# Dimensional reduction of the massless limit of the linearized ‘New Massive Gravity’

## Abstract

The so-called ‘New Massive Gravity’ in \(D=2+1\) consists of the Einstein–Hilbert action (with minus sign) plus a quadratic term in the curvature (\(K\)-term). Here we perform the Kaluza–Klein dimensional reduction of the linearized \(K\)-term to \(D=1+1\). We end up with fourth-order massive electrodynamics in \(D=1+1\), described by a rank-2 tensor. Remarkably, there appears a local symmetry in \(D=1+1\), which persists even after gauging away the Stueckelberg fields of the dimensional reduction. It plays the role of a \(U(1)\) gauge symmetry. Although of higher order in the derivatives, the new 2D massive electrodynamics is ghost free, as we show here. It is shown, via a master action, to be dual to the Maxwell–Proca theory with a scalar Stueckelberg field.

### Keywords

Dimensional Reduction Particle Content Massive Gravity Maxwell Theory Proca Theory## 1 Introduction

The authors of [1] have suggested an invariant theory under general coordinate transformations which describes a massive spin-2 particle (graviton) in \(D=2+1\). The model contains the Einstein–Hilbert theory and an extra term of fourth order in the derivatives and quadratic in the curvatures, the so-called \(K\)-term, which has been analyzed in [2], see also [3]. Since massless particles in \(D\) dimensions have the same number of degrees of freedom as massive particles in \(D-1\) dimensions, one might wonder whether the ‘New Massive Gravity’ (NMG) theory might be regarded as a dimensional reduction of some fourth-order (in the derivatives) massless spin-2 model in \(D=3+1\), which would be certainly interesting from the point of view of renormalizable quantum gravity in \(D=3+1\). As far as we know there has been given no positive answer to that question so far.^{1} As an attempt to gain more insight in that question we investigate here the dimensional reduction of the massless part of the linearized NMG theory, the linearized \(K\)-term. We show here that the linearized \(K\)-term is reduced to a kind of higher-derivative massive 2D electrodynamics, which is in agreement with the fact that the linearized \(K\)-term is dual to the Maxwell theory in 3D as shown in [5]; see also [2] and [6]. However, it is remarkable that a new local symmetry shows up after dimensional reduction and plays the role of a \(U(1)\) symmetry not broken by the mass term. We also derive in Sect. 4 a master action interpolating between the new (higher-order) massive 2D electrodynamics and the usual Maxwell–Proca theory with a Stueckelberg field. We emphasize that throughout this work we only deal with quadratic (linearized) free theories.

## 2 From \(2+1\) to \(1+1\)

## 3 Gauge invariant massive electrodynamics in \(D=1+1\)

### 3.1 Equations of motion

^{2}. From (29), (32), (33), and (35), (34) becomes

### 3.2 Propagator and absence of ghosts

## 4 Master action and duality

Now two remarks are in order. First, if we take \(\Box _{D-1} \rightarrow \Box _D + m^2 \) in (16), (18), and (22) we derive the corresponding massless higher-dimensional theories (15), (17), and (2) (for \(D=3\)), respectively. However, if we try the same (non-rigorous) inverse dimensional reduction with the linearized ‘new massive gravity’ of [1] it turns out that we do not get rid of the \(m^2\) in the corresponding \(4D\) theory. Moreover, we have a tachyon at \(k^2=m^2\). Thus, there seems to be no simple Kaluza–Klein reduction of a fourth-order spin-2 massless model in \(4D\) which might lead to the ‘new massive gravity’ of [1]; see, however, the footnote in the introduction. The results of [13] and [14] suggest that one should try to obtain [1] from the dimensional reduction of an extra discrete dimension, see also [15, 16]. This is under investigation. Second, it is worth commenting that, although the K-term has a nonlinear gravitational completion in \(D=2+1\), see (1), there is no such completion for \(S_{2\mathrm{D}}\) since there is no local vector symmetry whatsoever in \(S_{2\mathrm{D}}\).

## 5 Conclusion

By performing a Kaluza–Klein dimensional reduction of the massless limit of the linearized ‘new massive gravity’ (linearized \(K\)-term) we have obtained a new 2D massive electrodynamics of fourth order in the derivatives. This is in agreement with the equivalence of the linearized \(K\)-term with the Maxwell theory [5], see also [2] and [6]. However, it is remarkable that the reduced 2D theory, although massive, has local \(U(1)\) gauge symmetry even after gauging away the Stueckelberg fields of the dimensional reduction. The \(U(1)\) symmetry (24) seems to be a consequence of the lack of a Stueckelberg version of the fundamental field \(h_{\mu \nu }\) invariant under both linearized reparametrizations and Weyl transformations, see the comment after (24).

We have also noticed that the dimensional reduction of the K-term follows the same simple pattern of the usual spin-1 (Maxwell to Maxwell–Proca) and spin-2 (massless Fierz–Pauli to massive Fierz–Pauli) cases, namely, we have the practical rule \(\Box _D \rightarrow \Box _{D-1} - m^2\) altogether with the replacement of the fundamental field by its Stueckelberg version \(h_{MN} \rightarrow h_{\mu \nu } + (\partial _{\mu }\phi _{\nu } + \partial _{\nu } \phi _{\mu })/m - \partial _{\mu }\partial _{\nu }H/m^2 \).

We have made a classical and quantum analysis of the particle content of the reduced theory, confirming that, although of fourth order in the derivatives, is ghost free and contains only a massive vector field in the spectrum. In particular, we have found a master action interpolating between the new 2D massive electrodynamics and the Maxwell–Proca theory with a scalar Stueckelberg field and we identified a dual map between gauge invariant vector fields in both theories, see (60). A possible non-Abelian extension of the new 2D electrodynamics and the issue of unitarity in the context of the Schwinger mass generation when coupled to fermions are under investigation.

## Footnotes

- 1.
See, however, [4], which shows that a Kaluza–Klein dimensional reduction of the usual (

**second-order**) massless Fierz–Pauli theory followed by an unconventional elimination of fields and a dualization procedure leads to the linearized NMG theory. - 2.
The identity (35) corresponds to the linearized version of the Einstein equation \(R_{\mu \nu } = g_{\mu \nu }R/2\) against a flat background \(g_{\mu \nu } = \eta _{\mu \nu } + h_{\mu \nu }\). Recall that the Einstein equation is a trivial identity in \(D=1+1\) without any dynamic content.

## Notes

### Acknowledgments

The work of D.D. is supported by CNPq (304238/2009-0) and FAPESP (2013/00653-4). The work of H.A.B and G.B.de G. is supported by CNPq (507064/2010-0). We thank Dr. Karapet Mkrtchyan for reminding us of [4].

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