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

Orientation problems and gauge theory

Markus Upmeier1,2

SPP2026 Conference “Geometry at Infinity” Münster, April 3, 2019

1University of Oxford 2Universität 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 G2-instantons with gauge group SU(m) or U(m)

Outline Introduction G2-geometry Moduli space of G2-instantons Orientation and index problems Flag structures Main theorem

Problem (Donaldson–Segal programme)

Let E → X be a G-principal bundle over a G2-manifold (X, φ). To define counting invariants using the moduli space of G2-instantons Mirr

E := {A ∈ Airr E | ∗(FA ∧ φ) = −FA}

• 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 G2-manifold. A flag structure F on X determines, for every principal SU(n)-bundle E → X, a canonical

• rientation of the moduli space Mirr

E of G2-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

• rientation of H0(X) ⊕ H1(X) ⊕ H+(X) determines a canonical
• rientation of the ASD moduli space ∗FA = −FA.

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

• rientation of the flat connection moduli space FA = 0.

Theorem (Walpuski 2013)

Let E → X be a G-principal bundle over a closed G2-manifold X. For G = U(m), SU(m) the G2-instanton moduli space Mirr

E is

• rientable.

Definition

φ ∈ Λ3V ∗ on a 7-dimensional vector space is non-degenerate if ιXφ ∧ ιXφ ∧ φ = 0 ∀X ∈ V \ {0}. A G2-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 gφ and orientation on V such that gφ(X, Y ) volgφ = ιXφ ∧ ιY φ ∧ φ.

Proof.

Take 0 = ω ∈ Λ7(V ∗). Define g(X, Y )ω := ιXφ ∧ ιY φ ∧ φ (*) with g positive definite (pass to −ω). Then volg = c · ω for c > 0 defines the orientation. Rescale g → λg and solve λ−5/2 = c.

The cross product gφ(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 = R7 we have 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.

Definition

A G2-structure φ ∈ Ω3(X) on X 7 is torsion-free : ⇐ ⇒ ∇LCgφφ = 0 ⇐ ⇒ dφ = 0 and dψ = 0 for ψ := ∗gφφ ∈ Ω4(X).

Example (Relation to other geometries)

We have embeddings of Lie groups SU(3) → G2 → Spin(7).

• 1. A G2-structure (X, φ) is equivalent to a spin structure (X, g)

with a preferred non-vanishing real spinor 1 ∈ S = R ⊕ TX. The G2-structure is torsion-free ⇐ ⇒ the spinor 1 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 (M, ω1, ω2, ω3) we get a

torsion-free G2-manifold R3 × M or S1 × S1 × S1 × M with φ = dx123 − dx1 ∧ ω1 − dx2 ∧ ω2 − dx3 ∧ ω3.

G2-instantons

Definition

A connection A on a principal G-bundle E → X over a G2-manifold (X, φ3, ψ4 = ∗φφ) is a G2-instanton if FA ∧ ψ = 0 ( ⇐ ⇒ ∗(FA ∧ φ) = −FA)

Example

• 1. The Levi-Civita connection on the tangent bunde of a

torsion-free G2-manifold. This is because R ∈ Λ2T ∗M ⊗ g2.

• 2. ASD-connections ∗FA = −FA on a hyperkähler 4-manifold M

(ωi self-dual): ∗X(FA ∧ φ) = ∗X

• FA ∧ dx123

= ∗MFA = −FA.

• 3. Hermitian Yang–Mills connections ΛFA = 0, F 0,2

A

= 0 on a Calabi–Yau 3-fold Z.

Infinitesimal moduli space For a G2-instanton FA ∧ ψ = 0 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 .)

Atiyah–Hitchin deformation complex The deformation complex Ω0(X; gE)

dA

− → Ω1(X; gE)

dA∧ψ

− − − → Ω6(X; gE)

dA

− → Ω7(X; gE) 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 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{ /

DA}A∈AE .

Determinant line bundles

Definition

Let {Dy}y∈Y be a Y -family of real Fredholm operators. The Quillen determinant line bundle is Det{Dy} :=

• y∈Y

Λtop(Ker Dy) ⊗ Λtop(Coker Dy)∗ ց Y .

Definition

The orientation cover is (represents π1FredR = Z2) Or{Dy}y∈Y := (Det{Dy} \ {zero section})

• R>0

ց Y . 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).

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 AE-family. The

• rientation torsor of E is OrE := C ∞(AE, Or{ /

Dsu(A)}A∈AE ). Elements of OrE are orientations of Ker / Dsu(E) ⊕ Coker / Dsu(E) (arbitrary connection)

Proposition

OrE⊕F ∼ = OrE ⊗Z2 OrF canonically.

Proof.

su(E ⊕ F) = su(E) ⊕ su(F) ⊕ R ⊕ HomC(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

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

Or(Φ,s,s′)

− − − − − − → OrE ′ցX ′ .

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

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.

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).

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]. Here D(s0, s1) is defined as follows. Let ∂Z = [Y1] − [Y0]. Then D(s0, s1) = Z • ([s0] − [s1]) is the intersection number of Z with perturbations of Y0 and Y1 in direction of s0 and 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.

Proposition

Flag structures are a (non-empty) torsor over H3(X; Z2). (Does this have to do with w2

2 + w4 = 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 ′), dφ(s′)).

Proposition

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.

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

To be continued ...

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)
• = (F/φ∗F′)(c2(E|U, s)) · oF′(E ′).