bar{Y}^a\frac{\delta }{\delta Y^a}\nonumber \\&+\,\lambda ^a\frac{\delta }{\delta \lambda ^a}- \bar{\lambda }^a\frac{\delta }{\delta \bar{\lambda }^a}+ \theta ^a\frac{\delta }{\delta \theta ^a}-\bar{\theta }^a \frac{\delta }{\delta \bar{\theta }^a}\nonumber \\&+\,\rho ^a\frac{\delta }{\delta \rho ^a}-\bar{\rho }^a \frac{\delta }{\delta \bar{\rho }^a}+\gamma ^a\frac{\delta }{\delta \gamma ^a}-\bar{\gamma }^a\frac{\delta }{\delta \gamma ^a}. \end{aligned}$$(4.20)
- Field equations:$$\begin{aligned}&\frac{\delta \Sigma }{\delta \bar{\lambda }_a}=\sigma ^a-m e^a, \end{aligned}$$(4.21)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \lambda _a}=\bar{\sigma }^a-m e^a,\end{aligned}$$(4.22)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \bar{\theta }_a}-\frac{\delta \Sigma }{\delta \bar{Y}_a}=0,\end{aligned}$$(4.23)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \theta _a}-\frac{\delta \Sigma }{\delta Y_a}=0,\end{aligned}$$(4.24)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \bar{\gamma }_a}+\frac{\delta \Sigma }{\delta \bar{X}_a}=0,\end{aligned}$$(4.25)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \gamma _a}+\frac{\delta \Sigma }{\delta X_a}=0,\end{aligned}$$(4.26)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \rho _a}=\eta ^a,\end{aligned}$$(4.27)$$\begin{aligned}&\frac{\delta \Sigma }{\delta \rho _a}=\bar{\eta }^a. \end{aligned}$$(4.28)
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 terms^{7} in (2.1), only interacting terms in the fields \(\omega \) and e. This is a problem if one plans to quantize the LC action because this property ruins the well-established 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 \(\Sigma \) instead is that the model can be interpreted as a typical gauge theory for the gauge field \(\omega \) 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 \(\omega \) as the spin connection is natural.
A quantum version of such model would also be highly non-perturbative since there are no quadratic terms of \(\omega \) in \(\Sigma \). Moreover, the terms in the constraint action \(S_c\) 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 \(\omega \), 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.
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 (4.214.224.234.244.254.264.27) are exact or linearly broken, the validity of constraints (4.1) at quantum level is ensured.
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.
Footnotes
- 1.
The fields directly determine the spacetime dynamics.
- 2.
Eventually, they will be associated with the Lovelock–Cartan parameters \(z_i\) appearing in (2.1).
- 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 tree-level propagators). Hence, a perturbative expansion around a free theory is not at our disposal. In fact, all non-vanishing terms in (3.5) are interacting terms. A theory of this type is said to be highly non-perturbative. Of course, one can always define background configurations and enforce a perturbative regime around these configurations.
- 4.
This parameter will, eventually, be identified with \(\mu \) appearing in (2.1).
- 5.
Although we are at classical level, we mean that the vierbein would remain classical in a possible quantum scenario.
- 6.
The symmetry \(R^{(4)}\) is quadratically broken and, thus, it is not an interesting identity for the model. As a consequence, generalized versions of \(Q^{(2)}\) and \(\bar{Q}_0\) are not at our disposal in the full model. Obviously, since there is no generalization of the \(\bar{Q}_0\) symmetry, there is no place for generalizing \(Q^{(i)}_{\mathrm{{eff}}}\) as well.
- 7.
This is a problem that also appears in the case of the pure Einstein–Hilbert action.
- 8.
See the previous section.
- 9.
The expectation values are taken with respect to the functional measure \(D\Phi \) as defined in Sect. 5.1.
Notes
Acknowledgements
The Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) are acknowledge for financial support.
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