# Constrained gauge-gravity duality in three and four dimensions

## Abstract

The equivalence between Chern–Simons and Einstein–Hilbert actions in three dimensions established by Achúcarro and Townsend (Phys Lett B 180:89, 1986) and Witten (Nucl Phys B 311:46, 1988) is generalized to the off-shell case. The technique is also generalized to the Yang–Mills action in four dimensions displaying de Sitter gauge symmetry. It is shown that, in both cases, we can directly identify a gravity action while the gauge symmetry can generate spacetime local isometries as well as diffeomorphisms. The price we pay for working in an off-shell scenario is that specific geometric constraints are needed. These constraints can be identified with foliations of spacetime. The special case of spacelike leafs evolving in time is studied. Finally, the whole set up is analyzed under fiber bundle theory. In this analysis we show that a traditional gauge theory, where the gauge field does not influence in spacetime dynamics, can be (for specific cases) consistently mapped into a gravity theory in the first order formalism.

## 1 Introduction

In [1], Achúcarro and Townsend and later Witten [2] were able to show that Chern–Simons (CS) theory for the de Sitter and Poincaré groups are equivalent to three dimensional gravity with and without cosmological constant, respectively. In [1] the equivalence is discussed under a supersymmetric scenario while in [2] the non-supersymmetric scenario is also studied. Moreover, these results strongly depend on the validity of the classical field equations (on-shell cases). Essentially, the mapping between a CS theory and a gravity one consists on the equivalence between, for instance, the groups *SU*(2) and *SO*(3). Furthermore, an important feature is that the gauge symmetry contains local Lorentz isometries as well as diffeomorphisms, provided the validity of the classical field equations. In the case of the Poincaré group *ISO*(3), the resulting gravity is the pure Einstein–Hilbert (EH) action while for the de Sitter group *SO*(4) the resulting gravity action contains also the cosmological constant term. For a complete review on Chern–Simons theories and three-dimensional gravity we refer to [3, 4, 5, 6, 7] and references therein.

It is possible to look at Gauge-Gravity Equivalence (GGE) [2] discussed in the previous paragraph as a mechanism to generate an effective gravity theory from a traditional gauge theory in flat spacetime.^{1} Essentially, we can state that the CS action for the group \(ISO(3)\equiv SU(2)\times {\mathbb {R}}^3\) (or \(SO(4)\equiv SU(2)\times S(3)\), where *S*(3) describes gauge pseudo-translations. See Sect. 2.2.) is only a gauge theory, where the gauge group has no relation with spacetime dynamics. The spacetime itself can be taken as a flat manifold (Euclidean or Minkowskian), but this is not a requirement, only a simplification. Then, by making use of the homomorphism^{2} \(SU(2)\longmapsto SO(3)\), the gauge field can be mapped into the spin-connection and the dreibein. The sectors \({\mathbb {R}}^3\) and *S*(3) in each case are identified with diffeomorphisms. The result is that the original spacetime is deformed into an effective manifold. For the general idea, see for instance [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Formally, this mechanism can be understood as a bundle map [19, 20, 21, 22] between a gauge principal bundle and a coframe bundle (See Sect. 5). Intuitively, one can think of the gauge fields as being “absorbed” by the spacetime.

In this work, besides the reinterpretation of the GGE [2] above described, we develop two main generalizations. First, we generalize such equivalence to the case where the field equations are no longer required to be obeyed (off-shell generalization). The main difference is that we have to consider a pair of constraints on the curvature and torsion 2-forms. Following [2], we study the cases of the Poincaré (*ISO*(3)) and de Sitter (*SO*(4)) groups in three dimensions. Second, we generalize the GGE to four dimensions where we consider de Sitter group (*SO*(5)) as the gauge group of a Yang–Mills theory and employ the same ideas of [2] under the scenario described in the previous paragraph. We show that the gauge group can generate local Lorentz isometries as well as diffeomorphisms. The latter are originated from the coset *S*(4) just like diffeomorphisms emerge from the coset *S*(3) in the three-dimensional de Sitter case. Nevertheless, constraints are also demanded. We remark that, with little careful, everything can be systematically generalized to the general gauge group *SO*(*m*, *n*) with \(m+n=5\) in four dimensions. Nevertheless, for simplicity, we restrict ourselves to the case \(m=0\), i.e., *SO*(5).

An important exigence of the mechanism is that a mass parameter is always required. This is evident if we realize that the gauge field is mass dimensionful while the vielbein is dimensionless. Thus, without a mass parameter, the identification between the gauge field and the vielbein may never be realized. In three dimensions, the coupling parameter is already dimensionful. Thus, there is a mass scale inherent to the theory. In contrast, in four dimensions, the coupling parameter is dimensionless. Therefore, it is necessary to impose the existence of a mass parameter to the theory. Fortunately, it is widely known that mass parameters are plenty in Yang–Mills theories [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. On the other hand, in most models where gravity is generated from gauge theories [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35], a symmetry breaking, or/and an Inönü-Wigner contraction [36], is required. In the present approach, no breaking mechanism is needed although a mass parameter existence is mandatory.

Furthermore, we also discuss the geometrical meaning of these constraints and argue that they are equivalent to spacetime foliations. The specific case of the spacelike leafs evolving in time is then discussed. We find that, for the constraints to be fulfilled at the off-shell cases, the diffeomorphism symmetry must be broken to a smaller diffeomorphic group.

Finally, we provide a study of the GGE we develop in terms of fiber bundle theory [19, 20, 21, 22]. A gauge theory is defined over a principal bundle whose structure group is the gauge group and does nor affect the base space (spacetime). Gravity, on the other hand, is constructed over a coframe bundle. The coframe bundle is also a principal bundle but with the special property that the structure group (gauge group) is identified with the base space isometries (local isometries of the spacetime). Essentially, we discuss how to identify a gauge principal bundle with a coframe bundle.

This work is organized as follows: In Sect. 2 we review the three-dimensional case as first realized by Witten and generalize it to the off-shell case. In Sect. 3, the four-dimensional case for the Yang–Mills action is discussed. In Sect. 4 the constraints are discussed from viewpoint of a specific foliation where spacelike leafs evolve in time. Then, in Sect. 5 we formally discuss how to reinterpret Witten’s GGE in terms of fiber bundles. Finally, the conclusions and perspectives are displayed in Sect. 6.

## 2 Constrained gravity in three dimensions

In this section we stick to three dimensions. We review and generalize the non-supersymmetric situation discussed in [2].

### 2.1 Poincaré gauge symmetry and gravity

*Y*is the 1-form connection for the Poincaré group in the representation \(ISO(3)\equiv SU(2)\times {\mathbb {R}}^3\), which is the gauge group that leaves the action (2.1) invariant. In this representation, the following conventions for the algebra are assumed,

*su*(2) sector of the algebra.

^{3}is determined by \(G=\dfrac{1}{8\varsigma \kappa ^2}\). Needless to say, the action (2.9) is the three dimensional Einstein–Hilbert (EH) action. The respective field equations are simply \(R=T=0\), where \(T={\widetilde{D}}e\) is the torsion 2-form and \({\widetilde{D}}=d+\omega \) is the covariant derivative with respect to the

*SO*(3) sector.

*SO*(3) sector with local Lorentz transformations and the sector \({\mathbb {R}}^3\) with diffeomorphisms. In fact, diffeomorphisms can be obtained from the Lie derivative [19, 37],

^{4}The subindex

*v*characterizes an element in a vector space \({\mathcal {V}}({\mathbb {M}}^3)\), where \({\mathbb {M}}^3\) is the spacetime manifold. Then, \(v^\mu \) is identified with the diffeomorphism parameters. The Greek indices are world indices and vary as \(\{0,1,2\}\). Therefore, it is quite simple to show that,

- (i)
\(R^{ij}= 0\); \(T^i=0\), which is the on-shell case discussed in [2]. This case displays full diffeomorphism symmetry but the classical vacuum solutions must be imposed. We remark that, if matter is included then this situation does not rely on the on-shell case anymore.

- (ii)
\(v^\nu = 0\); which means that diffeomorphism symmetry is entirely broken. Obviously, this is not an interesting situation if diffeomorphisms are desired as a property of the final result.

- (iii)The most general case is to consider the constraints (2.18) in a situation different from (i) and (ii). In fact, these constraints formally define a spacetime regular foliation [39, 40] with partial breaking of the diffeomorphisms, i.e., an intermediate situation between (i) and (ii). Hence, in the present case, the full diffeomorphism symmetry is then broken to a smaller group,$$\begin{aligned} \mathrm {Diff}(2)_{\mathrm {fol}}=\{v\in \mathrm {Diff}(3)\;\big |\;i_vR^{ij}=0\;\;;\;\;i_vT^i=0\}.\nonumber \\ \end{aligned}$$(2.19)

### 2.2 de Sitter gauge symmetry and gravity

*S*(3) are pseudo-translations. The translational sector of the algebra (2.2) is modified by \([P_{i},P_{j}]=\epsilon _{ijk} J^{k}\) while the invariant quadratic forms (2.3) are exactly the same. Employing again decomposition (2.4), but with the algebra of the

*SO*(4) group, to the Chern–Simons action (2.1), we find

*SO*(4) in the representation \(SO(3)\times S(3)\).

*A*and \(\theta \), namely,

*S*(3) with diffeomorphisms. For that, we consider once again the relations (2.14). The gauge transformations (2.23), restricted to the

*S*(3) sector, read now

## 3 Constrained gravity in four dimensions

*SO*(5) over a four-dimensional manifold with Euclidean signature. The gauge group can be decomposed according to \(SO(5)\equiv SO(4)\times S(4)\). The respective algebra splits as

^{5}Hence, the Yang–Mills action (3.3) assumes the form

*SO*(5) theory are

*SO*(4). Using the identifications (3.4), it is easy to show that these transformations can be rewritten as

*S*(4) sector of the transformations, we achieve

*S*(4) part. Thus, following the prescription of the three dimensional case, we can obtain the diffeomorphic transformations of the fields from the Lie derivative, namely,

*S*(4) sector of the gauge algebra can be identified with diffeomorphisms.

- (i)
de Sitter curvature and vanishing torsion (\(R^{ab}=\dfrac{\Lambda ^2}{3}e^ae^b\) and \(T^a=0\)): This case displays full diffeomorphism. However, this is not the on-shell case. Instead, this geometry sets \(S_{{4d(grav)}}=0\).

- (ii)
\(v^\nu = 0\); meaning that diffeomorphism symmetry is entirely broken and the geometry is totally free.

- (iii)The most general case is the intermediate situation between (i) and (ii) where (3.13) still holds and the full diffeomorphism symmetry is then broken to a smaller diffeomorphic group,$$\begin{aligned}&\mathrm {Diff}(3)_{\mathrm {fol}}\nonumber \\&\quad =\left\{ v\in \mathrm {Diff}(4)\;\big |\;i_v\left( R^{ab} {-}\dfrac{\Lambda ^2}{3}e^ae^b\right) {=}0;\;\;i_vT^a{=}0\right\} \;.\nonumber \\ \end{aligned}$$(3.14)

## 4 Analysis of the constraints in foliated spacetime

Given the generic constraints in three and four dimensions (cases (iii)), namely (2.18), (2.26) and (3.13), it is our intent to verify their consistency. The analysis we perform in this section is quite general, but we rather be closer to Physics and think of a spacetime manifold is foliated as \({\mathbb {M}}^d={\mathbb {M}}^{(d-1)}\times {\mathbb {R}}_t\), where \({\mathbb {R}}_t\) and \({\mathbb {M}}^{(d-1)}\) are the temporal and spatial parts, respectively. In particular, this choice is consistent with the Arnowitt–Deser–Misner (ADM) foliation formalism [41, 42, 43, 44, 45, 46, 47]. In this approach the dynamics of the gravitational field can be seen as the time evolution of these hypersurfaces along time.

*d*-dimensional curvature of the spacetime

*R*and the extrinsic curvature \({K^I}_{\mu }\) of the foliation, is given by the Gauss equation

*d*-dimensional torsion tensor

*T*in the leaf is given by

### 4.1 Three dimensions

In the case of three dimensions we refer to the constraints (2.18) and (2.26) for Poincaré and de Sitter symmetries, respectively. For the sake of simplicity, let us consider some special cases before the general ones.

#### 4.1.1 Poincaré symmetry

- (I)
**On-shell torsion and general curvature**

*rhs*in (4.4) vanishes due to the constraint (2.18), yielding

- (II)
**On-shell curvature and general torsion**

*rhs*in (4.10) vanishes, hence,

- (III)
**General torsion and general curvature**

#### 4.1.2 de Sitter symmetry

- (I)
**On-shell torsion and general curvature**

- (II)
**On-shell de Sitter curvature and general torsion**

- (III)
**General torsion and general curvature**

### 4.2 Four dimensions

- (I)
**Vanishing torsion and general curvature**

^{6}resulting in

- (II)
**General torsion and de Sitter curvature**

- (III)
**General torsion and general curvature**

## 5 Principal bundles, gauge theories and gravity

In this section we provide a quick mathematical analysis of our results. Specifically, we discuss the mapping of a principal bundle (gauge theory) into a coframe bundle (gravity as geometrodynamics) from the point of view of fiber bundle theory [19, 21, 22, 38, 51, 52]. This analysis serve as a mathematical support of the previous sections.

The principal bundle for which a gauge connection is generated [21] is denoted by \(\{P,{\mathbb {G}},{\mathbb {M}}^n,\pi \}\) where \(\pi :{P}\longmapsto {\mathbb {M}}^n\). In short notation, we write simply \({P}({\mathbb {M}}^n,{{\mathbb {G}}})\). The fiber and the structure group are both a Lie group \({\mathbb {G}}\). The base space is a *n*-dimensional manifold \({\mathbb {M}}^n\), often recognized as the spacetime. The total space is the non-trivial product \(P={\mathbb {M}}^n\times {\mathbb {G}}\). The projection map \(\pi \) is a continuous surjective map and the local triviality condition of *P* is guaranteed by the homeomorphism \(\phi _i:{\pi }^{-1}(\{{\mathbb {M}}^n\}_i)\longmapsto \{{\mathbb {M}}^n\}_i\times {\mathbb {G}}\), where \(\{{\mathbb {M}}^n\}_i\) are open sets covering \({\mathbb {M}}^n\). The definition of *P* can be regarded as a formal way to describe the localization of the Lie group \({\mathbb {G}}\) over the *n*-dimensional spacetime, associating to each point \(x\in {\mathbb {M}}^n\) different values for the elements \(u(x)\in {\mathbb {G}}\). The gauge connection appears in the definition of parallel transport in *P*. Gauge transformations are associated with coordinates changes in *P* by keeping the coordinates of the base space fixed, i.e., a translation along the fiber \({\pi }^{-1}\). Furthermore, for any connection 1-form *A* there is a curvature 2-form defined over *P* given by \(F=dA+AA\) which is recognized as the field strength in gauge theories.

Gravity theories can also be described in terms of principal bundles and, consequently, as gauge a theory. Initially, we will take a manifold^{7} \(\widetilde{{\mathbb {M}}}^n\) and the respective collection of tangent spaces \(T_X(\widetilde{{\mathbb {M}}}^n)\) at all points \(X\in \widetilde{{\mathbb {M}}}^n\). The collection of all tangent spaces defines the tangent bundle \({\mathbb {T}}\). In this bundle, \(\widetilde{{\mathbb {M}}}^n\) is the base space, the fiber is denoted by \(T_X(\widetilde{{\mathbb {M}}}^n)\) and the structure group is the \(GL(n,{\mathbb {R}}^n)\). Similarly the cotangent bundle \({\mathbb {T}}^{*}\) is defined as union of all cotangent spaces in \(\widetilde{{\mathbb {M}}}^n\). The fundamental structure for gravity theories is, for instance, the coframe bundle. The coframe bundle is associated with the cotangent bundle. A typical fiber at \(X\in \widetilde{{\mathbb {M}}}^n\) is the set of all local coframes, which are represented by vielbeins *e* defined in \(T^*_X ({\mathbb {M}}^n)\). Consequently, given a fixed coframe at \(T^*_X (\widetilde{{\mathbb {M}}}^n)\), there will be an infinite number of equivalent coframes related to the former by transformations of \(GL(n,{\mathbb {R}}^n)\). Obviously, the fibers coincide with the group. We can define the coframe bundle as \(P^*=(GL(n,{\mathbb {R}}^n),\widetilde{{\mathbb {M}}}^n)\). Note that \(GL(n,{\mathbb {R}}^n)\) can be trivially reduced to orthogonal group *SO*(*n*) which leads to bundle \(P'^*(SO(n),\widetilde{{\mathbb {M}}}^n)\) that coincides with the mathematical structure of Einstein–Cartan gravity. See [20, 53]. In \(P'^*\) the *SO*(*n*) algebra-valued connection is the spin-connection 1-form \(\omega \). The curvature 2-form in \(P'^*\) is defined as \(R=d\omega +\omega \omega \) while the torsion 2-form is \(T=de+\omega e\). It is useful to understand that the only difference between the coframe bundle \(P'^*\) and the gauge principal bundle *P* is that the structure group (and fiber) *SO*(*n*) is also the group of local isometries. Hence, gravity can be visualized as a gauge theory where the gauge group is identified with the spacetime local isometries.

In order to connect a gauge theory with a gravity theory in the light of the principal bundles above described, let us start with the Poincaré three-dimensional case. First, we assume that the original Chern–Simons action (2.1) is constructed over a flat manifold, say, the three dimensional Euclidean spacetime. It is possible then to reinterpret the transition from action (2.1) to (2.9) as a kind of duality between a Chern–Simons action in Euclidean spacetime and a gravity action over a general manifold. The action (2.1) is a standard Chern–Simons gauge theory and, in principle, has nothing to do with gravity. Essentially, the field *A* is a connection over the principal bundle \((SU(2)\times {\mathbb {R}}^3,{\mathbb {R}}^3)\) where the base space is the three dimensional Euclidean space \({\mathbb {R}}^3\) and the structure group (and fiber) is the gauge group. On the other hand, action (2.9) is a gravity action describing the geometrodynamics of a three-dimensional manifold \(\widetilde{{\mathbb {M}}}^3\). Thus, the spin-connection \(\omega \) is a connection over a coframe bundle \((SO(3),\widetilde{{\mathbb {M}}}^3)\) where the base space is \(\widetilde{{\mathbb {M}}}^3\), the structure group is *SO*(3) and a typical fiber at a point \(X\in \widetilde{{\mathbb {M}}}^3\) is the collection of all coframes that can be defined at the tangent space \(T_X(\widetilde{{\mathbb {M}}}^3)\). Obviously the coframes are identified with the dreibein *e*.

In [16], a map between a gauge theory over a Euclidean spacetime and a geometrodynamics gravity theory was proposed in four dimensions. However, an Inönü-Wigner contraction and symmetry breaking were required. In the present case, neither of these effects is needed. In fact, the formal map \(f:(SU(2)\times {\mathbb {R}}^3,{\mathbb {R}}^3)\longmapsto (SO(3),\widetilde{{\mathbb {M}}}^3)\) can be understood as a set of maps \(f_i\) as follows: An isomorphism between base spaces \(f_0:{\mathbb {R}}^3\longmapsto \widetilde{{\mathbb {M}}}^3\). The *SU*(2) sector of the gauge group is mapped into the local isometry group *SO*(3) of \(\widetilde{{\mathbb {M}}}^3\), while the gauge sector \({\mathbb {R}}^3\) is mapped into the diffeomorphisms group, namely, \(f_1: SU(2)\longmapsto SO(3)\) and \(f_2: {\mathbb {R}}^3\longmapsto \mathrm {Diff}(3)\), respectively. To \(f_2\) be properly defined, one has to impose the relations (2.17). We also demand that the space of *p*-forms in \({\mathbb {R}}^3\) is identified with the space of *p*-forms in \(\widetilde{{\mathbb {M}}}^3\), namely \(\Omega ^p\) and \({\widetilde{\Omega }}^p\), by means of \(f_3:\Omega ^p\longmapsto {\widetilde{\Omega }}^p\). Finally, by imposing the relation (2.6) as an extra condition to \(f_3\), the coframe bundle \((SO(3),\widetilde{{\mathbb {M}}}^3)\) can be constructed from \((SU(2)\times {\mathbb {R}}^3,{\mathbb {R}}^3)\).

*L*is the matrix that transforms the components of forms \(\Omega ^p\) into forms \({\widetilde{\Omega }}^p\); \(g=|\det g_{\mu \nu }|\) and \({\tilde{g}}=|\det {\tilde{g}}_{\mu \nu }|\); \(g_{\mu \nu }\) e \({\tilde{g}}_{\mu \nu }\) are respectively original and target metrics. Obviously, for the present case we have \(g_{\mu \nu }=\delta _{\mu \nu }\) and \(n=3\). Thus,

*e*and \(\omega \). This means that \(f_2\) is not a direct equivalence between \({\mathbb {R}}^3\) and \(\mathrm {Diff}(3)\), but a special case where (2.17) is imposed. Remarkably, all geometrical properties of the principal bundle \((SU(2)\times {\mathbb {R}}^3,{\mathbb {R}}^3)\) can be used to construct the coframe bundle \((SO(3),\widetilde{{\mathbb {M}}}^3)\), including the diffeomorphic transition functions.

In the case where the cosmological term is included, a similar map can be defined. The difference with respect to the previous case is that \(f_2:S(3)\longmapsto \mathrm {Diff}(3)\). In this case, (2.17) has a stronger appeal because \(f_2\) generate diffeomorphisms from a non-Abelian sector of the gauge group which is not even a subgroup, but a symmetric coset space.

Once again, we remark that the identification (2.6) is only possible if a mass parameter is at our disposal. In three dimensions, we have used the coupling constant \(\kappa \). Since this parameter is inherent to the theory, this Chern–Simons gravity duality can be performed at any scale of the theory, a feature that is not possible in four dimensions. Moreover, it is important to be understood that the action (2.1) is a theory where the gauge symmetry has nothing to do with the geometrical properties of the spacetime. Only after the mapping, the gauge symmetry is identified with the geometrical properties of the base space and the fields with the geometrical objects that describe the dynamics of this geometry.

The previous discussion can easily be extended to the four-dimensional case. We assume that the Yang–Mills action (3.3) is constructed over a flat four-dimensional Euclidean manifold. The transition from action (3.3) to (3.5) can be taken as a duality between a Yang–Mills action in Euclidean spacetime and a gravity action over a general manifold. The field *Y* is a connection over the principal bundle \((SO(5),{\mathbb {R}}^4)\). The gravity action (3.5) describes the geometrodynamics of a four-dimensional manifold \(\widetilde{{\mathbb {M}}}^4\). Thus, the spin-connection \(\omega \) is a connection over a coframe bundle \((SO(4),\widetilde{{\mathbb {M}}}^4)\). Again, the concepts of Inönü-Wigner contraction and symmetry breaking are not required. The formal map \(f:(SO(5),{\mathbb {R}}^4)\longmapsto (SO(4),\widetilde{{\mathbb {M}}}^4)\) is a set of maps \(f_i\) as follows: An isomorphism between base spaces \(f_0:{\mathbb {R}}^4\longmapsto \widetilde{{\mathbb {M}}}^4\). The *SO*(4) sector of the gauge group is mapped into the local isometry group *SO*(4) of \(\widetilde{{\mathbb {M}}}^4\), while the gauge sector \(S(4)=SO(5)/SO(4)\) is mapped into the diffeomorphisms group, namely, \(f_1: SO(4)\longmapsto SO(4)\) and \(f_2: S(4)\longmapsto \mathrm {Diff}(4)\), respectively. Moreover, the space of *p*-forms in \({\mathbb {R}}^4\) is identified with the space of *p*-forms in \(\widetilde{{\mathbb {M}}}^4\) by means of \(f_3:\Omega ^p\longmapsto {\widetilde{\Omega }}^p\). Finally, by imposing the relation (3.4) to \(f_3\), the coframe bundle \((SO(4),\widetilde{{\mathbb {M}}}^4)\) is constructed from the principal bundle \((SO(5),{\mathbb {R}}^4)\). We stress out that \(f_0\) must be an isomorphism due to the same reasons of the three-dimensional case. Also, \(f_2\) is not a direct equivalence between *S*(4) and \(\mathrm {Diff}(4)\), but a special case where (3.12) is imposed. Once again, we remark that the identification (3.4) is only possible if a mass parameter is at our disposal. In four dimensions (in contrast to the three-dimensional case), the action (3.3) is massless. Hence, the mapping depends whenever the theory develops mass parameters. Fortunately, Yang–Mills theories develop a few non-perturbative effects that lead to the appearance of mass scales [25, 26, 28, 29, 30]. Thus, the mapping would only hold for low a low energy regime.

To end this section, we wish to recall that alternative mappings that could be applied in the same context can be found in [8] and references therein.

## 6 Conclusions

In this work, we were able to provide a reinterpretation of Witten’s three-dimensional equivalence result [2] by showing how one can construct an equivalent gravity theory originated from a Chern–Simons theory in Euclidean spacetime. Such equivalence required a mass parameter (coupling parameter) and a set of geometric constraints ((2.18) for Poincaré gauge theory and; (2.26) for de Sitter gauge theory). The method is valid at any scale since the coupling parameter already carries mass dimension. The method was also applied to the four dimensional case for the Yang–Mills action with \(SO(5)\equiv SO(4)\times S(4)\) gauge symmetry and considering flat spacetime. In this case, the coupling parameter is dimensionless and another parameter is required. Fortunately, there are a few mass parameters at our disposal in four-dimensional Yang–Mills theories at lower energy regime. For instance, the Gribov parameter is a nice candidate due to the recent discovery that it is a gauge invariant quantity [28, 29, 30]. A set of geometric constraints were also required, namely (3.13). In both cases (three and four dimensions), the gauge field can be associated with geometrical objects, namely, the vielbein and spin-connection, while the gauge group can be identified with local isometries and diffeomorphisms. Hence, the gauge field degrees of freedom can be somehow seen as being absorbed by spacetime. It is worth mentioning that the present method is comparable with the mechanism described in [16, 18, 35] for Yang–Mills theory. The advantage here is the fact that we do not depend on the running of the mass parameters to perform the mapping nor on an Inönü-Wigner contraction mechanism. The present mechanism merely depends on the existence of a mass parameter and not on its behavior with respect to the energy scale.

The usual Witten on-shell conditions which lead to gravity theories displaying full diffeomorphism symmetry. However, vacuum geometry must be imposed.

On-shell torsion and general curvature: The result is a Riemannian Einstein–Hilbert gravity ((2.9) for Poincaré symmetry and (2.21) for de Sitter symmetry) with restricted diffeomorphism symmetry, see (4.8) for the Poincaré symmetry and (4.21) for the de Sitter symmetry.

On-shell curvature and general torsion: The result is a Weitzenböck Einstein–Hilbert gravity ((2.9) for Poincaré symmetry and (2.21) for de Sitter symmetry) with restricted diffeomorphism symmetry, see (4.14) for both, Poincaré and de Sitter symmetries.

General torsion and curvature: The result is a Riemann–Cartan Einstein–Hilbert gravity ((2.9) for Poincaré symmetry and (2.21) for de Sitter symmetry) with restricted diffeomorphism symmetry, see (4.15) for the Poincaré symmetry and (4.23) for the de Sitter symmetry.

Vanishing torsion and general curvature: The result is a Riemannian gravity (action (3.5) with \(T=0\)) with restricted diffeomorphism symmetry, see (4.26).

de Sitter curvature and general torsion: The result is a Riemann-Cartan gravity (action (3.5) with curvature given by \(R^{ab}=\dfrac{\Lambda ^2}{3}e^ae^b\)) with restricted diffeomorphism symmetry, see (4.28).

General torsion and curvature: The result is a Riemann–Cartan Einstein–Hilbert gravity (action (2.9)) with restricted diffeomorphism symmetry, see (4.29).

Finally, we wish to point out two possibly interesting analysis that are left for future investigation: First, the quantum consistency of the mechanism. In three dimensions, this may be helped by the fact that Chern–Simons theories are finite [54, 55]. It is appropriate to dedicate a few words about the 3D quantum gravity scenario explored in [3] where the author scrutinize both, the canonical and functional quantization methods. The ADM decomposition, geometric structures, covariant phase space are studied in second and first order formalisms. See also [56, 57] for more recent developments. In four dimensions, *SO*(5) Yang–Mills theories are known to be renormalizable and unitary [58]. Remembering that in both cases the constraints must be taken into account. Second, the inclusion of matter fields (bosonic and fermionic) may produce interesting effects to be studied. Possibly extra constraints may be required. It is clear then that the results of the present work generalizes the classical results of [3] and opens the possibility to account for other spacetime foliations than the ADM decomposition. Moreover, as mentioned before, the whole setup is also generalized to four dimensions for the Yang–Mills theory.

## Footnotes

- 1.
Throughout this work the term spacetime will always be associated with a Hausdorff differentiable manifold (or simply manifold).

- 2.
The

*SU*(2) group is a double cover of*SO*(3) and there is a \(2\longmapsto 1\) surjective homomorphism from*SU*(2) to*SO*(3). - 3.
The quotation marks are employed because gravity theory in three dimensions does not have a Newtonian limit. Nevertheless, since

*G*appear in front of the three-dimensional Einstein–Hilbert action, it can be associated with the four-dimensional Newton’s constant by analogy. - 4.
The interior derivative acts as \(i_v(dx^\nu ):=v^\nu \). See, for instance, [38].

- 5.
- 6.
In the three-dimensional case we had also three free indices but no three-index Levi-Civita symbol.

- 7.
We included a “tilde” in the manifold of gravity to emphasize that this manifold is not the manifold without the “tilde” of the gauge theory. This distinction was not made before because there was no need until now.

## Notes

### Acknowledgements

The authors are grateful to Conselho Nacional de Desenvolvimento Científico e Tecnológico, The Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and the Pró-Reitoria de Pesquisa, Pós-Graduação e Inovação (PROPPI-UFF) are acknowledge for financial support.

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