Yang--Mills sphalerons in all even spacetime dimensions $d=2k$, $k>2$ : $k$=3,4

TL;DR

This paper investigates classical Yang--Mills solutions in higher even-dimensional spacetimes, identifying their instability as sphalerons through numerical and analytical methods, focusing on dimensions 6, 7, and 8.

Contribution

It provides a detailed analysis of the stability and characteristics of Yang--Mills sphalerons in all even spacetime dimensions, extending understanding beyond previously studied cases.

Findings

Yang--Mills solutions in even dimensions are always unstable sphalerons.

Constructed noncontractible loops and calculated Chern--Simons charges.

Numerically identified negative modes indicating instability.

Abstract

The classical solutions to higher dimensional Yang--Mills (YM) systems, which are integral parts of higher dimensional Einstein--YM (EYM) systems, are studied. These are the gravity decoupling limits of the fully gravitating EYM solutions. In odd spacetime dimensions, depending on the choice of gauge group, these are either topologically stable or unstable. Both cases are analysed, the latter numerically only. In even spacetime dimensions they are always unstable, describing saddle points of the energy, and can be described as {\it sphalerons}. This instability is analysed by constructing the noncontractible loops and calculating the Chern--Simons (CS) charges, and also perturbatively by numerically constructing the negative modes. This study is restricted to the simplest YM system in spacetime dimensions $d=6,7,8$, which is amply illustrative of the generic case.