Gauge theory on Aloff–Wallach spaces

Ball, Gavin; Oliveira, Goncalo

doi:10.2140/gt.2019.23.685

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Gauge theory on Aloff–Wallach spaces

Gavin Ball and Goncalo Oliveira

Geometry & Topology 23 (2019) 685–743

DOI: 10.2140/gt.2019.23.685

Abstract

For gauge groups $U (1)$ and $SO (3)$ we classify invariant $G_{2}$ –instantons for homogeneous coclosed $G_{2}$ –structures on Aloff–Wallach spaces $X_{k, l}$ . As a consequence, we give examples where $G_{2}$ –instantons can be used to distinguish between different strictly nearly parallel $G_{2}$ –structures on the same Aloff–Wallach space. In addition to this, we find that while certain $G_{2}$ –instantons exist for the strictly nearly parallel $G_{2}$ –structure on $X_{1, 1}$ , no such $G_{2}$ –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 $G_{2}$ –instantons that, as the structure varies, merge into the same reducible and obstructed one and $G_{2}$ –instantons on nearly parallel $G_{2}$ –manifolds that are not locally energy-minimizing.

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Keywords

G2 geometry, gauge theory, instantons, Aloff–Wallach spaces, tri-Sasakian, nearly parallel, cocalibrated

Mathematical Subject Classification 2010

Primary: 53C07, 53C29, 53C38, 57R57

References

Publication

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

Authors

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 Niterói Rio de Janeiro-RJ Brazil

Volume 23, issue 2 (2019)