The Bogomolny Equations and Solutions for Einstein-Yang-Mills-Dilaton- $\sigma$ Models

TL;DR

This paper derives Bogomolny equations for an Einstein-Yang-Mills-dilaton-$\sigma$ model, linking solutions to flat conformally scaled three-metrics and super-covariantly constant spinors, and explores axially symmetric solutions for SU(2).

Contribution

It introduces Bogomolny equations for the EYMD-$\sigma$ model and characterizes static solutions via flat conformally scaled three-metrics and spinor conditions.

Findings

Derived Bogomolny equations for EYMD-$\sigma$ model.

Connected Einstein equations to flat conformally scaled three-metrics.

Explored axially symmetric solutions for SU(2) gauge group.

Abstract

We derive Bogomolny equations for an Einstein-Yang-Mills-dilaton-$\sigma$ model (EYMD-$\sigma$) on a static spacetime, showing that the Einstein equations are satisfied if and only if the associated (conformally scaled) three-metric is flat. These are precisely the static metrics for which super-covariantly constant spinors exist. We study some general properties of these equations and then consider the problem of obtaining axially symmetric solutions for the gauge group SU(2).