In the classification of Riemannian holonomy groups, the exceptional holonomy groups are G2 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 G2 and Spin(7)

Joyce, Dominic

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ORA - Oxford University Research Archive

Compact manifolds with exceptional holonomy

Dominic Joyce

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Compact Manifolds with Exceptional Holonomy

Joyce, D

Oxford University Research Archive (ORA)

Mathematical Institute Eprints Archive

Compact manifolds with exceptional holonomy
Dominic Joyce, Lincoln College, Oxford
published as pages 245–252 in J.E. Andersen, J. Dupont, H.
Pedersen and A. Swann, editors, ‘Geometry and Physics’,
Lecture notes in pure and applied math. vol. 184, Marcel
Dekker, New York, 1997.
In the theory of riemannian holonomy groups, perhaps the most myste-
rious are the two exceptional cases, the holonomy group G2 in 7 dimensions
and the holonomy group Spin(7) in 8 dimensions. In a series of three re-
cent papers [5], [6], [7] the author constructed the first known examples of
compact 7- and 8-manifolds with metrics of holonomy G2 and Spin(7). This
article will give a brief and informal description of the construction.
Our aim is to introduce G2 and Spin(7), give a little background material,
and help the reader understand the main results of [5]-[7] and a taste of their
proofs. We will not be able to state the results in full, to give any rigorous
proofs, or to give due credit to the many authors on whose work these results
rely. For these we refer the reader to the papers [5]-[7], and to Salamon’s
book [9], which is a good introduction to holonomy groups.
The article is in three sections. Section 1 gives Berger’s classification of
holonomy groups, defines G2 and Spin(7), and explains a little about them.
Section 2 describes the construction of [5]-[7], and Section 3 discusses the
results.
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 TxM , TyM at x and y.
1
Definition 1.1 The holonomy group Hol(g) of g is the group of isometries
of TxM generated by parallel transport around closed loops based at x inM .
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 possible holonomy groups was achieved by Berger
[1] in 1955, although Alekseevskii later eliminated the group Spin(9) from
Berger’s list.
Theorem 1.2 [1] Let M be a simply-connected, n-dimensional manifold,
and g an irreducible, nonsymmetric riemannian metric onM . 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) = G2, or
(v) n = 8 and Hol(g) = Spin(7).
NowG2 and Spin(7) are the exceptional cases in this classification, so they
are called the exceptional holonomy groups. Here is a brief introduction to
G2 and Spin(7). All unattributed results can be found in [9, §§11 & 12].
Facts about the holonomy group G2
Let R7 have coordinates (x1, . . . , x7). Define a 3-form ϕ0 on R7 by
ϕ0 = dx1∧dx2∧dx7 + dx1∧dx3∧dx6 + dx1∧dx4∧dx5 + dx2∧dx3∧dx5
−dx2∧dx4∧dx6 + dx3∧dx4∧dx7 + dx5∧dx6∧dx7. (1)
The subgroup of GL(7,R) preserving ϕ0 is the exceptional Lie group G2.
This group also preserves the euclidean metric g0 = dx
2
1 + · · · + dx27. It is a
compact, semisimple, 14-dimensional Lie group, a subgroup of SO(7).
A G2-structure on a 7-manifold M gives rise to a 3-form ϕ and a metric
g on M , such that each tangent space of M admits an isomorphism with R7
identifying ϕ and g with ϕ0 and g0 respectively.
Proposition 1.3 Let M be a 7-manifold and (ϕ, g) a G2-structure on M .
Then the following are equivalent:
2
(i) Hol(g) ⊆ G2, 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 .
The quantity ∇ϕ is called the torsion of the G2-structure (ϕ, g). If ∇ϕ =
0 then the G2-structure is called torsion-free. If g has holonomy Hol(g) ⊆ G2,
then g is ricci-flat. Here is a result about compact 7-manifolds with holonomy
G2, taken from [6, Prop. 1.1.1] and [5, Thm. C].
Proposition 1.4 Let M be a compact 7-manifold, and suppose that (ϕ, g)
is a torsion-free G2-structure on M . Then Hol(g) = G2 if and only if pi1(M)
is finite. In this case the moduli space of metrics with holonomy G2 on
M , up to diffeomorphisms isotopic to the identity, is a smooth manifold of
dimension b3(M).
Facts about the holonomy group Spin(7)
Let R8 have coordinates (x1, . . . , x8). Define a 4-form Ω0 on R8 by
Ω0 = ϕ0 ∧ dx8 + ∗ϕ0, (2)
where ϕ0 is the 3-form defined in (1), and ∗ϕ0 is the Hodge star of ϕ0 using
the euclidean metric on R7. The subgroup of GL(8,R) preserving Ω0 is the
holonomy group Spin(7). This group also preserves the orientation on R8
and the euclidean metric g0 = dx
2
1 + · · · + dx28. It is a compact, semisimple,
21-dimensional Lie group, a subgroup of SO(8).
A Spin(7)- structure on an 8-manifold M gives rise to a 4-form Ω and a
metric g on M , such that each tangent space of M admits an isomorphism
with R8 identifying Ω and g with Ω0 and g0 respectively.
Proposition 1.5 Let M be a compact 8-manifold and (Ω, g) a Spin(7)-
structure on M . Then the following are equivalent:
(i) Hol(g) ⊆ Spin(7), and Ω is the induced 4-form,
(ii) ∇Ω = 0 on M , where ∇ is the Levi-Civita connection of g, and
(iii) dΩ = 0 on M .
3
Again, ∇Ω is the torsion of the Spin(7)-structure (Ω, g), and (Ω, g) is
torsion-free if ∇Ω = 0. If g has holonomy Hol(g) ⊆ Spin(7), then g is ricci-
flat. Here is a result on compact 8-manifolds with holonomy Spin(7), taken
from [7, Thms C & D].
Proposition 1.6 Let M be a compact 8-manifold, and suppose that (Ω, g)
is a torsion-free Spin(7)-structure on M . Then Hol(g) = Spin(7) if and only
if M is simply-connected, and b3(M) + b4+(M) = b
2(M) + 2b4−(M) + 25.
In this case the moduli space of metrics with holonomy Spin(7) on M , up
to diffeomorphisms isotopic to the identity, is a smooth manifold of dimen-
sion 1 + b4−(M).
For some time after Berger’s classification, the holonomy groups G2 and
Spin(7) 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, completemetrics with holonomy
G2 and Spin(7) on noncompact manifolds. Recently, the author constructed
examples of metrics with holonomy G2 and Spin(7) on compact manifolds [5],
[6], [7]. We will now describe this construction.
2. A ‘Kummer construction’ for 7- and 8-manifolds
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/Z2, where Z2
acts on T 4 with 16 fixed points. This is called the Kummer construction, and
is described in [8]. 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 G2 case only, but the Spin(7) case is very similar.
Step 1. Let T 7 be the 7-torus. Let (ϕ0, g0) be a flat G2-structure on T
7.
Choose a finite group Γ of isometries of T 7 preserving (ϕ0, g0).
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 sin-
gularities of T 7/Γ in a natural way, using complex geometry. We
get a nonsingular, compact 7-manifold M , together with a map
pi :M → T 7/Γ, the resolving map.
4
Step 3. OnM , we explicitly write down a 1-parameter family ofG2-structures
(ϕt, gt) depending on a real variable t ∈ (0, ²). These G2-structures
are not torsion-free, but when t is small, they have small torsion.
As t → 0, the G2-structure (ϕt, gt) converges to the singular G2-
structure pi∗(ϕ0, g0).
Step 4. We prove using analysis that for all sufficiently small t, the G2-
structure (ϕt, gt) on M , with small torsion, can be deformed to a
G2-structure (ϕ˜t, g˜t), with zero torsion. Finally, we show that g˜t is
a metric with holonomy G2 on the compact 7-manifold M .
We will now explain the steps in greater detail.
Steps 1-2
We begin with an example of a suitable group Γ. Let (x1, . . . , x7) be coordi-
nates on T 7 = R7/Z7, where xi ∈ R/Z. Let (ϕ0, g0) be the flat G2-structure
on T 7 defined by (1). Let α, β and γ be the involutions of T 7 defined by
α((x1, . . . , x7)) = (−x1,−x2,−x3,−x4, x5, x6, x7), (3)
β((x1, . . . , x7)) = (−x1, 12 − x2, x3, x4,−x5,−x6, x7), (4)
γ((x1, . . . , x7)) = (
1
2
− x1, x2, 12 − x3, x4,−x5, x6,−x7). (5)
By inspection, α, β and γ preserve (ϕ0, g0), because of the careful choice of
exactly which signs to change. Also, α2 = β2 = γ2 = 1, and α, β and γ
commute. Thus they generate a group Γ = 〈α, β, γ〉 ∼= Z32 of isometries of
T 7 preserving the flat G2-structure (ϕ0, g0). The following Lemma is proved
in [5, §2.1].
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 × C2/{±1}.
Our goal is to resolve the singular set S of T 7/Γ to get a compact 7-
manifold M with holonomy G2. How can we do this? In general we cannot,
5
because we have no idea of how to resolve general orbifold singularities with
holonomy G2. However, there is a good theory of how to resolve complex
singularities like C2/G, with holonomy SU(2). Suppose that we find Γ such
that S is a disjoint union of connected components, each component is a copy
of T 3, and the singularities are each modelled on the product of T 3 with a
complex singularity C2/G, for G a finite subgroup of SU(2). (This happens
in the example above, with G = {±1}.) Then there is a natural method for
resolving T 7/Γ.
Let X be the resolution of C2/G coming from algebraic geometry, so
that X has holonomy SU(2) and pi : X → C2/G is the resolving map.
For G = {±1}, X is called the Eguchi-Hanson space, with metric given
explicitly in [4]. Each component of S is modelled on the singularity of
T 3×C2/G, and we resolve by replacing this with T 3×X, using the resolving
map pi : T 3 × X → T 3 × C2/G to glue in the patches T 3 × X. In this way
we get a compact, nonsingular 7-manifold M .
Step 3
It turns out thatX admits not just one, but a 1-parameter family {ht : t > 0}
of metrics with holonomy SU(2). Let h be the euclidean metric on C2/G,
and r the distance from the origin. Then ht satisfies
ht = pi
∗(h) +O(t4pi∗(r)−4), (6)
so that ht is asymptotically locally euclidean, and ht is isometric to t
2h1 after
an automorphism of X. Let hT 3 be a flat metric on T
3. Since SU(2) is a
subgroup of G2 and Hol(ht) = SU(2), gˆt = hT 3 + ht is a metric on T
3 × X
with holonomy contained in G2. Let ϕˆt be the associated G2-structure. Then
(ϕˆt, gˆt) is a torsion-free G2-structure on T
3 ×X.
The idea is to make a G2-structure (ϕt, gt) on M by gluing together the
torsion-free G2-structures (ϕˆt, gˆt) on the patches T
3 × X, and (ϕ0, g0) 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 (ϕt, gt) is not
torsion-free.
The size of the torsion ∇ϕt depends on the difference ϕˆt−ϕ0 in the region
where the partition of unity changes. Using (6) to estimate this, we find that
∇ϕt = O(t4). However, as ht is isometric to t2h1, the injectivity radius of ht
is proportional to t, and ‖R(ht)‖C0 is proportional to t−2. Similar estimates
6
therefore apply to gt. The following result gives the estimates on (ϕt, gt) that
we need.
Theorem A There is a mathematical construction, which can be written
entirely explicitly in coordinates, that gives a compact 7-manifold M and a
1-parameter family of G2-structures (ϕt, gt) on M depending on t ∈ (0, ²).
These G2-structures satisfy the following bounds:
• the torsion ∇ϕt satisfies ‖∇ϕt‖C1 ≤ C1t4,
• the injectivity radius δ(gt) satisfies δ(gt) ≥ C2t,
• the riemann curvature R(gt) satisfies ‖R(gt)‖C0 ≤ C3t−2,
• the volume vol(M) satisfies C4 ≤ vol(M) ≤ 2C4, and
• the diameter diam(M) satisfies diam(M) ≤ C5.
Here C1, . . . , C5 are positive constants independent of t.
Step 4
We prove the following analysis result.
Theorem B Let C1, . . . , C5 be positive constants. Then there exist positive
constants λ,K depending only on C1, . . . , C5, such that for every t ∈ (0, λ],
the following is true.
LetM be a compact 7-manifold, and (ϕ, g) a G2-structure onM . Suppose
that dϕ = 0, that ψ is a 3-form on M with d∗ψ = d∗ϕ, and that these five
conditions hold:
(i) ‖ψ‖C2 ≤ C1t4,
(ii) the injectivity radius δ(g) satisfies δ(g) ≥ C2t,
(iii) the riemann curvature R(g) of g satisfies ‖R(g)‖C0 ≤ C3t−2,
(iv) the volume vol(M) satisfies C4 ≤ vol(M) ≤ 2C4, and
(v) the diameter diam(M) satisfies diam(M) ≤ C5.
Then there exists a smooth, torsion-free G2-structure (ϕ˜, g˜) on M
with ‖ϕ˜− ϕ‖C0 ≤ Kt1/2.
Basically, this result says that if (ϕ, g) is a G2-structure on M , and the
torsion ∇ϕ is sufficiently small, then we can deform to a nearby G2-structure
7
(ϕ˜, g˜) that is torsion-free. Regard ψ as a first integral of ∇ϕ. Thus the
norm ‖ψ‖C2 is a measure of the torsion ∇ϕ, approximately equal to ‖∇ϕ‖C1 .
Then parts (i)-(v) say that the torsion ∇ϕ must be small compared to the
injectivity radius, riemann curvature, volume and diameter of M .
Now the hypotheses of Theorem B are, intentionally, nearly the same as
the conclusions of Theorem A. With a little work we can close the gap be-
tween them. Therefore, applying Theorems A and B together for sufficiently
small t, we see that the compact 7-manifold M constructed in Step 2 admits
torsion-free G2-structures (ϕ˜, g˜). Applying Proposition 1.4, Hol(g˜) = G2 if
and only if pi1(M) is finite. In this way we construct compact riemannian
7-manifolds with holonomy G2.
The proof of Theorem B is not easy, and it represents most of the hard
work in [5]. Here is a sketch, ignoring several technical points. We have a
3-form ϕ with dϕ = 0 and d∗ϕ = d∗ψ, ψ small, and we wish to construct a
nearby 3-form ϕ˜ with dϕ˜ = 0 and d˜∗ϕ˜ = 0. Set ϕ˜ = 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), (7)
where F is nonlinear, satisfying F (dη) = O(|dη|2).
We solve (7) by iteration, introducing a sequence {ηj}∞j=0 with η0 = 0,
satisfying the inductive equations
d∗dηj+1 = −d∗ψ + d∗F (dηj), d∗ηj+1 = 0. (8)
If such a sequence exists and converges to η, then taking the limit in (8) shows
that η satisfies (7), giving us the solution we want. The key to proving this is
an inductive estimate: one must show that if ηj, ηj+1 satisfy (8) and ‖ηj‖B ≤
C then ‖ηj+1‖B ≤ C, where ‖ . ‖B is the norm in some suitable Banach space
B, and C is a positive constant. Then by induction the sequence {ηj}∞j=0 is
bounded in B, and convergence follows by standard techniques.
The inductive estimate we use has two ingredients. The first is
‖dηj+1‖L2 ≤ ‖ψ‖L2 + cst.‖dηj‖L2‖dηj‖C0 , (9)
obtained from (8) by taking the L2-inner product with ηj+1 and integrating
by parts. (The 3-form ψ was introduced solely to achieve this inequality.)
The second is the inequality
8
‖dηj+1‖C1,1/2 ≤ cst.‖ψ‖C1,1/2 + cst.‖dηj‖C1,1/2‖dηj‖C0 + cst.‖dηj+1‖L2 , (10)
which is an elliptic regularity estimate. Here the terms ‘cst.’ are positive
constants depending on C2, . . . , C5 and t. We show that if t ≤ λ, for λ > 0
depending on C1, . . . , C5, then (9) and (10) combine to give the necessary
inductive estimate, and the remainder of the proof is straightforward. For
the Spin(7) case, the proof is different, and more difficult.
3. Presentation of results
By considering different groups Γ acting on T 7 and T 8, we are able to find
metrics with holonomy G2 and Spin(7) on many topologically distinct 7- and
8-manifolds. It also happens that the same orbifold T k/Γ can admit several
topologically distinct resolutions, and this increases the number of examples.
Table 1 is a graph of known examples of compact 7-manifolds M with
holonomyG2, from [5] and [6]. We have plotted b
2(M) against b3(M), as these
are the important betti numbers. The symbols ‘•’, ‘∗’ and ‘+’ denote the betti
numbers of one or more 7-manifolds with holonomy G2. Here ‘•’ means a
simply-connected manifold, ‘∗’ means a non-simply-connected manifold, and
‘+’ means both a simply-connected and a non-simply-connected manifold.
0 5 10 15 20 25 30 35 40 45 50 b3(M)
0
5
10
15
20
•••••••••
∗∗∗∗∗∗∗∗∗
∗∗∗∗
∗
∗∗∗∗∗ +++++++++
∗∗∗∗∗∗∗
•
∗•
• •
∗•
++
••
•
•
b2(M) Table 1.
The total number of distinct 7-manifolds known is at least 68.
9
Similarly, we shall plot a graph of betti numbers b2(M) against b3(M) of
compact 8-manifolds M with holonomy Spin(7), and this appears in Table
2. Such manifolds are always simply-connected. However, we can use the
cup product in cohomology to distinguish 8-manifolds with the same betti
numbers. Therefore, in the graph below, the data points are figures, each
figure being the number of different 8-manifolds with these betti numbers.
All known examples satisfy b4+ = 103− b2 + b3 and b4− = 39− b2 + b3, but as
far as we know, these are not necessary topological conditions.
0 5 10 15 20 b2(M)
0
5
10
15
20
1 2 3 4 5 5 5 5 5 4 3 2 1
1 2 4 5 7 7 7 5 4 2 1
1 1 1 1 1
b3(M) Table 2.
The number of distinct 8-manifolds known is at least 95. Although this is
greater than the G2 case, the variety is less, as all these examples are derived
from only 3 groups Γ, giving the 3 horizontal lines on the graph. It seems to
be harder to find suitable groups Γ in the Spin(7) case.
Finally, we consider the connections with physics. After many discus-
sions with C. Vafa, S. Shatashvili, E. Witten and M. Rocˇek, it appears that
much of the ‘mirror symmetry’ story for holonomy SU(3) also applies to the
holonomy groups G2 and Spin(7). In particular, Vafa and Shatashvili claim
that the reason that the examples are clustered in families with b2 + b3 con-
stant in Table 1, and with b3 and 2b2 + b4 constant in Table 2, is because
the examples in each family are mirrors of each other. Also, compact rie-
mannian 7-manifolds with holonomy G2 may become important in physics if
11-dimensional theories come back into fashion, because they are needed to
reduce the number of observable dimensions to 4.
10
References
[1] M. Berger, ‘Sur les groupes d’holonomie homoge`ne des varie´te´s a` connexion
affines et des varie´te´s riemanniennes’, Bull. Soc. Math. France 83 (1955) 279-
330.
[2] R.L. Bryant, ‘Metrics with exceptional holonomy’, Ann. Math. 126 (1987)
525-576.
[3] R.L. Bryant & S.M. Salamon, ‘On the construction of some complete metrics
with exceptional holonomy’, Duke Math. J. 58 (1989) 829-850.
[4] T. Eguchi & A.J. Hanson, ‘Asymptotically flat solutions to Euclidean gravity’,
Phys. Lett. 74B (1978) 249-251.
[5] D.D. Joyce, ‘Compact riemannian 7-manifolds with holonomy G2. I’, to ap-
pear in Journal of Differential Geometry.
[6] D.D. Joyce, ‘Compact riemannian 7-manifolds with holonomy G2. II’, to ap-
pear in Journal of Differential Geometry.
[7] D.D. Joyce, ‘Compact riemannian 8-manifolds with holonomy Spin(7)’,
preprint (1995).
[8] D.N. Page, ‘A physical picture of the K3 gravitational instanton’, Phys. Lett.
80B (1978), 55-57.
[9] S.M. Salamon, ‘Riemannian geometry and holonomy groups’, Pitman Res.
Notes in Math. 201, Longman, Harlow, 1989.
11


Oxford University Research Archive

Compact manifolds with exceptional holonomyDominic Joyce, Lincoln College, Oxfordpublished as pages 245–252 in J.E. Andersen, J. Dupont, H.Pedersen and A. Swann, editors, ‘Geometry and Physics’,Lecture notes in pure and applied math. vol. 184, MarcelDekker, New York, 1997.In the theory of riemannian holonomy groups, perhaps the most myste-rious are the two exceptional cases, the holonomy group G2 in 7 dimensionsand the holonomy group Spin(7) in 8 dimensions. In a series of three re-cent papers [5], [6], [7] the author constructed the first known examples ofcompact 7- and 8-manifolds with metrics of holonomy G2 and Spin(7). Thisarticle will give a brief and informal description of the construction.Our aim is to introduce G2 and Spin(7), give a little background material,and help the reader understand the main results of [5]-[7] and a taste of theirproofs. We will not be able to state the results in full, to give any rigorousproofs, or to give due credit to the many authors on whose work these resultsrely. For these we refer the reader to the papers [5]-[7], and to Salamon’sbook [9], which is a good introduction to holonomy groups.The article is in three sections. Section 1 gives Berger’s classification ofholonomy groups, defines G2 and Spin(7), and explains a little about them.Section 2 describes the construction of [5]-[7], and Section 3 discusses theresults.1. Riemannian holonomy groupsLet M be a connected n-dimensional manifold, let g be a riemannian metricon M , and let ∇ be the Levi-Civita connection of g. Let x, y be points in Mjoined by a smooth path γ. Then parallel transport along γ using ∇ definesan isometry between the tangent spaces TxM , TyM at x and y.1Definition 1.1 The holonomy group Hol(g) of g is the group of isometriesof TxM 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 byelements of O(n). Then Hol(g) is independent of the base point x in M .The classification of possible holonomy groups was achieved by Berger[1] in 1955, although Alekseevskii later eliminated the group Spin(9) fromBerger’s list.Theorem 1.2 [1] 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) = G2, or(v) n = 8 and Hol(g) = Spin(7).Now G2 and Spin(7) are the exceptional cases in this classification, so theyare called the exceptional holonomy groups. Here is a brief introduction toG2 and Spin(7). All unattributed results can be found in [9, §§11 & 12].Facts about the holonomy group G2Let R7 have coordinates (x1, . . . , x7). Define a 3-form ϕ0 on R7 byϕ0 = dx1∧dx2∧dx7 + dx1∧dx3∧dx6 + dx1∧dx4∧dx5 + dx2∧dx3∧dx5−dx2∧dx4∧dx6 + dx3∧dx4∧dx7 + dx5∧dx6∧dx7. (1)The subgroup of GL(7,R) preserving ϕ0 is the exceptional Lie group G2.This group also preserves the euclidean metric g0 = dx21 + · · · + dx27. It is acompact, semisimple, 14-dimensional Lie group, a subgroup of SO(7).A G2-structure on a 7-manifold M gives rise to a 3-form ϕ and a metricg on M , such that each tangent space of M admits an isomorphism with R7identifying ϕ and g with ϕ0 and g0 respectively.Proposition 1.3 Let M be a 7-manifold and (ϕ, g) a G2-structure on M .Then the following are equivalent:2(i) Hol(g) ⊆ G2, 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 .The quantity ∇ϕ is called the torsion of the G2-structure (ϕ, g). If ∇ϕ =0 then the G2-structure is called torsion-free. If g has holonomy Hol(g) ⊆ G2,then g is ricci-flat. Here is a result about compact 7-manifolds with holonomyG2, taken from [6, Prop. 1.1.1] and [5, Thm. C].Proposition 1.4 Let M be a compact 7-manifold, and suppose that (ϕ, g)is a torsion-free G2-structure on M . Then Hol(g) = G2 if and only if π1(M)is finite. In this case the moduli space of metrics with holonomy G2 onM , up to diffeomorphisms isotopic to the identity, is a smooth manifold ofdimension b3(M).Facts about the holonomy group Spin(7)Let R8 have coordinates (x1, . . . , x8). Define a 4-form Ω0 on R8 byΩ0 = ϕ0 ∧ dx8 + ∗ϕ0, (2)where ϕ0 is the 3-form defined in (1), and ∗ϕ0 is the Hodge star of ϕ0 usingthe euclidean metric on R7. The subgroup of GL(8,R) preserving Ω0 is theholonomy group Spin(7). This group also preserves the orientation on R8and the euclidean metric g0 = dx21 + · · · + dx28. It is a compact, semisimple,21-dimensional Lie group, a subgroup of SO(8).A Spin(7)- structure on an 8-manifold M gives rise to a 4-form Ω and ametric g on M , such that each tangent space of M admits an isomorphismwith R8 identifying Ω and g with Ω0 and g0 respectively.Proposition 1.5 Let M be a compact 8-manifold and (Ω, g) a Spin(7)-structure on M . Then the following are equivalent:(i) Hol(g) ⊆ Spin(7), and Ω is the induced 4-form,(ii) ∇Ω = 0 on M , where ∇ is the Levi-Civita connection of g, and(iii) dΩ = 0 on M .3Again, ∇Ω is the torsion of the Spin(7)-structure (Ω, g), and (Ω, g) istorsion-free if ∇Ω = 0. If g has holonomy Hol(g) ⊆ Spin(7), then g is ricci-flat. Here is a result on compact 8-manifolds with holonomy Spin(7), takenfrom [7, Thms C & D].Proposition 1.6 Let M be a compact 8-manifold, and suppose that (Ω, g)is a torsion-free Spin(7)-structure on M . Then Hol(g) = Spin(7) if and onlyif M is simply-connected, and b3(M) + b4+(M) = b2(M) + 2b4−(M) + 25.In this case the moduli space of metrics with holonomy Spin(7) on M , upto diffeomorphisms isotopic to the identity, is a smooth manifold of dimen-sion 1 + b4−(M).For some time after Berger’s classification, the holonomy groups G2 andSpin(7) remained a mystery. In 1987, Bryant [2] used the theory of exteriordifferential systems to show that locally there exist many metrics with theseholonomy groups, and gave some explicit, incomplete examples. Then in1989, Bryant and Salamon [3] found explicit, complete metrics with holonomyG2 and Spin(7) on noncompact manifolds. Recently, the author constructedexamples of metrics with holonomy G2 and Spin(7) on compact manifolds [5],[6], [7]. We will now describe this construction.2. A ‘Kummer construction’ for 7- and 8-manifoldsIt is well known that metrics with holonomy SU(2) on the K3 surface canbe obtained by resolving the 16 singularities of the orbifold T 4/Z2, where Z2acts on T 4 with 16 fixed points. This is called the Kummer construction, andis described in [8]. Our construction is motivated by and modelled on this.It can be divided into four steps. Here is a summary of each. For simplicitywe will describe the G2 case only, but the Spin(7) case is very similar.Step 1. Let T 7 be the 7-torus. Let (ϕ0, g0) be a flat G2-structure on T7.Choose a finite group Γ of isometries of T 7 preserving (ϕ0, g0).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 sin-gularities of T 7/Γ in a natural way, using complex geometry. Weget a nonsingular, compact 7-manifold M , together with a mapπ : M → T 7/Γ, the resolving map.4Step 3. On M , we explicitly write down a 1-parameter family of G2-structures(ϕt, gt) depending on a real variable t ∈ (0, ε). These G2-structuresare not torsion-free, but when t is small, they have small torsion.As t → 0, the G2-structure (ϕt, gt) converges to the singular G2-structure π∗(ϕ0, g0).Step 4. We prove using analysis that for all sufficiently small t, the G2-structure (ϕt, gt) on M , with small torsion, can be deformed to aG2-structure (ϕ̃t, g̃t), with zero torsion. Finally, we show that g̃t isa metric with holonomy G2 on the compact 7-manifold M .We will now explain the steps in greater detail.Steps 1-2We begin with an example of a suitable group Γ. Let (x1, . . . , x7) be coordi-nates on T 7 = R7/Z7, where xi ∈ R/Z. Let (ϕ0, g0) be the flat G2-structureon T 7 defined by (1). Let α, β and γ be the involutions of T 7 defined byα((x1, . . . , x7)) = (−x1,−x2,−x3,−x4, x5, x6, x7), (3)β((x1, . . . , x7)) = (−x1, 12 − x2, x3, x4,−x5,−x6, x7), (4)γ((x1, . . . , x7)) = (12− x1, x2, 12 − x3, x4,−x5, x6,−x7). (5)By inspection, α, β and γ preserve (ϕ0, g0), because of the careful choice ofexactly which signs to change. Also, α2 = β2 = γ2 = 1, and α, β and γcommute. Thus they generate a group Γ = 〈α, β, γ〉 ∼= Z32 of isometries ofT 7 preserving the flat G2-structure (ϕ0, g0). The following Lemma is provedin [5, §2.1].Lemma 2.1 The elements βγ, γα, αβ and αβγ of Γ have no fixed points onT 7. The fixed points of α, β, γ are each 16 copies of T 3. The singular set Sof 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 × C2/{±1}.Our goal is to resolve the singular set S of T 7/Γ to get a compact 7-manifold M with holonomy G2. How can we do this? In general we cannot,5because we have no idea of how to resolve general orbifold singularities withholonomy G2. However, there is a good theory of how to resolve complexsingularities like C2/G, with holonomy SU(2). Suppose that we find Γ suchthat S is a disjoint union of connected components, each component is a copyof T 3, and the singularities are each modelled on the product of T 3 with acomplex singularity C2/G, for G a finite subgroup of SU(2). (This happensin the example above, with G = {±1}.) Then there is a natural method forresolving T 7/Γ.Let X be the resolution of C2/G coming from algebraic geometry, sothat X has holonomy SU(2) and π : X → C2/G is the resolving map.For G = {±1}, X is called the Eguchi-Hanson space, with metric givenexplicitly in [4]. Each component of S is modelled on the singularity ofT 3×C2/G, and we resolve by replacing this with T 3×X, using the resolvingmap π : T 3 × X → T 3 × C2/G to glue in the patches T 3 × X. In this waywe get a compact, nonsingular 7-manifold M .Step 3It turns out that X admits not just one, but a 1-parameter family {ht : t > 0}of metrics with holonomy SU(2). Let h be the euclidean metric on C2/G,and r the distance from the origin. Then ht satisfiesht = π∗(h) + O(t4π∗(r)−4), (6)so that ht is asymptotically locally euclidean, and ht is isometric to t2h1 afteran automorphism of X. Let hT 3 be a flat metric on T3. Since SU(2) is asubgroup of G2 and Hol(ht) = SU(2), ĝt = hT 3 + ht is a metric on T3 × Xwith holonomy contained in G2. Let ϕ̂t be the associated G2-structure. Then(ϕ̂t, ĝt) is a torsion-free G2-structure on T3 ×X.The idea is to make a G2-structure (ϕt, gt) on M by gluing together thetorsion-free G2-structures (ϕ̂t, ĝt) on the patches T3 × X, and (ϕ0, g0) onT 7/Γ. The gluing is done using a partition of unity. Naturally, the firstderivative of the partition of unity introduces ‘errors’, so that (ϕt, gt) is nottorsion-free.The size of the torsion ∇ϕt depends on the difference ϕ̂t−ϕ0 in the regionwhere the partition of unity changes. Using (6) to estimate this, we find that∇ϕt = O(t4). However, as ht is isometric to t2h1, the injectivity radius of htis proportional to t, and ‖R(ht)‖C0 is proportional to t−2. Similar estimates6therefore apply to gt. The following result gives the estimates on (ϕt, gt) thatwe need.Theorem A There is a mathematical construction, which can be writtenentirely explicitly in coordinates, that gives a compact 7-manifold M and a1-parameter family of G2-structures (ϕt, gt) on M depending on t ∈ (0, ε).These G2-structures satisfy the following bounds:• the torsion ∇ϕt satisfies ‖∇ϕt‖C1 ≤ C1t4,• the injectivity radius δ(gt) satisfies δ(gt) ≥ C2t,• the riemann curvature R(gt) satisfies ‖R(gt)‖C0 ≤ C3t−2,• the volume vol(M) satisfies C4 ≤ vol(M) ≤ 2C4, and• the diameter diam(M) satisfies diam(M) ≤ C5.Here C1, . . . , C5 are positive constants independent of t.Step 4We prove the following analysis result.Theorem B Let C1, . . . , C5 be positive constants. Then there exist positiveconstants λ,K depending only on C1, . . . , C5, such that for every t ∈ (0, λ],the following is true.Let M be a compact 7-manifold, and (ϕ, g) a G2-structure on M . Supposethat dϕ = 0, that ψ is a 3-form on M with d∗ψ = d∗ϕ, and that these fiveconditions hold:(i) ‖ψ‖C2 ≤ C1t4,(ii) the injectivity radius δ(g) satisfies δ(g) ≥ C2t,(iii) the riemann curvature R(g) of g satisfies ‖R(g)‖C0 ≤ C3t−2,(iv) the volume vol(M) satisfies C4 ≤ vol(M) ≤ 2C4, and(v) the diameter diam(M) satisfies diam(M) ≤ C5.Then there exists a smooth, torsion-free G2-structure (ϕ̃, g̃) on Mwith ‖ϕ̃− ϕ‖C0 ≤ Kt1/2.Basically, this result says that if (ϕ, g) is a G2-structure on M , and thetorsion ∇ϕ is sufficiently small, then we can deform to a nearby G2-structure7(ϕ̃, g̃) that is torsion-free. Regard ψ as a first integral of ∇ϕ. Thus thenorm ‖ψ‖C2 is a measure of the torsion ∇ϕ, approximately equal to ‖∇ϕ‖C1 .Then parts (i)-(v) say that the torsion ∇ϕ must be small compared to theinjectivity radius, riemann curvature, volume and diameter of M .Now the hypotheses of Theorem B are, intentionally, nearly the same asthe conclusions of Theorem A. With a little work we can close the gap be-tween them. Therefore, applying Theorems A and B together for sufficientlysmall t, we see that the compact 7-manifold M constructed in Step 2 admitstorsion-free G2-structures (ϕ̃, g̃). Applying Proposition 1.4, Hol(g̃) = G2 ifand only if π1(M) is finite. In this way we construct compact riemannian7-manifolds with holonomy G2.The proof of Theorem B is not easy, and it represents most of the hardwork in [5]. Here is a sketch, ignoring several technical points. We have a3-form ϕ with dϕ = 0 and d∗ϕ = d∗ψ, ψ small, and we wish to construct anearby 3-form ϕ̃ with dϕ̃ = 0 and d̃∗ϕ̃ = 0. Set ϕ̃ = ϕ + dη, where η is asmall 2-form. Then η must satisfy a nonlinear p.d.e., which we write asd∗dη = −d∗ψ + d∗F (dη), (7)where F is nonlinear, satisfying F (dη) = O(|dη|2).We solve (7) by iteration, introducing a sequence {ηj}∞j=0 with η0 = 0,satisfying the inductive equationsd∗dηj+1 = −d∗ψ + d∗F (dηj), d∗ηj+1 = 0. (8)If such a sequence exists and converges to η, then taking the limit in (8) showsthat η satisfies (7), giving us the solution we want. The key to proving this isan inductive estimate: one must show that if ηj, ηj+1 satisfy (8) and ‖ηj‖B ≤C then ‖ηj+1‖B ≤ C, where ‖ . ‖B is the norm in some suitable Banach spaceB, and C is a positive constant. Then by induction the sequence {ηj}∞j=0 isbounded in B, and convergence follows by standard techniques.The inductive estimate we use has two ingredients. The first is‖dηj+1‖L2 ≤ ‖ψ‖L2 + cst.‖dηj‖L2‖dηj‖C0 , (9)obtained from (8) by taking the L2-inner product with ηj+1 and integratingby parts. (The 3-form ψ was introduced solely to achieve this inequality.)The second is the inequality8‖dηj+1‖C1,1/2 ≤ cst.‖ψ‖C1,1/2 + cst.‖dηj‖C1,1/2‖dηj‖C0 + cst.‖dηj+1‖L2 , (10)which is an elliptic regularity estimate. Here the terms ‘cst.’ are positiveconstants depending on C2, . . . , C5 and t. We show that if t ≤ λ, for λ > 0depending on C1, . . . , C5, then (9) and (10) combine to give the necessaryinductive estimate, and the remainder of the proof is straightforward. Forthe Spin(7) case, the proof is different, and more difficult.3. Presentation of resultsBy considering different groups Γ acting on T 7 and T 8, we are able to findmetrics with holonomy G2 and Spin(7) on many topologically distinct 7- and8-manifolds. It also happens that the same orbifold T k/Γ can admit severaltopologically distinct resolutions, and this increases the number of examples.Table 1 is a graph of known examples of compact 7-manifolds M withholonomy G2, from [5] and [6]. We have plotted b2(M) against b3(M), as theseare the important betti numbers. The symbols ‘•’, ‘∗’ and ‘+’ denote the bettinumbers of one or more 7-manifolds with holonomy G2. Here ‘•’ means asimply-connected manifold, ‘∗’ means a non-simply-connected manifold, and‘+’ means both a simply-connected and a non-simply-connected manifold.0 5 10 15 20 25 30 35 40 45 50 b3(M)05101520•••••••••∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ +++++++++∗∗∗∗∗∗∗•∗•• •∗•++••••b2(M) Table 1.The total number of distinct 7-manifolds known is at least 68.9Similarly, we shall plot a graph of betti numbers b2(M) against b3(M) ofcompact 8-manifolds M with holonomy Spin(7), and this appears in Table2. Such manifolds are always simply-connected. However, we can use thecup product in cohomology to distinguish 8-manifolds with the same bettinumbers. Therefore, in the graph below, the data points are figures, eachfigure being the number of different 8-manifolds with these betti numbers.All known examples satisfy b4+ = 103− b2 + b3 and b4− = 39− b2 + b3, but asfar as we know, these are not necessary topological conditions.0 5 10 15 20 b2(M)051015201 2 3 4 5 5 5 5 5 4 3 2 11 2 4 5 7 7 7 5 4 2 11 1 1 1 1b3(M) Table 2.The number of distinct 8-manifolds known is at least 95. Although this isgreater than the G2 case, the variety is less, as all these examples are derivedfrom only 3 groups Γ, giving the 3 horizontal lines on the graph. It seems tobe harder to find suitable groups Γ in the Spin(7) case.Finally, we consider the connections with physics. After many discus-sions with C. Vafa, S. Shatashvili, E. Witten and M. Roček, it appears thatmuch of the ‘mirror symmetry’ story for holonomy SU(3) also applies to theholonomy groups G2 and Spin(7). In particular, Vafa and Shatashvili claimthat the reason that the examples are clustered in families with b2 + b3 con-stant in Table 1, and with b3 and 2b2 + b4 constant in Table 2, is becausethe examples in each family are mirrors of each other. Also, compact rie-mannian 7-manifolds with holonomy G2 may become important in physics if11-dimensional theories come back into fashion, because they are needed toreduce the number of observable dimensions to 4.10References[1] M. Berger, ‘Sur les groupes d’holonomie homogène des variétés à connexionaffines et des variétés riemanniennes’, Bull. Soc. Math. France 83 (1955) 279-330.[2] R.L. Bryant, ‘Metrics with exceptional holonomy’, Ann. Math. 126 (1987)525-576.[3] R.L. Bryant & S.M. Salamon, ‘On the construction of some complete metricswith exceptional holonomy’, Duke Math. J. 58 (1989) 829-850.[4] T. Eguchi & A.J. Hanson, ‘Asymptotically flat solutions to Euclidean gravity’,Phys. Lett. 74B (1978) 249-251.[5] D.D. Joyce, ‘Compact riemannian 7-manifolds with holonomy G2. I’, to ap-pear in Journal of Differential Geometry.[6] D.D. Joyce, ‘Compact riemannian 7-manifolds with holonomy G2. II’, to ap-pear in Journal of Differential Geometry.[7] D.D. Joyce, ‘Compact riemannian 8-manifolds with holonomy Spin(7)’,preprint (1995).[8] D.N. Page, ‘A physical picture of the K3 gravitational instanton’, Phys. Lett.80B (1978), 55-57.[9] S.M. Salamon, ‘Riemannian geometry and holonomy groups’, Pitman Res.Notes in Math. 201, Longman, Harlow, 1989.11

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Compact manifolds with exceptional holonomy

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