Compact Manifolds with Exceptional Holonomy

Dominic Joyce

Abstract. In the classification of Riemannian holonomy groups, the exceptional holonomy groups are G_{2} in 7 dimensions, and Spin(7) in 8 dimensions. We outline the construction of the first known examples of compact 7- and 8-manifolds with holonomy G_{2} and Spin(7).

In the case of G_{2}, we first choose a finite group Γ of automorphisms of the torus T^{7} and a flat Γ-invariant G_{2}-structure on T^{7}, so that T^{7}/Γ is an orbifold. Then we resolve the singularities of T^{7}/Γ to get a compact 7-manifold M. Finally we use analysis, and an understanding of Calabi-Yau metrics, to construct a family of metrics with holonomy G_{2} on M, which converge to the singular metric on T^{7}/Γ.

1991 Mathematics Subject Classification: 53C15, 53C25, 53C80, 58G30. Keywords and Phrases: exceptional holonomy, G_{2}, Spin(7), Ricci-flat.

In the theory of Riemannian holonomy groups, perhaps the most mysterious are the two exceptional cases, the holonomy group G_{2} in 7 dimensions and the holonomy group Spin(7) in 8 dimensions. We shall describe the construction of the first known examples of compact 7-manifolds with holonomy G_{2}. There is a very similar construction for compact 8-manifolds with holonomy Spin(7), which we will not discuss because of lack of space. All the details can be found in the author’s papers [5], [6], [7] and the forthcoming book [8]. A good reference on Riemannian holonomy groups, and G_{2} and Spin(7) in particular, is the book by Salamon [13].

1 Riemannian holonomy groups

Let M be a connected n-dimensional manifold, let g be a Riemannian metric on M, and let ∇ be the Levi-Civita connection of g. Let x, y be points in M joined by a smooth path γ. Then parallel transport along γ using ∇ defines an isometry between the tangent spaces T_{x}M, T_{y}M at x and y.

Definition 1.1 The holonomy group Hol(g) of g is the group of isometries of T_{x}M generated by parallel transport around closed loops based at x in M. We consider Hol(g) to be a subgroup of O(n), defined up to conjugation by elements of O(n). Then Hol(g) is independent of the base point x in M.

The classification of holonomy groups was achieved by Berger [1] in 1955.

Theorem 1.2 Let M be a simply-connected, n-dimensional manifold, and g an irreducible, nonsymmetric Riemannian metric on M. Then either

(i) Hol(g) = SO(n),

(ii) n = 2m and Hol(g) = SU(m) or U(m), (iii) n = 4m and Hol(g) = Sp(m) or Sp(m)Sp(1), (iv) n = 7 and Hol(g) = G_{2}, or

(v) n = 8 and Hol(g) = Spin(7).

Now G_{2} and Spin(7) are the exceptional cases in this classification, so they are called the exceptional holonomy groups. For some time after Berger’s classification, the exceptional holonomy groups remained a mystery. In 1987, Bryant [2] used the theory of exterior differential systems to show that locally there exist many metrics with these holonomy groups, and gave some explicit, incomplete examples. Then in 1989, Bryant and Salamon [3] found explicit, complete metrics with holonomy G_{2} and Spin(7) on noncompact manifolds. In 1994-5 the author constructed examples of metrics with holonomy G_{2} and Spin(7) on compact manifolds [5, 6, 7, 8], and these are the subject of this article.

We now introduce the holonomy group G_{2}. Let (x_{1}, …, x_{7}) be coordinates on ℝ^{7}. Define a metric g_{0} and a 3-form φ_{0} on ℝ^{7} by

g_{0} =dx_{1}^{2} + ⋯ + dx_{7}^{2},  (1)
\begin{matrix}φ_{0} = & dx_{1}∧dx_{2}∧dx_{7} + dx_{1}∧dx_{3}∧dx_{6} + dx_{1}∧dx_{4}∧dx_{5} + dx_{2}∧dx_{3}∧dx_{5} \\ − dx_{2}∧dx_{4}∧dx_{6} + dx_{3}∧dx_{4}∧dx_{7} + dx_{5}∧dx_{6}∧dx_{7}. \end{matrix} (2)

The subgroup of GL(7, ℝ) preserving φ_{0} is the exceptional Lie group G_{2}. This group also preserves g_{0} and the orientation on ℝ^{7}. It is a compact, semisimple, 14-dimensional Lie group, a subgroup of SO(7).

A G_{2}-structure on a 7-manifold M is a principal subbundle of the frame bundle of M, with structure group G_{2}. Each G_{2}-structure gives rise to a 3-form φ and a metric g on M, such that every tangent space of M admits an isomorphism with ℝ^{7} identifying φ and g with φ_{0} and g_{0} respectively. By an abuse of notation, we will refer to (φ, g) as a G_{2}-structure.

Proposition 1.3 Let M be a 7-manifold and (φ, g) a G_{2}-structure on M. Then the following are equivalent:

(i) Hol(g) ⊆ G_{2}, and φ is the induced 3-form, (ii) ∇φ = 0 on M, where ∇ is the Levi-Civita connection of g, and (iii) dφ = d^{∗}φ = 0 on M.

We call ∇φ the torsion of the G_{2}-structure (φ, g), and when ∇φ = 0 the G_{2}-structure is torsion-free. If (φ, g) is torsion-free, then g is Ricci-flat.

Proposition 1.4 Let M be a compact 7-manifold, and suppose that (φ, g) is a torsion-free G_{2}-structure on M. Then Hol(g) = G_{2} if and only if π_{1}(M) is finite. In this case the moduli space of metrics with holonomy G_{2} on M, up to diffeomorphisms isotopic to the identity, is a smooth manifold of dimension b^{3}(M).

2 A ‘Kummer construction’ for a 7-manifold

It is well known that metrics with holonomy SU(2) on the K3 surface can be obtained by resolving the 16 singularities of the orbifold T^{4}/ℤ_{2}, where ℤ_{2} acts on T^{4} with 16 fixed points. This is called the Kummer construction. Our construction is motivated by and modelled on this. It can be divided into four steps. Here is a summary of each. For simplicity we will describe the G_{2} case only, but the Spin(7) case is very similar.

Step 1. Let T^{7} be the 7-torus. Let (φ_{0}, g_{0}) be a flat G_{2}-structure on T^{7}. Choose a finite group Γ of isometries of T^{7} preserving (φ_{0}, g_{0}). Then the quotient T^{7}/Γ is a singular, compact 7-manifold.

Step 2. For certain special groups Γ there is a method to resolve the singularities of T^{7}/Γ in a natural way, using complex geometry. We get a nonsingular, compact 7-manifold M, together with a map π : M → T^{7}/Γ, the resolving map.

Step 3. On M, we explicitly write down a 1-parameter family of G_{2}-structures (φ_{t}, g_{t}) depending on a real variable t ∈ (0, ϵ). These G_{2}-structures are not torsion-free, but when t is small, they have small torsion. As t → 0, the G_{2}-structure (φ_{t}, g_{t}) converges to the singular G_{2}-structure π^{∗}(φ_{0}, g_{0}).

Step 4. We prove using analysis that for all sufficiently small t, the G_{2}-structure (φ_{t}, g_{t}) on M, with small torsion, can be deformed to a G_{2}-structure (φ̃_{t}, g̃_{t}), with zero torsion. Finally, we show that g̃_{t} is a metric with holonomy G_{2} on the compact 7-manifold M.

We will now explain the steps in greater detail.

Step 1

Here is an example of a suitable group Γ. Let (x_{1}, …, x_{7}) be coordinates on T^{7} = ℝ^{7}/ℤ^{7}, where x_{i} ∈ ℝ/ℤ. Let (φ_{0}, g_{0}) be the flat G_{2}-structure on T^{7} defined by (2). Let α, β and γ be the involutions of T^{7} defined by

α((x_{1}, …, x_{7})) = (−x_{1}, −x_{2}, −x_{3}, −x_{4}, x_{5}, x_{6}, x_{7}),  (3)
β((x_{1}, …, x_{7})) = (−x_{1}, \frac{1}{2} − x_{2}, x_{3}, x_{4}, −x_{5}, −x_{6}, x_{7}),  (4)
γ((x_{1}, …, x_{7})) = (\frac{1}{2} − x_{1}, x_{2}, \frac{1}{2} − x_{3}, x_{4}, −x_{5}, x_{6}, −x_{7}).  (5)

By inspection, α, β and γ preserve (φ_{0}, g_{0}), because of the careful choice of exactly which signs to change. Also, α^{2} = β^{2} = γ^{2} = 1, and α, β and γ commute. Thus they generate a group Γ = ⟨α, β, γ⟩ ≅ ℤ_{2}^{3} of isometries of T^{7} preserving the flat G_{2}-structure (φ_{0}, g_{0}).

Lemma 2.1 The elements βγ, γα, αβ and αβγ of Γ have no fixed points on T^{7}. The fixed points of α, β, γ are each 16 copies of T^{3}. The singular set S of T^{7}/Γ is a disjoint union of 12 copies of T^{3}, 4 copies from each of α, β, γ. Each component of S is a singularity modelled on that of T^{3} × ℂ^{2}/{±1}.

Thus the singular set splits into a disjoint union of connected components, and each component is very simple. This is helpful because we can desingularize each connected component independently, and simple singularities are easier to resolve.

Step 2

Our goal is to resolve the singular set S of T^{7}/Γ to get a compact 7-manifold M with holonomy G_{2}. How can we do this? In general we cannot, because we have no idea of how to resolve general orbifold singularities with holonomy G_{2}. However, suppose we can arrange that every connected component of S is locally isomorphic to either

(a) T^{3} × ℂ^{2}/G, for G a finite subgroup of SU(2), or

(b) S^{1} × ℂ^{3}/G, for G a finite subgroup of SU(3) acting freely on ℂ^{3} \ 0.

In this case we can use complex algebraic geometry to find a natural resolution X of ℂ^{2}/G or Y of ℂ^{3}/G, and then T^{3} × X or S^{1} × Y gives a local model for how to resolve the corresponding component of S in T^{7}/Γ.

In case (a), X must have a Kähler metric h with holonomy SU(2) that is asymptotic to the flat Euclidean metric on ℂ^{2}/G. Such metrics are called Asymptotically Locally Euclidean (ALE). They have been classified by Kronheimer [10, 11], and they exist for every finite subgroup G ⊂ SU(2). The point is that if X has holonomy SU(2), then the product 7-manifold T^{3} × X has holonomy {1} × SU(2). But {1} × SU(2) is a subgroup of G_{2}, and so T^{3} × X has a torsionfree G_{2}-structure by Proposition 1.3. Hence, T^{3} × X gives a local model for how to resolve the singularity T^{3} × ℂ^{2}/G with holonomy G_{2}.

In case (b), Y is a crepant resolution of ℂ^{3}/G, and carries an ALE Kähler metric h with holonomy SU(3). Such resolutions and metrics exist for all finite G ⊂ SU(3), by work of Roan [12] and the author [8]. Since {1} × SU(3) ⊂ G_{2}, if (Y, h) has holonomy SU(3) then S^{1} × Y has a torsion-free G_{2}-structure, and provides a local model for how to resolve the singularity S^{1} × ℂ^{3}/G with holonomy G_{2}.

Suppose that all the singularities of T^{7}/Γ are of type (a) or (b). Then we can construct a compact, nonsingular 7-manifold M by resolving each singularity T^{3} × ℂ^{2}/G using T^{3} × X, and resolving each singularity S^{1} × ℂ^{3}/G using S^{1} × Y, as above. In the example this means gluing 12 copies of T^{3} × X into T^{7}/Γ, where X is the blow-up of ℂ^{2}/{±1} at its singular point.

Step 3

For each resolution X of ℂ^{2}/G in case (a), and Y of ℂ^{3}/G in case (b), we can find a 1-parameter family {h_{t} : t > 0} of metrics with the properties

(a) h_{t} is a Kähler metric on X with Hol(h_{t}) = SU(2). Its injectivity radius satisfies δ(h_{t}) = O(t), its Riemann curvature satisfies ∥R(h_{t})∥_{C^{0}} = O(t^{−2}), and h_{t} = h + O(t^{4}r^{−4}) for large r, where h is the Euclidean metric on ℂ^{2}/G, and r the distance from the origin.

(b) h_{t} is Kähler on Y with Hol(h_{t}) = SU(3), satisfying δ(h_{t}) = O(t), ∥R(h_{t})∥_{C^{0}} = O(t^{−2}), and h_{t} = h + O(t^{6}r^{−6}) for large r.

In fact we can choose h_{t} to be isometric to t^{2}h_{1}, and the properties above are easy to prove.

Suppose one of the components of the singular set S of T^{7}/Γ is locally modelled on T^{3}×ℂ^{2}/G. Then T^{3} has a natural flat metric h_{T^{3}}. Let X be the resolution of ℂ^{2}/G and let {h_{t} : t > 0} satisfy property (a). Then ĝ_{t} = h_{T^{3}} + h_{t} is a metric on T^{3} × X with holonomy {1} × SU(2), which is contained in G_{2}. Thus there is an associated torsion-free G_{2}-structure (φ̂_{t}, ĝ_{t}) on T^{3} × X. Similarly, if a component of S is modelled on S^{1} × ℂ^{3}/G, we get a family of torsion-free G_{2}-structures (φ̂_{t}, ĝ_{t}) on S^{1} × Y.

The idea is to make a G_{2}-structure (φ_{t}, g_{t}) on M by gluing together the torsion-free G_{2}-structures (φ̂_{t}, ĝ_{t}) on the patches T^{3}×X and S^{1}×Y, and (φ_{0}, g_{0}) on T^{7}/Γ. The gluing is done using a partition of unity. Naturally, the first derivative of the partition of unity introduces ‘errors’, so that (φ_{t}, g_{t}) is not torsion-free. The size of the torsion ∇φ_{t} depends on the difference φ̂_{t} − φ_{0} in the region where the partition of unity changes. On the patches T^{3} × X, since h_{t} − h = O(t^{4}r^{−4}) and the partition of unity has nonzero derivative when r = O(1), we find that ∇φ_{t} = O(t^{4}). Similarly ∇φ_{t} = O(t^{6}) on the patches S^{1} × Y, and so ∇φ_{t} = O(t^{4}) on M.

For small t, the dominant contributions to the injectivity radius δ(g_{t}) and Riemann curvature R(g_{t}) are made by those of the metrics h_{t} on X and Y, so we expect δ(g_{t}) = O(t) and ∥R(g_{t})∥_{C^{0}} = O(t^{−2}) by properties (a) and (b) above. In this way we prove the following result, which gives the estimates on (φ_{t}, g_{t}) that we need.

Theorem A On the compact 7-manifold M described above, and on many other 7-manifolds constructed in a similar fashion, one can write down the following data explicitly in coordinates:

• Positive constants A_{1}, A_{2}, A_{3} and ϵ,

• A G_{2}-structure (φ_{t}, g_{t}) on M with dφ_{t} = 0 for each t ∈ (0, ϵ), and

• A 3-form ψ_{t} on M with d^{∗}ψ_{t} = d^{∗}φ_{t} for each t ∈ (0, ϵ).

These satisfy three conditions:

(i) ∥ψ_{t}∥_{L^{2}} ≤ A_{1}t^{4} and ∥d^{∗}ψ_{t}∥_{L^{14}} ≤ A_{1}t^{4}, (ii) the injectivity radius δ(g_{t}) satisfies δ(g_{t}) ≥ A_{2}t, (iii) the Riemann curvature R(g_{t}) of g_{t} satisfies ∥R(g_{t})∥_{C^{0}} ≤ A_{3}t^{−2}.

Here the operator d^{∗} and the norms ∥ . ∥_{L^{2}}, ∥ . ∥_{L^{14}} and ∥ . ∥_{C^{0}} depend on g_{t}.

Here one should regard ψ_{t} as a first integral of the torsion ∇φ_{t} of (φ_{t}, g_{t}). Thus the norms ∥ψ_{t}∥_{L^{2}} ≤ A_{1}t^{4} and ∥d^{∗}ψ_{t}∥_{L^{14}} ≤ A_{1}t^{4} are measures of ∇φ_{t}. So parts (i)-(iii) say that the torsion ∇φ_{t} must be small compared to the injectivity radius and Riemann curvature of (M, g_{t}).

Step 4

We prove the following analysis result.

Theorem B In the situation of Theorem A there are constants κ, K > 0 depending only on A_{1}, A_{2}, A_{3} and ϵ, such that for each t ∈ (0, κ] there exists a smooth, torsion-free G_{2}-structure (φ̃_{t}, g̃_{t}) on M with ∥φ̃_{t} − φ_{t}∥_{C^{0}} ≤ Kt^{1/2}.

Basically, this result says that if (φ, g) is a G_{2}-structure on M, and the torsion ∇φ is sufficiently small, then we can deform to a nearby G_{2}-structure (φ̃, g̃) that is torsion-free. Here is a sketch of the proof of Theorem B, ignoring several technical points. The proof is that given in [8], which is an improved version of the proof in [5]. For simplicity we omit the subscripts t.

We have a 3-form φ with dφ = 0 and d^{∗}φ = d^{∗}ψ for small ψ, and we wish to construct a nearby 3-form φ̃ with dφ̃ = 0 and d̃^{∗}φ̃ = 0. Set φ̃ = φ + dη, where η is a small 2-form. Then η must satisfy a nonlinear p.d.e., which we write as

d^{∗}dη = −d^{∗}ψ + d^{∗}F (dη),  (6)

where F is nonlinear, satisfying F (dη) = O(|dη|^{2}).

We solve (6) by iteration, introducing a sequence {η_{j}}_{j=0}^{∞} with η_{0} = 0, satisfying the inductive equations

d^{∗}dη_{j+1} = −d^{∗}ψ + d^{∗}F (dη_{j}),  d^{∗}η_{j+1} = 0.  (7)

If such a sequence exists and converges to η, then taking the limit in (7) shows that η satisfies (6), giving us the solution we want.

The key to proving this is an inductive estimate on the sequence {η_{j}}_{j=0}^{∞}. The inductive estimate we use has three ingredients, the equations

∥dη_{j+1}∥_{L^{2}} ≤ ∥ψ∥_{L^{2}} + C_{1}∥dη_{j}∥_{L^{2}}∥dη_{j}∥_{C^{0}},  (8)
∥∇dη_{j+1}∥_{L^{14}} ≤ C_{2}(∥d^{∗}ψ∥_{L^{14}} + ∥∇dη_{j}∥_{L^{14}}∥dη_{j}∥_{C^{0}} + t^{−4}∥dη_{j+1}∥_{L^{2}}),  (9)
∥dη_{j}∥_{C^{0}} ≤ C_{3}(t^{1/2}∥∇dη_{j}∥_{L^{14}} + t^{−7/2}∥dη_{j}∥_{L^{2}}).  (10)

Here C_{1}, C_{2}, C_{3} are positive constants independent of t. Equation (8) is obtained from (7) by taking the L^{2}-inner product with η_{j+1} and integrating by parts. Using the fact that d^{∗}φ = d^{∗}ψ and ψ is O(t^{4}), we get a powerful a priori estimate of the L^{2}-norm of dη_{j+1}.

Equation (9) is derived from an elliptic regularity estimate for the operator d + d^{∗} acting on 3-forms on M. Equation (10) follows from the Sobolev embedding theorem, since L_{1}^{14}(M) embeds in C^{0}(M). Both (9) and (10) are proved on small balls of radius O(t) in M, using parts (ii) and (iii) of Theorem A, and this is where the powers of t come from.

Using (8)-(10) and part (i) of Theorem A we show that if

∥dη_{j}∥_{L^{2}} ≤ C_{4}t^{4}, ∥∇dη_{j}∥_{L^{14}} ≤ C_{5}, and ∥dη_{j}∥_{C^{0}} ≤ Kt^{1/2},  (11)

where C_{4}, C_{5} and K are positive constants depending on C_{1}, C_{2}, C_{3} and A_{1}, and if t is sufficiently small, then the same inequalities (11) apply to dη_{j+1}. Since η_{0} = 0, by induction (11) applies for all j and the sequence {dη_{j}}_{j=0}^{∞} is bounded in the Banach space L_{1}^{14}(Λ^{3}T^{∗}M). One can then use standard techniques in analysis to prove that this sequence converges to a smooth limit dη. This concludes the sketch proof of Theorem B.

From Theorems A and B we see that the compact 7-manifold M constructed in Step 2 admits torsion-free G_{2}-structures (φ̃, g̃). Proposition 1.4 then shows that Hol(g̃) = G_{2} if and only if π_{1}(M) is finite. In the example above M is simplyconnected, and so π_{1}(M) = {1} and M has metrics with holonomy G_{2}, as we want.

By considering different groups Γ acting on T^{7}, and also by finding topologically distinct resolutions M_{1}, …, M_{k} of the same orbifold T^{7}/Γ, we can construct many compact Riemannian 7-manifolds with holonomy G_{2}. Here is a graph of the Betti numbers b^{2}(M) and b^{3}(M) of the 68 examples found in [5, 6]. More examples will be given in [8].

[figure: 0; 5; 10; 15; 20; 25; 30; 35; 40; 45; 50; 0; 5; 10; 15; 20; b^{3}(M); b^{2}(M)]

Betti numbers of known compact 7-manifolds with holonomy G_{2}

On this graph the symbol ‘•’ denotes the Betti numbers of a simply-connected 7-manifold, ‘∘’ denotes a non-simply-connected manifold, and ‘+’ denotes both a simply-connected and a non-simply-connected manifold.

So far we have discussed only the holonomy group G_{2}. There is a very similar construction for compact manifolds with holonomy Spin(7), described in [7] and [8]. Here are some of the similarities and differences in the two cases. The holonomy group Spin(7) is a subgroup of SO(8), a compact 21-dimensional Lie group isomorphic to the double cover of SO(7). It is the subgroup of GL(8, ℝ) preserving a certain 4-form Ω_{0} on ℝ^{8}, and also preserves the Euclidean metric g_{0} on ℝ^{8}.

Thus a Spin(7)-structure on an 8-manifold M is equivalent to a pair (Ω, g), where Ω is a 4-form and g a Riemann metric that are pointwise isomorphic to Ω_{0} and g_{0}. Riemannian manifolds with holonomy Spin(7) are Ricci-flat. Compact manifolds M with holonomy Spin(7) are simply-connected spin manifolds, and computing the index of the Dirac operator shows that their Betti numbers must satisfy b^{3}(M) + b_{+}^{4}(M) = b^{2}(M) + b_{−}^{4}(M) + 25.

We can construct compact 8-manifolds with holonomy Spin(7) by resolving the singularities of orbifolds T^{8}/Γ. The construction is more difficult than the G_{2} case in two ways. Firstly, it seems to be more difficult to find suitable orbifolds T^{8}/Γ, and it is necessary to consider more complicated kinds of orbifold singularities. Secondly, the analysis is more difficult, and one has to try harder to make the sequence converge. In [7] we find at least 95 topologically distinct compact 8-manifolds with holonomy Spin(7), realizing 29 distinct sets of Betti numbers, and [8] will give more examples.

Note that compact manifolds with holonomy G_{2} and Spin(7) are examples of compact Ricci-flat Riemannian manifolds. In fact, compact manifolds with holonomy G_{2} are the only known source of odd-dimensional examples of compact, simply-connected Ricci-flat Riemannian manifolds.

3 Directions for future research

Here are four areas in which I hope to see interesting developments soon.

• Other constructions of compact manifolds with exceptional holonomy. The author has extended the constructions of [5]-[7] to include resolutions of more general quotient singularities, in particular non-isolated quotient singularities ℂ^{m}/G for G a finite subgroup of SU(m) and m = 3 or 4, and the results will be published in [8]. Another promising possibility is to try to replace the orbifold T^{7}/Γ by (S^{1} × W)/Γ, where W is a Calabi-Yau 3-fold.

• Harvey and Lawson’s theory of calibrated geometry [4] singles out three classes of special submanifolds in manifolds of exceptional holonomy: associative 3-folds and coassociative 4-folds in G_{2}-manifolds, and Cayley 4-folds in Spin(7)-manifolds. They are minimal submanifolds, and have good properties under deformation. Compact examples can be constructed as the fixed point sets of isometries, as in [6].

It would be interesting to study families of compact manifolds of these types, to understand the way singularities develop in such families, and whether a compact G_{2} or Spin(7)-manifold can be fibred by coassociative or Cayley 4-manifolds, with some singular fibres.

• Gauge theory on compact Spin(7)-manifolds. Let M be a compact 8manifold with holonomy Spin(7), let E be a vector bundle or principal bundle over M, and let A be a connection on E. Then the curvature F_{A} of A is a 2-form with values in ad(E). Now the Spin(7)-structure induces a splitting Λ^{2}T^{∗}M = Λ_{7}^{2} ⊕ Λ_{21}^{2}, where Λ_{7}^{2}, Λ_{21}^{2} are vector bundles over M with fibre ℝ^{7}, ℝ^{21} respectively. We call A a Spin(7)-instanton if the component of F_{A} in ad(E) ⊗ Λ_{7}^{2} is zero.

It turns out that Spin(7)-instantons have many properties in common with instantons in 4 dimensions, that are studied in Donaldson theory. Christopher Lewis and the author [9] have proved an existence theorem for Spin(7)- instantons with gauge group SU(2) on certain compact 8-manifolds with holonomy Spin(7). In 4 dimensions a sequence of instantons can ‘bubble’ at a finite number of points. In 8 dimensions we expect ‘bubbling’ to occur instead around a compact Cayley 4-manifold, and we construct families of instantons in which this happens.

• Connections with String Theory. String Theory is a branch of high-energy theoretical physics that aims to unify quantum theory and gravity by modelling particles as 1-dimensional objects called strings. One of its features is that it prescribes the dimension of space-time. This depends on the details of the theory, but the most popular model, supersymmetric string theory, gives dimension 10. To explain the discrepancy between this and the 4 space-time dimensions that we observe, it is supposed that the universe looks locally like ℝ^{4} × M^{6}, where M^{6} is a compact 6-manifold with very small radius, of order 10^{−33}cm.

In supersymmetric string theory, M must be a Calabi-Yau 3-fold. So string theorists are interested in Calabi-Yau 3-folds, and have contributed many ideas to the subject, including that of Mirror Symmetry. However, if instead we consider ℝ^{3} × M^{7}, corresponding to an observable universe with 3 space-time dimensions, then by work of Vafa and Shatashvili M^{7} must be a compact 7-manifold with holonomy G_{2}. Similarly, if we consider ℝ^{2} × M^{8}, so that the observable universe has 2 space-time dimensions, then M^{8} is a compact 8-manifold with holonomy Spin(7).

Recently, string theorists have begun to seriously consider the possibility that the universe may have 11 dimensions (‘M theory’) or even 12 dimensions (‘F theory’). To reduce to 4 observable space-time dimensions in these theories will require a manifold of dimension 7 or 8, and it seems likely that compact manifolds with exceptional holonomy will play a rôle in this.

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[13] S.M. Salamon, ‘Riemannian geometry and holonomy groups’, Pitman Res. Notes in Math. 201, Longman, Harlow, 1989.

Dominic Joyce Lincoln College Oxford OX1 3DR England