Spin(7) and generalized SO(8) instantons in eight dimensions

TL;DR

This paper derives explicit formulas for topologically nontrivial instanton solutions in eight dimensions related to Spin(7) and SO(8) gauge groups, revealing new types of instantons with dual topological charges.

Contribution

It introduces explicit expressions for Spin(7) and generalized SO(8) instantons in eight dimensions, highlighting their topological charges and relation to higher-dimensional gauge theories.

Findings

Explicit formulas for Spin(7) instantons are provided.

Two distinct topological charges are identified for SO(8) instantons.

Calculations include standard and new type instantons in 8D.

Abstract

We present a simple compact formula for a topologically nontrivial map $S^7 \to Spin(7)$ associated with the fiber bundle $Spin(7) \stackrel{G_2}{\to} S^7$. The homotopy group $\pi_7[Spin(7)] = \mathbb{Z}$ brings about the topologically nontrivial 8-dimensional gauge field configurations that belong to the algebra $spin(7)$. The instantons are special such configurations that minimize the functional $\int {\rm Tr} \{F\wedge F \wedge \star(F \wedge F)\} $ and satisfy non-linear self-duality conditions, $ F \wedge F \ =\ \pm \star (F\wedge F)$. $Spin(7) \subset SO(8)$, and $Spin(7)$ instantons represent simultaneously $SO(8)$ instantons of a new type. The relevant homotopy is $\pi_7[SO(8)] = \mathbb{Z} \times \mathbb{Z}$, which implies the existence of two different topological charges. This also holds for all groups $SO(4n)$ with integer $n$. We present explicit expressions for two topological charges and calculate their values for the conventional 4-dimensional and 8-dimensional instantons and also for the 8-dimensional instantons of the new type. Similar constructions for other algebras in different dimensions are briefly discussed.