Orientation Problems in 7-dimensional Gauge Theory Markus Upmeier - - PowerPoint PPT Presentation

Orientation Problems in 7-dimensional Gauge Theory

Markus Upmeier

University of Oxford Talk based on: 1) M. Upmeier, A categorified excision principle for elliptic symbol families (soon) 2) D. Joyce, Y. Tanaka and M. Upmeier, On orientations for gauge-theoretic moduli spaces, arXiv:1811.01096. 3) D. Joyce and M. Upmeier, Canonical orientations for moduli spaces of G2-instantons with gauge group SU(m), arXiv:1811.02405.

January 28, 2019

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Outline

Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions

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Outline

Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions

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Twisted Dirac Operators

Setup

• 1. Compact 7-dimensional spin manifold (X, g)
• 2. Real spinor bundle /

S ց X, connection ∇/

S

• 3. Clifford multiplication c : TX × /

S → / S

• 4. Lie group G
• 5. G-principal bundle P ց X
• 6. Ad P := P ×G g ց X

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Twisted Dirac Operators

Setup

• 1. Compact 7-dimensional spin manifold (X, g)
• 2. Real spinor bundle /

S ց X, connection ∇/

S

• 3. Clifford multiplication c : TX × /

S → / S

• 4. Lie group G
• 5. G-principal bundle P ց X
• 6. Ad P := P ×G g ց X

Definition

Let ∇P ∈ Ω1(P; g) be a connection on P. The twisted Diracian is / D∇Ad P : C ∞(/ S ⊗R Ad P) − → C ∞(/ S ⊗R Ad P), s − →

7

• i=1

c(ei, ∇/

S⊗Ad P ei

s)

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Example in 7D: Manifolds with G2-Structure

Definition

A topological G2-structure on (X 7, g) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1, 0): c : O × O

bilinear

− − − − → O, v · w = v · w.

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Example in 7D: Manifolds with G2-Structure

Definition

A topological G2-structure on (X 7, g) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1, 0): c : O × O

bilinear

− − − − → O, v · w = v · w. Adjoint of c|TX is φ ∈ Ω3(X); Let ψ := ∗φ ∈ Ω4(X).

• 1. Every manifold with G2-structure is spin /

S := O.

• 2. Clifford multiplication is c.
• 3. For a torsion-free G2-structure ∇/

S = ∇R ⊕ ∇LC.

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Example in 7D: Manifolds with G2-Structure

Definition

A topological G2-structure on (X 7, g) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1, 0): c : O × O

bilinear

− − − − → O, v · w = v · w. Adjoint of c|TX is φ ∈ Ω3(X); Let ψ := ∗φ ∈ Ω4(X).

• 1. Every manifold with G2-structure is spin /

S := O.

• 2. Clifford multiplication is c.
• 3. For a torsion-free G2-structure ∇/

S = ∇R ⊕ ∇LC.

General 7-dimensional spin manifold, with preferred spinor.

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Example in 7D: Continued

Connection ∇P induces d∇P : Ωk(X, Ad P) − → Ωk+1(X, Ad P).

Proposition

Assume ∇LCφ = 0. Then the twisted Diracian / D∇Ad P equals L∇P = d∗

∇P

d∇P ∗(ψ ∧ d∇P)

• Ω0(X, Ad P) ⊕ Ω1(X, Ad P) −

→ Ω0(X, Ad P) ⊕ Ω1(X, Ad P).

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Example in 7D: Continued

Connection ∇P induces d∇P : Ωk(X, Ad P) − → Ωk+1(X, Ad P).

Proposition

Assume ∇LCφ = 0. Then the twisted Diracian / D∇Ad P equals L∇P = d∗

∇P

d∇P ∗(ψ ∧ d∇P)

• Ω0(X, Ad P) ⊕ Ω1(X, Ad P) −

→ Ω0(X, Ad P) ⊕ Ω1(X, Ad P).

Corollary

The tangent space at ∇P of the moduli space of G2-instantons F ∇Ad P ∧ ψ = 0 is described by the Diracian / D∇Ad P . Ω0(X; gP)

d∇P

− − → Ω1(X; gP)

d∇P ∧ψ

− − − − → Ω6(X; gP)

d∇P

− − → Ω7(X; gP)

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Today’s Problem

Let AP := {connections ∇P on P → X}. By twisting / D using each ∇P ∈ AP get an AP-family of differential operators on X.

Questions

◮ Equivariant orientability of ∇P∈AP Det /

D∇Ad P ց AP? − → Can be answered using index theory.

◮ How do we pick orientations, canonically, fixing perhaps some

topological data on X?

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Today’s Problem

Let AP := {connections ∇P on P → X}. By twisting / D using each ∇P ∈ AP get an AP-family of differential operators on X.

Questions

◮ Equivariant orientability of ∇P∈AP Det /

D∇Ad P ց AP? − → Can be answered using index theory.

◮ How do we pick orientations, canonically, fixing perhaps some

topological data on X?

Theorem (Joyce–U. 2018)

Let (X, φ3, ψ4 = ∗φφ) be a closed G2-manifold. A flag structure F

• n X determines, for every principal SU(n)-bundle P → X, an
• rientation of the moduli space Mirr

P of G2-instantons

{A ∈ Airr

P | FA ∧ ψ = 0}

• Aut(P).

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Outline

Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions

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Determinant Line Bundle and Orientations

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 .

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Determinant Line Bundle and Orientations

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 Or{Dy}y∈Y := (Det{Dy} \ {zero section})

• R>0

ց Y . Represents π1FredR = Z2

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Determinant Line Bundle and Orientations

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 Or{Dy}y∈Y := (Det{Dy} \ {zero section})

• R>0

ց Y . Represents π1FredR = Z2

Today’s problem

Given G ֒ → P ։ X 7, trivialize Or{ / D∇Ad P }∇P∈AP ց AP, canonically in terms of data on X.

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Index Theory

Definition

• 1. For D Fredholm, ind D := dimR Ker D − dimR Coker D ∈ Z

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Index Theory

Definition

• 1. For D Fredholm, ind D := dimR Ker D − dimR Coker D ∈ Z
• 2. For Y -family {Dy}y∈Y , ind D ∈ KO0(Y ). Up to isomorphism

w1(ind D) = [Or{Dy}] ∈ H1(Y ; Z2)

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Index Theory

Definition

• 1. For D Fredholm, ind D := dimR Ker D − dimR Coker D ∈ Z
• 2. For Y -family {Dy}y∈Y , ind D ∈ KO0(Y ). Up to isomorphism

w1(ind D) = [Or{Dy}] ∈ H1(Y ; Z2)

Properties

◮ Natural in Y ◮ If {Dt}t∈[0,1] : D0 ≃ D1 through Fred, then

ind D0 = i∗

0 ind D = i∗ 1 ind D = ind D1 ◮ ind(D1 ⊕ D2) = ind D1 + ind D2 ◮ ind D† = − ind D

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Elliptic Symbol Families

Definition

Family of Elliptic (ψ)DOs Dy : C ∞(X, Ey) → C ∞(X, Fy) on X determined, up to convex choice, by elliptic symbol family pξ,y = σξ,y(D): Ey

∼ =

− − → Fy, pλ·ξ,y = λmpξ,y, for all 0 = ξ ∈ T ∗X, y ∈ Y , λ > 0. Here m ∈ R is the order.

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Elliptic Symbol Families

Definition

Family of Elliptic (ψ)DOs Dy : C ∞(X, Ey) → C ∞(X, Fy) on X determined, up to convex choice, by elliptic symbol family pξ,y = σξ,y(D): Ey

∼ =

− − → Fy, pλ·ξ,y = λmpξ,y, for all 0 = ξ ∈ T ∗X, y ∈ Y , λ > 0. Here m ∈ R is the order.

Example

E ց X × Y vector bundle, X 7 spin, c : TX ⊗ / S → / S Clifford

• multiplication. Let pξ,y := cξ ⊗ idEy for y ∈ Y .

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Elliptic Symbol Families

Definition

Family of Elliptic (ψ)DOs Dy : C ∞(X, Ey) → C ∞(X, Fy) on X determined, up to convex choice, by elliptic symbol family pξ,y = σξ,y(D): Ey

∼ =

− − → Fy, pλ·ξ,y = λmpξ,y, for all 0 = ξ ∈ T ∗X, y ∈ Y , λ > 0. Here m ∈ R is the order.

Example

E ց X × Y vector bundle, X 7 spin, c : TX ⊗ / S → / S Clifford

• multiplication. Let pξ,y := cξ ⊗ idEy for y ∈ Y .

Further properties

◮ ind p = ind D well-defined, for any σ(D) = p ◮ i : U ֒

→ X open embedding, p compactly supported on U = ⇒ i!(ind p) = ind i!p

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Categorical Index Calculus

For Y -family of elliptic symbols p = {pξ,y}y∈Y on X have object Or p := lim

σ(D)=p Or D ց Y

in Covgr

Z2(Y )

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Categorical Index Calculus

For Y -family of elliptic symbols p = {pξ,y}y∈Y on X have object Or p := lim

σ(D)=p Or D ց Y

in Covgr

Z2(Y )

Properties become structure maps in Covgr

Z2(Y )

• 1. {pt}: p0 ≃ p1 =

⇒ Or p0 → Or q1

• 2. Or(p ⊕ q) → (Or p) ⊗ (Or q), Or p† → (Or p)∗
• 3. If φ: X− → X+ diffeomorphism with φ∗p+ = p−, then

Or(φ): Or p− − → Or p+

• 4. For i : U ֒

→ X open embedding, p compactly supported on U i! : Or(p) − → Or(i!p)

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Categorical Index Calculus

For Y -family of elliptic symbols p = {pξ,y}y∈Y on X have object Or p := lim

σ(D)=p Or D ց Y

in Covgr

Z2(Y )

Properties become structure maps in Covgr

Z2(Y )

• 1. {pt}: p0 ≃ p1 =

⇒ Or p0 → Or q1

• 2. Or(p ⊕ q) → (Or p) ⊗ (Or q), Or p† → (Or p)∗
• 3. If φ: X− → X+ diffeomorphism with φ∗p+ = p−, then

Or(φ): Or p− − → Or p+

• 4. For i : U ֒

→ X open embedding, p compactly supported on U i! : Or(p) − → Or(i!p) All these maps are compatible.

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Excision

X− U− L− Ξ− : p− ∼ = q− p−, q− (φ, Π, K) X+ U+ L+ Ξ+ : p+ ∼ = q+ p+, q+

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Excision

X− U− L− Ξ− : p− ∼ = q− p−, q− (φ, Π, K) X+ U+ L+ Ξ+ : p+ ∼ = q+ p+, q+

◮ Pair of symbol families p±, q± on X±, isomorphic outside

compact subsets L± of U±

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Excision

X− U− L− Ξ− : p− ∼ = q− p−, q− (φ, Π, K) X+ U+ L+ Ξ+ : p+ ∼ = q+ p+, q+

◮ Pair of symbol families p±, q± on X±, isomorphic outside

compact subsets L± of U±

◮ Identification Π, K of pairs over diffeomorphism φ: U− → U+

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Excision

X− U− L− Ξ− : p− ∼ = q− p−, q− (φ, Π, K) X+ U+ L+ Ξ+ : p+ ∼ = q+ p+, q+

◮ Pair of symbol families p±, q± on X±, isomorphic outside

compact subsets L± of U±

◮ Identification Π, K of pairs over diffeomorphism φ: U− → U+

This data induces an excision isomorphism in Covgr

Z2(Y )

Ex(φ, Π, K): Or(p−)∗ ⊗ Or(q−) − → Or(p+)∗ ⊗ Or(q+) compatible with all structure maps.

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Outline

Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions

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Effect of Global Diffeomorphisms on Orientations

Definition

For P ց X principal G-bundle over spin manifold X, Clifford multiplication c : TX ⊗ / S → / S, define OrP := Or(c ⊗ Ad P)∗ ⊗ Or(c ⊗ su(n)) = Or( / DAd P)∗ ⊗ Or( / Dsu(n))

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Effect of Global Diffeomorphisms on Orientations

Definition

For P ց X principal G-bundle over spin manifold X, Clifford multiplication c : TX ⊗ / S → / S, define OrP := Or(c ⊗ Ad P)∗ ⊗ Or(c ⊗ su(n)) = Or( / DAd P)∗ ⊗ Or( / Dsu(n))

Theorem (APS)

Let Φ: P → P be an SU(n)-isomorphism over a spin diffeomorphism φ: X → X. Then we have Or(Φ) = (−1)δ(Φ)·idOrP, δ(Φ) :=

• Xφ

ˆ A(TXφ)

• ch(P∗

Φ ⊗ PΦ) − n2

, where PΦ = P ×Z R ց Xφ = X ×Z R are the mapping tori.

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Effect of Global Diffeomorphisms on Orientations

Definition

For P ց X principal G-bundle over spin manifold X, Clifford multiplication c : TX ⊗ / S → / S, define OrP := Or(c ⊗ Ad P)∗ ⊗ Or(c ⊗ su(n)) = Or( / DAd P)∗ ⊗ Or( / Dsu(n))

Theorem (APS)

Let Φ: P → P be an SU(n)-isomorphism over a spin diffeomorphism φ: X → X. Then we have Or(Φ) = (−1)δ(Φ)·idOrP, δ(Φ) :=

• Xφ

ˆ A(TXφ)

• ch(P∗

Φ ⊗ PΦ) − n2

, where PΦ = P ×Z R ց Xφ = X ×Z R are the mapping tori. δ(Φ) ≡ 1 2

• Xφ

p1(TXφ)c2(PΦ) ≡

• Xφ

c2(PΦ) ∪ c2(PΦ) mod 2

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Flag Structures

Definition

Let X 7 be oriented, Y 3 ⊂ X 7 compact oriented submanifold

◮ A flag on Y ⊂ X is a non-vanishing normal section

s : Y → NY /X.

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Flag Structures

Definition

Let X 7 be oriented, Y 3 ⊂ X 7 compact oriented submanifold

◮ A flag on Y ⊂ X is a non-vanishing normal section

s : Y → NY /X.

◮ For flags s0, s1 define degree of s0 w.r.t. s1 as

d(s0, s1) := [Y ] • {ts1(y) + (1 − t)s0(y) | t ∈ [0, 1], y ∈ Y }.

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Flag Structures

Definition

Let X 7 be oriented, Y 3 ⊂ X 7 compact oriented submanifold

◮ A flag on Y ⊂ X is a non-vanishing normal section

s : Y → NY /X.

◮ For flags s0, s1 define degree of s0 w.r.t. s1 as

d(s0, s1) := [Y ] • {ts1(y) + (1 − t)s0(y) | t ∈ [0, 1], y ∈ Y }.

◮ s0 ∼ s1 : ⇐

⇒ d(s0, s1) ∈ 2Z. Let Flag(Y ⊂ X) := {s}/ ∼.

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Flag Structures

Definition

Let X 7 be oriented, Y 3 ⊂ X 7 compact oriented submanifold

◮ A flag on Y ⊂ X is a non-vanishing normal section

s : Y → NY /X.

◮ For flags s0, s1 define degree of s0 w.r.t. s1 as

d(s0, s1) := [Y ] • {ts1(y) + (1 − t)s0(y) | t ∈ [0, 1], y ∈ Y }.

◮ s0 ∼ s1 : ⇐

⇒ d(s0, s1) ∈ 2Z. Let Flag(Y ⊂ X) := {s}/ ∼.

◮ Flag structure for Y ⊂ X is a choice of base-point

F : Flag(Y ⊂ X)

∼ =

− − → Z2.

◮ Flag structure is a flag structure for all Y 3 ⊂ X, where

F(Y1, s1) = (−1)D((Y1,s1),(Y2,s2)) · F(Y2, s2) if [Y1] = [Y2]. Define a torsor over H3(X; Z2).

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Flags for Trivial Normal Bundle

Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. d(s0, s1) =

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Flags for Trivial Normal Bundle

Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. Set S(y, t) := (1 − t)s0(y) + ts1(y). d(s0, s1) = s0 s1

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Flags for Trivial Normal Bundle

Example

Let Y := Y0 = Y1 with trivializable normal bundle. For s0, s1 : Y → H unit length write s1 = q · s0 with q : Y → S3. Set S(y, t) := (1 − t)s0(y) + ts1(y). S(y, t) = [(1 − t) + qt] · s0(y) = 0 ⇐ ⇒ t = 1 2 and q(y) = −1. = ⇒ d(s0, s1) = degree(q : Y → S3). s0 s1 ×

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Flag Structures: Continued

Definition

Oriented manifolds X±, orientation-preserving diffeomorphism φ: X− → X+. Pullback of flag structure F+ is (φ∗F+)

• Y−

s−

− → NY−⊂X−

• = F+
• dφ ◦ s− ◦ φ−1|Y−
• .

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Flag Structures: Continued

Definition

Oriented manifolds X±, orientation-preserving diffeomorphism φ: X− → X+. Pullback of flag structure F+ is (φ∗F+)

• Y−

s−

− → NY−⊂X−

• = F+
• dφ ◦ s− ◦ φ−1|Y−
• .

Proposition

If φ: X → X or. diffeomorphism, Y 3 ⊂ X, φ|Y = idY , then F/φ∗F[Y ] = (−1)[Y ×S1]•[Y ×S1] in the mapping torus X 8

φ = (X × [0, 1])

• (x, 1) ∼ (φ(x), 0).

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Statement of Main Theorem: Slide 1/2

A flag structure F (on Y 3 ⊂ X 7) determines an orientation

• F(P) ∈ OrP := Or( /

DAd P) ⊗ Or( / Dsu(n))∗ for every SU(n)-bundle P ց X (with [Y ] Poincar´ e dual to c2(P)). This association is uniquely determined by the following properties:

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Statement of Main Theorem: Slide 1/2

A flag structure F (on Y 3 ⊂ X 7) determines an orientation

• F(P) ∈ OrP := Or( /

DAd P) ⊗ Or( / Dsu(n))∗ for every SU(n)-bundle P ց X (with [Y ] Poincar´ e dual to c2(P)). This association is uniquely determined by the following properties:

• 1. (Normalization.) For P = SU(n) trivial: ∃ canonical

base-point otriv(P) ∈ OrP. For every flag structure F:

• F(P) = otriv(P).

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Statement of Main Theorem: Slide 1/2

A flag structure F (on Y 3 ⊂ X 7) determines an orientation

• F(P) ∈ OrP := Or( /

DAd P) ⊗ Or( / Dsu(n))∗ for every SU(n)-bundle P ց X (with [Y ] Poincar´ e dual to c2(P)). This association is uniquely determined by the following properties:

• 1. (Normalization.) For P = SU(n) trivial: ∃ canonical

base-point otriv(P) ∈ OrP. For every flag structure F:

• F(P) = otriv(P).
• 2. (Stabilization.) Via the isomorphism

stab: OrP×SU(n)SU(m) ∼ = OrP ⊗Z2 OrSU(m) ∼ = OrP we have

• F(P ×SU(n) SU(m)) = oF(P).

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Statement of Main Theorem: Slide 2/2

• 3. (Natural.)

◮ Let P± ց X± be SU(n)-bundles, F± flag structures on X±.

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Statement of Main Theorem: Slide 2/2

• 3. (Natural.)

◮ Let P± ց X± be SU(n)-bundles, F± flag structures on X±. ◮ Let ρ± be sections of P± outside open subsets U± ⊂ X±.

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Statement of Main Theorem: Slide 2/2

• 3. (Natural.)

◮ Let P± ց X± be SU(n)-bundles, F± flag structures on X±. ◮ Let ρ± be sections of P± outside open subsets U± ⊂ X±. ◮ Let Φ: P−|U− → P+|U+ be an SU(n)-isomorphism over a spin

diffeomorphism φ: U− → U+ preserving ρ±.

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Statement of Main Theorem: Slide 2/2

• 3. (Natural.)

◮ Let P± ց X± be SU(n)-bundles, F± flag structures on X±. ◮ Let ρ± be sections of P± outside open subsets U± ⊂ X±. ◮ Let Φ: P−|U− → P+|U+ be an SU(n)-isomorphism over a spin

diffeomorphism φ: U− → U+ preserving ρ±. Under excision Ex(φ, Φ): OrP− → OrP+ we then have Ex(φ, Φ)

• F−(P−)
• = (F−/φ∗F+)(c2(P−)) · oF+(P+).

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Illustration of Excision Axiom

P− X− f r a m i n g ρ− P+ X+ framing ρ+ Y+

framed isomorphism φ: U− − → U+

Y−

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

• 3. Embed i : Y ֒

→ S7 and use i−1 to define spin structure on Y ′.

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

• 3. Embed i : Y ֒

→ S7 and use i−1 to define spin structure on Y ′.

• 4. Pick a spin isomorphism Φ: NY ′⊂S7 → NY ⊂X, gives spin

diffeomorphism S7 ⊃ U′ φ − → U ⊂ X.

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

• 3. Embed i : Y ֒

→ S7 and use i−1 to define spin structure on Y ′.

• 4. Pick a spin isomorphism Φ: NY ′⊂S7 → NY ⊂X, gives spin

diffeomorphism S7 ⊃ U′ φ − → U ⊂ X.

• 5. Set P′ := φ∗P|U ∪φ∗s SU(2) ց S7.

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

• 3. Embed i : Y ֒

→ S7 and use i−1 to define spin structure on Y ′.

• 4. Pick a spin isomorphism Φ: NY ′⊂S7 → NY ⊂X, gives spin

diffeomorphism S7 ⊃ U′ φ − → U ⊂ X.

• 5. Set P′ := φ∗P|U ∪φ∗s SU(2) ց S7.

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Proof of Uniqueness for SU(2)-Bundles P ց X 7

• 1. Pick a transverse section s of E := P ×SU(2) C2 with zero set

Y 3 = s−1(0).

• 2. Then ds : NY ∼

= E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y .

• 3. Embed i : Y ֒

→ S7 and use i−1 to define spin structure on Y ′.

• 4. Pick a spin isomorphism Φ: NY ′⊂S7 → NY ⊂X, gives spin

diffeomorphism S7 ⊃ U′ φ − → U ⊂ X.

• 5. Set P′ := φ∗P|U ∪φ∗s SU(2) ց S7.

By excision axiom (F7 unique flag structure on S7.): OrP′ցS7 OrPցX

• F7(P′)

(−1)(F7/φ∗F)[Y ] · oF(P) Ex ∈ ∈

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Proof of Uniqueness for SU(2)-Bundles: Continued

Since π6 (SU(4)) = {1}, P′ ×SU(2) SU(4) is trivializable on S7.

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Proof of Uniqueness for SU(2)-Bundles: Continued

Since π6 (SU(4)) = {1}, P′ ×SU(2) SU(4) is trivializable on S7. By stabilization and normalization axioms: OrP′×SU(2)SU(4)ցS7 OrP′ցS7

• triv = oF7(P′ ×SU(2) SU(4))
• F7(P′)

stab ∈ ∈

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Proof of Uniqueness for SU(2)-Bundles: Continued

Since π6 (SU(4)) = {1}, P′ ×SU(2) SU(4) is trivializable on S7. By stabilization and normalization axioms: OrP′×SU(2)SU(4)ցS7 OrP′ցS7

• triv = oF7(P′ ×SU(2) SU(4))
• F7(P′)

stab ∈ ∈ So uniquely determined by axioms:

• F(E) = Ex ◦ stab((−1)(F7/φ∗F)[Y ] · otriv)

(∗)

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Proof of Existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

• F(E) := (−1)(F7/φ∗F)[Y ] · Ex ◦ stab(otriv)

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Proof of Existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

• F(E) := (−1)(F7/φ∗F)[Y ] · Ex ◦ stab(otriv)

◮ Dependence on φ reduced to a model calculation for the

automorphism of Estd(NY ) ց S(NY ⊕ R) induced by a spin automorphism ψ: NY → NY . We have calculated Or(ψ) = (F/ψ∗F)[Y ].

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Proof of Existence

◮ Show that (∗) is independent of the choices s, φ

(i and the tubular neighborhoods are unique up to isotopy).

• F(E) := (−1)(F7/φ∗F)[Y ] · Ex ◦ stab(otriv)

◮ Dependence on φ reduced to a model calculation for the

automorphism of Estd(NY ) ց S(NY ⊕ R) induced by a spin automorphism ψ: NY → NY . We have calculated Or(ψ) = (F/ψ∗F)[Y ].

◮ Let s0, s1 be transverse sections of E ց X. The two

corresponding excision isomorphisms can be deformed into each other = ⇒ equal by discreteness.

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s−1

1 (0)

X s−1

0 (0)

t

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Summary

◮ Orientation covers Or(p) ∈ CovZ2(Y ) behave exactly like

ind(p) ∈ KO0(Y ), one categorical level up. Here p is an elliptic symbol family on X parameterized by Y .

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Summary

◮ Orientation covers Or(p) ∈ CovZ2(Y ) behave exactly like

ind(p) ∈ KO0(Y ), one categorical level up. Here p is an elliptic symbol family on X parameterized by Y .

◮ A flag structure solves the ‘orientation problem’ for

7-dimensional twisted Diracians / DAd P when G = SU(n).

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Summary

◮ Orientation covers Or(p) ∈ CovZ2(Y ) behave exactly like

ind(p) ∈ KO0(Y ), one categorical level up. Here p is an elliptic symbol family on X parameterized by Y .

◮ A flag structure solves the ‘orientation problem’ for

7-dimensional twisted Diracians / DAd P when G = SU(n).

◮ Canonical orientations defined by comparing the ‘orientation

problem’ on X to that on S7 via excision.

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