G2 and Spin(7) Manifolds

Introduction

The holonomy groups of an irreducible Riemannian manifold that are not among the classical families are exactly two, and they occur in dimensions seven and eight: the exceptional Lie group $G_2$, of dimension $14$, contained in $SO(7)$, and its eight-dimensional relative $\mathrm{Spin}(7)$, of dimension $21$, contained in $SO(8)$. A Riemannian manifold with holonomy in $G_2$ is seven-dimensional, one with holonomy in $\mathrm{Spin}(7)$ is eight-dimensional, and in both cases the holonomy reduction is equivalent to the existence of a parallel differential form: a $3$-form $\varphi$ in dimension seven, a $4$-form $\Phi$ in dimension eight. These are the exceptional holonomy geometries, the last two entries of Berger's classification, and the analogues in dimensions seven and eight of the Kähler and Calabi–Yau geometries of even complex dimension.

Both geometries are governed by the octonions: $G_2$ is the automorphism group of the octonion algebra, and $\mathrm{Spin}(7)$ is generated by $G_2$ and the multiplication by a unit octonion. Both holonomy groups give Ricci-flat metrics, both admit a notion of calibrated submanifold — the associative and coassociative submanifolds of a $G_2$-manifold, the Cayley submanifolds of a $\mathrm{Spin}(7)$-manifold — and both have a well-behaved deformation theory in which the harmonic forms of middle degree control the moduli. Compact examples are difficult and recent: the first constructions are due to Joyce, by resolving and deforming quotients of the flat seven- and eight-torus, and the modern constructions by twisted connected sums produce large families of compact $G_2$-manifolds.

The boundaries of the article. The Riemannian metric, the Levi-Civita connection, holonomy and Berger's classification are those of Riemannian Geometry. The Clifford algebras, the spinor module and the groups $\mathrm{Spin}(n)$ are those of Part II's Clifford Algebras, Spin Representations and Clifford Modules with Inner Conjugation and The Clifford, Pin and Spin Groups with Signed Inner Conjugation; the group $\mathrm{Spin}(7)$ introduced here is the group of that article, and its double cover of $SO(7)$ and its spinor representation are used without re-derivation. The complex and Kähler structures, the fundamental form and the Chern connection are those of Hermitian Geometry and Almost Complex Structures and Kähler Geometry; the Ricci-flat and Calabi–Yau conditions and Yau's theorem are those of Calabi–Yau Manifolds, and the quaternionic group $Sp(n)$ and its geometry those of Quaternionic Geometry and Hyperkähler Geometry. The octonion algebra is introduced in what follows, as standard mathematics with a standard citation, because the corpus's ownlies later in the menu than this article and is not available here. Manifolds, bundles, differential forms and the Hodge star are those of Smooth Manifolds and Differential Geometry, Differential Forms and Stokes' Theorem and Fibre Bundles, Connections and Curvature, all. The existence of the compact examples rests on the elliptic and parabolic analysis of the deformation theory of the obstruction, which belongs to Part III, where the measure and the limit are available; the construction is stated here and cited. The base field is $\mathbb{R}$ and no physics is invoked.

The Octonions and the Exceptional Groups

Definition. The octonion algebra $\mathbb{O}$ is the real vector space with basis $e_0 = 1, e_1, \ldots, e_7$, made into a non-associative algebra by the Cayley–Dickson doubling $\mathbb{O} = \mathbb{H}\oplus\mathbb{H}$ with multiplication

$$ (a,b)(c,d) = (ac - d^{\natural}b,\ da + b c^{\natural}), $$

and equipped with the norm $N(x) = x x^{\natural} = x^{\natural} x \in \mathbb{R}$, which is positive definite and multiplicative: $N(xy) = N(x)N(y)$. Every nonzero octonion is invertible, with $x^{-1} = x^{\natural}/N(x)$, and $\mathbb{O}$ is the last of the four normed division algebras; the automorphism group of $\mathbb{O}$ is the exceptional group $G_2$, a compact simple Lie group of dimension $14$.

Remark. The construction and the uniqueness of the octonions among normed division algebras are the Hurwitz theorem, and their proper development — the multiplication table, the non-associativity of the associator, the relation to the exceptional Jordan algebra — belongs, which lies's order than this article and is not available here, and to Normed Division Algebras and the Hurwitz Theorem, which belongs to Part I and is, where the Cayley–Dickson construction and the Hurwitz theorem are stated. What is used here is only the norm, the conjugation, the multiplication and the fact that the automorphism group is $G_2$.

Definition. Let $V = \mathbb{R}^7$ be the imaginary octonions, that is, the orthogonal complement of $1$ in $\mathbb{O}$ under the norm. The associative 3-form is

$$ \varphi_0(u,v,w) = \langle u, vw\rangle, \qquad u, v, w \in \mathbb{R}^7, $$

where $\langle\,\cdot\,,\,\cdot\,\rangle$ is the inner product polarising the norm and $vw$ is the octonion product; equivalently $\varphi_0 = \sum_{i

$$ G_2 = \{A\in GL(7,\mathbb{R}) : A^*\varphi_0 = \varphi_0\}, $$

and this identifies $G_2$ as a subgroup of $SO(7)$, since $\varphi_0$ determines the metric.

Proposition. The form $\varphi_0$ is definite: at each point the orbit $GL(7,\mathbb{R})\cdot\varphi_0$ is open in $\Lambda^3(\mathbb{R}^7)^*$, of dimension $49 - 14 = 35$, equal to the dimension of $\Lambda^3(\mathbb{R}^7)^*$; so the $3$-forms equivalent to $\varphi_0$ form an open cone, the cone of definite or positive $3$-forms. The $3$-form $\varphi_0$ determines an orientation and a metric by the identity

$$ g_\varphi(u,v)\,\mathrm{vol}_\varphi = \tfrac16\,\iota_u\varphi\wedge\iota_v\varphi\wedge\varphi, $$

and $\psi_0 = *_\varphi\varphi_0\in\Lambda^4(\mathbb{R}^7)^*$ is the coassociative 4-form, which is also $G_2$-invariant.

Proof. The orbit dimension is the dimension of $GL(7)$ minus the dimension of the stabiliser, and the stabiliser is $G_2$ of dimension $14$; $49-14=35=\dim\Lambda^3(\mathbb{R}^7)^*$, so the orbit is open. The formula for the metric is the standard identity expressing $g$ in terms of $\varphi$; the invariance of $\psi_0$ follows from the $G_2$-invariance of $\varphi_0$ and of the Hodge star defined by the metric and the orientation that $\varphi_0$ induces.

Definition. The Cayley 4-form on $\mathbb{R}^8 = \mathbb{R}^7\oplus\mathbb{R}$ is the $4$-form

$$ \Phi_0 = \varphi_0\wedge e^7 + \psi_0, $$

for $e^7$ the $1$-form of the last coordinate and a suitable choice of orientation and normalisation, equivalently the $4$-form of the octonion multiplication whose stabiliser in $GL(8,\mathbb{R})$ is $\mathrm{Spin}(7)$. The Cayley form is self-dual for the orientation it induces, and its stabiliser $\mathrm{Spin}(7)$ is contained in $SO(8)$ and has dimension $21$; the orbit $GL(8,\mathbb{R})\cdot\Phi_0$ has dimension $64-21=43$, so the Cayley forms form a $43$-dimensional cone in the $70$-dimensional space $\Lambda^4(\mathbb{R}^8)^*$.

Proposition. The groups fit into the chain of inclusions

$$ SU(3) \subset G_2 \subset \mathrm{Spin}(7) \subset SO(8), \qquad G_2 \subset SO(7)\subset SO(8), $$

with $\dim SU(3) = 8$, $\dim G_2 = 14$, $\dim \mathrm{Spin}(7)=21$ and $\dim SO(8) = 28$; moreover $\mathrm{Spin}(7)/G_2 = S^7$ and $G_2/SU(3) = S^6$, so that $G_2$ acts transitively on the six-sphere and $\mathrm{Spin}(7)$ on the seven-sphere.

Proof. The dimensional statements are the dimensions of the standard groups. The homogeneous space statements follow from the description of the stabilisers: $G_2$ acts transitively on the imaginary unit octonions, which are the unit sphere $S^6\subseteq\mathbb{R}^7$, with stabiliser the subgroup preserving a preferred imaginary unit, which is $SU(3)$; likewise $\mathrm{Spin}(7)$ acts transitively on the unit sphere of the spinor representation $\mathbb{R}^8$ with stabiliser $G_2$.

$G_2$ Structures on a Seven-Manifold

Definition. A $G_2$-structure on an oriented seven-manifold $M$ is a $3$-form $\varphi\in\Omega^3(M)$ such that at every point $p$ there is an isomorphism $T_pM\cong\mathbb{R}^7$ pulling $\varphi_p$ back to the model form $\varphi_0$; equivalently, $\varphi$ belongs to the definite cone at every point, or equivalently the structure group of $TM$ reduces from $SO(7)$ to $G_2$ along $\varphi$. Such a $\varphi$ determines a Riemannian metric $g_\varphi$, a volume form $\mathrm{vol}_\varphi$ and a coassociative $4$-form $\psi = *_\varphi\varphi$.

Definition. The torsion of a $G_2$-structure is the tensor $\nabla\varphi$, where $\nabla$ is the Levi-Civita connection of $g_\varphi$; it decomposes under $G_2$ into four irreducible components,

$$ \nabla\varphi \in W_1\oplus W_2\oplus W_3\oplus W_4, $$

which are described by the exterior derivatives: $d\varphi\in W_1\oplus W_3\oplus W_4$ and $d\psi\in W_2\oplus W_4$. A $G_2$-structure is

Class	Condition	Name
$W_1$	$d\varphi = \lambda\,\psi$, $\lambda\neq0$ (then $d\psi = 0$)	nearly parallel
$W_2$	$d\varphi = 0$	closed
$W_3$	$d\psi = 0$ and $d\varphi\neq0$	coclosed
$W_4$	neither $d\varphi$ nor $d\psi$ vanishes	the general class
$\tau = 0$	$d\varphi = 0$ and $d\psi = 0$	torsion-free, holonomy in $G_2$

Theorem (Fernández–Gray; the torsion-free case). A $G_2$-structure $\varphi$ on a seven-manifold is torsion-free, that is $\nabla\varphi = 0$, if and only if $d\varphi = 0$ and $d\psi = 0$. In that case the holonomy group of $g_\varphi$ is contained in $G_2$, the metric is Ricci-flat, and the structure is called a $G_2$-holonomy manifold or a $G_2$-manifold when the holonomy equals $G_2$.

Proof sketch. The equivalence of $\nabla\varphi=0$ with the pair of closedness conditions is the general principle of the Gray–Hervella type classifications: the covariant derivative of the defining form lies in a finite sum of irreducible modules, and each module is detected by one of the exterior derivatives of $\varphi$ and $\psi$; the two conditions together force $\nabla\varphi=0$. The holonomy statement follows because $\varphi$ is parallel, so the holonomy group preserves it and lies in its stabiliser $G_2$. Ricci-flatness is the theorem of Bonan: the holonomy representation is irreducible, so the curvature tensor lies in the finite-dimensional space of $G_2$-invariant algebraic curvature tensors, and every such tensor has vanishing Ricci contraction, whence the metric is Ricci-flat.

Proposition (Betti number constraints). Let $M$ be a compact $G_2$-manifold with holonomy exactly $G_2$. Then $b_1(M) = b_6(M) = 0$, $b_2(M)=b_5(M)$, $b_3(M)=b_4(M)$, and the Euler characteristic vanishes: $\chi(M) = 0$.

Proof. A harmonic $1$-form on a Ricci-flat manifold is parallel, and a parallel $1$-form would be a nonzero vector fixed by the holonomy group $G_2$, which has none; so $b_1=0$. Poincaré duality gives $b_2=b_5$ and $b_3=b_4$, and the Euler characteristic computes as

$$ \chi = 1 - 0 + b_2 - b_3 + b_4 - b_5 + b_6 - 1 = b_2 - b_3 + b_3 - b_2 = 0, $$

because $b_6=b_1=0$.

Definition. Let $(M,\varphi)$ be a $G_2$-holonomy manifold. A $3$-dimensional submanifold $L\subseteq M$ is associative if $\varphi|_L = \mathrm{vol}_L$, and a $4$-dimensional submanifold $N\subseteq M$ is coassociative if $\psi|_N = \mathrm{vol}_N$.

Proposition. Associative submanifolds are calibrated by $\varphi$ and coassociative submanifolds by $\psi = *\varphi$; hence both are volume-minimising in their homology classes. A coassociative submanifold carries a natural deformation theory in which the moduli space is the space of closed self-dual $2$-forms on $N$, of dimension $b^2_+(N)$.

Proof. The calibration inequality is the pointwise statement $\varphi|_V \leq \mathrm{vol}_V$ for every oriented $3$-plane $V$, with equality exactly for the associative ones, and likewise for $\psi$ and the coassociative ones; the deformation statement is the classical McLean theorem for coassociative $4$-folds.

Remark. The associative and coassociative submanifolds are the analogues for $G_2$ geometry of the complex curves and the special Lagrangian submanifolds of a Calabi–Yau threefold, and the coassociative fibrations of a compact $G_2$-manifold — a fibration whose total space is $G_2$ and whose fibres are coassociative — are the tool by which the twisted connected sum construction of compact examples is carried out. The detailed theory of the calibrated submanifolds and of their moduli is the subject of the calibrated geometry literature cited below.

$\mathrm{Spin}(7)$ Structures and Manifolds

Definition. A $\mathrm{Spin}(7)$-structure on an oriented eight-manifold $M$ is a $4$-form $\Phi\in\Omega^4(M)$ pointwise equivalent to the Cayley form $\Phi_0$; equivalently the structure group reduces from $SO(8)$ to $\mathrm{Spin}(7)$ along $\Phi$. The $4$-form determines a Riemannian metric $g_\Phi$, an orientation and a volume form, and the companion form $\Psi = *_\Phi\Phi\in\Omega^4(M)$.

Definition. The torsion of a $\mathrm{Spin}(7)$-structure decomposes into two components,

$$ \nabla\Phi \in W_1\oplus W_2, $$

with $d\Phi\in W_1$ and $d\Psi\in W_2$. A $\mathrm{Spin}(7)$-structure is torsion-free if $d\Phi = 0$, equivalently if $\nabla\Phi=0$; then $\Psi$ is also closed and the holonomy group of $g_\Phi$ is contained in $\mathrm{Spin}(7)$.

Theorem ($\mathrm{Spin}(7)$ holonomy). For a $\mathrm{Spin}(7)$-structure $\Phi$ on an eight-manifold the following are equivalent: $d\Phi = 0$; $\nabla\Phi = 0$; the holonomy group of $g_\Phi$ is contained in $\mathrm{Spin}(7)$. In this case the metric $g_\Phi$ is Ricci-flat, and the structure is a $\mathrm{Spin}(7)$-manifold when the holonomy equals $\mathrm{Spin}(7)$.

Proof sketch. As in the $G_2$ case, the components of $\nabla\Phi$ are detected by $d\Phi$ and $d\Psi$, and the vanishing of $d\Phi$ already forces the vanishing of $d\Psi$ for an eight-manifold with a $\mathrm{Spin}(7)$-structure by a dimensional and representation-theoretic accident; the holonomy then lies in the stabiliser $\mathrm{Spin}(7)$. Ricci-flatness is again the theorem of Bonan, by the same invariant-curvature-tensor argument as in the $G_2$ case.

Proposition (Betti number constraints). Let $M$ be a compact $\mathrm{Spin}(7)$-manifold with holonomy exactly $\mathrm{Spin}(7)$. Then $b_1(M)=b_7(M)=0$, $b_2(M)=b_6(M)$, $b_3(M)=b_5(M)$, and

$$ \chi(M) = 2 + 2b_2(M) - 2b_3(M) + b_4(M). $$

Proof. The vanishing of $b_1$ is as in the $G_2$ case: a parallel $1$-form would be fixed by the holonomy group, and $\mathrm{Spin}(7)$ fixes no vector. The remaining identities are Poincaré duality, and the Euler characteristic is the alternating sum with $b_0=b_8=1$ and $b_1=b_7=0$.

Definition. A $4$-dimensional submanifold $N\subseteq M$ of a $\mathrm{Spin}(7)$-manifold is a Cayley submanifold if $\Phi|_N = \mathrm{vol}_N$.

Proposition. Cayley submanifolds are calibrated by $\Phi$ and are volume-minimising in their homology classes; their deformation theory is elliptic and the moduli space is a smooth manifold of finite dimension for a generic Cayley submanifold.

Proof. The calibration inequality is pointwise as in the $G_2$ case, with equality exactly on the Cayley planes; the deformation theory is that of the Cayley calibration and is the analogue of the associative case.

Remark. Cayley submanifolds are the eight-dimensional analogue of the complex surfaces in a Calabi–Yau fourfold and of the associative $3$-folds in a $G_2$-manifold. The four-cycle condition $\Phi|_N=\mathrm{vol}_N$ places them in the middle dimension, and the counting of Cayley submanifolds is governed by the same kind of enumerative invariants — the Donaldson–Thomas and Gromov–Witten theories for the even-dimensional holonomy geometries — as the curve counting in Calabi–Yau threefolds.

Examples

Example (flat space). On $\mathbb{R}^7$ with the constant $3$-form $\varphi_0$ the Levi-Civita connection is flat, so $\varphi_0$ is parallel and the holonomy is trivial, contained in $G_2$; likewise $\Phi_0$ on $\mathbb{R}^8$ is parallel with holonomy contained in $\mathrm{Spin}(7)$. Quotients by lattices preserving the forms give compact examples with trivial holonomy, hence not full holonomy; the holonomy of a non-flat $G_2$-manifold is exactly $G_2$.

Example (the seven-sphere). On $S^7$ the round metric is nearly parallel: the $3$-form induced by the octonion multiplication satisfies $d\varphi = c\,\psi$ with $c\neq0$, so the structure lies in the class $W_1$ and the metric is Einstein but not Ricci-flat. The holonomy is $SO(7)$, and the seven-sphere is the standard example showing that the nearly parallel class is nonempty and genuinely weaker than the torsion-free class. The octonionic almost complex structure on $S^6$ discussed in Hermitian Geometry and Almost Complex Structures is the nearly Kähler structure of complex dimension three, and the nearly parallel class plays the same intermediate role one dimension higher.

Example (products with a circle and with Calabi–Yau factors). If $X$ is a Calabi–Yau threefold with holonomy $SU(3)$, then $X\times S^1$ carries a torsion-free $G_2$-structure: $\varphi = \omega\wedge d\theta + \mathrm{Im}\,\Omega$, where $\Omega$ is the holomorphic volume form of $X$ (a complex $(3,0)$-form, in the sense of Calabi–Yau Manifolds), $\omega$ its Kähler form and $d\theta$ the circle form, is closed because $d\omega = 0$ and the holomorphic $\Omega$ is closed, and $\psi = *\varphi$ is closed as well. The holonomy of the product metric is $SU(3)$, contained in $G_2$ but not equal to it, and the example is the prototype of the twisted products: if the circle direction is allowed to vary over $X$, the structure remains torsion-free under a first-order system on the twisting function, and the twisted products produce manifolds with full holonomy $G_2$ and non-compact complete ends. In one dimension higher the product of a $G_2$-manifold with a circle carries a torsion-free $\mathrm{Spin}(7)$-structure, because $G_2\subseteq\mathrm{Spin}(7)$, and this is the standard route from dimension seven to dimension eight.

Example (the Bryant–Salamon manifolds). The total space of the bundle $\Lambda^2_-S^4$ of anti-self-dual $2$-forms over the four-sphere, of real dimension $7$, carries a complete $G_2$-holonomy metric, and the total space of the spinor bundle of $S^4$, of real dimension $8$, carries a complete $\mathrm{Spin}(7)$-holonomy metric. These are the Bryant–Salamon metrics; they are the first complete examples with full exceptional holonomy, they are asymptotic to cones over the homogeneous spaces of the previous section, and they are the local models for the ends of more general examples.

Example (compact $G_2$ manifolds by resolution). The first compact $G_2$-manifolds were constructed by Joyce as desingularisations of orbifolds $T^7/\Gamma$ for finite groups $\Gamma\subseteq G_2$ acting on the flat seven-torus with isolated singularities; the resolution replaces each singularity by a local model with a known holonomy and the resulting smooth seven-manifold carries a torsion-free $G_2$-structure whose existence is proved by a deformation of the approximately torsion-free form and an implicit function theorem. The Betti numbers $b_2$ and $b_3$ of the resulting manifolds are large, and the construction produces the first examples with full holonomy in dimension seven. The same method, applied to $T^8/\Gamma$, produces compact $\mathrm{Spin}(7)$-manifolds.

Example (compact $G_2$ manifolds by twisted connected sums). The twisted connected sum construction glues two asymptotically cylindrical Calabi–Yau threefolds with a circle factor along a common cross-section, with a hyperkähler rotation of the two circle directions, and produces a compact $G_2$-manifold with full holonomy; the construction has produced many thousands of compact examples with computable Betti numbers, and the analogous construction in dimension eight produces compact $\mathrm{Spin}(7)$-manifolds. The gluing analysis is the analysis of the obstruction as an elliptic problem, and it belongs to the same circle of ideas as the deformation theory of Part III.

Example (Calabi–Yau fourfolds as $\mathrm{Spin}(7)$-manifolds). A Calabi–Yau fourfold with holonomy $SU(4)$ carries a torsion-free $\mathrm{Spin}(7)$-structure: its holonomy is contained in $\mathrm{Spin}(7)$, and the real part of the holomorphic volume form is a Cayley form with $d\Phi=0$, because the holomorphic volume form is closed and its real part is the $\mathrm{Spin}(7)$-invariant $4$-form of the metric. The holonomy is $SU(4)\subsetneq\mathrm{Spin}(7)$, so such a manifold is a $\mathrm{Spin}(7)$-manifold in the weak sense of the holonomy inclusion, and it supplies a large supply of examples of the torsion-free $\mathrm{Spin}(7)$-structures whose holonomy is not full. The same remark applies to $K3\times K3$ and to the hyperkähler fourfolds of Hyperkähler Geometry.

The Exceptional Holonomy Geometries in Berger's List

Theorem (Berger; the classification of holonomy groups). A connected, simply connected, irreducible Riemannian manifold that is not locally symmetric has holonomy group one of

$$ SO(n), \qquad U(n), \qquad SU(n), \qquad Sp(n), \qquad Sp(n)\cdot Sp(1), \qquad G_2, \qquad \mathrm{Spin}(7), $$

acting on $\mathbb{R}^n$ in the standard way.

Proof sketch. The listed groups are exactly the closed connected subgroups of $SO(n)$ acting irreducibly on $\mathbb{R}^n$ that act transitively on the unit sphere and are not the isotropy representations of locally symmetric spaces; the classification is the theorem of Berger, with the locally symmetric cases handled by the theorem of Simons.

Remark. The two exceptional entries are $G_2$ in dimension seven and $\mathrm{Spin}(7)$ in dimension eight, and both are characterised by the existence of a parallel form of odd or even degree: the $3$-form $\varphi$ and the $4$-form $\Phi$. The parallel form is the geometric datum, the holonomy group is its stabiliser, and the torsion-free condition is the closedness of the form; in every entry of the list the Kähler-type geometries of even complex dimension are those with a parallel form of degree two, and the exceptional entries are those in which the parallel form has degree three or four. In this sense a $G_2$-manifold is the seven-dimensional analogue of a Calabi–Yau threefold, and a $\mathrm{Spin}(7)$-manifold the eight-dimensional analogue of a Calabi–Yau fourfold; the relation is made precise by the presence of the subgroups $SU(3)\subset G_2$ and $SU(4)\subset\mathrm{Spin}(7)$, which turn every Calabi–Yau threefold with a circle into a $G_2$-manifold and every Calabi–Yau fourfold into a $\mathrm{Spin}(7)$-manifold.

Remark (the deformation theory and the counting of examples). The torsion-free structures on a fixed compact seven-manifold are parametrised locally by the harmonic $3$-forms, so the moduli space of $G_2$-structures has local dimension $b_3(M)$; the moduli of $\mathrm{Spin}(7)$-structures is similarly governed by the harmonic $4$-forms. The problem of constructing compact manifolds with these structures is thus a problem of solving an elliptic equation with a topological constraint, and the known constructions — Joyce's resolutions and the twisted connected sums — produce finitely many topological types with large Betti numbers, while the classification of the compact examples remains open. The details of the existence proofs and of the gluing analysis are the subject of the literature cited below.

Summary

The exceptional holonomy groups are $G_2$, of dimension $14$, in dimension seven and $\mathrm{Spin}(7)$, of dimension $21$, in dimension eight; both arise from the octonion algebra, $G_2$ as its automorphism group and $\mathrm{Spin}(7)$ as the group generated with the unit octonions. A $G_2$-structure is a definite $3$-form $\varphi$ on a seven-manifold, determining a metric $g_\varphi$, an orientation and a coassociative $4$-form $\psi=*\varphi$; its torsion decomposes into the four classes $W_1,\ldots,W_4$, and the structure is torsion-free — equivalently $d\varphi=0$ and $d\psi=0$ — exactly when the holonomy of $g_\varphi$ lies in $G_2$. A $\mathrm{Spin}(7)$-structure is a Cayley $4$-form $\Phi$ on an eight-manifold; its torsion has two components, and $d\Phi=0$ is equivalent to the holonomy lying in $\mathrm{Spin}(7)$. In both cases the torsion-free metric is Ricci-flat.

The calibrated submanifolds are the associative $3$-folds and coassociative $4$-folds of a $G_2$-manifold and the Cayley $4$-folds of a $\mathrm{Spin}(7)$-manifold; the coassociative fibrations and the Cayley cycles drive the constructions. For a compact $G_2$-manifold with full holonomy $b_1=0$ and $\chi=0$; for a compact $\mathrm{Spin}(7)$-manifold with full holonomy $b_1=0$ and $\chi = 2+2b_2-2b_3+b_4$.

The examples are the flat spaces with trivial holonomy, the seven-sphere with its nearly parallel $G_2$-structure, the products and twisted products with Calabi–Yau factors, the complete Bryant–Salamon metrics on $\Lambda^2_-S^4$ and the spinor bundle of $S^4$, and the compact examples constructed by Joyce from resolved torus orbifolds and by the twisted connected sum of asymptotically cylindrical Calabi–Yau threefolds. These are the last two entries of Berger's classification, and the four-dimensional parts of the theory — the calibrated geometry and the enumerative invariants — connect them to the Calabi–Yau and hyperkähler geometries of the preceding articles.

Summary of Notation

Symbol	Meaning
$\mathbb{O}$	Octonions; norm $N(x)=x x^{\natural}$, conjugation $x^{\natural}$; automorphism group $G_2$
$G_2$	Exceptional compact simple Lie group, $\dim 14$, stabiliser of $\varphi_0$ in $GL(7,\mathbb{R})$
$\mathrm{Spin}(7)$	$\dim 21$, stabiliser of $\Phi_0$ in $GL(8,\mathbb{R})$, double cover of $SO(7)$
$\varphi$, $\varphi_0$	Associative (definite) $3$-form on a $7$-manifold and its flat model
$\psi = *_\varphi\varphi$	Coassociative $4$-form
$g_\varphi$, $\mathrm{vol}_\varphi$	Metric and volume form determined by $\varphi$
$\Phi$, $\Phi_0$	Cayley (self-dual) $4$-form on an $8$-manifold and its flat model
$W_1,W_2,W_3,W_4$	Torsion classes of a $G_2$-structure; nearly parallel $=W_1$, closed $=W_2$, coclosed $=W_3$
$W_1, W_2$	Torsion classes of a $\mathrm{Spin}(7)$-structure
Torsion-free	$\nabla\varphi=0$ ($d\varphi=d\psi=0$), resp. $\nabla\Phi=0$ ($d\Phi=0$); holonomy in $G_2$, resp. $\mathrm{Spin}(7)$
Associative, coassociative	$3$- and $4$-submanifolds calibrated by $\varphi$, $\psi=*\varphi$
Cayley	$4$-submanifold calibrated by $\Phi$
$\chi$, $b_i$	Euler characteristic and Betti numbers; $\chi=0$ for compact $G_2$-manifolds with full holonomy
Bryant–Salamon	Complete $G_2$- and $\mathrm{Spin}(7)$-metrics on $\Lambda^2_-S^4$ and the spinor bundle of $S^4$