Background Geometry in Gauge Gravitation Theory

TL;DR

This paper explores the role of background geometry and spin structures in gauge gravitation theory, proposing solutions for symmetry breaking issues related to tetrad fields and extending Logunov's gravity theory to include dynamic connections and fermions.

Contribution

It introduces two models for handling spin structures in gauge gravitation theory and extends Logunov's Relativistic Theory of Gravity to incorporate dynamic connections and fermion fields.

Findings

Existence of a universal spin structure compatible with various tetrads.

Background tetrad fields can be fixed with deviations described by tensor fields.

Gauge transformations can preserve background tetrads while transforming effective fields.

Abstract

Dirac fermion fields are responsible for spontaneous symmetry breaking in gauge gravitation theory because the spin structure associated with a tetrad field is not preserved under general covariant transformations. Two solutions of this problem can be suggested. (i) There exists the universal spin structure $S\to X$ such that any spin structure $S^h\to X$ associated with a tetrad field $h$ is a subbundle of the bundle $S\to X$. In this model, gravitational fields correspond to different tetrad (or metric) fields. (ii) A background tetrad field $h$ and the associated spin structure $S^h$ are fixed, while gravitational fields are identified with additional tensor fields $q^\la{}_\m$ describing deviations $\wt h^\la_a=q^\la{}_\m h^\m_a$ of $h$. One can think of $\wt h$ as being effective tetrad fields. We show that there exist gauge transformations which keep the background tetrad field $h$ and act on the effective fields by the general covariant transformation law. We come to Logunov's Relativistic Theory of Gravity generalized to dynamic connections and fermion fields.