# Notes on the octonions

@article{Salamon2010NotesOT, title={Notes on the octonions}, author={Dietmar A. Salamon and Thomas Walpuski}, journal={arXiv: Rings and Algebras}, year={2010} }

This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in $G_2$ and ${\rm Spin}(7)$.

## 38 Citations

Associative submanifolds in Joyce's generalised Kummer constructions

- Mathematics
- 2022

This article constructs examples of associative submanifolds in 2–manifolds obtained by resolving 2–orbifolds using Joyce’s generalised Kummer construction. As the 2– manifolds approach the…

$G_2$-instantons, associative submanifolds, and Fueter sections

- Mathematics
- 2017

We give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle…

Cayley deformations of compact complex surfaces

- MathematicsJ. Lond. Math. Soc.
- 2019

It is proved that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.

Computing $\nu$-invariants of Joyce's compact $G_2$-manifolds

- Mathematics
- 2020

Crowley and Nordstrom introduced an invariant of $G_2$-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this…

Computing ν-invariants of Joyce’s compact G2-manifolds

- Mathematics
- 2020

Crowley and Nordström introduced an invariant of G2-structures on the tangent bundle of a closed 7-manifold, taking values in the integers modulo 48. Using the spectral description of this invariant…

G2-instantons on generalised Kummer constructions

- Mathematics
- 2011

In this article we introduce a method to construct $\rm{G}_2$-instantons on $\rm{G}_2$-manifolds arising from Joyce's generalised Kummer construction. The method is based on gluing ASD instantons…

Spin(9) and almost complex structures on 16-dimensional manifolds

- Mathematics
- 2011

For a Spin(9)-structure on a Riemannian manifold M16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant…

On the Moduli Space of the Octonionic Nahm's Equations

- Mathematics
- 2020

In this paper, we study some basic properties of the octonionic Nahm's equations over $[0,1]$. We prove that the moduli space of the smooth solutions to the octonionic Nahm's equations over $[0,1]$…

## References

SHOWING 1-10 OF 40 REFERENCES

Gauge Theory in higher dimensions, II

- Mathematics
- 2009

The main aim of the paper is to develop the "Floer theory" associated to Calabi-Yau 3-folds, exending the analogy of Thomas' "holomorphic Casson invariant". The treatment in the body of the paper is…

Some Notes on G_2 and Spin(7) Geometry

- Physics
- 2006

We collect together various facts about G_2 and Spin(7) geometry which are likely well known but which do not seem to have appeared explicitly in the literature before. These notes should be useful…

Introduction to Lie Algebras and Representation Theory

- Mathematics
- 1973

Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-…

Twisted connected sums and special Riemannian holonomy

- Mathematics
- 2000

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial…

Metrics with exceptional holonomy

- Mathematics
- 1987

It is proved that there exist metrics with holonomy G2 and Spin(7), thus settling the remaining cases in Berger's list of possible holonomy groups. We first reformulate the "holonomy H" condition as…

G2-instantons on generalised Kummer constructions

- Mathematics
- 2011

In this article we introduce a method to construct $\rm{G}_2$-instantons on $\rm{G}_2$-manifolds arising from Joyce's generalised Kummer construction. The method is based on gluing ASD instantons…

Some remarks on G2-structures

- Mathematics
- 2006

This article consists of loosely related remarks about the geometry of G2structures on 7-manifolds, some of which are based on unpublished joint work with two other people: F. Reese Harvey and Steven…

Riemannian manifolds with structure groupG2

- Mathematics
- 1982

SummaryRiemannian manifolds with structure group G2are 7-dimensional and have a distinguished 3-form. In this paper such manifolds are treated as analogues of almost Hermitian manifolds. Thus S7has…

Compact Manifolds with Special Holonomy

- Mathematics
- 2000

The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence…

Stable forms and special metrics

- Mathematics
- 2001

We show how certain diffeomorphism-invariant functionals on differential forms in dimensions 6,7 and 8 generate in a natural way special geometrical structures in these dimensions: metrics of…