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

TL;DR

This paper explores topological indices in four-dimensional space-time for general relativity and Yang-Mills theory using Chern-Weil theory, introducing novel methods to define indices in Minkowski space-time.

Contribution

It introduces a unified framework using Chern-Weil theory and a theta-metric space to analyze topological indices in Minkowski space-time, extending traditional elliptic theory approaches.

Findings

Topological indices are defined in Minkowski space-time using theta-metric space.

A duplex superstructure appears in the bundle with Z2-grading operator.

The Dirac operator yields topological indices via Atiyah-Singer theorem.

Abstract

This report investigates general relativity and the Yang-Mills theory in four-dimensional space-time using a 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 by twisting several principal bundles with the gauge symmetry. In addition to the principal connection, we introduce the Hodge-dual connection into the Lagrangian to make gauge fields have dynamics independent from the Bianchi identity. We show that the duplex superstructure appears in the bundle when a Z2-grading operator exists in the total space of the bundle in general. The Dirac operator appears in the secondary superspace using the one-dimensional Clifford algebra, and it provides topological indices from the Atiyah-Singer index theorem. Though the topological index is usually discussed in the elliptic-type manifold, this report treats it in the hyperbolic-type space-time manifold using a novel method, the theta-metric space. The theta-metric treats the Euclidean and Minkowski spaces simultaneously and defines the topological index in the Minkowski space-time.