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Gauge theory on Aloff–Wallach spaces

Gavin Ball and Goncalo Oliveira

Geometry & Topology 23 (2019) 685–743

For gauge groups U(1) and SO(3) we classify invariant G2 –instantons for homogeneous coclosed G2 –structures on Aloff–Wallach spaces Xk,l. As a consequence, we give examples where G2 –instantons can be used to distinguish between different strictly nearly parallel G2 –structures on the same Aloff–Wallach space. In addition to this, we find that while certain G2 –instantons exist for the strictly nearly parallel G2 –structure on X1,1, no such G2 –instantons exist for the 3–Sasakian one. As a further consequence of the classification, we produce examples of some other interesting phenomena, such as irreducible G2 –instantons that, as the structure varies, merge into the same reducible and obstructed one and G2 –instantons on nearly parallel G2 –manifolds that are not locally energy-minimizing.

G2 geometry, gauge theory, instantons, Aloff–Wallach spaces, tri-Sasakian, nearly parallel, cocalibrated
Mathematical Subject Classification 2010
Primary: 53C07, 53C29, 53C38, 57R57
Received: 23 June 2017
Revised: 31 May 2018
Accepted: 18 September 2018
Published: 8 April 2019
Proposed: Simon Donaldson
Seconded: Tobias H Colding, Tomasz Mrowka
Gavin Ball
Department of Mathematics
Duke University
Durham, NC
United States
Goncalo Oliveira
Departamento de Matemàtica Aplicada, Instituto de Matemática e Estatística
Universidade Federal Fluminense
Rio de Janeiro-RJ