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

A. V. Smilga

doi:10.1016/j.nuclphysb.2022.115666

paper

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

arXiv:2102.07415 · doi:10.1016/j.nuclphysb.2022.115666

Abstract

We present a simple compact formula for a topologically nontrivial map $S^{7} \to S p in (7)$ associated with the fiber bundle $S p in (7) \to G_{2} S^{7}$ . The homotopy group $π_{7} [S p in (7)] = Z$ brings about the topologically nontrivial 8-dimensional gauge field configurations that belong to the algebra $s p in (7)$ . The instantons are special such configurations that minimize the functional $\int Tr {F \land F \land ⋆ (F \land F)}$ and satisfy non-linear self-duality conditions, $F \land F = \pm ⋆ (F \land F)$ . $S p in (7) \subset S O (8)$ , and $S p in (7)$ instantons represent simultaneously $S O (8)$ instantons of a new type. The relevant homotopy is $π_{7} [S O (8)] = Z \times Z$ , which implies the existence of two different topological charges. This also holds for all groups $S O (4 n)$ 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.

New material concerning generalized SO(8) instantons and the second topological charge added

References in corpus (1)

ADHM Construction of (Anti-)Self-dual Instantons in $4 n$ Dimensions