nLab
associative 3-form

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

On the Cartesian space 7\mathbb{R}^7 the associative 3-form is the differential 3-form ωΩ 3( 7)\omega \in \Omega^3(\mathbb{R}^7) which is constant and whose value at the origin on three vectors is

ω(u,v,w)u,v×w, \omega(u,v,w) \coloneqq \langle u, v \times w\rangle \,,

where

  • ,\langle -,-\rangle is the canonical bilinear form on 7\mathbb{R}^7;

  • ()×()(-)\times (-) is the cross product in 7\mathbb{R}^7.

This are the structure constants of the unit octonions.

The Hodge dual of the associative 3-form is sometimes called the co-associative 4-form.

Properties

Relation to G 2G_2

The group of linear diffeomorphisms of 7\mathbb{R}^7 which preserve this form is the exceptional Lie group G2.

classification of special holonomy manifolds by Berger's theorem:

\,G-structure\,\,special holonomy\,\,dimension\,\,preserved differential form\,
\,\mathbb{C}\,\,Kähler manifold\,\,U(n)\,2n\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
\,hyper-Kähler manifold\,\,Sp(n)\,4n\,4n\,ω=aω 2 (1)+bω 2 (2)+cω 2 (3)\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\, (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\,\mathbb{O}\,\,Spin(7) manifold\,\,Spin(7)\,\,8\,\,Cayley form\,
\,G2 manifold\,\,G2\,7\,7\,\,associative 3-form\,

References

(…)

Last revised on February 19, 2018 at 01:18:17. See the history of this page for a list of all contributions to it.