# Gauge symmetry and constraints structure for topologically massive AdS gravity: a symplectic viewpoint

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## Abstract

By applying the Faddeev–Jackiw symplectic approach we systematically show that both the local gauge symmetry and the constraint structure of topologically massive gravity with a cosmological constant \(\Lambda \), elegantly encoded in the zero-modes of the symplectic matrix, can be identified. Thereafter, via a suitable partial gauge-fixing procedure, the time gauge, we calculate the quantization bracket structure (generalized Faddeev–Jackiw brackets) for the dynamic variables and confirm that the number of physical degrees of freedom is one. This approach provides an alternative to explore the dynamical content of massive gravity models.

## 1 Introduction

Fundamental issues in modern cosmology, such as inflation, dark matter, dark energy and the accelerating expansion of our universe [1, 2], have long motivated to propose alternative gravity theories beyond original Einstein’s General Relativity, both in the ultraviolet (UV) and the infrared (IR) regimes. According to Lovelock’s theorem [3, 4] any modification of General Relativity requires at least one of the following ingredients: (1) extra dimensions, (2) extra degrees of freedom, (3) higher-derivatives terms, and iv) non-locality. Massive gravity theories are an example of the type-ii ingredients, in which the graviton will acquire a non-zero mass (see e.g. [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]). Along these lines, it has long been known that the first massive gravity theory was introduced by Pauli and Fierz [6], where they presented a linear action for a spin-2 field on a four dimensional flat background. Moreover, the Fierz-Pauli theory describes the propagation of a field with five degrees of freedom of positive energy whereas General Relativity theory has two degrees of freedom. However, Boulware and Deser studied some specific fully non-linear massive gravity theories and pointed out that a general non-linear theory of massive gravity generically contains six propagating degrees of freedom [19]. While the linear theory has five degrees of freedom, the non-linear theories studied by these authors turned out to have an extra degree of freedom, which however is unphysical since it has a negative kinetic energy and renders the whole theory unstable: it was therefore called the Boulware-Deser ghost. After a great effort, a non-linear theory free of such a ghost field was finally obtained by de Rham, Gabadadze and Tolley (dRGT) [7, 8, 9]. The advantage of the dRGT model is that it contains two dynamical constraints that eliminate both the ghost field and its canonically-conjugate momentum. The absence of the Boulware-Deser ghost was shown explicitly by performing the counting of the physical degrees of freedom in the framework of the Hamiltonian formalism [10, 11, 12, 17, 18]. Unfortunately, the Hamiltonian analysis of these models is a difficult task to develop, and its symmetry properties have not been studied yet via a complete analysis of its constraints. On the other hand, in the study of massive gravity models, it is always useful to consider toy models that share the conceptual foundations of the four-dimensional theories, but at the same time are free of technical difficulties; this is particularly true in three-dimensional (3D) gravity. In this work, we focus on the simplest 3D version of a massive gravity theory; in this manner, in order to obtain a realistic 3D-Einstein gravity as compared to the higher-dimensional theory, regarding the local propagating modes, one can modify the theory by adding up higher-derivative curvature terms in the Einstein-Hilbert (EH) action, which leads to the simplest 3D-massive gravity theory known as Topologically Massive Gravity (TMG). This theory consists of an EH term, with or without a consmological constant \(\Lambda \), plus a parity-violating gravitational Chern-Simons (CS) term with coefficient \(\frac{1}{\mu }\) [20, 21, 22, 23, 24, 25]. At the linear level, without a cosmological constant this theory describes a single massive state of helicity \(\pm \, 2\) (depending on the relative sign between the EH and CS terms) on a Minkowski background^{1} [26], and defines an unitary irreducible representation of the 3D Poincaré group [27].

However, while a linearized analysis usually allows us a reliable counting of the physical degrees of freedom, it can yield misleading results in some cases. A Lagrangian/Hamiltonian formulation should provide a way to count the number of local physical degrees of freedom without resorting to linearization, that is, taking into account all the physical constraints and gauge invariance. In this sense, the identification of the physical degrees of freedom can be addressed by a direct application of Dirac’s method for constrained Hamiltonian systems [28], which systematically separates all constraints into first and second-class ones [29, 30]. As a consequence, the physical degrees of freedom can be separated from the gauge degrees of freedom, and a generator of the gauge symmetry can be constructed out of a combination of first-class constraints [31]. Furthermore, the Dirac brackets to quantize a gauge system can be obtained once the second-class constraints are removed. In the case of the massive gravity theories, however, the separation between first-and second-class constraints is a delicate issue, and the system considered in this paper is not an exception [32, 33, 34, 35]. In particular, in Ref. [32] the Hamiltonian structure of TMG was further analyzed via the Dirac formalism. Indeed, these authors obtain the secondary first-class-constraint structure of this model with the help of the Dirac conjecture: *“If *\(\phi \)* is a first-class constraint, then* \(\{\phi , H^{T}\}\) *is also a first-class constraint”*. Nevertheless, this treatment is quite involved and unsatisfactory. On the other hand, the authors of Refs. [34, 35] present a fully Lagrangian analysis, but the right number of physical degrees of freedom in the configuration space can only be obtained once an ad hoc extra constraint on the basic variables is invoked. Hence, it is worth exploring whether all the necessary constraints can be systematically obtained via a Lagrangian formulation, therefore, the analysis of the constraints and the gauge symmetry of massive gravity models, still is missing in the literature.

As an alternative approach to Dirac’s method, Faddeev and Jackiw [36] proposed a new framework, which is geometrically well motivated and is based on the symplectic structure for constrained systems. This approach, the so-called Faddeev–Jackiw (F–J) symplectic formalism (for a detailed account see [37, 38, 39, 40, 41, 42, 43, 44]), is useful to obtain in an elegant way several essential elements of a particular physical theory, such as the physical constraints, the local gauge symmetry, the quantization bracket structure and the number of physical degrees of freedom. It turns out that the F–J approach does not require to classify the constraints into first- and second-class ones as in Dirac’s approach is done. Even more, it does not invoke Dirac’s conjecture; rather, in this approach, all relevant information can be ontained through a symplectic matrix. For a gauge system, the symplectic matrix remains singular unless a gauge-fixing procedure is introduced. In addition, the generators of the gauge symmetries are given in terms of the zero-modes of the symplectic matrix, thus, the F–J symplectic method provides an effective tool for dealing with gauge theories.

In this manner, the purpose of this article is to present a detailed F–J analysis of three-dimensional topologically massive AdS gravity in a completely different context to that presented in Refs. [32, 33, 34, 35]. In particular, we study the nature of the physical constraints and we obtain the gauge symmetry, as well as its generators, under which all the physical quantities must be invariant. Afterwards, we obtain both the fundamental quantization brackets and the number of physical degrees of freedom by introducing an appropriate gauge-fixing procedure.

The remainder of this paper is structured as follows. In Sect. 2 we briefly review the topologically massive AdS gravity action. Section 3 is devoted to explore the nature of the constraints within the Faddeev–Jackiw symplectic framework and derive the corresponding symplectic matrix. The full set of physical constraints of the theory are also obtained. In Sect. 4, the gauge symmetry and its generators are obtained via the zero-modes of the symplectic matrix. In Sect. 5 we introduce gauge-fixing conditions in order to obtain both the quantization bracket structure and the number of physical degrees of freedom. In Sect. 6, we conclude with a brief discussion of our results.

## 2 Action and equations of motion of topologically massive gravity

*G*the 3D Newton’s constant, and \(\Lambda \) is a cosmological constant such that \(\Lambda =-1/l^{2}\), where

*l*is the AdS radius [26]. Furthermore, the fundamental fields of this action are: the dreibein 1-form \(e^{i}=e_{\mu }^{i}dx^{\mu }\) that determines a space-time metric via \(g_{\mu \nu }=e_{\mu }{^{i}}e_{\nu }{^{j}}\eta _{ij}\); the auxiliary field 1-form \(\lambda ^{i}\) that ensures that the torsion vanishes \(T_{i}=0\) [45, 46]; and the dualized spin-connection \(A^{i}=A_{\mu }{^{i}}dx^{\mu }\) valued on the adjoint representation of the Lie group

*SO*(2, 2), so that, it admits an invariant totally anti-symmetric tensor \(f_{ijk}\). The connection acts on internal indices and defines a derivative operator:

*f*-tensor, \(A_{\mu }{^{ij}}=-f^{ij}{_{k}}A_{\mu }{^{k}}\). Moreover, by inserting Eq. (6) into Eq. (5), one can solve for the Lagrangian multiplier \(\lambda _{\mu }{^{i}}\) in terms of the 3D Schouten tensor of the manifold \({\mathcal {M}}\):

## 3 The nature of the constraints in the Faddeev–Jackiw symplectic framework

## 4 Gauge transformations

*SO*(2, 2) transformations [32]. In these transformations the diffeomorphisms have not been found explicitly. However, it is well known that an appropriate choice of the gauge parameters does generate the diffeomorphism (on-shell) [47, 48, 50]. Let us redefine the gauge parameters as

## 5 The Faddeev–Jackiw brackets and degrees of freedom counting

## 6 Conclusions and discussions

In the present paper, the nature of the constraints and gauge structure of the topologically massive AdS gravity theory was studied from the perspective of the Faddeev–Jackiw symplectic approach. The whole set of independent physical constraints was identified through the consistency condition and the zero-modes. It was shown that even when all physical constraints are found, the symplectic matrix still has zero-modes, that is, when the zero-modes are orthogonal to the gradient of the symplectic potential on the surface of the constraints, one is led to deduce that the theory has a local gauge symmetry. Therefore, the zero-modes straightforwardly generate the local gauge symmetry under which all physical quantities are invariant. By mapping the gauge parameters appropriately we have also obtained the Poincaré transformations and the diffeomorphism symmetry. Additionally, we have shown that the time-gauge fixing of the density Lagrangian renders the non-degenerate symplectic matrix \(f_{IJ}\). We have then identified the quantizaion bracket (F–J brackets) structure and have proved that there is one physical degree of freedom. It is worth remarking that all the results presented here can be applied to the study of the physical content of models such as massive gravity and bigravity theories in \(2+1\) dimensions, in which secondary, tertiary, or higher-order constraints are present. Such problems are under study and will be published elsewhere [57]. Another line for further research is the application of the procedure used here to explore conceptual and technical issues of gravity models in \(3+1\) dimensions.

## Footnotes

- 1.
In the presence of a cosmological constant, Minkowski space-time is no longer a vacuum solution and the new maximally symmetric solutions are de Sitter (dS) space-time for positive \(\Lambda \) ( dS has

*SO*(3, 1) isometry) and anti-de Sitter (AdS) space-time for negative \(\Lambda \) (AdS has*SO*(2, 2) isomety). In this respect, the*SO*(2, 2) group can be seen as a \(\Lambda \)-deformed Poincaré group [56], if \(\Lambda \rightarrow 0\) the AdS algebra contracts to the usual Poincaré algebra.

## Notes

### Acknowledgements

This work has been partially supported by CONACyT under grand number CB-2014-01/240781. We would like to thank G. Tavares-Velasco for reading a draft version of this paper and alerting us to various typos.

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