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 G2-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 = NY . (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, φ3, ψ4 = ∗φφ) be a closed G2-manifold. A flag structure on X determines, for every principal SU(n)-bundle E → X, an

• rientation of the moduli space of G2-instantons

Mirr

E = {A ∈ Airr E | ∗ (FA ∧ φ) = −FA ⇐

⇒ FA ∧ ψ = 0}/ Aut(E). Theorem 2.2 (Walpuski 2013). The moduli space of G2-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 H1(M) ⊕ H+(M) determines an

• rientation of the ASD-moduli space of connections on E.

1

2 MARKUS UPMEIER (JOINT WITH DOMINIC JOYCE)

• 3. Orientation problem

For a G2-instanton A 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. The solutions a span the tangent space at A to the space of G2-instantons. For a G2-instanton A we have the deformation complex (1) Ω0(X; gE)

dA

− − → Ω1(X; gE)

dA∧ψ

− − − − → Ω6(X; gE)

dA

− − → Ω7(X; gE), 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{LA}A∈AE. Using / SX = R ⊕ T ∗X for G2-manifolds, the principal symbols of LA and of the twisted Diracian / DA 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 {Pt}t∈T be a T-family of elliptic operators. The Quillen determinant line bundle is Det{Pt} :=

• t∈T

Λtop(Ker Pt)∗ ⊗ Λtop Coker Pt ց T. Let Or{Pt}t∈T ց T be its double cover of orientations. Up to canonical iso- morphism, this only depends on principal symbols (limit exists): Or{pt} → Or{Pt}. These covers categorify w1 (ind{Pt}t∈T ∈ KO(T)) ∈ H1(T; Z2). Example 4.2. Since the principal symbols icξ ⊗ idgE of LA and / DA agree, the

• rientation 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 OrE := Or

• /

S ⊗ gE

cξ⊗1

− − − → / S ⊗ gE

• ∼

=Or / DgE

⊗Z2 Or

• /

S ⊗ su(n)

cξ⊗1

− − − → / S ⊗ su(n)

• ∼

=Z2 ∗

Proposition 4.4. OrE⊕F ∼ = OrE ⊗Z2 OrF 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 X ⊃ U

φ

− → U ′ ⊂ X′. (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 OrEցX

O(Φ,s,s′)

− − − − − − → OrE′ցX′ . The families index for real operators is rather intractable. An important excep- tion are S1-families {Pt}t∈S1 of self-adjoint operators (Atiyah–Patodi–Singer). In this case, the holonomy of the determinant line bundle w1

• ind Pt ∈ KO0(S1)
• ∈ Z2

equals the index of a single operator

∂ ∂t + Pt on the space X × S1, which is com-

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) · idOE, δ(Φ) :=

• Xφ

ˆ A(TXφ)

• ch(E∗

Φ ⊗ EΦ) − rk(EΦ)2

. where EΦ = E ×Z R ց Xφ = X ×Z R are the mapping tori. In dimension 7: δ(Φ) ≡ 1 2

• Xφ

p1(TXφ)c2(EΦ) ≡

• Xφ

c2(EΦ) ∪ c2(EΦ) mod 2. In particular, O(Φ) = id for every gauge transformation φ = idX (Walpuski). The second formula is a self-intersection in X8

φ of a manifold Poincaré dual to c2(EΦ).

• 5. Flag structures

Let Y 3 ⊂ X7 be an spin submanifold of a spin manifold. Then we have a global Spin(4)-framing of NY . (any O(4)-framing should suffice) Definition 5.1. For s0, s1 : Y → NY non-vanishing sections write s0 = ϕ · s1, ϕ: Y → H∗. Define d(s0, s1) = (−1)degree(ϕ) ∈ Z2. This definition can be extended to s0 : Y0 → NY0 \ {0} and s1 : Y1 → NY1 \ {0} with Y0 and Y1 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(Y0, s0) = (−1)d(s0,s1)F(Y1, s1). Flag structures help pick out normal framings s of Y . Proposition 5.3. Flag structures are a (non-empty) torsor over Hom(H3(X; Z), Z2). Corollary 5.4. Every manifold with H3(X) = {0} has a unique flag structure. Given a preferred set of submanifold generators [Yi] ∈ H3(X) with preferred normal sections si, we have a unique flag structure with F(Yi, si) := 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 : H3(X; Z) → Z2.

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

• F(E) ∈ OrE

with the following properties: (1) (Normalization) For E = Ck trivial, evaluation defines OrE = Z2. Let

• flat(E) ∈ OrE be the image of 1 ∈ Z2. Then
• F(E) = oflat(E).

(2) (Stabilization) Under OrE⊕Ck = OrE ⊗Z2 OrCk = OrE we have

• F(E ⊕ Ck) = oF(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 Or(Φ, s, s′)

• F(E ց X)
• = (F/φ∗F′)[Y ] · oF(E′ ց X′),

where [Y ] ∈ H3(U; Z) is the homology class Poincaré dual to the relative Chern class c2(P|U, s) ∈ H4

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 Cn over X′. We have an excision isomorphism O(can, φ∗s, s): OE′ցX′ → OEցX. 7.2. Standard model. Let U, U ′ be tubular neighborhoods of spin submanifolds Y, Y ′. Let Φ: NY ′ → NY 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 X7 be an SU(2)-bundle. Pick a transverse section s with zero set Y 3 = s−1(0). Then ds: NY ∼ = E|Y , which defines an SU(2)-structure and hence a spin structure on NY and Y . Embed i: Y ֒ → S7 with image Y ′ let φ◦ = i−1 and use φ◦ to define the spin structure on Y ′. Since dim Y = 3 we may pick a spin isomorphism Φ: NY ′ → NY . Let F7 be the unique flag structure on S7. Set E′ := φ∗E|U ∪φ∗s Cn ց S7. By the excision axiom OE′ցS7 OEցX

• F7(E′)

(−1)(F7/φ∗F)[Y ] · oF(E) O(can, φ∗s, s) ∈ ∈ Since π6 (SU(4)) = {1} the bundle E′⊕C2 is trivializable on S7. By the stabilization and normalization axiom

ORIENTATIONS FOR MODULI SPACES IN SEVEN-DIMENSIONAL GAUGE THEORY 5

OE′⊕C2ցS7 OE′ցS7

• flat = oF7(E′ ⊕ C2)
• F7(E′)

stab ∈ ∈ Therefore (∗)

• F(E) = O(can, φ∗s, s) ◦ stab((−1)(F7/φ∗F)[Y ] · oflat)

is uniquely determined by the axioms.

• Proof of existence. Show that (∗) is independent of the choices s, φ (i and the

tubular neighborhoods are unique up to isotopy). The dependence on φ can be modelled by the automorphism of Estd(NY ) ց S(NY ⊕ R) induced by a spin automorphism ψ: NY → NY . Since this is global, we can calculate O(ψ) = (F/ψ∗F)[Y ] · idOEstd(NY ). This effect is balanced precisely by the flag structure in (∗). Let s0, s1 be transverse sections of E ց X. Since the excision isomorphisms O(can, φ∗

0s0, s0), O(can, φ∗ 1s1, s1) can be deformed into each other, they are equal

by discreteness. The deformation is by excision isomorphisms as in the basic step, but not coming from a submanifold: