**2.2 Contraction of principal bundles**

We now discuss some relevant results concerning gauge bundles:

**Theorem 2.1.** *Let HR* = (*H*, *R*) *be a reduced gauge bundle obtained from a former gauge bundle GR* = (*G*, *R*)*, where G* = *H* ⊗ *K induces a Lie algebra decomposition G* = *H* ⊕ *K. If G <sup>R</sup> is endowed with a connection form Y* = *A* + *B, where A* ∈ *H and B* ∈ *K, then A defines a connection on H <sup>R</sup> if, and only if, H is a stability group of G.*

*Comment.* This theorem3 is a standard result [17, 23]. The formal proof can be found in [17]. It follows from the fact that a gauge transformation on a fiber *π*−1(*x*) will always keep *A* as a connection and *B* as an element of *K* as it can be seen from the decomposition of the gauge transformation in *GR*. Obviously, this is a direct consequence of the stability of *H*. This result establishes that the original bundle imposes a connection on the reduced bundle, independently of the mechanism that led to *HR*.

**Corollary 2.2.** *The space K defines an associated bundle KR* = (*H*, *R*, *K*) ≡ *HR* × *K.*

<sup>1</sup> By independent we mean the set of gauge connections that cannot be related to each other through a gauge transformation, *i.e.*, they do not belong to the same equivalence class.

<sup>2</sup> We adopt the standard fiber bundle notation where *<sup>π</sup>* : **<sup>Y</sup>** �−→ Y is the projection map.

<sup>3</sup> From now on the conditions for the validity of this theorem are assumed to hold.

2 Will-be-set-by-IN-TECH

1-form will be called gauge connection, or simply, connection. The gauge transformations are associated with coordinates changing of the total space with fixed base space coordinates,

*<sup>Y</sup>* there is a curvature 2-form defined over *GR*, namely *<sup>F</sup>* <sup>=</sup> <sup>∇</sup><sup>2</sup> <sup>=</sup> <sup>d</sup>*<sup>Y</sup>* <sup>+</sup> *YY*, where <sup>∇</sup> is the covariant derivative, ∇ = d + *Y*, and d is the exterior derivative in *R*. The covariant derivative is defined from the parallel transport between fibers and the curvature is obviously

The gauge connection does not belong to the former structure of the gauge bundle, it originates from a unique choice for the decomposition of the tangent space *Tq*(*GR*), in a point *q* ∈ *GR*, into vertical and horizontal spaces. The mathematical structure that describes the dynamics of *Y* must contain all possible gauge connections that can be defined in *GR* as well as the information of gauge transformations as the definition of equivalence classes for gauge connections. This task is achieved through the moduli bundle **Y** = (*GR*, Y), see for instance [14, 18, 20–22]. In **Y**, the fiber and structure group are the local Lie group *GR* and the base space <sup>Y</sup> is the space of all independent connection 1-forms<sup>1</sup> *<sup>Y</sup>*, the so called moduli space. The typical fiber<sup>2</sup> *<sup>π</sup>*−1(*Y*) is a gauge orbit obtained from a configuration *<sup>Y</sup>*(*x*) ∈ Y and all of its possible gauge transformations *Y<sup>g</sup>* = *g*−1(d + *Y*)*g*. Thus, the total space **Y** can be understood

The interpretation of the gauge and moduli principal bundles is as follows: The gauge bundle provides the localization of a Lie group and the existence of a gauge connection. To give dynamics for the connection one should consider all possible connections (together with a minimizing principle for a classical theory or a path integral measure for a quantum one [14,

**Theorem 2.1.** *Let HR* = (*H*, *R*) *be a reduced gauge bundle obtained from a former gauge bundle GR* = (*G*, *R*)*, where G* = *H* ⊗ *K induces a Lie algebra decomposition G* = *H* ⊕ *K. If G <sup>R</sup> is endowed with a connection form Y* = *A* + *B, where A* ∈ *H and B* ∈ *K, then A defines a connection on H <sup>R</sup> if,*

*Comment.* This theorem3 is a standard result [17, 23]. The formal proof can be found in [17]. It follows from the fact that a gauge transformation on a fiber *π*−1(*x*) will always keep *A* as a connection and *B* as an element of *K* as it can be seen from the decomposition of the gauge transformation in *GR*. Obviously, this is a direct consequence of the stability of *H*. This result establishes that the original bundle imposes a connection on the reduced bundle,

<sup>1</sup> By independent we mean the set of gauge connections that cannot be related to each other through a

**Corollary 2.2.** *The space K defines an associated bundle KR* = (*H*, *R*, *K*) ≡ *HR* × *K.*

<sup>2</sup> We adopt the standard fiber bundle notation where *<sup>π</sup>* : **<sup>Y</sup>** �−→ Y is the projection map. <sup>3</sup> From now on the conditions for the validity of this theorem are assumed to hold.

gauge transformation, *i.e.*, they do not belong to the same equivalence class.

as the union of all gauge orbits which determine the equivalence classes in **Y**.

22]). This dynamics is provided by the infinite dimensional moduli bundle.

We now discuss some relevant results concerning gauge bundles:

) = *<sup>f</sup>* <sup>−</sup>1(*x*)(<sup>d</sup> <sup>+</sup> *<sup>Y</sup>*(*x*, *<sup>g</sup>*)) *<sup>f</sup>*(*x*), where *<sup>g</sup>*� <sup>=</sup> *g f* and {*g*�

recognized as the field strength in gauge theories.

**2.2 Contraction of principal bundles**

*and only if, H is a stability group of G.*

independently of the mechanism that led to *HR*.

), which corresponds to a translation along the fiber, providing *Y*(*x*, *g*) �−→

, *g*, *f* } ⊂ *G*. To every connection

(*x*, *g*) → (*x*, *g*�

*Y*(*x*, *g*�

*Proof.* The coset *K* is an invariant subspace with respect to the stability group *H* and thus a homogeneous space, which is the requirement for *K* to be the fiber of an associated bundle [20]. From Theorem 2.1 it is clear that a point *q* ∈ *GR* will split *q* = (*u*, *k*) where *u* ∈ *HR* and *<sup>k</sup>* <sup>∈</sup> *<sup>K</sup>*. Thus, we define the action of *<sup>H</sup>* on *HR* <sup>×</sup> *<sup>K</sup>* by (*u*, *<sup>k</sup>*) �−→ (*uh*, *<sup>h</sup>*−1*k*) by taking the transitions functions to act on the fiber *K* while an element of the group suffers its own action from the right as allowed by the principal bundle nature of *HR*. Ever since the point *x* ∈ *R* is general, the proof holds for the entire bundle *KR*.

*Comment.* From Corollary 2.2 it is clear that the field *B* is a section over *KR* [17, 18]. Thus, the component *B* ∈ *K* of the connection *Y* migrates to the sector of matter fields on *HR*.

We consider now moduli bundles:

**Theorem 2.3.** *Let* **Y** = (*GR*, Y) *be a moduli bundle constructed from GR* = (*G*, *R*)*. Then the reduction GR* −→ *HR induces a reduction on* **Y** *according to* **Y** −→ **A** *where the reduced moduli bundle is* **A** = (*HR*, C)*. The base space* C = A×B *is the decomposed moduli space of stable connections A* ∈ *H and independent sections B* ∈ *K on K <sup>R</sup> while the fiber is the decomposed gauge orbit:*

$$A^h = h^{-1}(\mathbf{d} + A)h$$

$$B^h = h^{-1}Bh \,. \tag{1}$$

*Proof.* Since *GR* is the fiber of **Y** its reduction to *HR* is equivalent to a split on the gauge orbit (1). Thus, the gauge orbit is reduced to the first of (1) where, A⊂Y, represented by independent elements *A* ∈ *H*, define the reduced moduli space of connections. The space B = Y/A, on the other hand, is the set of all fields *B* that cannot be related through a gauge transformation. Thus, a point in the base space can be defined as *C* = (*A*, *B*) and the fiber is constructed by the action of *<sup>H</sup>* as *<sup>C</sup>* �−→ *<sup>C</sup><sup>h</sup>* = (*Ah*, *<sup>B</sup>h*). The reduced total space is the union of all reduced gauge orbits. The stable character of *H* ensures that there will be no mixing between the spaces **A** and **B** = **Y**/**A** along any gauge orbit.

*Comment.* The infinite dimensional space B is equivalent to the set of all independent sections *B*(*x*) that can be defined in *KR*. Thus, the space **B** is the collection of all possible sections in *KR*. The space **B** can be also understood as the fiber bundle **B** = (Σ(*B*), *HR*, B) where the base space is B and a fiber Σ(*B*) is the collection of all equivalent sections for a given *B* ∈ B.

**Corollary 2.4.** *Define a composite field θ, which is an invariant representation of H, that can be constructed from the original set of connections. For each base space point C there is only one field θ*(*C*)*. If an equivalence class C<sup>h</sup> is defined then θ<sup>h</sup>* = *θ*(*Ch*) *is on the same equivalence class of θ*(*C*) *where θ<sup>h</sup>* = *hθ.*

*Proof.* The field *θ* is, by construction, an invariant representation of *H*, thus, it transforms as *<sup>θ</sup>* �−→ *<sup>θ</sup><sup>h</sup>* <sup>=</sup> *<sup>h</sup>θ*. The last expression defines the equivalence class for *<sup>θ</sup>*. Now, since *<sup>θ</sup>* <sup>=</sup> *θ*(*C*), then *θ*(*C*� ), constructed in another point *C*� , belongs to the same equivalence class of the original field if *θ*(*C*� ) = *hθ*(*C*). However, the transformation of *θ* is induced by the action of the group on its dependence on *C*. Thus, *θ*(*C*� ) = *<sup>θ</sup>*(*Cg*) for an element *<sup>g</sup>* <sup>∈</sup> *<sup>H</sup>*. Using again the definition of *θ* as an invariant representation, we have that *θ*(*Cg*) = *gθ*(*C*). Thus, *g* = *h*.

gauge orbit is then

**4. Effective geometries**

**4.1 Gauge and coframe bundles**

(*SO*(*d* − *n*, *n*), *M*) *if and only if*

the fiber at a cotangent space *T*∗

*e* ∈ *T*<sup>∗</sup>

*T*∗

*1. The base spaces R and M are isomorphic;*

*fundamental representation of SO*(*d* − *n*, *n*)*.*

moduli bundles.

*ω<sup>g</sup>* = *g*−1(d + *ω*)*g* ,

Fiber Bundles, Gauge Theories and Gravity 83

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*.

We now discuss the possibility of a gauge theory to be mapped into a gravity theory. We first discuss the map between gauge and coframe bundles and then we generalize the results for

**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* =

*2. The structure groups H and SO*(*d* − *n*, *n*) *are related, at least, by a surjective homomorphism; 3. A composite field θ, which is an invariant representation of H, can be identified with an invariant*

*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

*<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

*<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

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

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*).

the field *θ*, in the fiber *HR*, can be defined as the cotangent 1-form *e* ∈ *T*<sup>∗</sup>

correctly along the fiber *π*−1(*X*) under the action of the local isometries in *T*<sup>∗</sup>

*<sup>X</sup>*(*M*) in each point *X* ∈ *M* we need two quantities: a coframe

*<sup>X</sup>*(*M*), recognized as a

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

*e<sup>g</sup>* = *ge* , (2)

*Comment.* The field *θ* is a one to one map *θ* : *C* �−→ *θ*(*C*) which establishes that for at each point *C* there will be only one *θ* such that if *C* ∼ *C*� then *θ* ∼ *θ*� . In other words, in each fiber *C<sup>h</sup>* there will be only one equivalence class for *θ*.
