Abstract
We present a simple compact formula for a topologically nontrivial map S7→Spin(7) associated with the fiber bundle Spin(7)→G2S7. The homotopy group π7[Spin(7)]=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 ∫Tr{F∧F∧⋆(F∧F)} and satisfy non-linear self-duality conditions, F∧F=±⋆(F∧F).Spin(7)⊂SO(8), and Spin(7) instantons represent simultaneously SO(8) instantons of a new type. The relevant homotopy is π7[SO(8)]=Z×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.
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