**2.1 Principal bundles for gauge theories**

First we define two classes of principal bundles within gauge theories can be formally described. The first one is the principal bundle which localizes a gauge group [18] *GR* = (*G*, *R*) where *G* is a Lie group characterizing the fiber and structure group while *R* is the base space, a differential manifold with *do* dimensions identified with spacetime. The total space *GR* describes the localization of the Lie group *G* in the manifold *R*, assembling to each point *x* ∈ *R* a different value for the elements of *G*. We shall refer to *GR* as *gauge bundle*.

It is assumed that *GR* is endowed with a connection 1-form *Y*. The connection 1-form is recognized as the gauge field, the fundamental field of gauge theories. The connection

*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

Fiber Bundles, Gauge Theories and Gravity 81

*Comment.* From Corollary 2.2 it is clear that the field *B* is a section over *KR* [17, 18]. Thus, the

**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*

*A<sup>h</sup>* = *h*−1(d + *A*)*h* ,

*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

*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*)

*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>

definition of *θ* as an invariant representation, we have that *θ*(*Cg*) = *gθ*(*C*). Thus, *g* = *h*.

) = *hθ*(*C*). However, the transformation of *θ* is induced by the action of

*B<sup>h</sup>* = *h*−1*Bh* . (1)

, belongs to the same equivalence class of the

) = *<sup>θ</sup>*(*Cg*) for an element *<sup>g</sup>* <sup>∈</sup> *<sup>H</sup>*. Using again the

component *B* ∈ *K* of the connection *Y* migrates to the sector of matter fields on *HR*.

general, the proof holds for the entire bundle *KR*.

between the spaces **A** and **B** = **Y**/**A** along any gauge orbit.

), constructed in another point *C*�

the group on its dependence on *C*. Thus, *θ*(*C*�

We consider now moduli bundles:

*orbit:*

*where θ<sup>h</sup>* = *hθ.*

*θ*(*C*), then *θ*(*C*�

original field if *θ*(*C*�

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, (*x*, *g*) → (*x*, *g*� ), which corresponds to a translation along the fiber, providing *Y*(*x*, *g*) �−→ *Y*(*x*, *g*� ) = *<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*� , *g*, *f* } ⊂ *G*. To every connection *<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 recognized as the field strength in gauge theories.

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

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