# Constructing compact 8-manifolds with holonomy Spin(7)

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1 Constructing compact 8-manifolds with holonomy Spin(7) Dominic Joyce, Oxford University Simons Collaboration meeting, Imperial College, June Based on Invent. math. 123 (1996), ; J. Diff. Geom. 53 (1999), ; and Compact Manifolds with Special Holonomy, OUP, These slides available at joyce/. 1 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Plan of talk: Apology This talk contains no new work since 2000, except open problems. 2 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

2 1. in 8 dimensions is one of the exceptional cases G 2, Spin(7) in Berger s classification. The action of Spin(7) on R 8 preserves the Euclidean metric g 0 = dx dx 8 2, the orientation, and the 4-form Ω 0 = dx dx dx dx 1357 dx 1368 dx 1458 dx 1467 dx 2358 dx 2367 dx dx dx dx dx 5678, where dx ijkl = dx i dx j dx k dx l. If (X, g) is a Riemannian 8-manifold with holonomy Spin(7) then X has a natural 4-form Ω with Ω = 0 isomorphic to Ω 0 at each x X, and g is Ricci-flat. We call a pair (Ω, g) a Spin(7)-structure on X if at each x X there is an isomorphism T x X = R 8 identifying (Ω x, g x ) with (Ω 0, g 0 ). We call (Ω, g) torsion-free if Ω = 0 for the Levi-Civita connection of g, or equivalently if dω = 0 (though this is apparently weaker). Then Hol(g) Spin(7). 3 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) One difference with G 2 is that Spin(7)-forms Ω are not generic. Call a 4-form Ω on X admissible if it is pointwise isomorphic to Ω 0, and write AX Λ 4 T X for the bundle of admissible forms. Then AX X has fibre GL(8, R)/ Spin(7) with dimension = 43, but Λ 4 T X X has fibre of dimension ( 8 4) = 70, so AX has codimension = 27 in Λ 4 T X. So to construct holonomy Spin(7) metrics we need to find sections Ω of the nonlinear subbundle AX Λ 4 T M satisfying dω = 0, which is superficially a more complicated problem than the G 2 case. Let (X, Ω, g) be a Spin(7)-manifold. Then we have natural decompositions of exterior forms Λ 1 T X = Λ 1 8, Λ 2 T X = Λ 2 7 Λ 2 21, Λ 3 T X = Λ 3 8 Λ 3 48, Λ 4 T X = Λ 4 + Λ 4, Λ 4 + = Λ 4 1 Λ 4 7 Λ 2 27, Λ 4 = Λ 4 35, Λ 5 T X = Λ 5 8 Λ 5 48, Λ 6 T X = Λ 6 7 Λ 6 21, Λ 7 T X = Λ 7 8, where Λ k l is a vector bundle of rank l. 4 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

3 We have inclusions of holonomy groups SU(2) SU(2) Sp(2) SU(4) Spin(7), SU(3) G 2 Spin(7). Thus Calabi Yau 4-folds and hyperkähler 8-manifolds are torsion-free Spin(7)-manifolds. If Y is a G 2 -manifold, and Z is a Calabi Yau 3-fold, then Y S 1, Y R, Z T 2, Z R 2 are torsion-free Spin(7)-manifolds. Suppose (X, Ω, g) is a compact, torsion-free Spin(7)-manifold. Then the Â-genus of X, a characteristic class of X, satisfies 24Â(X ) = 1 + b 1 b 2 + b 3 + b 4 + 2b 4. If X is simply-connected then there are four cases: (i) Hol(g) = Spin(7) and Â(X ) = 1; (ii) Hol(g) = SU(4) and Â(X ) = 2; (iii) Hol(g) = Sp(2) and Â(X ) = 3; and (iv) Hol(g) = SU(2) SU(2) and Â(X ) = 4. Conversely, if g has any of these holonomy groups then X is simply-connected. Thus, Hol(g) = Spin(7) if and only if π 1 (X ) = {1} and Â(X ) = 1. 5 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) The moduli space of torsion-free Spin(7)-structures Theorem 1 (Bryant Harvey?; Joyce) Let X be a compact, oriented 8-manifold. Then the moduli space M of oriented, torsion-free Spin(7)-structures (Ω, g) on X, modulo diffeomorphisms isotopic to the identity, is a smooth manifold of dimension dim M = Â(X ) + b 1 (X ) + b 4 (X ), so that dim M=1+b 4 (X ) if X admits metrics of holonomy Spin(7). The map ι : M H 4 (X ; R) taking ι : [(Ω, g)] [Ω] is an immersion. It seems to be difficult to compute the submanifold ι(m) H 4 (X ; R) (the period domain) in examples. For the analogous problem for Calabi Yau manifolds, there is a deep theory which allows you to do this. 6 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

4 In 1987, Robert Bryant proved the local existence of many metrics with holonomy Spin(7), using EDS. In 1989, Robert Bryant and Simon Salamon produced explicit, complete examples of holonomy Spin(7) manifolds. In 1996 and 2000 I gave two constructions of compact 8-manifolds with holonomy Spin(7), which I will explain today. Alexei Kovalev has a third construction (this conference). Let (X, Ω, g) be a torsion-free Spin(7)-manifold. A Cayley 4-fold C in X is a 4-submanifold C X calibrated w.r.t. Ω. They are minimal submanifolds. McLean (1998) studied the deformation theory of compact Cayley 4-folds C, and found it is elliptic, and obstructed, and the Cayley moduli space M C has virtual dimension b 0 (C) b 1 (C) + b 2 +(C) [C] [C]. I gave examples of compact Cayley 4-folds in compact 8-manifolds with holonomy Spin(7), as fixed points of involutions. 7 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) If (X, Ω, g) is a Spin(7)-manifold, there is a natural splitting Λ 2 T X = Λ 2 7 Λ2 21 into vector subbundles of rank 7,21. Let P X be a principal bundle, and A a connection on P with curvature F A. We call (P, A) a Spin(7)-instanton if π7 2(F A) = 0, where π7 2 is the projection to ad(p) Λ2 7 ad(p) Λ2 T X. Spin(7)-instantons have elliptic deformation theory, and so form well-behaved moduli spaces. These should fit into the Donaldson Thomas / Donaldson Segal programme for defining invariants, though I m not sure if there are particular conjectures. Borisov Joyce use Derived Algebraic Geometry to prove existence of deformation-invariant Donaldson Thomas style invariants counting stable coherent sheaves on a Calabi Yau 4-fold. Morally speaking, the B J construction treats moduli spaces of Hermitian Yang Mills connections as moduli spaces of Spin(7)-instantons in order to define the virtual cycle. So this is some evidence that there may be interesting deformation-invariant counting information in Spin(7)-instanton moduli spaces. 8 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

5 2. My first construction of compact 8-manifolds with holonomy Spin(7) (1996, 2000) worked, as in the G 2 case, by starting with a torus T 8 = R 8 /Λ with flat Spin(7)-structure (Ω 0, g 0 ), choosing a finite group Γ of automorphisms of (T 8, Ω 0, g 0 ), taking the quotient orbifold T 8 /Γ, and resolving the singularities by gluing in ALE and Quasi-ALE Calabi Yau 4-folds. This is more difficult than the G 2 case for several reasons: All orbifold singularities of C Y 3-folds admit crepant resolutions. But many C Y 4-fold orbifold singularities do not, C 4 /{±1} for instance. So, it is a lot harder to find orbifolds T 8 /Γ all of whose singularities can be resolved. In the G 2 case, we can work with orbifolds T 7 /Γ whose only singularities look like T 3 C 2 /G or S 1 C 3 /G. In the Spin(7) case, because of the Â-genus, it is necessary that T 8 /Γ has orbifold strata intersecting in points. 9 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) As Spin(7) forms are not generic, we need a different method of proof to deform small torsion to zero torsion. The general method of proof for both G 2 and Spin(7) needs (codimension of singularities) > 1 2 (total dimension). ( ) For G 2 this is 4 > For Spin(7) it is As equality holds in the Spin(7) case, we have to work harder to define a Spin(7)-structure with small enough torsion to deform to torsion-free. Here ( ) holds as if c is the codimension of singularities and d the total dimension, and G = G 2 or Spin(7), then roughly we expect to construct a G-structure Ω with torsion τ satisfying τ L 2 = O(t c ), where t is the injectivity radius. Then we solve L Ω (δω) = τ, where δω is an infinitesimal change in Ω and L Ω is the linearized torsion. We find that δω C 0 Ct d/2 τ L 2 = O(t c d/2 ). So if c > d/2 then taking t small is enough. But if c = d/2 then we need to add in extra correction terms to make τ L 2 = o(t c ). 10 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

6 An example Let T 8 = R 8 /Z 8 with standard Spin(7)-structure (Ω 0, g 0 ). Let Γ = α, β, γ, δ = Z 4 2, where α, β, γ, δ are involutions acting by α(x 1,..., x 8 ) = ( x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8 ), β(x 1,..., x 8 ) = (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8 ), γ(x 1,..., x 8 ) = ( 1 2 x 1, 1 2 x 2, x 3, x 4, 1 2 x 5, 1 2 x 6, x 7, x 8 ), δ(x 1,..., x 8 ) = ( x 1, x 2, 1 2 x 3, x 4, 1 2 x 5, x 6, 1 2 x 7, x 8 ). The singular set of T 8 /Γ consists of: (a) 4 copies of T 4 /{±1} from the fixed points of α. (b) 4 copies of T 4 /{±1} from the fixed points of β. (c) 2 copies of T 4 from the fixed points of γ. (d) 2 copies of T 4 from the fixed points of δ. Here (a),(b) intersect in 64 points, from the fixed points of αβ. 11 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Let Y be the Eguchi Hanson space, ALE C Y 2-fold asymptotic to C 2 /{±1}. We desingularize T 8 /Γ by gluing in T 4 /{±1} Y at each T 4 /{±1} from (a),(b), and gluing in T 4 Y at each T 4 from (c),(d), and gluing in Y Y at each point of intersection of (a),(b). This gives a compact, simply-connected 8-manifold X with b 2 (X ) = 12, b 3 (X ) = 16, b 4 +(X ) = 107 and b 4 (X ) = 43. We then write down a family of Spin(7)-structures (Ω t, g t ) on X for t (0, ɛ], which have small torsion when t is small. We do this by shrinking the C Y 2 structure on Y by a factor t > 0, using this to give Spin(7) structures on T 4 /{±1} Y, T 4 Y, Y Y, and gluing these to (Ω 0, g 0 ) with a partition of unity. We glue the forms either as Spin(7)-structures (Ω t, g t ), or as closed 4-forms Ω t. Then φ t = Ω t Ω t is a small 4-form. The obvious definitions would give φ t L 2 = O(t 4 ), which is not small enough. But by solving an equation on T 8 /Γ to cancel out the leading order errors, we can achieve φ t L 2 = O(t 9/2 ), which is small enough. 12 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

7 We then prove the following analytic theorem: Theorem 2 Let λ, µ, ν > 0. Then there exist κ, K > 0 depending only on λ, µ, ν such that whenever 0 < t κ, the following holds. Suppose (X, Ω t, g t ) is a compact Spin(7)-manifold, and φ t is a 4-form on X with dω t + dφ t = 0, and: (i) φ t L 2 λt 13/3, (ii) dφ t L 2 λt 7/5, (iii) the injectivity radius δ(g t ) satisfies δ(g t ) µt, and (iv) the Riemann curvature R(g t ) satisfies R(g t ) C 0 νt 2. Then there exists a torsion-free Spin(7)-structure (ˆΩ t, ĝ t ) on X with ˆΩ t Ω t C 0 Kt 1/3. We show (i) (iv) hold in our examples for some A 1,..., A 4 > 0 independent of t (0, ɛ], and thus Theorem 2 shows that we can deform (Ω t, g t ) to a torsion-free Spin(7)-structure (ˆΩ t, ĝ t ) for small t > 0. As in 1, a topological criterion, that π 1 (X ) = {1} and Â(X ) = 1, then implies that Hol(ĝ t ) = Spin(7). 13 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Note that we do not know the cohomology class [ˆΩ t ] H 4 (X ; R) (in particular, it need not be [Ω t + φ t ]). This is necessary, since as in Theorem 1 we don t know the possible cohomology classes of torsion-free Spin(7)-structures in advance. To prove Theorem 2, we first show that for any C 0 -small 4-form η t on X which lies in Λ 4 T X w.r.t. g t, there is a unique decomposition Ω t + η t = ˆΩ t + χ t for ˆΩ t a Spin(7)-form which is C 0 -close to Ω t, and χ t a C 0 -small 4-form which lies in Λ 4 27 w.r.t. ˆΩ t. We regard ˆΩ t, χ t as smooth nonlinear functions of η t. So we can write dˆω t = 0 as a nonlinear p.d.e. in η t, and we show that it has a unique solution η t which is L 2 -orthogonal to the harmonic anti-self-dual 4-forms H. 4 This works as our equation involves a small nonlinear perturbation of the isomorphism d : { anti-self-dual 4-forms L 2 -orthogonal to H 4 } = } {exact 5-forms. 14 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

8 3. In 2000 I gave a second construction of compact 8-manifolds with holonomy Spin(7), which starts not with T 8 /Γ, but with a Calabi Yau 4-orbifold Y. Suppose Y is a compact complex 4-orbifold with c 1 (Y ) = 0, admitting Kähler metrics, such that the only singularities of Y are p 1,..., p k locally modelled on C 4 / i, where i = Z 4 acts on C 4 by multiplication in the obvious way. Suppose σ : Y Y is an antiholomorphic involution whose only fixed points are p 1,..., p k. Then Y / σ is a compact 8-orbifold with orbifold points p 1,..., p k locally modelled on R 8 /G, for G a nonabelian group of order 8, which may be thought of as {±1, ±i, ±j, ±k} H. By the Calabi Conjecture (orbifold version) there exist Ricci-flat Kähler metrics on Y, which will be σ-invariant if the Kähler class is σ-invariant. So as SU(4) Spin(7), we get σ-invariant torsion-free Spin(7)-structures on Y, which descend to Y / σ. 15 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Thus, we have a compact orbifold Y / σ with torsion-free Spin(7)-structure (Ω, g). We resolve Y / σ to an 8-manifold X by gluing in an ALE Spin(7)-manifold Z asymptotic to R 8 /G at each of p 1,..., p k. Then we define a family of Spin(7)-structures (Ω t, g t ) on X for t (0, ɛ], with small torsion for small t, and use Theorem 2 to deform them to torsion-free Spin(7)-structures (ˆΩ t, ĝ t ) as for the T 8 /Γ case, which have Hol(ĝ t ) = Spin(7). To define the ALE Spin(7)-manifold Z, note that G acts on C 4 = T pi Y as α, β, where α : (z 1,..., z 4 ) (iz 1, iz 2, iz 3, iz 4 ), β : (z 1,..., z 4 ) ( z 2, z 1, z 4, z 3 ), where α 4 = β 4 = 1, α 2 = β 2, αβ = βα / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

9 The crepant resolution of C 4 / α is the blow-up W of C 4 / α at 0, the line bundle O( 4) CP 3, and it carries an explicit ALE C Y 4 metric due to Calabi. The action of β on C 4 / α lifts to a free action of β = Z 2 on W, so W / β is an ALE Spin(7)-manifold asymptotic to C 4 /G. Now if we did the blow-up of C 4 / α using the given complex structure on Y, then the Spin(7)-manifold X we end up with would not have π 1 (X ) = {1} and holonomy Spin(7), but we would have X = X / β where X is a Calabi Yau 4-fold with a free action of β = Z 2, and metrics on X with holonomy Z 2 SU(4). However, the action of G on R 8 has a kind of hyperkähler twist : we can use a different complex structure on R 8 which swaps round α and β. That is, there are two different ways to glue W / β into Y / σ at each p i. As long as we use the twisted gluing for at least one i = 1,..., k, we get π 1 (X ) = {1}, and holonomy Spin(7) metrics on X. 17 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) An example Work in the weighted projective space CP 5 1,1,1,1,4,4. Let Y be the hypersurface Y = { [z 0,..., z 5 ] CP 5 1,1,1,1,4,4 : z z z z z 3 4 +z 3 5 = 0 }. Then Y is a Calabi Yau 4-orbifold with three singular points p 1 =[0, 0, 0, 0, 1, 1], p 2 =[0, 0, 0, 0, 1, e πi/3 ], p 3 =[0, 0, 0, 0, 1, e πi/3 ], each of which is locally modelled on C 4 / i. Define an antiholomorphic involution σ : Y Y by σ : [z 0,..., z 5 ] [ z 1, z 0, z 3, z 2, z 5, z 4 ]. Then the only fixed points of σ are p 1, p 2, p 3. The construction yields a compact 8-manifold X with holonomy Spin(7), which has b 1 = b 2 = b 3 = 0, b 4 + = 1639 and b 4 = 807. The family of holonomy Spin(7) metrics on X has dimension / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

10 4. Finding compact 8-manifolds with holonomy Spin(7) Problem 1 Use the known constructions to of compact 8-manifolds with holonomy Spin(7) find as many new examples as you can. Presumably this could be done by some kind of computer search, of the kind that some String Theorists are very good at. The examples I found, I did with a pencil and paper and counting on my fingers, and when I felt I had enough, I stopped. It is very possible that there are many more examples left to find. Problem 2 Can you find new ways to construct compact 8-manifolds with holonomy Spin(7)? 19 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Problems about Cayley submanifolds Problem 3 Study the singularities of Cayley 4-folds. Find more examples of model singularities in R 8. Consider how singularities form, e.g. by shrinking ALE Cayley 4-folds. Get some idea of which singularities can occur in low codimension amongst all Cayley 4-folds, e.g. codimension 1. Since special Lagrangian 4-folds and complex surfaces in Calabi Yau 4-folds, and associative 3-folds S 1, coassociative 4-folds {point} in G 2 -manifold S 1, all provide examples of Cayley 4-folds, we know quite a lot of examples already; and there is some work on Cayley 4-folds as well. 20 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

11 Problem 4 Construct examples of compact 8-manifolds with Spin(7), together with a fibration by Cayley 4-folds, including singular fibres. This is rather a nice problem, which I would like to see done during the Simons collaboration. One could hope to approach it in a similar way to a proposal by Alexei Kovalev in the G 2 case, in which the singularities of the fibration are locally modelled on singularities of holomorphic fibrations of Cayley 4-folds by surfaces. 21 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7) Problems about Spin(7) instantons We can ask similar questions about Spin(7) instantons to G 2 instantons, and others here know a lot more about this than I do. Study singularities of Spin(7) instantons, in particular, bubbling on Cayley 4-folds. Use to define Donaldson Segal style counting invariants of Spin(7)-manifolds? Use to define a Floer theory for G 2 instantons on a G 2 -manifold X, by considering Spin(7) instantons on X R? 22 / 22 Dominic Joyce, Oxford University Constructing compact 8-manifolds with holonomy Spin(7)

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