Topological indices of general relativity and Yang-Mills theory in four-dimensional space-time

Abstract

This report investigates general relativity and the Yang-Mills theory in four-dimensional space-time using the common mathematical framework, the Chern-Weil theory for principal bundles. The whole theory is described owing to the fibre bundle with the GL(4) symmetry, a model of the space-time, and twisting several principal bundles with the gauge symmetry that are a model of force and matter fields.We introduce the Hodge-dual connection in addition to the principal connection into the Lagrangian to make gauge fields have dynamics independent from the Bianchi identity. The duplex superstructure appears in the bundle when a Z2-grading operator exists in a total space of the bundle in general. The Dirac operator appears in the secondary superspace using the one-dimensional Clifford algebra. Thus, it provides topological indices owing to the Atiyah-Singer index theorem. We introduce a novel method, namely the theta-metric space, to discuss topological indices in the hyperbolic space-time manifold. The theta-metric treats the Euclidean and Minkowski spaces simultaneously and defines the topological index in the Minkowski space-time.In this framework, we show several examples of topological indices appearing in the field theory and clarify the physical significance of these indices in the Minkowski space-time.