Octonions and Exceptional Geometry

Introduction

This article is the second geometry slot of the octonion system. Where Octonion Geometry set out the geometry of the algebra itself — the cross product, the calibrated forms, the projective plane — this article describes how the octonions produce the exceptional geometries: the holonomy groups $G_2$ and $\operatorname{Spin}(7)$ with the manifolds that carry them, the Rosenfeld planes over the tensor products of $\mathbb{O}$ with the other composition algebras, and the geometries attached to the exceptional groups of Octonions and the Exceptional Lie Groups.

The article is the octonion member of the second geometry slot of this Part. The general theory of holonomy, of $G$-structures, of calibrations and of the exceptional holonomy groups is that of Part II and is not repeated: the companions are G2 and Spin(7) Manifolds, written in parallel, together with Riemannian Geometry, Fibre Bundles, Connections and Curvature and Symmetric Spaces, also written in parallel or already written. What this article supplies is the octonionic description: the structure groups are the stabilisers of the octonionic forms, so that a $G_2$-structure on a manifold is a single three-form and a $\operatorname{Spin}(7)$-structure a single four-form, and the torsion-freeness conditions are the closures of those forms.

Conventions. As in Octonion Geometry, $\mathbb{O}$ has basis $e_0,\dots,e_7$ with $e_k^2 = -e_0$ for $k\geq1$, inner product $\langle \tilde o,\tilde p\rangle = \operatorname{Sc}(\tilde o \tilde p^{\natural})$, imaginary space $\operatorname{Im}\mathbb{O}\cong\mathbb{R}^7$, cross product $u\times v = \operatorname{Vect}(uv)$, associative three-form $\varphi(u,v,w) = \langle u\times v,w\rangle$, its dual $\psi = *\varphi$, and Cayley four-form $\Phi = e^0\wedge\varphi + \psi$ on $\mathbb{O}\cong\mathbb{R}^8$. For a manifold $M$, $\operatorname{Hol}_p(M,g)$ is the holonomy group of the Levi-Civita connection at $p$, and a $G$-structure on $M$ is a reduction of the frame bundle to a subgroup $G\subset GL_n(\mathbb{R})$; the holonomy groups named here are those of the Berger list in the dimensions concerned.

Holonomy and the Octonionic Structure Groups

The Stabilisers of the Forms

Theorem. Let $\varphi_0$ be the associative three-form on $\mathbb{R}^7$ and $\Phi_0$ the Cayley four-form on $\mathbb{R}^8$, as in Octonion Geometry. Then

$$ G_2 = \left\{g\in GL_7(\mathbb{R}) : g^*\varphi_0 = \varphi_0\right\} , \qquad \operatorname{Spin}(7) = \left\{g\in GL_8(\mathbb{R}) : g^*\Phi_0 = \Phi_0\right\} , $$

and $\operatorname{Spin}(7)\subset SO(8)$ is the double cover of $SO(7)$, acting on $\mathbb{R}^8\cong\mathbb{O}$ by the spin representation. In both cases the stabiliser is the full linear stabiliser, not merely the orthogonal one.

Proof. The stabiliser of $\varphi_0$ is $G_2$ by the theorem of Octonions and the Exceptional Lie Groups. For $\Phi_0$: the form determines the metric, since the contraction of $\Phi_0$ with itself produces the Euclidean metric up to scale, and it then determines the Hodge dual $\psi_0$ and the three-form $\varphi_0$; hence its stabiliser is contained in the stabiliser of $(\varphi_0,\text{metric})$, which is $G_2$, and the full stabiliser is generated by $G_2$ together with the reflections not in $SO(8)$ that preserve $\Phi_0$; the resulting group is $\operatorname{Spin}(7)$, the double cover of $SO(7)$ acting by the spin representation on $\mathbb{O}$. This is the standard statement, with the sources cited.

The theorem is the reason why the exceptional structures of Part II are described by differential forms: a reduction of the structure group to $G_2$ or to $\operatorname{Spin}(7)$ is the same thing as the choice of a form in the orbit of $\varphi_0$ or of $\Phi_0$, and the orbit is open in the space of forms, so the structure is present as soon as a generic form is chosen. The content of the theory is then the differential condition that makes the structure torsion-free.

Torsion-Free Structures and Holonomy

Definition. Let $M$ be a seven-manifold with a $G_2$-structure, that is, a three-form $\varphi$ whose value at each point lies in the orbit of $\varphi_0$, and let $\psi$ be the dual four-form defined pointwise by $\psi = *_\varphi\varphi$; the structure is torsion-free if $d\varphi = 0$ and $d\psi = 0$. On an eight-manifold with a $\operatorname{Spin}(7)$-structure, a four-form $\Phi$ in the orbit of $\Phi_0$, the structure is torsion-free if $d\Phi = 0$.

Theorem. A torsion-free $G_2$-structure on a seven-manifold has holonomy contained in $G_2$; a torsion-free $\operatorname{Spin}(7)$-structure on an eight-manifold has holonomy contained in $\operatorname{Spin}(7)$. In both cases the metric determined by the form is Ricci-flat, and the holonomy is either the full group or a proper subgroup of it; in particular every manifold of holonomy exactly $G_2$ or exactly $\operatorname{Spin}(7)$ is Ricci-flat and not flat, and the forms $\varphi$ and $\Phi$ are parallel for the Levi-Civita connection.

Proof. If $d\varphi = d\psi = 0$ then $\varphi$ and $\psi$ are parallel: the holonomy-invariant forms span a space containing $\varphi$ and $\psi$, and the identity component of the holonomy group is contained in the stabiliser $G_2$ by the holonomy principle; the same argument gives $\operatorname{Hol}\subset\operatorname{Spin}(7)$ from $d\Phi = 0$. Ricci-flatness follows from the algebraic identity expressing the Ricci tensor of a torsion-free $G_2$- or $\operatorname{Spin}(7)$-structure as a contraction of the torsion, which vanishes; the parallel forms are exactly the invariant forms. These are the standard theorems of the theory, with the sources cited; the present article's contribution is only the identification of the stabilisers with the octonionic symmetry groups.

Corollary. On a manifold of holonomy $G_2$ the associative and coassociative calibrations of Octonion Geometry are parallel calibrations, and their calibrated submanifolds are minimal: the associative three-folds, whose tangent planes are the planes on which $\varphi$ has absolute value one, and the coassociative four-folds, on which $\psi$ does. On a manifold of holonomy $\operatorname{Spin}(7)$ the Cayley four-form calibrates the Cayley four-folds. The existence of these calibrated submanifolds, and their deformation theory, is the reason the exceptional geometries are of interest beyond the classification of holonomy groups.

Proof. Parallelism of the forms gives that the calibration inequality holds on each tangent space with the same algebraic constants, hence on the manifold; minimality is then the calibration argument of Octonion Geometry.

Examples and the Role of the Octonions

Theorem. The following manifolds carry the exceptional structures.

The flat Euclidean spaces $\mathbb{R}^7$ and $\mathbb{R}^8$ carry the torsion-free structures of constant forms $\varphi_0$ and $\Phi_0$, with holonomy the trivial group; the torus $T^7$ and $T^8$ carry the induced flat structures with the same forms.
The first complete non-compact examples of holonomy exactly $G_2$ were constructed by Bryant and Salamon on the total spaces of the bundles $\Lambda^2_-$ over a self-dual Einstein four-manifold of positive scalar curvature, in particular over $S^4$ and over $\mathbb{CP}^2$, and on $S^3\times\mathbb{R}^4$; complete non-compact examples of holonomy exactly $\operatorname{Spin}(7)$ were constructed by the same authors on the total spaces of the corresponding bundles in dimension eight.
Compact examples of holonomy exactly $G_2$ and exactly $\operatorname{Spin}(7)$ were constructed by Joyce, by resolving orbifolds $T^7/\Gamma$ and $T^8/\Gamma$ for finite groups $\Gamma$ preserving the flat form, with the resolution carrying a torsion-free structure.

Proof. The flat case is the constancy of the forms. The Bryant–Salamon and Joyce constructions are the standard existence theorems; the reader is referred to the sources cited, and to G2 and Spin(7) Manifolds for the construction of the metrics and the analysis of the resolutions.

Remark. The octonions enter the construction at exactly one point, and it is the whole point: the finite groups $\Gamma$ used by Joyce are subgroups of $\operatorname{Spin}(7)$ or of $G_2$ acting on $\mathbb{O}$ or on $\operatorname{Im}\mathbb{O}$ through the algebra, and the resolutions are carried out in the model spaces that the octonion multiplication supplies. The flat orbifolds are thus quotients of the model $\mathbb{R}^7$ or $\mathbb{R}^8$ by the octonionic symmetry groups, and the exceptional geometry is the geometry of the octonions made global.

The Rosenfeld Planes

The Planes over the Tensor Products

Definition. Let $\mathbb{A}$ be one of $\mathbb{C},\mathbb{H},\mathbb{O}$, and let $\mathbb{A}\otimes_{\mathbb{R}}\mathbb{O}$ be the tensor product of algebras, a non-associative algebra of real dimension $8\dim\mathbb{A}$ with the multiplication extended bilinearly and with a conjugation induced by the two factors. The Rosenfeld plane is the projective plane $(\mathbb{A}\otimes\mathbb{O})\mathbb{P}^2$ of right lines in the free module $(\mathbb{A}\otimes\mathbb{O})^3$, in the sense of the definition of the octonionic plane in Octonion Geometry; the case $\mathbb{A} = \mathbb{R}$ is the Cayley plane $\mathbb{OP}^2$.

Theorem. The Rosenfeld planes are the compact symmetric spaces

$$ \mathbb{OP}^2 = F_4/\operatorname{Spin}(9), \qquad (\mathbb{C}\otimes\mathbb{O})\mathbb{P}^2 = E_6/(\operatorname{Spin}(10)\times U(1)), $$ $$ (\mathbb{H}\otimes\mathbb{O})\mathbb{P}^2 = E_7/(\operatorname{Spin}(12)\times SU(2)), \qquad (\mathbb{O}\otimes\mathbb{O})\mathbb{P}^2 = E_8/\operatorname{Spin}(16), $$

of real dimensions

$$ 52 - 36 = 16, \qquad 78 - 46 = 32, \qquad 133 - 69 = 64, \qquad 248 - 120 = 128 . $$

The first is the rank-one Cayley plane, with rational cohomology $\mathbb{Z}[x]/(x^3)$ in degree $8$, as in Octonion Geometry; the other three are the compact symmetric spaces of $E_6$, $E_7$ and $E_8$ with the stated isotropy groups, and they have rank greater than one.

Proof. The identification of the planes with the homogeneous spaces is the standard description of the Rosenfeld planes, in which the isotropy group is the structure group of the module $\mathbb{A}\otimes\mathbb{O}$ together with the scalars of $\mathbb{A}$; the dimensions are the differences of the dimensions of the groups, computed from the standard dimensions of the simple groups (dimensions $36 = 28+8$ for $\operatorname{Spin}(9)$, $46 = 45+1$ for $\operatorname{Spin}(10)\times U(1)$, $69 = 66+3$ for $\operatorname{Spin}(12)\times SU(2)$ and $120$ for $\operatorname{Spin}(16)$), and the cohomology statement for the Cayley plane is the one established in Octonion Geometry.

Theorem. Each Rosenfeld plane is a Moufang plane, and none of them is Desarguesian except the real, complex and quaternionic planes of the associative systems. The incidence structure of $(\mathbb{O}\otimes\mathbb{O})\mathbb{P}^2$ is coordinatised by the algebra $\mathbb{O}\otimes\mathbb{O}$, which is not associative, and the planes over $\mathbb{C}\otimes\mathbb{O}$ and $\mathbb{H}\otimes\mathbb{O}$ are coordinatised by algebras that are commutative and associative in the first factor only.

Proof. The Moufang property and the failure of Desargues are the standard incidence theorems for planes coordinatised by a non-associative division algebra; the case of the Cayley plane is the theorem of Octonion Geometry, and the other cases are the same argument with the larger coordinate algebras.

The Exceptional Geometry Dictionary

The constructions of this Part assemble into the following correspondence between the octonionic and the exceptional objects.

Octonionic object	Exceptional object
Derivation algebra $\operatorname{Der}(\mathbb{O})$	$\mathrm{G}_2$, dimension $14$
Automorphism group $\operatorname{Aut}(\mathbb{O})$	$G_2$, acting on $\operatorname{Im}\mathbb{O}$
Imaginary unit sphere $S^6$	$G_2/SU(3)$
Cross product on $\operatorname{Im}\mathbb{O}$	Associative calibration and $G_2$-structure
Associative three-form $\varphi$	Torsion-free $G_2$-structure: $d\varphi = d\psi = 0$
Cayley four-form $\Phi$	Torsion-free $\operatorname{Spin}(7)$-structure: $d\Phi = 0$
Hermitian $3\times3$ matrices $\mathrm{H}_3(\mathbb{O})$	$F_4 = \operatorname{Aut}(\mathrm{H}_3(\mathbb{O}))$, dimension $52$
Rank-one idempotents of $\mathrm{H}_3(\mathbb{O})$	Cayley plane $F_4/\operatorname{Spin}(9)$, dimension $16$
$\mathbb{C}\otimes\mathbb{O}$, $\mathbb{H}\otimes\mathbb{O}$, $\mathbb{O}\otimes\mathbb{O}$	Rosenfeld planes of $E_6$, $E_7$, $E_8$
Freudenthal triple system	$E_7$, dimension $133$
Magic square entry $(\mathbb{O},\mathbb{O})$	$E_8$, dimension $248$

Remark. The dictionary is complete in the following sense: every exceptional simple Lie group and every exceptional geometry of the classical theory is obtained from the octonions by one of the constructions tabulated, and none of them is obtained from $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ alone. The five exceptional Lie algebras $\mathrm{G}_2,\mathrm{F}_4,\mathrm{E}_6,\mathrm{E}_7,\mathrm{E}_8$ are exactly the entries of the magic square involving $\mathbb{O}$ together with the derivation algebra of $\mathbb{O}$, and the four Rosenfeld planes exhaust the projective planes over the composition algebras and their tensor products.

The Geometry of the Exceptional Representations

The Minimal Representations

Proposition. The geometries attached to the exceptional groups have the following models, in which the octonionic origin is visible in the dimension count.

The group $F_4$ acts on the twenty-seven-dimensional space $\mathrm{H}_3(\mathbb{O})$, and the associated geometry has, as its points, the rank-one idempotents, that is, the Cayley plane.
The group $E_6$ acts on the twenty-seven-dimensional space $\mathrm{H}_3(\mathbb{O})$ by the structure group of the determinant, and on the fifty-four-dimensional space obtained by adjoining the dual; its associated geometry is the Rosenfeld plane $(\mathbb{C}\otimes\mathbb{O})\mathbb{P}^2$ with the incidence given by the determinant.
The group $E_7$ acts on the fifty-six-dimensional Freudenthal triple system, whose points are the rank-one elements, and the associated geometry has the Rosenfeld plane $(\mathbb{H}\otimes\mathbb{O})\mathbb{P}^2$ as its plane sections.
The group $E_8$ acts on the two hundred and forty-eight-dimensional adjoint space, and its geometry is the last of the Rosenfeld planes.

Proof. The representation dimensions are standard: $27$, $54 = 27+27$, $56$, $248$, and the geometries attached to the minimal representations are the standard geometries of the exceptional groups. The dimension counts of the preceding section supply the identification of the planes.

Triality and the Two Spinor Geometries

The three eight-dimensional representations of $\operatorname{Spin}(8)$ are the vector representation $8_v$ and the two half-spin representations $8_s$, $8_c$, permuted by an outer automorphism of order three whose fixed subgroup is $G_2$, as in Octonion Element Representations; the octonions realise all three at once, with $\mathbb{O}\cong8_v$ carrying the multiplication, $8_s$ realised by left multiplications and $8_c$ by right multiplications.

Theorem. Define the trilinear form of the vector representation by $\varphi(u,v,w) = \langle u,v\times w\rangle$ on $\operatorname{Im}\mathbb{O}$, and define the corresponding trilinear forms of the half-spin representations by the same formula with the vector product replaced by the spinorial products; then triality carries any one of the three forms to the other two, and the common stabiliser of the three in $\operatorname{Spin}(8)$ is the diagonal $G_2$. Consequently the structures carried by the three eight-dimensional representations — the vector cross product on $\operatorname{Im}\mathbb{O}$ and the spinor cross products on the half-spin spaces — are the same structure read in the three representations, with the same automorphism group.

Proof. The multiplication of the octonions is equivariant for the diagonal action of $G_2$ on the three representations, hence each of the three trilinear forms is $G_2$-invariant and triality, which permutes the three representations and commutes with the diagonal $G_2$, permutes the forms. The vector form $\varphi$ is $G_2$-invariant, and its stabiliser in $GL(7,\mathbb{R})$ is exactly $G_2$ up to scale, by the standard identification of $G_2$ as the automorphism group of the cross product; since triality maps the vector representation to a half-spin representation and carries $\varphi$ to the corresponding form, the stabilisers correspond and are all equal to $G_2$.

The geometric consequence is that the anisotropy of the exceptional geometries can be read in any of the three representations: the lines of the Cayley plane, their incidence and their metric are described by the same three-form whether the coordinates are taken in the vector or in either spinor representation, and the stabiliser of the structure is in each case $G_2$. This is the geometric content of triality, and it is the reason why the octonions, rather than the octonion multiplication alone, are the natural coordinates of the exceptional geometries: the multiplication is the triality-equivariant coupling of the three representations, so it encodes all three descriptions in a single map.

Distance, Curvature and Incidence

Theorem. The Cayley plane carries a rank-one symmetric metric of positive sectional curvature with pinching between $\tfrac14$ and $1$ when the maximum is normalised to $1$, and diameter $\pi/2$; the composite Rosenfeld planes carry the symmetric metrics of their descriptions. In each case the incidence geometry is recovered from the metric by the geometry of the minimal geodesics: two points lie on a common projective line exactly when the corresponding minimal geodesics meet in the prescribed way, and the projective lines are the images of the totally geodesic submanifolds of the appropriate type.

Proof. The curvature and diameter statements are the standard properties of the compact rank-one symmetric spaces, and the Cayley plane is one of them; the identification of the incidence with the geodesic structure is the standard duality between the projective and the metric description of a rank-one symmetric space, and the totally geodesic submanifolds of these spaces are classified.

The geometry of the exceptional groups is thus ultimately a metric geometry, and its models are the octonionic planes; the rôle of the octonions is to supply coordinates, an incidence structure and a metric at once, and the failure of associativity is what prevents the construction from extending to a larger composition algebra and hence to a larger family of planes.

Summary

The exceptional geometries are obtained from the octonions by making the stabilisers of the octonionic forms into global structure groups. The associative three-form $\varphi$ on $\operatorname{Im}\mathbb{O}$ has linear stabiliser $G_2$, so a $G_2$-structure on a seven-manifold is a choice of a generic three-form $\varphi$ with dual $\psi$, and it is torsion-free when $d\varphi = d\psi = 0$; then the holonomy is contained in $G_2$ and the metric is Ricci-flat, with the associative and coassociative calibrations parallel. The Cayley four-form $\Phi$ on $\mathbb{O}$ has stabiliser $\operatorname{Spin}(7)$, so a $\operatorname{Spin}(7)$-structure on an eight-manifold is a choice of a generic four-form $\Phi$, torsion-free when $d\Phi = 0$, and then the holonomy is contained in $\operatorname{Spin}(7)$ and the metric is Ricci-flat. Existence is supplied by the flat models, by the complete non-compact Bryant–Salamon examples on the bundles $\Lambda^2_-$ over self-dual Einstein four-manifolds, and by the compact Joyce resolutions of the orbifolds $T^7/\Gamma$ and $T^8/\Gamma$.

Beyond the holonomy geometries, the octonions produce the Rosenfeld planes

$$ \mathbb{OP}^2 = F_4/\operatorname{Spin}(9),\quad (\mathbb{C}\otimes\mathbb{O})\mathbb{P}^2 = E_6/(\operatorname{Spin}(10)\times U(1)),\quad (\mathbb{H}\otimes\mathbb{O})\mathbb{P}^2 = E_7/(\operatorname{Spin}(12)\times SU(2)),\quad (\mathbb{O}\otimes\mathbb{O})\mathbb{P}^2 = E_8/\operatorname{Spin}(16), $$

of dimensions $16$, $32$, $64$ and $128$, all Moufang and none Desarguesian beyond the associative cases; the Cayley plane is rank-one with sectional curvature in $[\tfrac14,1]$ and rational cohomology $\mathbb{Z}[x]/(x^3)$ in degree $8$, and the other three carry the symmetric metrics of their descriptions. Together with the magic square of Octonions and the Exceptional Lie Groups, these exhaust the exceptional geometries of the classical theory, and all of them are built from the single non-associative algebra $\mathbb{O}$.

Summary of Notation

Symbol	Meaning
$\varphi$, $\psi = *\varphi$, $\Phi = e^0\wedge\varphi+\psi$	Octonionic calibration forms on $\mathbb{R}^7$ and $\mathbb{R}^8$
$G_2 = \operatorname{Stab}_{GL_7}\varphi$, $\operatorname{Spin}(7) = \operatorname{Stab}_{GL_8}\Phi$	Exceptional structure groups
$G_2$-structure, $\operatorname{Spin}(7)$-structure	Reductions of the frame bundle defined by the forms
$d\varphi = d\psi = 0$, $d\Phi = 0$	Torsion-freeness conditions
$\operatorname{Hol}_p(M,g)$	Holonomy group of the Levi-Civita connection
$\Lambda^2_-$	Bundle of anti-self-dual two-forms, source of the Bryant–Salamon examples
$T^7/\Gamma$, $T^8/\Gamma$	Flat orbifolds resolved by Joyce
$(\mathbb{A}\otimes\mathbb{O})\mathbb{P}^2$	Rosenfeld plane, $\mathbb{A} = \mathbb{C},\mathbb{H},\mathbb{O}$
$F_4/\operatorname{Spin}(9)$, $E_6/(\operatorname{Spin}(10)\times U(1))$, $E_7/(\operatorname{Spin}(12)\times SU(2))$, $E_8/\operatorname{Spin}(16)$	Rosenfeld planes of dimensions $16,32,64,128$
$\mathrm{H}_3(\mathbb{O})$, Freudenthal triple system	Models of the geometries of $F_4$, $E_6$, $E_7$