On the topological reduction from the affine to the orthogonal gauge theory of gravity

Abstract

Making use of the fibre bundle theory to describe metric-affine gauge theories of gravity we are able to show that metric-affine gauge theory can be reduced to the Riemann-Cartan one. The price we pay for simplifying the geometry is the presence of matter fields associated with the nonmetric degrees of freedom of the original setup. Also, a possible framework for the construction of a quantum gravity theory is developed along the text. * sobreiro@if.uff.br † vjose@cbpf.br

Induced gravity from gauge theories

“…In this approach, gravity is described by two fundamental 1-form fields, the vierbein 1 e a and the spin-connection ω a b . The geometric properties of spacetime are obtained from specific gauge invariant composite fields [15,16]. In particular, the metric tensor is obtained from…”

Fiber Bundles, Gauge Theories and Gravity

“…The mathematical structure that describes the dynamics of Y must contain all possible gauge connections that can be defined in G R 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 =( G R , Y ), see for instance [14,18,[20][21][22]. In Y, the fiber and structure group are the local Lie group G R and the base space Y is the space of all independent connection 1-forms 1 Y, the so called moduli space.…”

“…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.…”

“…This means that the coframe bundle can be naturally contracted down to SO M =( 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.…”

“…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][11][12][13][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].…”