**3. First order gravity**

Gravity can be mathematically defined as a coframe bundle [12, 13, 20, 21], *CM* = (*GL*(*d*, **R**), *M*), where *M* is a *d*-dimensional spacetime manifold. The structure group and fiber have a more deep meaning: In each point *X* ∈ *M* one can define the cotangent space *T*<sup>∗</sup> *<sup>X</sup>*(*M*). The fiber is the collection of all coframes *e* that can be defined in *T*∗ *<sup>X</sup>*(*M*) and which are related to each other through the action of the general linear group. As a consequence, the fiber is actually the group *GL*(*d*, **R**). In terms of Sect. 2, the coframe bundle is also a gauge bundle for the general linear group with the addend that the gauge group is identified with geometric properties of *M*. The action of the group from the right are local gauge transformations while the action of the group from the left are general coordinate transformations.

Geometrically, the gauge connection Γ is related to the parallel transport, in *M*, between two near cotangent spaces. The curvature 2-form is obtained from the double action of the covariant derivative, Ω = dΓ + ΓΓ while torsion, *T* = ∇*e*, is the minimal coupling of coframes. It is evident that, besides Γ, which is the gauge field of gravity, *e* is just as relevant. Moreover, the metric tensor *m* in the tangent space *T*(*M*) has to be introduced because the *GL*(*d*, **R**)/*SO*(*d* − *n*, *n*) sector of the general linear group does not preserve a flat metric. In practice *m* enters as an extra independent field. Thus, in *CM*, gravity possesses three fundamental fields, Γ, *e* and *m*, all relevant to determine spacetime geometry. A general theory of this type is a metric-affine gravity<sup>4</sup> [10–13].

Remarkably, the coframe bundle has a contractible piece *GL*(*d*, **R**)/*SO*(*d* − *n*, *n*) where *SO*(*d* − *n*, *n*) is obviously a stability group. This means that the coframe bundle can be naturally contracted down to *SOM* = (*SO*(*d* − *n*, *n*), *M*) [14, 17, 19]. The fact that the contraction is topologically favored has drastic consequences to the geometry, it means that every manifold *M* can assume a Riemannian metric, *i.e.*, the connection can always be chosen to be compatible with the metric. This means that the metric tensor can be set as a constant flat one, *m* = *η*, where the signature of *η* depends on *n*. As a consequence, a standard fiber at *X* is the set of all orthonormal coframes that can be obtained from an *SO*(*d* − *n*, *n*) transformation acting on a fixed coframe. The group *SO*(*d* − *n*, *n*) describes then the isometries in *T*<sup>∗</sup> *<sup>X</sup>*(*M*). From Theorem 2.1 the connection Γ = *ω* + *w* imposes an *SO*(*d* − *n*, *n*) connection *ω* ∈ *O*, where *O* is the algebra of *SO*(*d* − *n*, *n*) and *w* ∈ *GL* /*O*. A gravity theory constructed over *SOM* is a standard Einstein-Cartan gravity. In this work we shall deal strictly with *SOM*.

To construct a moduli bundle for gravity is not immediate as in pure gauge theories. If the coframe bundle is a gauge bundle then *e* is actually a matter field because it is a fundamental representation of the gauge group [24]. On the other hand, one can include the space of all independent *e* that can be defined in *T*∗ *<sup>X</sup>*(*M*) as the coframe moduli space E. Thus, defining the full moduli space as G = W×E, where W is the moduli space of spin-connections, the

<sup>4</sup> Metric-Affine gravities can be also generalized for the affine group *A*(*d*, **R**) = *GL*(*d*, **R**) **R***d*, however, the non-semi-simplicity of this group spoils the construction of an invariant action. We shall fix our attention to semi-simple groups.

gauge orbit is then

4 Will-be-set-by-IN-TECH

*Comment.* The field *θ* is a one to one map *θ* : *C* �−→ *θ*(*C*) which establishes that for at each

Gravity can be mathematically defined as a coframe bundle [12, 13, 20, 21], *CM* = (*GL*(*d*, **R**), *M*), where *M* is a *d*-dimensional spacetime manifold. The structure group and fiber have a more deep meaning: In each point *X* ∈ *M* one can define the cotangent space *T*<sup>∗</sup>

to each other through the action of the general linear group. As a consequence, the fiber is actually the group *GL*(*d*, **R**). In terms of Sect. 2, the coframe bundle is also a gauge bundle for the general linear group with the addend that the gauge group is identified with geometric properties of *M*. The action of the group from the right are local gauge transformations while

Geometrically, the gauge connection Γ is related to the parallel transport, in *M*, between two near cotangent spaces. The curvature 2-form is obtained from the double action of the covariant derivative, Ω = dΓ + ΓΓ while torsion, *T* = ∇*e*, is the minimal coupling of coframes. It is evident that, besides Γ, which is the gauge field of gravity, *e* is just as relevant. Moreover, the metric tensor *m* in the tangent space *T*(*M*) has to be introduced because the *GL*(*d*, **R**)/*SO*(*d* − *n*, *n*) sector of the general linear group does not preserve a flat metric. In practice *m* enters as an extra independent field. Thus, in *CM*, gravity possesses three fundamental fields, Γ, *e* and *m*, all relevant to determine spacetime geometry. A general

Remarkably, the coframe bundle has a contractible piece *GL*(*d*, **R**)/*SO*(*d* − *n*, *n*) where *SO*(*d* − *n*, *n*) is obviously a stability group. This means that the coframe bundle can be naturally contracted down to *SOM* = (*SO*(*d* − *n*, *n*), *M*) [14, 17, 19]. The fact that the contraction is topologically favored has drastic consequences to the geometry, it means that every manifold *M* can assume a Riemannian metric, *i.e.*, the connection can always be chosen to be compatible with the metric. This means that the metric tensor can be set as a constant flat one, *m* = *η*, where the signature of *η* depends on *n*. As a consequence, a standard fiber at *X* is the set of all orthonormal coframes that can be obtained from an *SO*(*d* − *n*, *n*) transformation acting

Theorem 2.1 the connection Γ = *ω* + *w* imposes an *SO*(*d* − *n*, *n*) connection *ω* ∈ *O*, where *O*

To construct a moduli bundle for gravity is not immediate as in pure gauge theories. If the coframe bundle is a gauge bundle then *e* is actually a matter field because it is a fundamental representation of the gauge group [24]. On the other hand, one can include the space of all

the full moduli space as G = W×E, where W is the moduli space of spin-connections, the

<sup>4</sup> Metric-Affine gravities can be also generalized for the affine group *A*(*d*, **R**) = *GL*(*d*, **R**) **R***d*, however, the non-semi-simplicity of this group spoils the construction of an invariant action. We shall fix our

on a fixed coframe. The group *SO*(*d* − *n*, *n*) describes then the isometries in *T*<sup>∗</sup>

standard Einstein-Cartan gravity. In this work we shall deal strictly with *SOM*.

. In other words, in each fiber

*<sup>X</sup>*(*M*) and which are related

*<sup>X</sup>*(*M*).

*<sup>X</sup>*(*M*). From

/*O*. A gravity theory constructed over *SOM* is a

*<sup>X</sup>*(*M*) as the coframe moduli space E. Thus, defining

point *C* there will be only one *θ* such that if *C* ∼ *C*� then *θ* ∼ *θ*�

The fiber is the collection of all coframes *e* that can be defined in *T*∗

the action of the group from the left are general coordinate transformations.

*C<sup>h</sup>* there will be only one equivalence class for *θ*.

theory of this type is a metric-affine gravity<sup>4</sup> [10–13].

is the algebra of *SO*(*d* − *n*, *n*) and *w* ∈ *GL*

independent *e* that can be defined in *T*∗

attention to semi-simple groups.

**3. First order gravity**

$$\begin{aligned} \omega^{\mathfrak{g}} &= \mathfrak{g}^{-1}(\mathfrak{d} + \omega)\mathfrak{g} \\ \mathfrak{e}^{\mathfrak{g}} &= \mathfrak{g}e \end{aligned} \tag{2}$$

with *g* ∈ *SO*(*d* − *n*, *n*) and *W* = (*ω*,*e*) ∈ G. The moduli coframe bundle is **O** = (*SOM*, G). This principal bundle is analogously equivalent to that described in Theorem 2.3. Thus, the space of all sections that can be defined over *SOM* is actually the functional space of coframes. This space is equivalent to the fiber bundle **E** = (Σ(*e*), *SOM*, E) where the fiber Σ(*e*) is the set of all equivalent sections that can be obtain from an element *e* ∈ E through the action of *SOM*.
