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Canonical Orientations for the Moduli Space of G 2 -instantons - - PowerPoint PPT Presentation

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Canonical Orientations for the Moduli Space of G 2 -instantons Markus Upmeier (joint with Dominic Joyce) Talk based on: Joyce, Upmeier - Orientations for moduli spaces of G 2 -instantons . ( Soon ) Joyce, Tanaka, Upmeier - On orientations for

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Canonical Orientations for the Moduli Space of G2-instantons

Markus Upmeier (joint with Dominic Joyce)

Talk based on: Joyce, Upmeier - Orientations for moduli spaces of G2-instantons. (Soon) Joyce, Tanaka, Upmeier - On orientations for gauge-theoretic moduli spaces http://people.maths.ox.ac.uk/joyce/JTU.pdf

Canonical Orientations for G2-instantons 1

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Outline Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 2

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Problem (Donaldson–Segal programme)

To define counting invariants for moduli of G2-instantons:

  • 1. Orientability and canonical orientations
  • 2. Compactifications
  • 3. Deformations

Similarly for other exceptional holonomies, e.g. Spin(7).

Canonical Orientations for G2-instantons 3

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Problem (Donaldson–Segal programme)

To define counting invariants for moduli of G2-instantons:

  • 1. Orientability and canonical orientations
  • 2. Compactifications
  • 3. Deformations

Similarly for other exceptional holonomies, e.g. Spin(7).

Theorem (Joyce–U. 2018)

Let (X, φ3, ψ4 = ∗φφ) be a closed G2-manifold. A flag structure F

  • n X determines, for every principal SU(n)-bundle E → X, an
  • rientation of the moduli space Mirr

E of G2-instantons

{A ∈ Airr

E | FA ∧ ψ = 0}

  • Aut(E).

Canonical Orientations for G2-instantons 3

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Theorem (Walpuski 2013)

The moduli space of G2-instantons Mirr

E is orientable.

Canonical Orientations for G2-instantons 4

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Theorem (Walpuski 2013)

The moduli space of G2-instantons Mirr

E is orientable.

Theorem (Donaldson 1987)

For ASD-connections on closed oriented Riemannian 4-manifolds, canonical orientations depend on an orientation of H1(M) ⊕ H+(M).

Canonical Orientations for G2-instantons 4

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Plan Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 5

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Definition

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 on each tangent space.

Canonical Orientations for G2-instantons 6

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Definition

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 on each tangent space.

Example

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 Then G2 := {A ∈ GL(7, R) | A∗φstd = φstd}.

Canonical Orientations for G2-instantons 6

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Definition

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 on each tangent space.

Example

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 Then 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.

Canonical Orientations for G2-instantons 6

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Lemma

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).

Canonical Orientations for G2-instantons 7

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Lemma

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).

Definition

A G2-structure φ is torsion-free : ⇐ ⇒ ∇gφ = 0 ⇐ ⇒ dφ = 0 and dψ = 0 for ψ := ∗gφ ∈ Ω4.

Canonical Orientations for G2-instantons 7

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Lemma

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).

Definition

A G2-structure φ is torsion-free : ⇐ ⇒ ∇gφ = 0 ⇐ ⇒ dφ = 0 and dψ = 0 for ψ := ∗gφ ∈ Ω4.

Definition

A connection A on a principal G-bundle E → X over a G2-manifold is a G2-instanton : ⇐ ⇒ FA ∧ ψ = 0 ( ⇐ ⇒ ∗(FA ∧ φ) = −FA).

Canonical Orientations for G2-instantons 7

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Example (Relation to other geometries)

We have inclusions SU(3) → G2 → Spin(7)

Canonical Orientations for G2-instantons 8

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Example (Relation to other geometries)

We have inclusions SU(3) → G2 → Spin(7)

  • 1. 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.

Canonical Orientations for G2-instantons 8

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Example (Relation to other geometries)

We have inclusions SU(3) → G2 → Spin(7)

  • 1. 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.
  • 2. 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 + ℜe(Ω).

Canonical Orientations for G2-instantons 8

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Example (Relation to other geometries)

We have inclusions SU(3) → G2 → Spin(7)

  • 1. 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.
  • 2. 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 + ℜe(Ω).

  • 3. 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.

Canonical Orientations for G2-instantons 8

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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.

Canonical Orientations for G2-instantons 9

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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.

Example (Simply connected examples)

  • 1. Non-compact examples with holonomy all of G2 were found

first and are due to Bryant (EDS) and Bryant–Salamon SS3, Λ+S4, Λ+CP2.

Canonical Orientations for G2-instantons 9

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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.

Example (Simply connected examples)

  • 1. Non-compact examples with holonomy all of G2 were found

first and are due to Bryant (EDS) and Bryant–Salamon SS3, Λ+S4, Λ+CP2.

  • 2. Compact examples were first found by Joyce.

Canonical Orientations for G2-instantons 9

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Moduli space infinitesimally For a G2-instanton A and deformation a ∈ Ω1(X; gE) the G2-instanton condition becomes 0 = FA+a ∧ ψ

Canonical Orientations for G2-instantons 10

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Moduli space infinitesimally 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,

Canonical Orientations for G2-instantons 10

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Moduli space infinitesimally 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. (spans tangent space at A to Mirr

E .)

Canonical Orientations for G2-instantons 10

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Moduli space infinitesimally 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. (spans tangent space at A to Mirr

E .) The deformation complex

Ω0(X; gE)

dA

− → Ω1(X; gE)

dA∧ψ

− − − → Ω6(X; gE)

dA

− → Ω7(X; gE) (1) has been made elliptic by adding the right-most term.

Canonical Orientations for G2-instantons 10

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Simplification of problem More generally, for any connection A, we may roll up the complex and define a self-adjoint elliptic operator LA = d∗

A

dA ∗(ψ ∧ dA)

  • : Ω0 ⊕ Ω1 → Ω0 ⊕ Ω1.

Canonical Orientations for G2-instantons 11

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Simplification of problem More generally, for any connection A, we may roll up the complex and define a self-adjoint elliptic operator LA = d∗

A

dA ∗(ψ ∧ dA)

  • : Ω0 ⊕ Ω1 → Ω0 ⊕ Ω1.

Hence the line bundle on Mirr

E we want to orient extends to

Airr

E / Aut(E) as the determinant line bundle Det{LA}A∈AE .

Canonical Orientations for G2-instantons 11

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Simplification of problem More generally, for any connection A, we may roll up the complex and define a self-adjoint elliptic operator LA = d∗

A

dA ∗(ψ ∧ dA)

  • : Ω0 ⊕ Ω1 → Ω0 ⊕ Ω1.

Hence the line bundle on Mirr

E we want to orient extends to

Airr

E / Aut(E) as the determinant line bundle Det{LA}A∈AE .

The principal symbols of LA and of the twisted Diracian / DA agree = ⇒ their orientation problems agree.

Canonical Orientations for G2-instantons 11

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Plan Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 12

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Determinant line bundles

Definition

Let {Pt}t∈T be a T-family of elliptic operators. The Quillen determinant line bundle is Det{Pt} :=

  • t∈T

Λtop(Ker Pt)∗ ⊗ Λtop(Coker Pt) ց T.

Canonical Orientations for G2-instantons 13

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Determinant line bundles

Definition

Let {Pt}t∈T be a T-family of elliptic operators. The Quillen determinant line bundle is Det{Pt} :=

  • t∈T

Λtop(Ker Pt)∗ ⊗ Λtop(Coker Pt) ց T. Its double cover of orientations is Or{Pt}t∈T := (Det{Pt} \ {zero section})

  • R>0

ց T.

Canonical Orientations for G2-instantons 13

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Determinant line bundles

Definition

Let {Pt}t∈T be a T-family of elliptic operators. The Quillen determinant line bundle is Det{Pt} :=

  • t∈T

Λtop(Ker Pt)∗ ⊗ Λtop(Coker Pt) ց T. Its double cover of orientations is Or{Pt}t∈T := (Det{Pt} \ {zero section})

  • R>0

ց T. Up to canonical isomorphism, depends only on principal symbols: Or{pt} → Or{Pt}. (limit exists)

Canonical Orientations for G2-instantons 13

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Determinant line bundles

Definition

Let {Pt}t∈T be a T-family of elliptic operators. The Quillen determinant line bundle is Det{Pt} :=

  • t∈T

Λtop(Ker Pt)∗ ⊗ Λtop(Coker Pt) ց T. Its double cover of orientations is Or{Pt}t∈T := (Det{Pt} \ {zero section})

  • R>0

ց T. Up to canonical isomorphism, depends only on principal symbols: Or{pt} → Or{Pt}. (limit exists) Categorifies w1 (ind{Pt}t∈T ∈ KO(T)) ∈ H1(T; Z2).

Canonical Orientations for G2-instantons 13

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Restriction to Diracians

Example

Since the principal symbols icξ ⊗ idgE of LA and / DA agree, so do the orientation problems.

Canonical Orientations for G2-instantons 14

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Restriction to Diracians

Example

Since the principal symbols icξ ⊗ idgE of LA and / DA agree, so do the orientation problems.

Definition

Let X be an odd-dimensional closed spin manifold. The orientation torsor of an SU(n)-bundle E → X is OrE := Or

  • /

S ⊗ gE

cξ⊗1

− − − → / S ⊗ gE

  • ∼

=Or / DgE

⊗Or

  • /

S ⊗ su(n)

cξ⊗1

− − − → / S ⊗ su(n) ∗

Canonical Orientations for G2-instantons 14

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Restriction to Diracians

Example

Since the principal symbols icξ ⊗ idgE of LA and / DA agree, so do the orientation problems.

Definition

Let X be an odd-dimensional closed spin manifold. The orientation torsor of an SU(n)-bundle E → X is OrE := Or

  • /

S ⊗ gE

cξ⊗1

− − − → / S ⊗ gE

  • ∼

=Or / DgE

⊗Or

  • /

S ⊗ su(n)

cξ⊗1

− − − → / S ⊗ su(n) ∗

Proposition

OrE⊕F ∼ = OrE ⊗Z2 OrF canonically.

Canonical Orientations for G2-instantons 14

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Special case of excision

Theorem (Excision)

Let E ց X, E ′ ց X ′ be SU(n)-bundles over closed spin manifolds.

  • 1. Let φ be a spin diffeomorphism of open subsets

X ⊃ U

φ

− → U′ ⊂ X ′.

  • 2. Let s and s′ be SU(n)-frames of E|X\K and E ′|X ′\K ′ defined
  • utside compact subsets K ⊂ U and K ′ ⊂ U′.
  • 3. Let Φ: E|U → φ∗E ′|U′ be an SU(n)-isomorphism with

Φ(s) = φ∗s′. Then we get an excision isomorphism OrEցX

O(Φ,s,s′)

− − − − − − → OrE ′ցX ′ .

Canonical Orientations for G2-instantons 15

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E X K ⊂ U f r a m i n g E ′ X ′ K ′ ⊂ U′ framing framed isomorphism φ = ⇒ OrEցX → OrE ′ցX ′ .

Canonical Orientations for G2-instantons 16

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Families index for real self-adjoint operators For families {Pt}t∈S1 of real self-adjoint operators (Ati-Pat-Si) w1

  • ind Pt ∈ KO0(S1)
  • ∈ Z2

=index of a single operator ∂ ∂t + Pt on the space X × S1 which is computable from local data (complexify).

Canonical Orientations for G2-instantons 17

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Families index for real self-adjoint operators For families {Pt}t∈S1 of real self-adjoint operators (Ati-Pat-Si) w1

  • ind Pt ∈ KO0(S1)
  • ∈ Z2

=index of a single operator ∂ ∂t + Pt on the space X × S1 which is computable from local data (complexify).

Theorem

Let X be a closed odd-dim. spin manifold, Φ: E → E an SU(n)-isomorphism over a spin diffeomorphism φ: X → X.

Canonical Orientations for G2-instantons 17

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Families index for real self-adjoint operators For families {Pt}t∈S1 of real self-adjoint operators (Ati-Pat-Si) w1

  • ind Pt ∈ KO0(S1)
  • ∈ Z2

=index of a single operator ∂ ∂t + Pt on the space X × S1 which is computable from local data (complexify).

Theorem

Let X be a closed odd-dim. spin manifold, Φ: E → E an SU(n)-isomorphism over a spin diffeomorphism φ: X → X.Then Or(Φ) = (−1)δ(Φ)·idOrE , δ(Φ) :=

  • Xφ

ˆ A(TXφ)

  • ch(E ∗

Φ ⊗ EΦ) − n2

, where EΦ = E ×Z R ց Xφ = X ×Z R are the mapping tori.

Canonical Orientations for G2-instantons 17

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Simplification of formula in 7D In dimension 7 for Or(Φ) = (−1)δ(Φ) : OrE → OrE we have δ(Φ) ≡ 1 2

  • Xφ

p1(TXφ)c2(EΦ)

Canonical Orientations for G2-instantons 18

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Simplification of formula in 7D In dimension 7 for Or(Φ) = (−1)δ(Φ) : OrE → OrE we have δ(Φ) ≡ 1 2

  • Xφ

p1(TXφ)c2(EΦ)

Proof.

Add to δ(Φ) = ind( / DgE ) − ind( / Dsu(n)) a suitable even multiple of the integer ind( / DE) to simplify the integral.

Canonical Orientations for G2-instantons 18

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Simplification of formula in 7D In dimension 7 for Or(Φ) = (−1)δ(Φ) : OrE → OrE we have δ(Φ) ≡ 1 2

  • Xφ

p1(TXφ)c2(EΦ)

Proof.

Add to δ(Φ) = ind( / DgE ) − ind( / Dsu(n)) a suitable even multiple of the integer ind( / DE) to simplify the integral.

Remark

◮ In particular Or(Φ) = id for every gauge transformation

φ = idX (Walpuski).

Canonical Orientations for G2-instantons 18

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Simplification of formula in 7D In dimension 7 for Or(Φ) = (−1)δ(Φ) : OrE → OrE we have δ(Φ) ≡ 1 2

  • Xφ

p1(TXφ)c2(EΦ) ≡

  • Xφ

c2(EΦ) ∪ c2(EΦ) mod 2.

Proof.

Add to δ(Φ) = ind( / DgE ) − ind( / Dsu(n)) a suitable even multiple of the integer ind( / DE) to simplify the integral.

Remark

◮ In particular Or(Φ) = id for every gauge transformation

φ = idX (Walpuski).

◮ The second formula is a self-intersection in X 8 φ of a homology

class Poincaré dual to c2(EΦ).

Canonical Orientations for G2-instantons 18

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Plan Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 19

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Flag structures

Definition

A flag structure on X 7 associates signs F(Y , s) to submanifolds Y 3 ⊂ X with non-vanishing normal sections s such that F(Y0, s0) = (−1)D(s0,s1)F(Y1, s1) ∀[Y0] = [Y1].

Canonical Orientations for G2-instantons 20

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Flag structures

Definition

A flag structure on X 7 associates signs F(Y , s) to submanifolds Y 3 ⊂ X with non-vanishing normal sections s such that F(Y0, s0) = (−1)D(s0,s1)F(Y1, s1) ∀[Y0] = [Y1]. For S : Z → NZ a transverse extension of s0, s1 over a bordism Z ⊂ X × [0, 1] from Y0 to Y1 we have D(s0, s1) ≡ #zeros of S mod 2.

Canonical Orientations for G2-instantons 20

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Flag structures

Definition

A flag structure on X 7 associates signs F(Y , s) to submanifolds Y 3 ⊂ X with non-vanishing normal sections s such that F(Y0, s0) = (−1)D(s0,s1)F(Y1, s1) ∀[Y0] = [Y1]. For S : Z → NZ a transverse extension of s0, s1 over a bordism Z ⊂ X × [0, 1] from Y0 to Y1 we have D(s0, s1) ≡ #zeros of S mod 2. This definition can be extended to chains Z in X as an intersection number D(s0, s1) = Z • ([s0] − [s1]).

Canonical Orientations for G2-instantons 20

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Flag structures

Definition

A flag structure on X 7 associates signs F(Y , s) to submanifolds Y 3 ⊂ X with non-vanishing normal sections s such that F(Y0, s0) = (−1)D(s0,s1)F(Y1, s1) ∀[Y0] = [Y1]. For S : Z → NZ a transverse extension of s0, s1 over a bordism Z ⊂ X × [0, 1] from Y0 to Y1 we have D(s0, s1) ≡ #zeros of S mod 2. This definition can be extended to chains Z in X as an intersection number D(s0, s1) = Z • ([s0] − [s1]). A flag structure F is a notion of parity for (Y , s). When NY has an SU(2)-structure, a flag structure reduces choices by helping to pick

  • ut a normal SU(2)-framing.

Canonical Orientations for G2-instantons 20

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Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. D(s0, s1) =

Canonical Orientations for G2-instantons 21

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Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. On Z = Y × [0, 1] set S(y, t) := (1 − t)s0(y) + ts1(y). D(s0, s1) = s0 s1

Canonical Orientations for G2-instantons 21

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Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. On Z = Y × [0, 1] set S(y, t) := (1 − t)s0(y) + ts1(y). S(y, t) = [(1 − t) + qt] · s0(y) = 0 ⇐ ⇒ t = 1 2 and q(y) = −1. = ⇒ D(s0, s1) = degree(q : Y → S3). s0 s1 ×

Canonical Orientations for G2-instantons 21

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Proposition

Flag structures are a (non-empty) torsor over H3(X; Z2).

Observation.

Ratio F(Y , s) : F′(Y , s) is independent of s (axiom for F).

Canonical Orientations for G2-instantons 22

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Proposition

Flag structures are a (non-empty) torsor over H3(X; Z2).

Observation.

Ratio F(Y , s) : F′(Y , s) is independent of s (axiom for F).

Definition

Let φ: X ′ → X be a diffeomorphism. The pullback of a flag structure F on X is (φ∗F)(Y ′, s′) := (φ(Y ′), dφ(s′)).

Canonical Orientations for G2-instantons 22

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Proposition

Flag structures are a (non-empty) torsor over H3(X; Z2).

Observation.

Ratio F(Y , s) : F′(Y , s) is independent of s (axiom for F).

Definition

Let φ: X ′ → X be a diffeomorphism. The pullback of a flag structure F on X is (φ∗F)(Y ′, s′) := (φ(Y ′), dφ(s′)).

Example

Let φ: X → X be an orientation-preserving isometry with φ|Y = idY . Then (F/φ∗F)[Y ] = F(Y , s) : F(Y , φ∗s) equals the self-intersection of Y × S1 in the mapping torus Xφ = X ×Z R.

Canonical Orientations for G2-instantons 22

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Flag structures

Corollary

Every manifold with H3(X; Z2) = {0} has a unique flag structure. A set of submanifold generators [Yi] ∈ H3(X; Z2) with preferred normal sections si determines a unique flag structure with F(Yi, si) := 1.

Example

X = S7 has a unique flag structure. Y 3 × S4 has a preferred one.

Canonical Orientations for G2-instantons 23

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Plan Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 24

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Theorem (Main theorem)

A flag structure F on a closed spin 7-manifold X induces uniquely, for every SU(n)-bundle E ց X, a canonical orientation

  • F(E) ∈ OrE

with the following properties:

Canonical Orientations for G2-instantons 25

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Theorem (Main theorem)

A flag structure F on a closed spin 7-manifold X induces uniquely, for every SU(n)-bundle E ց X, a canonical orientation

  • F(E) ∈ OrE

with the following properties:

  • 1. (Normalization) For E = Ck trivial, evaluation defines

OrE = Z2. Let otriv(E) ∈ OrE be the image of 1 ∈ Z2. Then

  • F(E) = otriv(E).

Canonical Orientations for G2-instantons 25

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Theorem (Main theorem)

A flag structure F on a closed spin 7-manifold X induces uniquely, for every SU(n)-bundle E ց X, a canonical orientation

  • F(E) ∈ OrE

with the following properties:

  • 1. (Normalization) For E = Ck trivial, evaluation defines

OrE = Z2. Let otriv(E) ∈ OrE be the image of 1 ∈ Z2. Then

  • F(E) = otriv(E).
  • 2. (Stabilization) Under OrE⊕Ck = OrE ⊗Z2 OrCk = OrE we have
  • F(E ⊕ Ck) = oF(E)

Canonical Orientations for G2-instantons 25

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SLIDE 61

Theorem (Main theorem)

A flag structure F on a closed spin 7-manifold X induces uniquely, for every SU(n)-bundle E ց X, a canonical orientation

  • F(E) ∈ OrE

with the following properties:

  • 1. (Normalization) For E = Ck trivial, evaluation defines

OrE = Z2. Let otriv(E) ∈ OrE be the image of 1 ∈ Z2. Then

  • F(E) = otriv(E).
  • 2. (Stabilization) Under OrE⊕Ck = OrE ⊗Z2 OrCk = OrE we have
  • F(E ⊕ Ck) = oF(E)

To be continued ...

Canonical Orientations for G2-instantons 25

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SLIDE 62

Main theorem (continued)

  • 3. (Excision) Let E, E ′ be SU(n)-bundles over closed spin

7-manifolds X, X ′ with flag structures F, F′.

Canonical Orientations for G2-instantons 26

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SLIDE 63

Main theorem (continued)

  • 3. (Excision) Let E, E ′ be SU(n)-bundles over closed spin

7-manifolds X, X ′ with flag structures F, F′. Let s, s′ be framings of E|X\K, E ′

X ′\K ′ outside compact subsets K, K ′.

Canonical Orientations for G2-instantons 26

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SLIDE 64

Main theorem (continued)

  • 3. (Excision) Let E, E ′ be SU(n)-bundles over closed spin

7-manifolds X, X ′ with flag structures F, F′. Let s, s′ be framings of E|X\K, E ′

X ′\K ′ outside compact subsets K, K ′. Let

Φ: E|U → E ′|U′ be an SU(n)-isomorphism covering a spin diffeomorphism φ: U → U′ mapping s to s′.

Canonical Orientations for G2-instantons 26

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SLIDE 65

Main theorem (continued)

  • 3. (Excision) Let E, E ′ be SU(n)-bundles over closed spin

7-manifolds X, X ′ with flag structures F, F′. Let s, s′ be framings of E|X\K, E ′

X ′\K ′ outside compact subsets K, K ′. Let

Φ: E|U → E ′|U′ be an SU(n)-isomorphism covering a spin diffeomorphism φ: U → U′ mapping s to s′. Under the excision isomorphism: Or(Φ, s, s′)

  • F(E ց X)
  • = (F/φ∗F′)[Y ] · oF′(E ′ ց X ′),

where [Y ] ∈ H3(U; Z) is the homology class Poincaré dual to the relative Chern class c2(E|U, s) ∈ H4

cpt(U; Z).

Canonical Orientations for G2-instantons 26

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SLIDE 66

E X K ⊂ U f r a m i n g E ′ X ′ K ′ ⊂ U′ framing Y ′ framed isomorphism φ Y F/φ∗F′[Y ]

Canonical Orientations for G2-instantons 27

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SLIDE 67

Plan Introduction G2-geometry Orientations in gauge theory Flag structures Main theorem Outline of proof

Canonical Orientations for G2-instantons 28

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SLIDE 68

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,

Canonical Orientations for G2-instantons 29

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SLIDE 69

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,

Canonical Orientations for G2-instantons 29

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SLIDE 70

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,

Canonical Orientations for G2-instantons 29

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SLIDE 71

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,

Canonical Orientations for G2-instantons 29

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SLIDE 72

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,
  • 5. an SU(n)-frame s of E outside a compact subset of U,

Canonical Orientations for G2-instantons 29

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SLIDE 73

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,
  • 5. an SU(n)-frame s of E outside a compact subset of U,

Canonical Orientations for G2-instantons 29

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SLIDE 74

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,
  • 5. an SU(n)-frame s of E outside a compact subset of U,

we let E ′ := φ∗E|U ∪φ∗s Cn over X ′. Excision: Or(can, φ∗s, s): OrE ′ցX ′ → OrEցX .

Canonical Orientations for G2-instantons 29

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SLIDE 75

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,
  • 5. an SU(n)-frame s of E outside a compact subset of U,

we let E ′ := φ∗E|U ∪φ∗s Cn over X ′. Excision: Or(can, φ∗s, s): OrE ′ցX ′ → OrEցX .

Canonical Orientations for G2-instantons 29

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SLIDE 76

Basic step Given

  • 1. E ց X an SU(n)-bundle over a closed spin manifold,
  • 2. X ′ a closed spin manifold,
  • 3. U ⊂ X and U′ ⊂ X ′ open,
  • 4. φ: U′ → U spin diffeomorphism,
  • 5. an SU(n)-frame s of E outside a compact subset of U,

we let E ′ := φ∗E|U ∪φ∗s Cn over X ′. Excision: Or(can, φ∗s, s): OrE ′ցX ′ → OrEցX .

Example

Let U, U′ be tubular neighborhoods of spin submanifolds Y , Y ′. Let Φ: NY ′ → NY be a spin isomorphism covering a spin diffeomorphism φ◦ : Y ′ → Y . This determines φ = φ(φ◦, Φ).

Canonical Orientations for G2-instantons 29

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SLIDE 77

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.

Canonical Orientations for G2-instantons 30

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SLIDE 78

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).

Canonical Orientations for G2-instantons 30

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SLIDE 79

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

Canonical Orientations for G2-instantons 30

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SLIDE 80

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

  • 4. Embed i : Y ֒

→ S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′.

Canonical Orientations for G2-instantons 30

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SLIDE 81

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

  • 4. Embed i : Y ֒

→ S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′.

  • 5. Pick a spin isomorphism Φ: NY ′ → NY .

Canonical Orientations for G2-instantons 30

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SLIDE 82

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

  • 4. Embed i : Y ֒

→ S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′.

  • 5. Pick a spin isomorphism Φ: NY ′ → NY .
  • 6. Let F7 be the unique flag structure on S7. Set

E ′ := φ∗E|U ∪φ∗s Cn ց S7.

Canonical Orientations for G2-instantons 30

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SLIDE 83

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

  • 4. Embed i : Y ֒

→ S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′.

  • 5. Pick a spin isomorphism Φ: NY ′ → NY .
  • 6. Let F7 be the unique flag structure on S7. Set

E ′ := φ∗E|U ∪φ∗s Cn ց S7.

Canonical Orientations for G2-instantons 30

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SLIDE 84

Proof of uniqueness

  • 1. Let E ց X 7 be an SU(2)-bundle.
  • 2. Pick a transverse section s with zero set Y 3 = s−1(0).
  • 3. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

  • 4. Embed i : Y ֒

→ S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′.

  • 5. Pick a spin isomorphism Φ: NY ′ → NY .
  • 6. Let F7 be the unique flag structure on S7. Set

E ′ := φ∗E|U ∪φ∗s Cn ց S7. By the excision axiom OrE ′ցS7 OrEցX

  • F7(E ′)

(−1)(F7/φ∗F)[Y ] · oF(E) Or(can, φ∗s, s) ∈ ∈

Canonical Orientations for G2-instantons 30

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SLIDE 85

Since π6 (SU(4)) = {1} the bundle E ′ ⊕ C2 is trivializable on S7.

Canonical Orientations for G2-instantons 31

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SLIDE 86

Since π6 (SU(4)) = {1} the bundle E ′ ⊕ C2 is trivializable on S7. By the stabilization and normalization axioms OrE ′⊕C2ցS7 OrE ′ցS7

  • triv = oF7(E ′ ⊕ C2)
  • F7(E ′)

stab ∈ ∈

Canonical Orientations for G2-instantons 31

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SLIDE 87

Since π6 (SU(4)) = {1} the bundle E ′ ⊕ C2 is trivializable on S7. By the stabilization and normalization axioms OrE ′⊕C2ցS7 OrE ′ցS7

  • triv = oF7(E ′ ⊕ C2)
  • F7(E ′)

stab ∈ ∈ Therefore

  • F(E) = Or(can, φ∗s, s) ◦ stab((−1)(F7/φ∗F)[Y ] · otriv)

(∗) is uniquely determined by the axioms.

Canonical Orientations for G2-instantons 31

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SLIDE 88

Proof of existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

Canonical Orientations for G2-instantons 32

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SLIDE 89

Proof of existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

◮ The dependence on φ can be reduced to a model calculation

for the automorphism of Estd(NY ) ց S(NY ⊕ R) induced by a spin automorphism ψ: NY → NY . Since this is global, we can calculate Or(ψ) = (F/ψ∗F)[Y ]. This effect is balanced precisely by the flag structure.

  • F(E) = Or(can, φ∗s, s) ◦ stab((−1)(F7/φ∗F)[Y ] · otriv)

Canonical Orientations for G2-instantons 32

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SLIDE 90

Proof of existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

◮ The dependence on φ can be reduced to a model calculation

for the automorphism of Estd(NY ) ց S(NY ⊕ R) induced by a spin automorphism ψ: NY → NY . Since this is global, we can calculate Or(ψ) = (F/ψ∗F)[Y ]. This effect is balanced precisely by the flag structure.

  • F(E) = Or(can, φ∗s, s) ◦ stab((−1)(F7/φ∗F)[Y ] · otriv)

◮ Let s0, s1 be transverse sections of E ց X. Since the excision

isomorphisms Or(can, φ∗

0s0, s0), Or(can, φ∗ 1s1, s1) can be

deformed into each other, they are equal by discreteness. The deformation is by excision isomorphisms as in the basic step, but not coming from a submanifold (example).

Canonical Orientations for G2-instantons 32

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SLIDE 91

s−1

1 (0)

X s−1

0 (0)

t

Canonical Orientations for G2-instantons 33

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SLIDE 92

Generalization to SU(n)-bundles

◮ To reconstruct E pick a generic s : Cn−1 → E. ◮ Let Y ⊂ X be the points where dim Ker s(y) = 1. ◮ The kernel defines a map φ: Y → CPn−2. ◮ We get a U(2)-structure ξ on π: NY → Y .

The reconstruction principle gives an isomorphism E|near Y ∼ =

  • Estd(NY ) ⊕ π∗φ∗TCPn−2

⊗ π∗φ∗O(−1)|on NY under which s corresponds to a framing outside Y defined by ξ. Now proceed similarly as before using Spinc in place of Spin.

Canonical Orientations for G2-instantons 34