A certain class of Einstein-Yang-Mills--systems

TL;DR

This paper constructs a class of Einstein-Yang-Mills systems on homogeneous spaces via dimensional reduction, providing explicit solutions including a sphaleron-type solution related to instantons and exploring connections to quantum state geometry.

Contribution

It introduces a new class of Einstein-Yang-Mills systems on homogeneous spaces derived from pure gravity, including explicit solutions for symmetric spaces and their relation to instantons and quantum state geometry.

Findings

Derived EYM systems on homogeneous spaces via dimensional reduction.

Obtained explicit solutions for symmetric spaces, including SU(2).

Connected solutions to instantons, sphalerons, and quantum state geometry.

Abstract

A class of $ G $-invariant Einstein-Yang-Mills (EYM) systems with cosmological constant on homogeneous spaces $ G / H $, where $ G $ is a semisimple compact Lie group, is presented. These EYM--systems can be obtained in terms of dimensional reduction of pure gravity. If $ G / H $ is a symmetric space, the EYM--system on $ G / H $ provides a static solution of the EYM--equations on spacetime $ {\Bbb R} \times G / H $. This way, in particular, a solution for an arbitrary Lie group $ F $, considered as a symmetric space, is obtained. This solution is discussed in detail for the case $ F = SU(2) $. A known analytical EYM--system on $ {\Bbb R} \times S^3 $ is recovered and it is shown - using a relation to the BPST instanton - that this solution is of sphaleron type. Finally, a relation to the distance of Bures and to parallel transport along mixed states is shown.