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

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 gauge-theoretic moduli spaces http://people.maths.ox.ac.uk/joyce/JTU.pdf Canonical Orientations for G 2 -instantons 1

Outline Introduction G 2 -geometry Orientations in gauge theory Flag structures Main theorem Outline of proof Canonical Orientations for G 2 -instantons 2

Problem (Donaldson–Segal programme) To define counting invariants for moduli of G 2 -instantons: 1. Orientability and canonical orientations 2. Compactifications 3. Deformations Similarly for other exceptional holonomies, e.g. Spin ( 7 ) . Canonical Orientations for G 2 -instantons 3

Problem (Donaldson–Segal programme) To define counting invariants for moduli of G 2 -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 , φ 3 , ψ 4 = ∗ φ φ ) be a closed G 2 -manifold. A flag structure F on X determines, for every principal SU ( n ) -bundle E → X , an orientation of the moduli space M irr E of G 2 -instantons � { A ∈ A irr E | F A ∧ ψ = 0 } Aut ( E ) . Canonical Orientations for G 2 -instantons 3

Theorem (Walpuski 2013) The moduli space of G 2 -instantons M irr E is orientable. Canonical Orientations for G 2 -instantons 4

Theorem (Walpuski 2013) The moduli space of G 2 -instantons M irr E is orientable. Theorem (Donaldson 1987) For ASD-connections on closed oriented Riemannian 4 -manifolds, canonical orientations depend on an orientation of H 1 ( M ) ⊕ H + ( M ) . Canonical Orientations for G 2 -instantons 4

Plan Introduction G 2 -geometry Orientations in gauge theory Flag structures Main theorem Outline of proof Canonical Orientations for G 2 -instantons 5

Definition φ ∈ Λ 3 V ∗ on a 7-dimensional vector space is non-degenerate if ι X φ ∧ ι X φ ∧ φ � = 0 ∀ X ∈ V \ { 0 } . A G 2 -structure on a 7-manifold X is a smooth 3-form φ that is non-degenerate on each tangent space. Canonical Orientations for G 2 -instantons 6

Definition φ ∈ Λ 3 V ∗ on a 7-dimensional vector space is non-degenerate if ι X φ ∧ ι X φ ∧ φ � = 0 ∀ X ∈ V \ { 0 } . A G 2 -structure on a 7-manifold X is a smooth 3-form φ that is non-degenerate on each tangent space. Example On V = R 7 we have φ ( X , Y , Z ) = � X × Y , Z � . In coordinates φ std = dx 123 + dx 1 � dx 45 + dx 67 � + dx 2 � dx 46 − dx 57 � + dx 3 � dx 47 + dx 56 � Then G 2 := { A ∈ GL ( 7 , R ) | A ∗ φ std = φ std } . Canonical Orientations for G 2 -instantons 6

Definition φ ∈ Λ 3 V ∗ on a 7-dimensional vector space is non-degenerate if ι X φ ∧ ι X φ ∧ φ � = 0 ∀ X ∈ V \ { 0 } . A G 2 -structure on a 7-manifold X is a smooth 3-form φ that is non-degenerate on each tangent space. Example On V = R 7 we have φ ( X , Y , Z ) = � X × Y , Z � . In coordinates φ std = dx 123 + dx 1 � dx 45 + dx 67 � + dx 2 � dx 46 − dx 57 � + dx 3 � dx 47 + dx 56 � Then G 2 := { A ∈ GL ( 7 , R ) | A ∗ φ std = φ std } . Since φ std encodes the multiplication table of the octonions we have G 2 ∼ = Aut ( O ) . This is a 14-dimensional simply connected Lie group. Canonical Orientations for G 2 -instantons 6

Lemma There exists a unique metric g and orientation on V such that g ( X , Y ) vol g = ι X φ ∧ ι Y φ ∧ φ. Moreover, G 2 -manifolds have a natural spin structure. Proof. G 2 ⊂ SO ( 7 ) is simply connected, so G 2 ⊂ Spin ( 7 ) . Canonical Orientations for G 2 -instantons 7

Lemma There exists a unique metric g and orientation on V such that g ( X , Y ) vol g = ι X φ ∧ ι Y φ ∧ φ. Moreover, G 2 -manifolds have a natural spin structure. Proof. G 2 ⊂ SO ( 7 ) is simply connected, so G 2 ⊂ Spin ( 7 ) . Definition ⇒ ∇ g φ = 0 A G 2 -structure φ is torsion-free : ⇐ ⇒ d φ = 0 and d ψ = 0 for ψ := ∗ g φ ∈ Ω 4 . ⇐ Canonical Orientations for G 2 -instantons 7

Lemma There exists a unique metric g and orientation on V such that g ( X , Y ) vol g = ι X φ ∧ ι Y φ ∧ φ. Moreover, G 2 -manifolds have a natural spin structure. Proof. G 2 ⊂ SO ( 7 ) is simply connected, so G 2 ⊂ Spin ( 7 ) . Definition ⇒ ∇ g φ = 0 A G 2 -structure φ is torsion-free : ⇐ ⇒ d φ = 0 and d ψ = 0 for ψ := ∗ g φ ∈ Ω 4 . ⇐ Definition A connection A on a principal G -bundle E → X over a G 2 -manifold is a G 2 -instanton : ⇐ ⇒ F A ∧ ψ = 0 ( ⇐ ⇒ ∗ ( F A ∧ φ ) = − F A ). Canonical Orientations for G 2 -instantons 7

Example (Relation to other geometries) We have inclusions SU ( 3 ) → G 2 → Spin ( 7 ) Canonical Orientations for G 2 -instantons 8

Example (Relation to other geometries) We have inclusions SU ( 3 ) → G 2 → Spin ( 7 ) 1. Any spin 7-manifold has a G 2 -structure given by choosing a never vanishing spinor. The G 2 -structure is torsion-free if and only if the spinor is parallel. Canonical Orientations for G 2 -instantons 8

Example (Relation to other geometries) We have inclusions SU ( 3 ) → G 2 → Spin ( 7 ) 1. Any spin 7-manifold has a G 2 -structure given by choosing a never vanishing spinor. The G 2 -structure is torsion-free if and only if the spinor is parallel. 2. If ( Z , ω, Ω) is a Calabi–Yau 3-fold then R × Z or S 1 × Z are torsion-free G 2 -manifolds with φ = dt ∧ ω + ℜ e (Ω) . Canonical Orientations for G 2 -instantons 8

Example (Relation to other geometries) We have inclusions SU ( 3 ) → G 2 → Spin ( 7 ) 1. Any spin 7-manifold has a G 2 -structure given by choosing a never vanishing spinor. The G 2 -structure is torsion-free if and only if the spinor is parallel. 2. If ( Z , ω, Ω) is a Calabi–Yau 3-fold then R × Z or S 1 × Z are torsion-free G 2 -manifolds with φ = dt ∧ ω + ℜ e (Ω) . 3. Given a hyperkähler surface ( S , ω 1 , ω 2 , ω 3 ) we get a torsion-free G 2 -manifold R 3 × S or S 1 × S 1 × S 1 × S with φ = dx 123 − dx 1 ∧ ω 1 − dx 2 ∧ ω 2 − dx 3 ∧ ω 3 . Canonical Orientations for G 2 -instantons 8

For a compact manifold the holonomy is all of G 2 if and only if the fundamental group is finite. Examples are very difficult to find. Canonical Orientations for G 2 -instantons 9

For a compact manifold the holonomy is all of G 2 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 G 2 were found first and are due to Bryant (EDS) and Bryant–Salamon S S 3 , Λ + S 4 , Λ + C P 2 . Canonical Orientations for G 2 -instantons 9

For a compact manifold the holonomy is all of G 2 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 G 2 were found first and are due to Bryant (EDS) and Bryant–Salamon S S 3 , Λ + S 4 , Λ + C P 2 . 2. Compact examples were first found by Joyce. Canonical Orientations for G 2 -instantons 9

Moduli space infinitesimally For a G 2 -instanton A and deformation a ∈ Ω 1 ( X ; g E ) the G 2 -instanton condition becomes 0 = F A + a ∧ ψ Canonical Orientations for G 2 -instantons 10

Moduli space infinitesimally For a G 2 -instanton A and deformation a ∈ Ω 1 ( X ; g E ) the G 2 -instanton condition becomes 0 = F A + a ∧ ψ = F A ∧ ψ + d A a ∧ ψ + a ∧ a ∧ ψ, Canonical Orientations for G 2 -instantons 10

Moduli space infinitesimally For a G 2 -instanton A and deformation a ∈ Ω 1 ( X ; g E ) the G 2 -instanton condition becomes 0 = F A + a ∧ ψ = F A ∧ ψ + d A a ∧ ψ + a ∧ a ∧ ψ, so the linearized G 2 -instanton equation is d A a ∧ ψ = 0 . (spans tangent space at A to M irr E .) Canonical Orientations for G 2 -instantons 10

Moduli space infinitesimally For a G 2 -instanton A and deformation a ∈ Ω 1 ( X ; g E ) the G 2 -instanton condition becomes 0 = F A + a ∧ ψ = F A ∧ ψ + d A a ∧ ψ + a ∧ a ∧ ψ, so the linearized G 2 -instanton equation is d A a ∧ ψ = 0 . (spans tangent space at A to M irr E .) The deformation complex d A d A ∧ ψ d A Ω 0 ( X ; g E ) → Ω 1 ( X ; g E ) → Ω 6 ( X ; g E ) → Ω 7 ( X ; g E ) − − − − − (1) has been made elliptic by adding the right-most term. Canonical Orientations for G 2 -instantons 10

Simplification of problem More generally, for any connection A , we may roll up the complex and define a self-adjoint elliptic operator � 0 � d ∗ : Ω 0 ⊕ Ω 1 → Ω 0 ⊕ Ω 1 . A L A = d A ∗ ( ψ ∧ d A ) Canonical Orientations for G 2 -instantons 11

Simplification of problem More generally, for any connection A , we may roll up the complex and define a self-adjoint elliptic operator � 0 � d ∗ : Ω 0 ⊕ Ω 1 → Ω 0 ⊕ Ω 1 . A L A = d A ∗ ( ψ ∧ d A ) Hence the line bundle on M irr E we want to orient extends to A irr E / Aut ( E ) as the determinant line bundle Det { L A } A ∈A E . Canonical Orientations for G 2 -instantons 11