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Gauge theory on Aloff-Wallach spaces. (English) Zbl 1418.53027

The Aloff-Wallach spaces \(X_{k,l}\) are 7-dimensional factor spaces \(\mathrm{SU}(3)/\mathrm{U}(1)\), where the embedding of \(\mathrm{U}(1)\) into \(\mathrm{SU}(3)\) is determined by two integers \(k,l\). The manifolds \(X_{k,l}\) give examples of infinite series of nondiffeomorphic 7-dimensional simply connected homogeneous spaces (for smaller dimensions there are only finitely many of such homogeneous spaces).
In this article \(G_2\)-structures on \(X_{k,l}\) are considered. On each \(X_{k,l}\) a real 4-dimensional family of \(G_2\)-structures, which contains exactly two nearly parallel \(G_2\)-structures, is investigated. A classification of invariant \(G_2\)-instantons for homogeneous coclosed \(G_2\)-structures on Aloff-Wallach spaces \(X_{k,l}\) are obtained. A \(G_2\)-instanton is a solution to a gauge theoretical equation that can be written for any oriented 7-dimensional manifold \(X_7\) equipped with a \(G_2\)-structure. The examples are given when \(G_2\)-instantons can be used to distinguish between different strictly nearly parallel \(G_2\)-structures on the same Aloff-Wallach space. Also many results and examples of some other interesting phenomena are presented.

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C29 Issues of holonomy in differential geometry
53C38 Calibrations and calibrated geometries
57R57 Applications of global analysis to structures on manifolds
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