ORIENTATIONS FOR MODULI SPACES IN SEVEN-DIMENSIONAL GAUGE THEORY - PDF document

ORIENTATIONS FOR MODULI SPACES IN SEVEN-DIMENSIONAL GAUGE THEORY MARKUS UPMEIER (JOINT WITH DOMINIC JOYCE) Contents 1. Plan of the talk 1 2. Preliminary statement of results. Donaldson–Segal programme 1 3. Orientation problem 1 4. Generalities on orientations 2 5. Flag structures 3 6. Main theorem 3 7. Canonical orientations 4 7.1. Basic comparison 4 7.2. Standard model 4 7.3. Proof of main theorem 4 1. Plan of the talk (1) Statement of G 2 -corollary. (2) Reduction of problem to twisted Diracians (as in introduction of paper). (3) Flag structures. (4) Statement of main theorem. (5) Orientation torsors and excision. (6) Definition of orientations by applying the main principle to the special case U = N Y . (7) Independence of choice of spin isomorphism and of transverse section s . 2. Preliminary statement of results. 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 2.1 (Joyce–U. 2018) . Let ( X, φ 3 , ψ 4 = ∗ φ φ ) be a closed G 2 -manifold. A flag structure on X determines, for every principal SU( n ) -bundle E → X , an orientation of the moduli space of G 2 -instantons M irr E = { A ∈ A irr E | ∗ ( F A ∧ φ ) = − F A ⇐ ⇒ F A ∧ ψ = 0 } / Aut( E ) . Theorem 2.2 (Walpuski 2013) . The moduli space of G 2 -instantons is orientable. To illustrate the dependence of orientations on additional structure of X , recall the following: Theorem 2.3 (Donaldson 1987) . Let E → M be an SU(2) -bundle over an oriented closed Riemannian 4 -manifold. An orientation of H 1 ( M ) ⊕ H + ( M ) determines an orientation of the ASD-moduli space of connections on E . 1

2 MARKUS UPMEIER (JOINT WITH DOMINIC JOYCE) 3. Orientation problem 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 span the tangent space at A to the space of G 2 -instantons. For a G 2 -instanton A we have 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) − − − − − − − − made elliptic by adding the right-most term. 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 . S X = R ⊕ T ∗ X for G 2 -manifolds, the principal symbols of L A and of the Using / twisted Diracian / D A agree. We shall see that this implies that their orientation problems agree (as families over all connections). 4. Generalities on orientations Definition 4.1. Let { P t } t ∈ T be a T -family of elliptic operators. The Quillen determinant line bundle is � Λ top (Ker P t ) ∗ ⊗ Λ top Coker P t ց T. Det { P t } := t ∈ T Let Or { P t } t ∈ T ց T be its double cover of orientations . Up to canonical iso- morphism, this only depends on principal symbols ( limit exists ) : Or { p t } → Or { P t } . These covers categorify w 1 (ind { P t } t ∈ T ∈ KO ( T )) ∈ H 1 ( T ; Z 2 ) . Example 4.2. Since the principal symbols ic ξ ⊗ id g E of L A and / D A agree, the orientation problems are the same. Definition 4.3. Let X be an odd-dimensional closed spin manifold. The orienta- tion cover of an SU( n ) -bundle E → X is � � � � ∗ c ξ ⊗ 1 c ξ ⊗ 1 / → / / → / Or E := Or S ⊗ g E − − − S ⊗ g E ⊗ Z 2 Or S ⊗ su ( n ) − − − S ⊗ su ( n ) � �� � � �� � ∼ ∼ =Or / = Z 2 D g E Proposition 4.4. Or E ⊕ F ∼ = Or E ⊗ Z 2 Or F canonically. Theorem 4.5 (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 → U ′ ⊂ X ′ . φ X ⊃ U − (2) Let s and s ′ be SU( n ) -frames of E | X \ K and E ′ | X ′ \ K ′ defined outside com- pact subsets K ⊂ U and K ′ ⊂ U ′ . (3) Let Φ: E | U → φ ∗ E ′ | U ′ be an SU( n ) -isomorphism with Φ( s ) = φ ∗ s ′ .

ORIENTATIONS FOR MODULI SPACES IN SEVEN-DIMENSIONAL GAUGE THEORY 3 Then we get an isomorphism O (Φ ,s,s ′ ) Or E ց X − − − − − − → Or E ′ ց X ′ . The families index for real operators is rather intractable. An important excep- tion are S 1 -families { P t } t ∈ S 1 of self-adjoint operators (Atiyah–Patodi–Singer). In this case, the holonomy of the determinant line bundle � � ind P t ∈ KO 0 ( S 1 ) w 1 ∈ Z 2 ∂t + P t on the space X × S 1 , which is com- ∂ equals the index of a single operator putable from local data (may complexify). Theorem 4.6. Let X be a closed odd-dimensional spin manifold. Let Φ: E → E be an SU( n ) -isomorphism covering some global spin diffeomorphism φ : X → X . Then � O (Φ) = ( − 1) δ (Φ) · id O E , � Φ ⊗ E Φ ) − rk( E Φ ) 2 � ˆ ch( E ∗ δ (Φ) := A ( TX φ ) . X φ where E Φ = E × Z R ց X φ = X × Z R are the mapping tori. In dimension 7 : � � δ (Φ) ≡ 1 p 1 ( TX φ ) c 2 ( E Φ ) ≡ c 2 ( E Φ ) ∪ c 2 ( E Φ ) mod 2 . 2 X φ X φ In particular, O (Φ) = id for every gauge transformation φ = id X ( Walpuski ) . The second formula is a self-intersection in X 8 φ of a manifold Poincaré dual to c 2 ( E Φ ) . 5. Flag structures Let Y 3 ⊂ X 7 be an spin submanifold of a spin manifold. Then we have a global Spin(4) -framing of N Y . (any O (4) -framing should suffice) Definition 5.1. For s 0 , s 1 : Y → N Y non-vanishing sections write ϕ : Y → H ∗ . s 0 = ϕ · s 1 , Define d ( s 0 , s 1 ) = ( − 1) degree( ϕ ) ∈ Z 2 . This definition can be extended to s 0 : Y 0 → N Y 0 \ { 0 } and s 1 : Y 1 → N Y 1 \ { 0 } with Y 0 and Y 1 homologous, as an intersection number. Definition 5.2. A flag structure on X associates a sign F ( Y, s ) to every subman- ifold Y 3 ⊂ X equipped with a non-vanishing normal section s . We require that F ( Y 0 , s 0 ) = ( − 1) d ( s 0 ,s 1 ) F ( Y 1 , s 1 ) . Flag structures help pick out normal framings s of Y . Proposition 5.3. Flag structures are a ( non-empty ) torsor over Hom( H 3 ( X ; Z ) , Z 2 ) . Corollary 5.4. Every manifold with H 3 ( X ) = { 0 } has a unique flag structure. Given a preferred set of submanifold generators [ Y i ] ∈ H 3 ( X ) with preferred normal sections s i , we have a unique flag structure with F ( Y i , s i ) := 1 . The pullback of F under a diffeomorphism φ : X ′ → X is ( φ ∗ F )( Y ′ , s ′ ) := ( φ ( Y ′ ) , dφ ( s ′ )) . When φ : X ′ = X → X we get F /φ ∗ F : H 3 ( X ; Z ) → Z 2 .

4 MARKUS UPMEIER (JOINT WITH DOMINIC JOYCE) 6. Main theorem Theorem 6.1. A flag structure F on a closed spin 7 -manifold X induces uniquely, for every SU( n ) -bundle E ց X , a canonical orientation o F ( E ) ∈ Or E with the following properties: (1) ( Normalization ) For E = C k trivial, evaluation defines Or E = Z 2 . Let o flat ( E ) ∈ Or E be the image of 1 ∈ Z 2 . Then o F ( E ) = o flat ( E ) . (2) ( Stabilization ) Under Or E ⊕ C k = Or E ⊗ Z 2 Or C k = Or E we have o F ( E ⊕ C k ) = o F ( E ) (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 � � = ( F /φ ∗ F ′ )[ Y ] · o F ( E ′ ց X ′ ) , Or(Φ , s, s ′ ) o F ( E ց X ) where [ Y ] ∈ H 3 ( U ; Z ) is the homology class Poincaré dual to the relative Chern class c 2 ( P | U , s ) ∈ H 4 cpt ( U ; Z ) . 7. Canonical orientations 7.1. Basic comparison. From (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 get E ′ := φ ∗ E | U ∪ φ ∗ s C n over X ′ . We have an excision isomorphism O (can , φ ∗ s, s ): O E ′ ց X ′ → O E ց X . 7.2. Standard model. Let U, U ′ be tubular neighborhoods of spin submanifolds Y, Y ′ . Let Φ: N Y ′ → N Y be a spin isomorphism covering a spin diffeomorphism φ ◦ : Y ′ → Y . This determines φ = φ ( φ ◦ , Φ) . 7.3. Proof of main theorem. Proof of uniqueness. Let E ց X 7 be an SU(2) -bundle. Pick a transverse section s with zero set Y 3 = s − 1 (0) . Then ds : N Y ∼ = E | Y , which defines an SU(2) -structure → S 7 with image Y ′ let and hence a spin structure on N Y and Y . Embed i : Y ֒ φ ◦ = i − 1 and use φ ◦ to define the spin structure on Y ′ . Since dim Y = 3 we may pick a spin isomorphism Φ: N Y ′ → N Y . Let F 7 be the unique flag structure on S 7 . Set E ′ := φ ∗ E | U ∪ φ ∗ s C n ց S 7 . By the excision axiom O (can , φ ∗ s, s ) O E ′ ց S 7 O E ց X ∈ ∈ ( − 1) ( F 7 /φ ∗ F )[ Y ] · o F ( E ) o F 7 ( E ′ ) Since π 6 (SU(4)) = { 1 } the bundle E ′ ⊕ C 2 is trivializable on S 7 . By the stabilization and normalization axiom