ORIENTING THE G 2 -INSTANTON MODULI SPACE MARKUS UPMEIER (JOINT WORK - - PDF document

ORIENTING THE G2-INSTANTON MODULI SPACE

MARKUS UPMEIER (JOINT WORK WITH DOMINIC JOYCE)

• 1. Donaldson–Segal programme

I shall discuss the following result, which is one of the steps in the Donaldson– Segal programme on gauge theory and special holonomy: Theorem 1.1 (Joyce–U. 2018). Let X be a G2-manifold and fix an SU(2)-bundle E → X. The moduli stack M of G2-instantons on E is orientable. A flag structure

• n X fixes a canonical orientation.

Very roughly, the idea in the Donaldson–Segal programme is that, when a cal- ibrated submanifold Y 3 ⊂ X7 is fixed, the geometry splits into 3 + 4 = 7 so that we may regard X as a family of 4-manifolds and study ASD-connections in the transverse direction. Problem 1.2. For the moduli space of G2-instantons, in order to define counting invariants, understand (1) Orientability / Orientations (2) Compactifications (3) Deformations There are similar questions for other exceptional holonomies, e.g. for Spin(7)- holonomy 8-manifolds. Theorem 1.3 (Donaldson 1987). To fix an orientation of the ASD-moduli space

• f a 4-manifold X, one needs to choose an orientation of H1(X) ⊕ H+(X).
• 2. G2-manifolds

Definition 2.1. φ ∈ Λ3V ∗ on a 7-dimensional vector space is non-degenerate if ιXφ ∧ ιXφ ∧ φ = 0 ∀X ∈ V \ {0}. A G2-structure on a 7-manifold X is a smooth 3-form φ that is non-degenerate

• n each tangent space. It is torsion-free if with ψ = ∗φ

dφ = 0, dψ = 0. Example 2.2. On V = R7 we have φ(X, Y, Z) = X × Y, Z. In coordinates φstd = dx123 + dx1 dx45 + dx67 + dx2 dx46 − dx57 + dx3 dx47 + dx56 Definition 2.3. The group G2 is the stabilizer of φstd, i.e. G2 = {A ∈ GL(7, R) | A∗φstd = φstd}. Since φstd encodes the multiplication table of the octonions we have G2 ∼ = Aut(O). This is a 14-dimensional simply connected Lie group. Remark 2.4. It follows that we may identify each tangent space of a G2-manifold with R7 equipped with φstd. This identification is only well-defined up to A ∈ G2.

1

2 MARKUS UPMEIER (JOINT WORK WITH DOMINIC JOYCE)

Lemma 2.5. There exists a unique metric g and orientation on V such that g(X, Y ) volg = ιXφ ∧ ιY φ ∧ φ. Moreover, G2-manifolds have a natural spin structure.

• Proof. G2 ⊂ SO(7) is simply connected, so G2 ⊂ Spin(7).
• Example 2.6 (Relation to other geometries). We have inclusions

SU(3) → G2 → Spin(7) Also SO(4) and SO(3) (see Salamon). Any spin 7-manifold has a G2-structure given by choosing a never vanishing spinor. The G2-structure is torsion-free if and

• nly if the spinor is parallel.

If (Z, ω, Ω) is a Calabi–Yau 3-fold then R × Z or S1 × Z are torsion-free G2- manifolds with φ = dt ∧ ω + ℜeΩ. Given a hyperkähler surface (S, ω1, ω2, ω3) we get a torsion-free G2-manifold R3 × S or S1 × S1 × S1 × S with φ = dx123 − dx1 ∧ ω1 − dx2 ∧ ω2 − dx3 ∧ ω3 Example 2.7 (Simply connected examples). For a compact manifold the holonomy is all of G2 if and only if the fundamental group is finite. Examples are very difficult to find (see Joyce). Non-compact examples with holonomy all of G2 were found first and are due to Bryant. Definition 2.8. Let X be a G2-manifold and let E → X be a principal G-bundle. A connection A ∈ Ω1(E; g) on E is a G2-instanton if ∗(FA ∧ φ) = −FA ∈ Ω2(X; gE). rep-theory ⇐ ⇒ FA ∧ ψ = 0

• Here we use the adjoint bundle

gE = E ×G,Ad g. For a G2-instanton A and deformation a ∈ Ω1(X; gE) the G2-instanton condition becomes 0 = FA+a ∧ ψ = FA ∧ ψ + dAa ∧ ψ + a ∧ a ∧ ψ so the linearized G2-instanton equation is dAa ∧ ψ = 0. The solutions a describe the tangent space at A to the space of G2-instantons. Remark 2.9. Dividing out the action of the gauge group G = Aut(E) should give a finite-dimensional moduli space of solutions to the G2-instanton equation. To have a good Fredholm theory, we want the linearized equations to be part of an elliptic ’deformation’ complex. To see which other conditions we could have imposed, recall that the G2-representation Λ2(R7)∗ splits into Λ2

14 = {α | α ∧ ψ = 0} ∼

= g2 Λ2

7 = {α | ∗(α ∧ φ) = 2α} −∧ψ

− − − → Λ6 so the invariant conditions are FA = 0, π7FA = 0, and π14FA = 0. In gauge theory we want the infinitesimal deformations to be governed by an elliptic complex beginning with gauge transformations, connections, curvature. The conditions FA = 0 and π14FA = 0 violate ellipticity and are thus ruled out.

ORIENTING THE G2-INSTANTON MODULI SPACE 3

Example 2.10. (1) The Levi-Civita connection on the tangent bunde of a torsion-free G2-manifold. This is because R ∈ Λ2T ∗M ⊗ g2. (2) An ASD-connection on a hyperkähler 4-manifold M (note ωi self-dual) ∗(FA ∧ φ) = ∗

• FA ∧ dx123

= ∗MFA = −FA. (3) A Hermitian Yang–Mills connection ΛFA = 0, F 0,2

A

= 0 on a Calabi– Yau 3-fold.

• 3. Instanton moduli space

We assume X is a closed connected G2-manifold. 3.1. Deformation complex. For a G2-instanton A we have the deformation complex (1) Ω0(X; gE)

dA

− − → Ω1(X; gE)

π7◦dA

− − − − → Ω2

7(X; gE) = Ω6(X; gE) dA

− − → Ω7(X; gE) which has been made elliptic by adding the right-most term. It describes the infinitesimal solutions, i.e. the tangent space to the moduli space. More generally, we may roll up the complex and define a self-adjoint elliptic operator LA =

• d∗

A

dA ∗(ψ ∧ dA)

• for any connection A.

3.2. Irreducible connections. Consider the action of G = Aut(E) on the space A of connections on E by pullback. The stabilizer StabA of this action at A ∈ A consists of all g ∈ Aut(E) with g∗A = A. It follows that G/Z(G) acts freely on the maximal (open) stratum. Definition 3.1. The moduli space of ’irreducible’ connections is the smooth man- ifold B∗ = A∗/(G/Z(G)) for B∗ = {A ∈ A | StabA = Z(G)} . The moduli space of irreducible G2-instantons is the topological subspace M∗ = {[A] ∈ B∗ | FA ∧ ψ = 0} .

• 4. Determinant line bundle. Orientations

Definition 4.1. For a family of elliptic operators Pa, a ∈ A, the determinant line bundle Det P is the union of the fibers Det Pa = Λtop(Ker Pa)∗ ⊗ Λtop Coker Pa. Even though the kernel and cokernels may jump individually, according to Atiyah– Singer, Bismut–Freed, or Segal, this combination is indeed a line bundle over A. Remark 4.2. The entire moduli space B = A/(G/Z(G)) can be regarded as a stack. Note that M∗ is not generally a manifold, since for that the equation needs to be cut

• ut transversely. It is a ’derived manifold’ whose bundle of top exterior forms is the

determinant of the deformation complex. This explains the relevance of Det L to

• ur orientation problem. Lacking a smooth structure on M∗ in general, we simply

define Det L|M∗ to be the bundle of top exterior forms. Our task is to orient this real line bundle. Definition 4.3. An orientation for the moduli space of G2-instantons is an orien- tation of Det L|M∗. Here we regard the G-equivariant real line bundle Det L → A∗ as a line bundle on B∗. Below we will orient Det L → B∗, instead of only its restriction to M∗.

4 MARKUS UPMEIER (JOINT WORK WITH DOMINIC JOYCE)

Theorem 4.4 (Walpuski 2013). Suppose that φ is a G2-structure for which all elements of M∗ are irreducible and unobstructed. Assume G = SO(n), SU(n). Then M∗ is a 0-dimensional manifold which can be ’naturally’ oriented in two ways. This is essentially an application of the Atiyah–Patodi–Singer index theorem. To select one of the two orientations is considerably more difficult. In fact, we will show the following generalization, which takes away focus from the special geometry. Theorem 4.5 (Joyce–Upmeier, work in progress). Let X be a spin 7-manifold, let E → X be an SU(2)-bundle. Consider the family {DA}A∈A∗ of twisted real Dirac

• perators. Its determinant line bundle descends to B∗ and is orientable. A flag

structure (defined below) on X fixes a canonical orientation. Since the principal symbols of LA and DA agree, we may use a straight-line in- terpolation to see that the orientation problems agree. Hence we obtain canonical

• rientations for the moduli space of G2-instantons.
• 5. Flag structures

Definition 5.1. Let X be an oriented 7-manifold. A flag on an oriented 3- submanifold Y ⊂ X is a nowhere vanishing section α of the normal bundle NY = TX|Y /TY → Y . We identify two flags α0, α1 if for the straight-line homotopy α: [0, 1]×Y → NY (or any other homotopy) between them with intersection number [α] • [Y ] = 0 ∈ Z. Then the set of flags becomes a Z-torsor. Definition 5.2. A flag structure is a map F : {flagged submanifolds (Y, α)} → Z2 satisfying the axioms (1) F(N, α) = F(N ′, α′) if (N, α) is a small perturbation of (N ′, α′). (2) F(N, α0) = (−1)k · F(N, α1) if [α] • [Y ] = k. (3) For disjoint submanifolds F(N1, α1) = (−1)D((N1,α1),(N2,α2))F(N2, α2) with [N1] = [N2]. (4) For N1, N2 disjoint F(N1 ⊔ N2, α1 ⊔ α2) = F(N1, α1) · F(N2, α2).

• 6. Proof sketch

We first explain how to orient under additional choices. Then I shall comment

• n how to prove the independence of these choices.

6.1. Choices. Fix a flag structure F on X. Let s: X → E be a transverse section of the SU(2)-bundle E. For the sub- manifold Y = s−1(0) we get an isomorphism ds: NY → E, in particular an SU(2)- structure on the normal bundle. Let α be a nowhere vanishing section of NY . Using the SU(2)-structure, α trivializes the normal bundle. Now we have trivialized E outside of Y and identified the compactified tubular neighborhood ν∞ with Y × S4. Under this identification, the bundle E can be reconstructed from Y × Estd where Estd → S4 denotes the standard SU(2)-bundle with c2(Estd) = 1.

ORIENTING THE G2-INSTANTON MODULI SPACE 5

E s X E

s

∼ = C2 E ∼ = Y × Estd Y = s−1(0) ν α Y × Estd Y S4 6.2. The orientation from choices. The twisted Diracians DE and DC2 agree

• utside a tubular neighborhood, provided we use a connection on E that is zero
• utside a neighborhood of Y .

The excision principle relates the ‘difference’ between the determinants of these operators to the ’difference’ between the corresponding determinants of the

• perators on the compactified tubular neighborhood Y × S4 (instead of X) on Estd

and C2. Det DX

E ⊗

• Det DX

C2

∗ ∼ = Det DY ×S4

Estd

⊗

• Det DY ×S4

C2

∗ The determinants DX

C2, DY ×S4 C2

with zero connection are canonically oriented since the operators are self-adjoint and and do not depend on a parameter.

6 MARKUS UPMEIER (JOINT WORK WITH DOMINIC JOYCE)

We may deform the metric on Y ×S4 so that it becomes a product (contractible choice) and there is a unique corresponding deformation of spin structures. Then the twisted Dirac operator splits (we had already split the bundle E into product form). This induces a decomposition of the determinant by the Künneth theorem for elliptic complexes Det DY ×S4

Estd

∼ = (Det DY )ind DS4

Estd ⊗ (Det DS4

Estd)ind DY

C2

The first index is −4, the second index is 0 since Y is odd-dimensional. It follows that Det DY ×S4

E

is canonically trivial. Putting the last two equations together, we get an orientation of Det DX

E .

Remark 6.1. What this means is that if we choose orientations of both of the cohomologies of DY and DS4

Estd we get an orientation of the tensor product elliptic

• complex. In favourable cases (depending on the index) the resulting orientation does

not depend on which choice we made. We modify the so obtained orientation by the flag structure F(Y, α) = ±1. This is needed to show independence of choices, which is the difficult part of the theorem. 6.3. Independence of choices. Two choices α1, α2 differ by a gauge transforma- tion g: Y → SU(2). Theorem 6.2. The action of g ∈ Aut(E) defines a map Det DA → Det Dg∗A which differs from fiber transport by the spectral flow of {At}t∈[0,1] for any choice

• f path from A to g∗A.

According to the Atiyah–Patodi–Singer index theorem, the spectral flow can be computed as the index on Eg → X ×S1. This can be used to show that the choices α1, α2 give orientations that differ by (−1)deg g. The independence of the choice s is more difficult. For it, we introduce a gen- eralization of the spectral flow that is defined for bordisms that are more general than cylinders. It is based on elliptic APS-boundary value problems. Some form of excision is needed.