Orientation problems and gauge theory Markus Upmeier 1 , 2 SPP2026 - PowerPoint PPT Presentation

Orientation problems and gauge theory Markus Upmeier 1 , 2 SPP2026 Conference “Geometry at Infinity” Münster, April 3, 2019 1 University of Oxford 2 Universität Augsburg 1) M. Upmeier, A categorified excision principle for elliptic symbol families 2) D. Joyce, Y. Tanaka and M. Upmeier, On orientations for gauge-theoretic moduli spaces 3) D. Joyce and M. Upmeier, Canonical orientations for moduli spaces of G 2 -instantons with gauge group SU ( m ) or U ( m )

Outline Introduction G 2 -geometry Moduli space of G 2 -instantons Orientation and index problems Flag structures Main theorem

Problem (Donaldson–Segal programme) Let E → X be a G -principal bundle over a G 2 -manifold ( X , φ ) . To define counting invariants using the moduli space of G 2 -instantons M irr E := { A ∈ A irr � E | ∗ ( F A ∧ φ ) = − F A } Aut ( E ) we need: 1. ( Canonical ) orientations 2. Compactification 3. Deformation invariance Similarly for other exceptional holonomies, e.g., Spin ( 7 ) . Theorem (Joyce–U. 2018) Let ( X , φ ) be a closed G 2 -manifold. A flag structure F on X determines, for every principal SU ( n ) -bundle E → X , a canonical orientation of the moduli space M irr E of G 2 -instantons. These are deformation-invariant, compatible with direct sums and excision.

Theorem (Donaldson 1987) Let E → X be an SU ( 2 ) -bundle over a closed spin 4 -manifold. An orientation of H 0 ( X ) ⊕ H 1 ( X ) ⊕ H + ( X ) determines a canonical orientation of the ASD moduli space ∗ F A = − F A . Theorem (Taubes 1990) Let E → X be an SU ( 2 ) -bundle over a closed Riemannian 3 -manifold. An orientation of H ∗ ( X ; R ) determines a canonical orientation of the flat connection moduli space F A = 0 . Theorem (Walpuski 2013) Let E → X be a G -principal bundle over a closed G 2 -manifold X . For G = U ( m ) , SU ( m ) the G 2 -instanton moduli space M irr E is orientable.

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 (possibly with torsion) on a 7-manifold X is a smooth 3-form φ that is non-degenerate on each tangent space. Lemma There exists a unique metric g φ and orientation on V such that g φ ( X , Y ) vol g φ = ι X φ ∧ ι Y φ ∧ φ. Proof. Take 0 � = ω ∈ Λ 7 ( V ∗ ) . Define g ( X , Y ) ω := ι X φ ∧ ι Y φ ∧ φ (*) with g positive definite (pass to − ω ). Then vol g = c · ω for c > 0 defines the orientation. Rescale g �→ λ g and solve λ − 5 / 2 = c .

The cross product g φ ( X × φ Y , Z ) := φ ( X , Y , Z ) on V extends to a normed algebra structure on R · 1 ⊕ V , so V ∼ = Im O . Example (prototype) In coordinates on V = R 7 we have dx 45 + dx 67 � dx 46 − dx 57 � dx 47 + dx 56 � φ std = dx 123 + dx 1 � + dx 2 � + dx 3 � 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. Definition A G 2 -structure φ ∈ Ω 3 ( X ) on X 7 is torsion-free : ⇐ ⇒ ∇ LC g φ φ = 0 ⇒ d φ = 0 and d ψ = 0 for ψ := ∗ g φ φ ∈ Ω 4 ( X ) . ⇐

Example (Relation to other geometries) We have embeddings of Lie groups SU ( 3 ) → G 2 → Spin ( 7 ) . 1. A G 2 -structure ( X , φ ) is equivalent to a spin structure ( X , g ) with a preferred non-vanishing real spinor 1 ∈ S = R ⊕ TX . The G 2 -structure is torsion-free ⇐ ⇒ the spinor 1 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 ( M , ω 1 , ω 2 , ω 3 ) we get a torsion-free G 2 -manifold R 3 × M or S 1 × S 1 × S 1 × M with φ = dx 123 − dx 1 ∧ ω 1 − dx 2 ∧ ω 2 − dx 3 ∧ ω 3 .

G 2 -instantons Definition A connection A on a principal G -bundle E → X over a G 2 -manifold ( X , φ 3 , ψ 4 = ∗ φ φ ) is a G 2 -instanton if F A ∧ ψ = 0 ( ⇐ ⇒ ∗ ( F A ∧ φ ) = − F A ) Example 1. The Levi-Civita connection on the tangent bunde of a torsion-free G 2 -manifold. This is because R ∈ Λ 2 T ∗ M ⊗ g 2 . 2. ASD-connections ∗ F A = − F A on a hyperkähler 4-manifold M ( ω i self-dual): F A ∧ dx 123 � � ∗ X ( F A ∧ φ ) = ∗ X = ∗ M F A = − F A . 3. Hermitian Yang–Mills connections Λ F A = 0, F 0 , 2 = 0 on a A Calabi–Yau 3-fold Z .

Infinitesimal moduli space For a G 2 -instanton F A ∧ ψ = 0 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 .)

Atiyah–Hitchin deformation complex The deformation complex d A ∧ ψ d A d A Ω 0 ( X ; g E ) → Ω 1 ( X ; g E ) → Ω 6 ( X ; g E ) → Ω 7 ( X ; g E ) − − − − − has been 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 { / D A } A ∈A E .

Determinant line bundles Definition Let { D y } y ∈ Y be a Y -family of real Fredholm operators. The Quillen determinant line bundle is � Λ top ( Ker D y ) ⊗ Λ top ( Coker D y ) ∗ Det { D y } := ց Y . y ∈ Y Definition The orientation cover is (represents π 1 Fred R = Z 2 ) � Or { D y } y ∈ Y := ( Det { D y } \ { zero section } ) ց Y . R > 0 Up to canonical isomorphism, depends only on principal symbols: Or { p t } − → Or { P t } . (limit exists) Categorifies w 1 ( ind { P t } t ∈ T ∈ KO ( T )) ∈ H 1 ( T ; Z 2 ) .

Restriction to Diracians Definition Let E → X be an SU ( n ) -bundle over a spin manifold. Twist / D by su ( E ) using induced connections su ( A ) to get an A E -family. The D su ( A ) } A ∈A E ) . orientation torsor of E is Or E := C ∞ ( A E , Or { / D su ( E ) ⊕ Coker / D su ( E ) Elements of Or E are orientations of Ker / (arbitrary connection) Proposition Or E ⊕ F ∼ = Or E ⊗ Z 2 Or F canonically. Proof. su ( E ⊕ F ) = su ( E ) ⊕ su ( F ) ⊕ R ⊕ Hom C ( E , F ) .

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 → U ′ ⊂ X ′ . φ X ⊃ U − 2. Let s and s ′ be SU ( n ) -frames of E | X \ K and E ′ | X ′ \ K ′ defined outside 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 Or (Φ , s , s ′ ) → Or E ′ ց X ′ . Or E ց X − − − − − −

E ′ E X ′ X g framing n i m K ⊂ U a r f K ′ ⊂ U ′ framed isomorphism φ ⇒ Or (Φ , s , s ′ ): Or E ց X − → Or E ′ ց X ′ . =

Families index for real self-adjoint operators For families { P t } t ∈ S 1 of real self-adjoint operators (Ati-Pat-Si) ind P t ∈ KO 0 ( S 1 ) � � w 1 ∈ Z 2 ≡ index of a single operator ∂ ∂ t + P t on the space X × S 1 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) · id Or E , δ (Φ) := ˆ ch ( E ∗ Φ ⊗ E Φ ) − n 2 � � A ( TX φ ) , X φ where E Φ = E × Z R ց X φ = X × Z R are the mapping tori.

Simplification of formula in 7D In dimension 7 for Or (Φ) = ( − 1 ) δ (Φ) : Or E → Or E we have 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 Proof. Add to δ (Φ) = ind ( / D g E ) − ind ( / D su ( n ) ) a suitable even multiple of the integer ind ( / D E ) to simplify the integral. Remark ◮ In particular Or (Φ) = id for every gauge transformation φ = id X ( Walpuski ) . ◮ The second formula is a self-intersection in X 8 φ of a homology class Poincaré dual to c 2 ( E Φ ) .

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 ( Y 0 , s 0 ) = ( − 1 ) D ( s 0 , s 1 ) F ( Y 1 , s 1 ) ∀ [ Y 0 ] = [ Y 1 ] . Here D ( s 0 , s 1 ) is defined as follows. Let ∂ Z = [ Y 1 ] − [ Y 0 ] . Then D ( s 0 , s 1 ) = Z • ([ s 0 ] − [ s 1 ]) is the intersection number of Z with perturbations of Y 0 and Y 1 in direction of s 0 and s 1 . A flag structure F is a notion of parity for ( Y , s ) . When N Y has an SU ( 2 ) -structure, a flag structure reduces choices by helping to pick out a normal SU ( 2 ) -framing. Proposition Flag structures are a ( non-empty ) torsor over H 3 ( X ; Z 2 ) . ( Does this have to do with w 2 2 + w 4 = 0 for TX 7 ? )

Definition Let φ : X ′ → X be a diffeomorphism. The pullback of a flag structure F on X is ( φ ∗ F )( Y ′ , s ′ ) := ( φ ( Y ′ ) , d φ ( s ′ )) . Proposition Let φ : X → X be an orientation-preserving isometry with φ | Y = id Y . Then ( F /φ ∗ F )[ Y ] = F ( Y , s ) : F ( Y , φ ∗ s ) equals the self-intersection of Y × S 1 in the mapping torus X φ = X × Z R .