**4.1 Gauge and coframe bundles**

**Theorem 4.1.** *Let HR* = (*H*, *R*) *be a stable reduced bundle obtained from the gauge bundle GR* = (*G*, *R*) *which is endowed with a connection Y* = *A* + *B. Then GR can define a geometry SOM* = (*SO*(*d* − *n*, *n*), *M*) *if and only if*


*Proof.* Condition *1* ensures that each point *x* ∈ *R* can define a unique point in *X* ∈ *M* while *M* will be entirely covered by the map with no overlapping points. Moreover, the algebraic structure defined in *R* will be preserved by the mapping. On the other hand, condition *2* ensures that the target group *SO*(*d* − *n*, *n*) will be entire covered by the mapping. To construct the fiber at a cotangent space *T*∗ *<sup>X</sup>*(*M*) in each point *X* ∈ *M* we need two quantities: a coframe *e* ∈ *T*<sup>∗</sup> *<sup>X</sup>*(*M*) and the isometries of the cotangent space. The use of conditions *1* and *2* ensures the existence of the isometries. Since there is a fiber *H* in each point *x* ∈ *R* and condition *2* ensures that *H* is at least homomorphic to *SO*(*d* − *n*, *n*), this fiber defines the cotangent space *T*∗ *<sup>X</sup>*(*M*) isometries. In addition, since there is one fiber for each point *x* ∈ *R* there will be only one set of isometries for each *X* ∈ *M*, as it is evident from condition *1*. Condition *3* ensures that the field *θ*, in the fiber *HR*, can be defined as the cotangent 1-form *e* ∈ *T*<sup>∗</sup> *<sup>X</sup>*(*M*), recognized as a coframe. Once more, the isomorphism of condition *1*, together with Corollary 2.4, ensures the uniqueness of *e* in *X*. Finally, a standard fiber in *SOR* is obtained by the action of *SO*(*d* − *n*, *n*) on *e*. A connection *ω* in *SOM* emerges naturally from *A*. Again, condition *1* establishes that at a point *X* there will be only one *ω* while the action of *H* on *A* ensures that *ω* will transform correctly along the fiber *π*−1(*X*) under the action of the local isometries in *T*<sup>∗</sup> *<sup>X</sup>*(*M*). More explicitly, In each fiber *π*−1(*x*) a connection *A* can be defined. This definition ensures the existence of an equivalence class along the fiber. Thus, a section *s*(*x*) : *R* �−→ *H*(*x*) is defined in such a way that *x* �−→ *q*, where *q* = (*x*, *g*) and *g* ∈ *G*. In each point *q* the connection *A*(*q*) can be identified with a connection *ω*(*Q*) in *SO*(*X*) at a point *Q* = (*X*, *u*) ∈ *SO*(*X*) and *u* ∈ *SO*(*d* − *n*, *n*) is the *SO*(*d* − *n*, *n*) equivalent of *g* such that *π*(*Q*) = *X*, *π*(*q*) = *x* and *x* �−→ *X*. Condition *1* ensures that there will be only one connection *ω*(*Q*) for each *A*(*q*).

*Proof.* In a principal principal bundle *GR* a connection *Y* can be defined. The collection of all possible connections *Y* defines the space **Y**. The separation of **Y** into equivalence classes organizes this space as the set of all gauge orbits over the moduli space C. According to Theorem 2.3, the gauge orbit splits as (1). Moreover, for each of these fibers one can associate a field *θ*, as allowed by Corollary 2.4. It is the pair (*A*, *θ*) that defines the geometric fields (*ω*,*e*) in *SOM*. Thus, to construct an **O** structure for gravity is an easy task to collect all possible pairs *W* = (*ω*,*e*) emerging from Theorem 4.1. In fact, each pair (*A*, *θ*) and the associated orbit define a fiber *<sup>W</sup><sup>g</sup>* <sup>∈</sup> **<sup>O</sup>**. That is ensured also by Condition *<sup>1</sup>* of Theorem 4.1. Thus, the fiber *HR* over a point (*ω*,*e*) ∈ G is obtained from *A* �−→ *ω* and *θ*(*A*, *B*) �−→ *e* and the respective action

Fiber Bundles, Gauge Theories and Gravity 85

The space **B** = (Σ(*B*), *HR*, B) is a dynamical space and survives the mapping. For the case **<sup>B</sup>** <sup>∩</sup> <sup>Θ</sup> <sup>=</sup> <sup>∅</sup> one can associate the moduli space with a set of independent fields <sup>B</sup> �−→ <sup>B</sup>˜ which are invariant representations of *SO*(*d* − *n*, *n*). Theorem 4.1 ensures that the structure group *HR* can be mapped into *SOM* while the fibers Σ(*B*) are identified with fibers Σ(*B*˜) over *B*˜. Thus, for each *<sup>B</sup>* <sup>∈</sup> **<sup>B</sup>** there will be a correspondent *<sup>B</sup>*˜ <sup>∈</sup> <sup>B</sup>˜ and the fiber <sup>Σ</sup>(*B*˜) is obtained from the action of *SO*(*<sup>d</sup>* <sup>−</sup> *<sup>n</sup>*, *<sup>n</sup>*). Thus, **<sup>B</sup>**˜ = (Σ(*B*˜), *SOM*, *<sup>B</sup>*˜). The proof for the case **<sup>B</sup>** <sup>∩</sup> <sup>Θ</sup> �<sup>=</sup> <sup>∅</sup> is totally

*Comment.* The final result is that of a gravity theory with an extra set of matter fields **B**˜ .

We have formally prove that a class of gauge theories can be deformed into a first order gravity theory and, possibly, with an extra set of matter fields. For that we have employed the theory of fiber bundles. The relevance (and motivation) of the present work is that it can be applied to quantum gravity models which are based on gauge theories that can generate an emergent gravity theory. The main problem in quantizing gravity is that the principles of general relativity are incompatible with those of quantum field theory. In fact, a quantum field theory can only be formulated in an Euclidean spacetime. For example, a quantum field is, by definition, an object that carries uncertainty fluctuations and is parametrized through spacetime coordinates, *i.e.*, a set of well defined real parameters. Now, if a coframe field is a quantum field5, *e*ˆ(*x*), and from the fact that it defines a mapping from tangent coordinates *x<sup>a</sup>* to world coordinates *xμ*, then quantum fluctuations of *e*ˆ will induce spacetime to fluctuate as

*<sup>a</sup> xa*. Thus, a paradox is encountered because *x* must be a set of parameters instead

On the other hand, if the starting gauge theory is constructed over an Euclidean manifold and it is renormalizable, then it can be an excellent candidate for a quantum gravity theory. All needed is that it emerges as a geometrodynamics at classical level. The class of theories that

A few practical examples are in order: In [15] a 4-dimensional *SU*(2) gauge theory generates a deformation of the 3-dimensional space. Time is left untouched by he mapping. In this example, the resulting theory contains the Einstein-Hilbert action for the extrinsic curvature and the solution of the equations of motion predicts not only curvature but also torsion. Another example can be found in [9], where a deformed 4-dimensional spacetime emerges from a de Sitter type gauge theory over an Euclidean spacetime. In this case, a dynamical mass scale is responsible for the separation between the gauge and gravity phases. In general,

fits on this program are determined essentially by theorems 4.1 and 4.3.

<sup>5</sup> The hat indicates the quantum nature of the field.

of *SO*(*d* − *n*, *n*). The uniqueness of this mapping is ensured by Corollary 2.4.

equivalent.

well, *x*ˆ*<sup>μ</sup>* = *e*ˆ

*μ*

of a fluctuating object.

**5. Final remarks**

Obviously, the reconstruction of the whole class of connections along a fiber is obtained from the action of the group on *ω*(*Q*). The final result is a mapping *GR* �−→ *SOR* which is actually a set of mappings

$$\begin{array}{l} R \longmapsto M \\ H \longmapsto SO(d-n,n) \\ \theta \longmapsto e \\ A \longmapsto \omega \end{array} \tag{3}$$

We remark that, in the mapping *GR* �−→ *SOR*, a contraction *GR* −→ *HR* is assumed.

*Comment.* If the map described in Theorem 4.1 is smooth and all fibers *π*−1(*x*) are mapped into fibers *π*−1(*X*) then this map is a bundle map. In that case, since each fiber *π*−1(*x*) is mapped into a fiber *<sup>π</sup>*−1(*X*) in a smooth way, a smooth map *<sup>R</sup>* �−→ *<sup>M</sup>* is induced [20].

*Comment.* We remark that dim *M* = dim *T*∗ *<sup>X</sup>*(*M*) and thus dim *R* = *do* is not necessary equal to dim *M* = *d*. Notwithstanding, the bound *d* ≤ *do* is a always valid. Furthermore, *d* is the dimension of the fundamental representation of *SO*(*d* − *n*, *n*), as a consequence it coincides with the dimension of the invariant representation of *H*, namely *θ*. The case *d* < *do* affects only a subsector of spacetime *R* ⊃ *Rsub* �−→ *M*, where dim *Rsub* = dim *M*. For instance, if *<sup>R</sup>* <sup>=</sup> **<sup>R</sup>***do* , then the resulting full manifold is then *<sup>M</sup>* <sup>×</sup> **<sup>R</sup>***do*−*d*. The case *<sup>d</sup>* <sup>=</sup> *do* deforms the entire spacetime. This case is more interesting because one can take the starting gauge theory as a description for quantum gravity. from now on, independently of the case, we shall call by *M* the full *do*-dimensional manifold formed by the deformed (*d*-dimensional subspace) and undeformed ((*do* − *d*)-dimensional subspace) sectors.

**Corollary 4.2.** *If the space of p-forms in R are directly mapped into the space of p-forms in M,* Π*<sup>p</sup> R* �−→ Π*p <sup>M</sup>, then the map can be explicitly computed and depends exclusively on the metric tensors of R and M.*

*Proof.* By duality the map Π*<sup>p</sup> <sup>R</sup>* �−→ <sup>Π</sup>*<sup>p</sup> <sup>M</sup>* induces a similar map for the Hodge dual space of (*<sup>d</sup>* <sup>−</sup> *<sup>p</sup>*)-forms, <sup>∗</sup>Π*<sup>p</sup> <sup>R</sup>* �−→ �Π*<sup>p</sup> <sup>M</sup>*, where ∗ is the Hodge operation in *R* while � is the Hodge operation in *M*. Thus, it is a straightforward exercise [9] to show that the map is given by

$$\frac{\partial X^{\nu}}{\partial x^{\mu}} = \left(\frac{\vec{m}}{m}\right)^{1/2d} \tilde{m}^{\nu \alpha} m\_{a\mu\nu} \,, \tag{4}$$

where *mμν* is the metric tensor in *R*, *m*˜ *μν* is the metric tensor in *M* and *m* and *m*˜ are the respective determinants. The determinants are assumed to be non-vanishing.

*Comment.* Since the mapping matrix (4) has an inverse, the geometry in *M* is unique.

### **4.2 Moduli bundles and gravity**

Theorem 4.1 can be generalized for moduli bundles:

**Theorem 4.3.** *If the map GR* �−→ *SOM exists then the map* **<sup>Y</sup>** �−→ **<sup>O</sup>** <sup>⊕</sup> **<sup>B</sup>**˜ *also exists. The space* **<sup>B</sup>**˜ *is the target space associated with the space* **B** *or* **B**/Θ *if* Θ ⊆ **B** *where* Θ *is the functional space of all possible θ that can be defined in* **Y***.*

6 Will-be-set-by-IN-TECH

Obviously, the reconstruction of the whole class of connections along a fiber is obtained from the action of the group on *ω*(*Q*). The final result is a mapping *GR* �−→ *SOR* which is actually

*H* �−→ *SO*(*d* − *n*, *n*) ,

*Comment.* If the map described in Theorem 4.1 is smooth and all fibers *π*−1(*x*) are mapped into fibers *π*−1(*X*) then this map is a bundle map. In that case, since each fiber *π*−1(*x*) is

to dim *M* = *d*. Notwithstanding, the bound *d* ≤ *do* is a always valid. Furthermore, *d* is the dimension of the fundamental representation of *SO*(*d* − *n*, *n*), as a consequence it coincides with the dimension of the invariant representation of *H*, namely *θ*. The case *d* < *do* affects only a subsector of spacetime *R* ⊃ *Rsub* �−→ *M*, where dim *Rsub* = dim *M*. For instance, if *<sup>R</sup>* <sup>=</sup> **<sup>R</sup>***do* , then the resulting full manifold is then *<sup>M</sup>* <sup>×</sup> **<sup>R</sup>***do*−*d*. The case *<sup>d</sup>* <sup>=</sup> *do* deforms the entire spacetime. This case is more interesting because one can take the starting gauge theory as a description for quantum gravity. from now on, independently of the case, we shall call by *M* the full *do*-dimensional manifold formed by the deformed (*d*-dimensional subspace) and

**Corollary 4.2.** *If the space of p-forms in R are directly mapped into the space of p-forms in M,* Π*<sup>p</sup>*

operation in *M*. Thus, it is a straightforward exercise [9] to show that the map is given by

 *m*˜ *m*

*Comment.* Since the mapping matrix (4) has an inverse, the geometry in *M* is unique.

*<sup>M</sup>, then the map can be explicitly computed and depends exclusively on the metric tensors of R and*

1/2*<sup>d</sup>*

where *mμν* is the metric tensor in *R*, *m*˜ *μν* is the metric tensor in *M* and *m* and *m*˜ are the

**Theorem 4.3.** *If the map GR* �−→ *SOM exists then the map* **<sup>Y</sup>** �−→ **<sup>O</sup>** <sup>⊕</sup> **<sup>B</sup>**˜ *also exists. The space* **<sup>B</sup>**˜ *is the target space associated with the space* **B** *or* **B**/Θ *if* Θ ⊆ **B** *where* Θ *is the functional space of all*

*A* �−→ *ω* . (3)

*<sup>X</sup>*(*M*) and thus dim *R* = *do* is not necessary equal

*<sup>M</sup>* induces a similar map for the Hodge dual space of

*m*˜ *ναmαμ* , (4)

*<sup>M</sup>*, where ∗ is the Hodge operation in *R* while � is the Hodge

*R* �−→

*R* �−→ *M* ,

*θ* �−→ *e* ,

We remark that, in the mapping *GR* �−→ *SOR*, a contraction *GR* −→ *HR* is assumed.

mapped into a fiber *<sup>π</sup>*−1(*X*) in a smooth way, a smooth map *<sup>R</sup>* �−→ *<sup>M</sup>* is induced [20].

a set of mappings

Π*p*

*M.*

*Proof.* By duality the map Π*<sup>p</sup>*

**4.2 Moduli bundles and gravity**

*possible θ that can be defined in* **Y***.*

(*<sup>d</sup>* <sup>−</sup> *<sup>p</sup>*)-forms, <sup>∗</sup>Π*<sup>p</sup>*

*Comment.* We remark that dim *M* = dim *T*∗

undeformed ((*do* − *d*)-dimensional subspace) sectors.

*<sup>R</sup>* �−→ �Π*<sup>p</sup>*

Theorem 4.1 can be generalized for moduli bundles:

*<sup>R</sup>* �−→ <sup>Π</sup>*<sup>p</sup>*

*∂X<sup>ν</sup> <sup>∂</sup>x<sup>μ</sup>* <sup>=</sup>

respective determinants. The determinants are assumed to be non-vanishing.

*Proof.* In a principal principal bundle *GR* a connection *Y* can be defined. The collection of all possible connections *Y* defines the space **Y**. The separation of **Y** into equivalence classes organizes this space as the set of all gauge orbits over the moduli space C. According to Theorem 2.3, the gauge orbit splits as (1). Moreover, for each of these fibers one can associate a field *θ*, as allowed by Corollary 2.4. It is the pair (*A*, *θ*) that defines the geometric fields (*ω*,*e*) in *SOM*. Thus, to construct an **O** structure for gravity is an easy task to collect all possible pairs *W* = (*ω*,*e*) emerging from Theorem 4.1. In fact, each pair (*A*, *θ*) and the associated orbit define a fiber *<sup>W</sup><sup>g</sup>* <sup>∈</sup> **<sup>O</sup>**. That is ensured also by Condition *<sup>1</sup>* of Theorem 4.1. Thus, the fiber *HR* over a point (*ω*,*e*) ∈ G is obtained from *A* �−→ *ω* and *θ*(*A*, *B*) �−→ *e* and the respective action of *SO*(*d* − *n*, *n*). The uniqueness of this mapping is ensured by Corollary 2.4.

The space **B** = (Σ(*B*), *HR*, B) is a dynamical space and survives the mapping. For the case **<sup>B</sup>** <sup>∩</sup> <sup>Θ</sup> <sup>=</sup> <sup>∅</sup> one can associate the moduli space with a set of independent fields <sup>B</sup> �−→ <sup>B</sup>˜ which are invariant representations of *SO*(*d* − *n*, *n*). Theorem 4.1 ensures that the structure group *HR* can be mapped into *SOM* while the fibers Σ(*B*) are identified with fibers Σ(*B*˜) over *B*˜. Thus, for each *<sup>B</sup>* <sup>∈</sup> **<sup>B</sup>** there will be a correspondent *<sup>B</sup>*˜ <sup>∈</sup> <sup>B</sup>˜ and the fiber <sup>Σ</sup>(*B*˜) is obtained from the action of *SO*(*<sup>d</sup>* <sup>−</sup> *<sup>n</sup>*, *<sup>n</sup>*). Thus, **<sup>B</sup>**˜ = (Σ(*B*˜), *SOM*, *<sup>B</sup>*˜). The proof for the case **<sup>B</sup>** <sup>∩</sup> <sup>Θ</sup> �<sup>=</sup> <sup>∅</sup> is totally equivalent.

*Comment.* The final result is that of a gravity theory with an extra set of matter fields **B**˜ .
