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**1. Introduction**

**2. Gauge theories**

**2.1 Principal bundles for gauge theories**

Motivated by the construction of a gravity theory independently of the metric structure of spacetime and on the stability of a quantum gravity theory many authors have developed schemes that allow a gauge theory to generate an effective metric, see for instance [1–9]. The models are constructed based on a gauge group *G* that possesses the Lorentz group *SO*(1, 3) as a stable subgroup. A symmetry breaking mechanism is imposed in order to *G* collapse to *SO*(1, 3). Mostly of these techniques are based on the de Sitter group and its variations. However, other groups are also considered such as the general linear and affine groups, see for instance [10–14], and also unitary groups [15]. The main motivation in the construction of a gauge theory of gravity that is metric independent is that the base space can be regarded as a flat one, and thus the standard quantization of gauge theories can be employed [16]. In fact,

**Fiber Bundles, Gauge Theories and Gravity** 

*UFF - Universidade Federal Fluminense, Instituto de Física,* 

**5**

*Brasil* 

Rodrigo F. Sobreiro

*Campus da Praia Vermelha, Niterói,* 

In the present work we consider the fiber bundle theory to describe gauge theories and gravity [17–20]. We then show that a gauge theory can be identified with a first order gravity if the principal bundle that describes the gauge theory can be identified with the principal bundle that describes gravity. We formally establish the conditions that the gauge theory must obey and the resulting gravity theory that emerges. The last is constructed from a mapping between

This work is organized as follows: In Sect. 2 we briefly review the fiber bundle description of gauge theories. Also in this section we enunciate some important results concerning reduction of principal bundles. The same approach to the first order gravity theories is displayed in Sect. 3. In Sect. 4 we discuss the emergent geometries that can be derived from a gauge theory

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

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

*x* ∈ *R* a different value for the elements of *G*. We shall refer to *GR* as *gauge bundle*.

the gauge principal bundle structures and the geometric setting of a gravity theory.

some of the cited works are in fact quantizable, at least perturbatively.

in terms of formal theorems. In sect. 5 we collect our final remarks.

February 18-22 2008, Shonan Village Center, Kanagawa, Japan. To be published in the volume Noncommutative Geometry and Physics III by World Scientific, arXiv: 0809.3029.

Tamura, I. (1992). *Topology of Foliations: An Introduction*, Vol. 97 of *Translations of Math. Monographs*, AMS, Providence.

Thurston, W. (1972). Noncobordant foliations of *S*3, *BAMS* 78: 511 – 514.

Witten, E. (1985). Global gravitational anomalies, *Commun. Math. Phys.* 100: 197–229.
