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

ORIENTING THE G 2 -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 G 2 -manifold and fix an SU(2) -bundle E → X . The moduli stack M of G 2 -instantons on E is orientable. A flag structure on X fixes a canonical orientation. Very roughly, the idea in the Donaldson–Segal programme is that, when a cal- ibrated submanifold Y 3 ⊂ X 7 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 G 2 -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 of a 4 -manifold X , one needs to choose an orientation of H 1 ( X ) ⊕ H + ( X ) . 2. G 2 -manifolds Definition 2.1. 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. It is torsion-free if with ψ = ∗ φ dφ = 0 , dψ = 0 . Example 2.2. 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 46 − dx 57 � dx 47 + dx 56 � + dx 2 � + dx 3 � Definition 2.3. The group G 2 is the stabilizer of φ std , i.e. 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. Remark 2.4. It follows that we may identify each tangent space of a G 2 -manifold with R 7 equipped with φ std . This identification is only well-defined up to A ∈ G 2 . 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 ) 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) . � Example 2.6 (Relation to other geometries) . We have inclusions SU(3) → G 2 → Spin(7) Also SO(4) and SO(3) (see Salamon). 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. 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 . 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 Example 2.7 (Simply connected examples) . 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 (see Joyce). Non-compact examples with holonomy all of G 2 were found first and are due to Bryant. Definition 2.8. Let X be a G 2 -manifold and let E → X be a principal G -bundle. A connection A ∈ Ω 1 ( E ; g ) on E is a G 2 -instanton if � rep-theory � ∗ ( F A ∧ φ ) = − F A ∈ Ω 2 ( X ; g E ) . ⇐ ⇒ F A ∧ ψ = 0 Here we use the adjoint bundle g E = E × G, Ad g . 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 . The solutions a describe the tangent space at A to the space of G 2 -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 G 2 -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 G 2 -representation Λ 2 ( R 7 ) ∗ splits into 14 = { α | α ∧ ψ = 0 } ∼ Λ 2 = g 2 −∧ ψ Λ 2 → Λ 6 7 = { α | ∗ ( α ∧ φ ) = 2 α } − − − so the invariant conditions are F A = 0 , π 7 F A = 0 , and π 14 F A = 0 . In gauge theory we want the infinitesimal deformations to be governed by an elliptic complex beginning with gauge transformations, connections, curvature. The conditions F A = 0 and π 14 F A = 0 violate ellipticity and are thus ruled out.

ORIENTING THE G 2 -INSTANTON MODULI SPACE 3 (1) The Levi-Civita connection on the tangent bunde of a Example 2.10. torsion-free G 2 -manifold. This is because R ∈ Λ 2 T ∗ M ⊗ g 2 . (2) An ASD-connection on a hyperkähler 4 -manifold M (note ω i self-dual) F A ∧ dx 123 � � ∗ ( F A ∧ φ ) = ∗ = ∗ M F A = − F A . (3) A Hermitian Yang–Mills connection Λ F A = 0 , F 0 , 2 = 0 on a Calabi– A Yau 3 -fold. 3. Instanton moduli space We assume X is a closed connected G 2 -manifold. 3.1. Deformation complex. For a G 2 -instanton A we have the deformation complex d A π 7 ◦ d A d A Ω 0 ( X ; g E ) → Ω 1 ( X ; g E ) → Ω 2 7 ( X ; g E ) = Ω 6 ( X ; g E ) → Ω 7 ( X ; g E ) (1) − − − − − − − − 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 � d ∗ � 0 L A = A d A ∗ ( ψ ∧ d A ) 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 Stab A 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 | Stab A = Z ( G ) } . The moduli space of irreducible G 2 -instantons is the topological subspace M ∗ = { [ A ] ∈ B ∗ | F A ∧ ψ = 0 } . 4. Determinant line bundle. Orientations Definition 4.1. For a family of elliptic operators P a , a ∈ A , the determinant line bundle Det P is the union of the fibers Det P a = Λ top (Ker P a ) ∗ ⊗ Λ top Coker P a . 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 out 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 our 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 G 2 -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 ∗ .