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Résumé |
I will start with briefly reminding the standard construction of 4-dimensional BPST instantons. They represent topologically nontrivial self-dual gauge field configurations realizing the minima of the Euclidean Yang-Mills action $S \propto \int_{R^4} \, F \wedge \star F$. The associated topology is $\pi_3[SU(2)] = \mathbb{Z}$. The mappings $S^3 \to SU(2)$ have the well-known rather simple explicit form.
Similar objects exist in higher dimensions, in particular, in eight dimensions. A special class of $8D$ instantons with the gauge group $SO(8)$ has been known since last millenium.
In that case, the associated mappings $S^7 \to SO(8)$ represented a rather straightforward generalization of the mappings $S^3 \to SU(2)$.
In this talk, I'll consider first the $Spin(7)$ gauge group and present a simple compact formula for a topologically nontrivial map $S^7 \to Spin(7)$ associated with the fiber bundle $Spin(7) \stackrel{G_2}{\to} S^7$. It reads
$$
g \ =\ \exp \left\{- \frac 12 \alpha_m f_{mnk} \Gamma_n \Gamma_k \right\}\,,
$$
where $\alpha_{m=1,\ldots,7}$ is a 7-dimensional vector of norm $0 \leq \|\alpha_m\| \leq \pi$ that parametrizes $S^7$, $\Gamma_m$ are the gamma matrices and $f_{mnk}$ are the structure constants of the octonion algebra,
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_{R^8} {\rm Tr} \{F\wedge F \wedge \star(F \wedge F)\} $ and satisfy {\it 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 {\it 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.
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