Orientations for gauge-theoretic moduli problems Yuuji Tanaka, - - PowerPoint PPT Presentation

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Orientations for gauge-theoretic moduli problems

Yuuji Tanaka, Oxford University Oxford, January 2019. Based on work with Dominic Joyce and Markus Upmeier in ‘On

• rientations for gauge-theoretic moduli spaces’, preprint, 2018,

arXiv:1811.01096. Funded by the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics.

1 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Plan of talk:

1 Gauge-theoretic moduli problem

–general picture, motivation

2 Anti-self-dual instanton moduli space

–Atiyah–Hitchin–Singer complex, Kuranishi model

3 Orientations

–orientability and orientations of gauge-theoretic moduli spaces

2 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

X: smooth manifold of real dimension n P → X: principal G-bundle over X, G: Lie group The gauge group GP := Aut(P) acts on AP, the space of all connections on P, by u(A) := A − (dAu)u−1, where u ∈ GP and A ∈ AP, GP is identified with Γ(P ×Ad G) and dA is the covariant derivative on Γ(P ×Ad G) induced by A. We denote the quotient AP/GP by BP. Gauge-theoretic equations (e.g. anti-self-dual instanton equations) assign a vector bundle E over BP and a section s of E. E

• MP ⊂ BP.

s

• We call MP := s−1(0) a gauge-theoretic moduli space.

3 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Gauge-theoretic moduli problem

Problem: construct a (virtual) fundamental cycle out of MP. Applications: intersection theory on the (virtual) fundamental cycles produces deformation invariants such as Donaldson invariants, Gromov–Witten, Seiberg–Witten, Donaldson–Thomas

• nes and so on. Furthermore, the generating functions of these

invariants typically have non-trivial properties such as modularity, which could indicate the origins of these theories perhaps. Issues: smoothness, orientability, compactness of MP. For smoothness: use Freed–Uhlenbeck perturbation, virtual techniques by Behrend–Fantechi et al., or invoke derived stacks. For compactness: take up Uhlenbeck, Gieseker compactifications for vector bundles/sheaves, or stable map compactification for pseudo-holomorphic curves.

4 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Anti-self-dual instantons

X: closed, oriented, smooth four-manifold P → X: principal G-bundle over X, G: Lie group Fix a Riemannian metric g on X, and consider the Hodge star

• perator ∗g on Λ2

X := (Λ2T ∗X). This satisfies ∗2 g = 1, so Λ2 X

decomposes as Λ2

X = Λ+ X ⊕ Λ− X.

Definition: A connection on P is said to be an anti-self-dual instanton, or ASD instanton for short, if the curvature FA of A satisfies F +

A := π+(FA) = 0, where π+ : Γ(gP ⊗ Λ2 X) → Γ(gP ⊗ Λ+ X)

is the projection and gP is the adjoint bundle of P. Consider MASD

P,g := {A ∈ AP : F + A = 0}/GP, the anti-self-dual

instanton moduli space. (The corresponding (E, s) in the earlier slide is given by E := AP ×GP Ω+

X(gP) → BP and s := F + A .)

5 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Atiyah–Hitchin–Singer complex: the infinitesimal deformation of an anti-self-dual instanton A is described by the following elliptic complex: 0 − → Γ(gP ⊗ Λ0

X) dA

− → Γ(gP ⊗ Λ1

X) d+

A

− − → Γ(gP ⊗ Λ+

X) −

→ 0, where d+

A := π+ ◦ dA.

We write its cohomology by Hi

A for i = 0, 1, 2.

Denote by ΓA := {u ∈ GP : u(A) = A} the stabilizer group of GP at [A] ∈ BP. Definition: A connection A of P is called irreducible if ΓA coincides with the centre of G and reducible otherwise.

6 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Kuranishi model

Theorem (Atiyah–Hitchin–Singer) Let A be an anti-self-dual

• instanton. Then there exists an open neighbourhood U of 0 in H1

A

and a differentiable map κ : U → H2

A with κ(0) = 0 and the first

derivative of κ vanishing at 0, which is ΓA-equivariant if A is reducible, such that the moduli space MASD

P,g around [A] is locally

modeled on κ−1(0)/ΓA Remark: One needs an appropriate Sobolev space setting to prove the above, for example in order to use an implicit function theorem in the infinite-dimensional setting. Note that H0

A = 0, if A is irreducible. Also, if H2 A = 0, then H1 A is

the tangent space at [A] ∈ MASD

P,g for A irreducible, so H2 A is the

• bstruction space to deforming the equivalence classes [A] of

irreducible connections in MASD

P,g .

7 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

In the analytic setting, if G = SU(2) or SO(3), then H2

A = 0 for a

generic choice of Riemannian metrics g on X. In addition, if b+

X ,

the dimension of maximal positive subspace for the intersection form on H2(X, Z), is positive, then there are no reducible connections other than the trivial one again for a generic metric g. Hence we obtain: Theorem (Atiyah–Hitchin–Singer, Freed–Uhlenbeck, Donaldson–Kronheimer) Let X be a closed, oriented, simply-connected, smooth four-manifold, and let P → X be a principal G-bundle over X. Take the structure group G of P to be SU(2) or SO(3), and assume that b+

X > 0. Then MASD P,g is a

smooth manifold of the expected dimension for a generic choice of metrics g on the underlying four-manifold. Remark: One can use topological stacks and derived manifolds when the above assumptions fail.

8 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Orientations

Finite-dimensional model: let X be a smooth n-manifold. a) X is orientable if the determinant bundle L := ΛnTX of TX is trivial. b) An orientation is a choice of trivialization of L. ASD instantons case: consider the family of elliptic operators parametrized by AP given by: δA := (d∗

A, d+ A ) : Γ(gP ⊗ Λ1 X) → Γ(gP ⊗ (Λ0 ⊕ Λ+ X)).

By the Fredholm property of elliptic operators, it has a well-defined determinant: LA := det(ind δA) := det(ker δA) ⊗ det(coker δA)∗. This defines a line bundle on AP, which descends to L → BP.

9 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

If there are no reducible connections (so H0

A = 0) and H2 A = 0 for

all [A] ∈ MASD

P,g , then ι∗(L) is isomorphic to the determinant line

bundle of the tangent bundle of MASD

P,g , where ι : MASD P,g ֒

→ BP is the natural inclusion. Theorem (i) Donaldson, ii) Donaldson–Kronheimer) Let X be a closed, oriented, smooth 4-manifold, and let P → X be a principal G-bundle over X. Assume that either i) the structure group G of P is U(m) or SU(m); or ii) X is simply-connected and G is a simlpy-connected, simple Lie group. Then a) L → BP is trivial, hence the smooth part of MASD

P

is

• rientable; and

b) a canonical orientation can be determined by choosing an

• rientation on H1(X) and H+(X).

10 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

In general, suppose we are given E0, E1 → X real vector bundles of the same rank over a compact manifold X, and D : Γ(E0) → Γ(E1), a linear elliptic operator. We write E• := (E0, E1, D). Let A ∈ AP. This induces a connection on gP → X. Then consider the elliptic linear operator twisted by A: DA : Γ(gP ⊗ E0) → Γ(gP ⊗ E1). As DA is elliptic on a compact manifold, we have that det(DA) = det(ker DA) ⊗ det(coker DA)∗ is a one-dimensional vector space. This defines a line bundle on AP, which descends to a line bundle LE•

P → BP, the determinant line bundle of BP.

We call OE•

P := (LE• P \ 0(BP))/(0, ∞) the orientation bundle of BP,

where 0(BP) is the zero section. This is a principal Z2-bundle.

11 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Definition: a) (BP, E•) is orientable if OE•

P

is isomorphic to the trivial principal Z2-bundle BP × Z2. b) An orientation on (BP, E•) is an isomorphism BP × Z2

∼ =

− → OE•

P

• f principal Z2-bundles.

Problems: a) Under what condition on X, G, P, E•, is (BP, E•) orientable? b) If it is orientable, can we construct a natural orientation, i.e. is there a way of choosing an orientation which is independent

• f extra data (e.g. Riemannian metrics ) on X?

[Joyce–T–Upmeier] solves these problems for various gauge-theoretic moduli spaces. One important technique is the excision theorem by Markus Upmeier, A categorified excision principle for elliptic symbol families, preprint, 2019.

12 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Methods for orientations

Excision theorems for the orientation bundles. Orientations from complex structures on E• = (E0, E1, D) or G, namely, if E0, E1 → X are complex vector bundles, and the symbol of D is complex linear, then we have a canonical trivialization OE•

P ∼ =

− → BP × Z2 coming from the complex structures. Relating orientations of moduli spaces for Lie subgroups H ⊂ G such as U(m1) × U(m2) ⊂ U(m1 + m2), U(m) ֒ → SU(m + 1), U(m) ֒ → Sp(m) and so on. Stabilization, e.g. consider the direct limit BP⊕C∞ := lim − →k→∞ BP⊕Ck for a principal U(m)-bundle via gluing maps BP⊕Ck → BP⊕Ck+1, and the direct limit of principal Z2-bundles OE•

P⊕C∞ → BP⊕C∞.

• etc. see [Joyce–T–Upmeier].

13 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

For the anti-self-dual instanton moduli spaces, we obtain: Theorem (Joyce–T–Upmeier) Let X be a closed, oriented, smooth four-manifold, and let P → X be a principal G-bundle over X, where G is a connected Lie group. 1) Choose an orientation on H0(X) ⊕ H1(X) ⊕ H+(X) and on g, and a Spinc-structure s on X. Then we can construct a canonical orientation on MASD

P

for all principal G-bundles P → X. 2) If G is simply-connected, or if G = U(m), then the orientation

• n MASD

P

in the above 1) is independent of the choice of Spinc-structure s. Part 1) is new, both the orientability of BP and the use of Spinc-structures in constructing canonical orientations.

14 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Structure of proof: Find a CW complex Y ⊂ X of dimension 2, and a trivialization P|X\Y → (X \ Y ) × G. Then choose an open neighbourhood U of Y in X such that U retract onto Y , an

• pen subset V ⊂ X with ¯

V ⊂ X \ Y and U ∪ V = X, and connection ˆ A on P, which is trivial over V ⊂ X \ Y . X does not necessarily have an almost complex structure, but by using a Spinc-structure, one can introduce an almost complex structure on the above U. Applying an excision theorem to these U, V etc., one obtains a choice for orientations at [ ˆ A], using the almost complex structure on E•|U. Then take paths from [ ˆ A] to other connections [A] in the space of connections on P in order to make choices at [A]. Prove that this orientation is independent of choices in the construction.

15 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

[Joyce–T–Upmeier] describes/solves the orientation problems also for flat connections on Riemann surfaces, flat connections and Casson invariants on 3-manifolds, Seiberg–Witten invariants, Vafa–Witten and Kapustin–Witten equations on 4-manifolds, Haydys–Witten equations on 5-manifolds, and Donaldson–Thomas instantons on compact symplectic 6-manifolds. For the orientation problems for G2-instantons, Spin(7)-instantons and DT4 moduli spaces, see

• D. Joyce and M. Upmeier, Canonical orientations for moduli

spaces of G2-instantons with gauge group SU(m) or U(m), arXiv:1811.02405, 2018.

• Y. Cao and D. Joyce, Orientability of moduli spaces of

Spin(7)-instantons, arXiv:1811.09658, 2018.

• Y. Cao, J. Gross and D. Joyce, On orientations for moduli

spaces of coherent sheaves on Calabi–Yau manifolds, in preparation, 2019.

16 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

Variations of the ASD instanton moduli space

• I. Kapustin–Witten equations on closed four-manifolds

Let X be a closed, oriented, smooth four-manifold, and let P → X be a principal G-bundle over X, where G is a connected Lie group. For (A, a) ∈ AP × Γ(gP ⊗ Λ1

X), we consider the following equations:

d∗

Aa = 0,

d−

A a = 0, and

F +

A + π+([a, a]) = 0,

where d−

A := π−(dA), and π− : Γ(gP ⊗ Λ2 X) → Γ(gP ⊗ Λ−) is the

projection. Remark: Kapustin and Witten introduced a one parameter family

• f equations. The above is obtained by specifying the parameter,

and corresponds to Simpson’s equations (i.e. (poly-)stable Higgs bundles) when the underlying manifold is a K¨ ahler surface.

17 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

The gauge group GP = Aut (P) acts on pairs (A, a) ∈ AP × Γ(gP ⊗ Λ1

X). We say that (A, a) is irreducible if the

stabilizer group in GP is trivial. Denote by MKW

P

the moduli space of gauge equivalence classes of irreducible solutions (A, a) to the Kapustin–Witten equations. The elliptic operator E• = (E0, E1, D) in this case is D : Γ(Λ1

X ⊕ Λ1 X) → Γ(Λ0 X ⊕ Λ+ X ⊕ Λ− X ⊕ Λ0 X),

where D := d∗ d+ d− d∗ T . So E• is the direct sum E• = E +

• ⊕ E −
• , where E +
• is as in the

anti-self-dual instantons case, and E −

• is E +
• for the opposite
• rientation on X. Thus OE•

P ∼

= OE +

• P

⊗Z2 OE −

• P .

18 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

The orientation bundle of MKW

P

is the pull-back of OE•

P → BP

under the forgetful map MKW

P

→ BP. Hence we obtain: Theorem (Joyce–T–Upmeier) Let X be a closed, oriented, smooth four-manifold, and let P → X be a principal G-bundle over X, where G is a connected Lie group. 1) Choose an orientation on H2(X) and on g, and a Spinc-structure s on X. Then we can construct a canonical

• rientation on MKW

P

for all principal G-bundles P → X, as a derived manifold. 2) If G is simply-connected, or if G = U(m), then the orientation

• n MKW

P

in the above 1) is independent of the choice of Spinc-structure s.

19 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

• II. Vafa–Witten equations on closed four-manifolds

Let X be a closed, oriented, smooth four-manifold, and let P → X be a principal G-bundle over X, where G is a connected Lie group. For (A, B, C) ∈ AP × Γ(gP ⊗ Λ+

X) × Γ(gP ⊗ Λ0 X), we consider the

following equations: d∗

AB + dAC = 0, and

F +

A + [B, C] + [B.B] = 0,

where [B.B] ∈ Γ(gP ⊗ Λ+

X).

The gauge group GP = Aut (P) acts on pairs (A, B, C) ∈ AP × Γ(gP ⊗ Λ+

X) × Γ(gP ⊗ Λ0 X). We say that

(A, B, C) is irreducible if the stabiliser group in GP is trivial. Denote by MVW

P

the moduli space of gauge equivalence classes of irreducible solutions (A, B, C) to the Vafa–Witten equations.

20 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems

Gauge-theoretic moduli problem Anti-self-dual instanton moduli space Orientations

The elliptic operator E• = (E0, E1, D) in this case is D : Γ(Λ0

X ⊕ Λ+ X ⊕ Λ1 X) → Γ(Λ1 X ⊕ Λ0 X ⊕ Λ+ X),

where D :=   d d∗ d∗ d+   . So, E• = E +

• ⊕ (E +
• )∗. Thus, the orientation bundle OE•

P

becomes OE•

P ∼

= OE +

• P

⊗Z2 O(E +

• )∗

P

∼ = OE +

• P

⊗Z2 (OE +

• P )∗ ∼

= BP × Z2, namely it is canonically trivial. As in the case of the Kapustin–Witten equations, the orientation bundle of MVW

P

is the pull-back of OE•

P → BP under the forgetful map MVW P

→ BP. Hence we

• btain:

Theorem (Joyce–T–Upmeier) The Vafa–Witten moduli spaces MVW

P

have canonical orientations for all G and principal G-bundles P → X, as a derived manifold.

21 / 21 Yuuji Tanaka, Oxford University Orientations for gauge-theoretic moduli problems