\(G_2\)-instantons, associative submanifolds and Fueter sections. (English) Zbl 1391.53029
This article is the outcome of work undertaken by the author for his PhD thesis at Imperial College London supervised by Simon Donaldson.
The main object of study in the thesis is a compact 7-manifold \(Y\) together with a positive 3-form \(\phi\) satisfying a certain non-linear partial differential equation. In a canonical way, the 3-form \(\phi\) equips \(Y\) with an orientation and a metric \(g\) for which the holonomy group \(\mathrm{Hol}(g)\) is contained in the exceptional Lie group \(G_2\). The pair \((Y, \phi)\) is commonly called a \(G_2\)-manifold, and the 3-form \(\phi\) a (torsion-free) \(G_2\)-structure on \(Y\).
In the main result of the thesis, the author gives sufficient conditions for a family of \(G_2\)-instantons to be born out of a Fueter section of a bundle of moduli spaces of ASD instantons over an associative submanifold. This phenomenon is one of the key difficulties in defining the conjectural \(G_2\) Casson invariant proposed by S. K. Donaldson and R. P. Thomas in the inspiring paper [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31–47 (1998; Zbl 0926.58003)].
The main object of study in the thesis is a compact 7-manifold \(Y\) together with a positive 3-form \(\phi\) satisfying a certain non-linear partial differential equation. In a canonical way, the 3-form \(\phi\) equips \(Y\) with an orientation and a metric \(g\) for which the holonomy group \(\mathrm{Hol}(g)\) is contained in the exceptional Lie group \(G_2\). The pair \((Y, \phi)\) is commonly called a \(G_2\)-manifold, and the 3-form \(\phi\) a (torsion-free) \(G_2\)-structure on \(Y\).
In the main result of the thesis, the author gives sufficient conditions for a family of \(G_2\)-instantons to be born out of a Fueter section of a bundle of moduli spaces of ASD instantons over an associative submanifold. This phenomenon is one of the key difficulties in defining the conjectural \(G_2\) Casson invariant proposed by S. K. Donaldson and R. P. Thomas in the inspiring paper [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31–47 (1998; Zbl 0926.58003)].
Reviewer: Vagn Lundsgaard Hansen (Lyngby)
MSC:
| 53C10 | \(G\)-structures |
| 53C38 | Calibrations and calibrated geometries |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 57M50 | General geometric structures on low-dimensional manifolds |
