Donaldson Invariants and of Yang-Mills

Jacob Gross

Lincoln College Oxford University

(slides posted at users.ox.ac.uk/ linc4221) ∼ The ASD Equation in Low Dimensions, 17 November 2017

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Moduli and Invariants

Invariants are gotten by cooking up a number (homology group, derived category, etc.) using auxiliary data (a metric, a polarisation, etc.) and showing independence of that initial choice.

Donaldson and Seiberg-Witten invariants count in moduli spaces of solutions of a pde (shown to be independent of conformal class of the metric).

Example (baby example) f :(M, g) R, the number of solutions of →

gradgf = 0

is independent of g (Poincare-Hopf).´

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Recall (Setup)

(X, g) smooth oriented Riemannian 4-manifold and a principal G-bundle π : P X over it. → Consider the Yang-Mills functional

YM(A) = F 2dμ. | A| ZM The corresponding Euler-Lagrange equations are

dA∗FA = 0.

Using the gauge group one generates lots of solutions from one. But one can fix aG gauge

dA∗a = 0

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Recall (ASD Equations)

2 2 2 In dimension 4, the Hodge star splits the 2-forms Λ = Λ+ Λ . This splits the curvature ⊕ −

+ FA = FA + FA−. Have

+ 2 2 + 2 2 κ(P) = F F − YM(A) = F + F − , k A k − k A k ≤ k A k k A k where κ(P) is a topological invariant (e.g. for SU(2)-bundles, 2 κ(P) = 8π c2(P)). This comes from Chern-Weil theory. − + Key Idea. Anti-self-dual connections, e.g. ones with FA = 0, minimise the Yang-Mills functional among connections on P.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Recall (ADHM Construction)

Atiyah-Drinfeld-Hitchin-Manin constructions describes all finite energy instantons over R4 and S4 in terms of algebraic data: a k-dimensional complex vector , self adjoint linear H operators Ti : , n-dim. vector space E , and a map H → H ∞ P : E S+ satisying the ADHM equations [ADHM]. ∞ → H ⊗

Example 4 Consider SU(2) bundle P R with c2(E) = 1. Then the → + ADHM matricesTi are 1 1 and P Hom(E , S ) is rank 1. × ∈ ∞ So the moduli space of YM instantons is R4 R+. ×

The Point. Instantons occur in moduli spaces.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons The Ambient Space

We begin with the space of all connections, modulo gauge.

Let G be SO(3) or SU(2). Let P X be a principal G-bundle over a smooth oriented Riemannian→ 4-manifold X.

infinite-dimensional affine space of connections on P A infinite-dimensional group of automorphisms of P, it acts on connectionsG via 1 u A := A ( u)u− . ∙ − ∇A Definition The space := / B A G (with quotient ) is the space of connections modulo gauge (on P).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Metrical Proporties

carries an L2-metric B 2 1/2 A B 2 := ( A B dμ) k − kL | − | ZM descends to a distance function

d([A], [B]) := infu A u B ∈Gk − ∙ k (clear, except for d([A], [B]) = 0 [A] = [B]; uses compactness of G) ⇒

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Local Models of the Orbit Space B Definition Fix A and  > 0. Define ∈ A 1 TA, = a Ω (gE ) d ∗a = 0, a 2 <  A L` 1 { ∈ | k k − } By Uhlenbeck’s gauge fixing theorem and the implicit function theorem, A + TA, meets every nearby orbit. Definition (/Lemma)

Write ΓA = Iso A. It is isomorphic CG(HA), where HA denotes the holonomy groupG of the connection A.

Theorem For small  > 0 the projection induces a A → B homeomorphism from T /Γ to a neighborhood of [A] in . A, A B

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Reducibility

Write for the subspace of with minimial -isotropy groups A∗ A G

∗ = A Γ = C(G) . A { ∈ A| A } These are called irreducible connections.

Theorem

The quotient ∗ = ∗/ is a smooth Banach manifold (the moduli spaceB of irreducibleA G connections).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Local Models of the ASD Moduli Space

Let MP (g) be the moduli space of ASD connections on P (X, g). Define a map ψ : T Ω+(g ) by → A, → E

ψ(a) = F +(A + a) and write 1 Z(ψ) := ψ− (0) T ⊂ A, Theorem The previous homeomorphism induces a homeomorphism from Z(ψ)/ΓA to a neighborhood of [A] in MP (g).

+ Have a bundle := ∗ /C( ) Ω (gE ) ∗. Taking the ASD E A ×G G → B portion of curvature yields a section Ψg : s.t. B∗ → E 1 Ψ− (0) = M∗ (g) ∗ P ⊂ B (the the moduli space of irreducible ASD connections).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Deformation-Obstruction Theory

Have a linear elliptic operator

d+ d 1 A A∗ 2 0 C∞(ad(P) Λ T ∗X) ⊕ C∞(ad(P) (Λ T ∗X) (Λ T ∗X)) ⊗ −−−−→ ⊗ + ⊕ its kernel is deformations of the ASD condition along gauge fixing. + ker (d d ∗) = T M A ⊕ A A and + coker (d d ∗) = O M. A ⊕ A A

Remark. It will turn out that for generic g, OAM = 0. This is a nice deformation-obstruction theory.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Deformation-Obstruction Theory

The deformation-obstruction theory of the ASD moduli problem is governed by an elliptic complex

d d+ Ω0 (g ) A Ω1 (g ) A Ω+(g ) X E −→ X E −−→ X E (d + d = 0 by ASD condition). A ◦ A Note H1 ker and H2 coker . A ∼= δA A ∼= δA The virtual dimension of the moduli problem is

8c2(E) 3(1 b1(X) + b (X)) − − + for SU(2) and

s = 2p1(E) 3(1 b1(X) + b (X)) − − − + for SO(3).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Analogy with other gauge theories

Flat connections (Casson theory)

d d d Ω0 A Ω1 A Ω2 A Ω3 −→ −→ −→ Holomorphic structures (Donaldson-Thomas theory)

∂ ∂ ∂ Ω0,0 A Ω0,1 A Ω0,2 A Ω0,3 −→ −→ −→

G2-instantons (Donaldson-Segal programme, ongoing)

d ψ d d Ω0 A Ω1 ∧ A Ω6 A Ω7 −→ −−−→ −→ Spin(7)-instantons (unsure of current state)

d π7 d Ω0 A Ω1 ◦ A Ω2 −→ −−−→ 7

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Finite-Dimensional Models

Theorem If A is an ASD connection over a neighborhood of X, a 1 neighborhood of [A] in M is modelled on a quotient f − (0)/ΓA, + + where f : ker (dA dA∗) coker (dA dA∗) is a ΓA-equivariant map. ⊕ → ⊕

Definition 2 An ASD connection is called regular if HA = 0 (unobstructed).

Definition An ASD moduli space is called regular if all of its points are regular.

If the moduli space if regular, 0 is a regular value, and Sard-Smale is applicable.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Transversality

We need some bits of Fredholm theory. Theorem (Sard-Smale) Let F : P Q be a smooth Fredholm operator between paracompact→ Banach manifolds. Then, the set of regular values of F is everywhere dense in Q. Moreover, if P is connected 1 then the fibre F − (y) P over any regular value y Q is a smooth submanifold of⊂ dimension equal to the Fredholm∈ index of F.

Theorem (Fredholm Transversality) Let F : P Q be a smooth Fredholm operator between paracompact→ Banach manifolds and let S be a finite-dimensional manifold equipped with a smooth map h : S Q. Then there is a (C ) arbitrarily close map → ∞ h : S Q, to h, such that h is transverse to F. 0 → 0

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Transversality

Write for the space of conformal classes of Riemannian metricsC on X. We obtain a parameterized bundle E → B∗ × C along with section Ψ: ∗ s.t. Ψ g = Ψg. C × B → E { }×B∗ Theorem (Uhlenbeck-Freed) Let π : P X be a principal G-bundle over a smooth oriented → 4-manifold X. Then the zero set of Ψ in is regular. B∗ × C Corollary There is a dense subset such that for [g] the moduli C0 ⊂ C ∈ C0 space MP∗ (g) is a regular (as the zero set of the section Ψg determined by [g]).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Compactness

Definition

An ideal ASD connection over X is a pair ([A], (x1, ..., x`)).A sequence of gauge equivalence classes [Aα] converges weakly to a limiting ideal connection ([A], (x1, ..., x`)) if 2 2 2 ` X f F(Aα) dμ X f F(A) dμ + 8π r=1 f (xr ) (the action| densities| → converge| as| measures) R R P there are bundle maps ρα : P` X x ,..,x Pk X x ,..,x s.t. | { 1 `} → | { 1 `} ρα∗ (Aα) converges (in C∞ on compact subsets of the punctured manifold) to A Write M for the set of all ideal connections (with a metrizable topology induced from the notion of weak covergence).

Theorem Any infinite seqeunce in M has a weakly convergent subsequence with a limit point in M.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Orientation

One can make sense of orientation data on all of . At each connection A ( or not) write elliptic operatorB

2 0 L : ad(E) T ∗X ad(E) (Λ Λ ) A ⊗ → ⊗ + ⊕ At each point form

top top 1 Λ (ker L ) (Λ (coker L ))− A ⊗ A as the fibre of a real line bundle L (orientation line bundle). → B

This induces an orientation on M . ⊂ B

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Invariants in dimension zero

First suppose a zero-dimensional moduli space ASD(g). Wish to show a count (with correct multiplicities) is independentM of g.

Example (Toy Example.) Take π : V B finite-rank and s section of π which is transverse→ to the zero section. Extract a homology class [Z(s)] H (B), ∈ d d := dim B rk V . This is the Euler class of V . −

For each choice of metric g, define an integer

q(g) = (A), A (g) ∈MXP where the sign  is determined by our orientation data L.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Invariants in dimension zero

Theorem

For any two generic metrics g0 and g1 on X,

q(g0) = q(g1).

Following the masters, we return to our toy example before proving this theorem.

Let C be a finite-dimensional manifold parameterising sections sc. Have a parameterized zero set Z B C. ⊂ ×

The number of zeros of some sc is the number of elements of the fibre π : Z C over c C–the degree of a proper map. → ∈

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Invariance of Degree

Theorem The degree of π : Z C is invariant of the choice in c C in the target. → ∈

Proof. Take a path γ : I C transverse to im(π). Define → W = (z, t) π(z) = γ(t) Z [0, 1] , { | ⊂ × } which is a smooth one-dimensional manifold with boundary. 1 The boundary component at 0 is π− (c0) and the boundary 1 component is π− (c1). And the total oriented boundary of a compact one-dimensional manifold is zero.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Invariance of Donaldson Invariants

Proof. Take a path γ :[0, 1] in the space of conformal classes of → C metrics which joins g0 and g1. Using Fredholm transversality, perturb it to be transverse to the Fredholm map . For M∗ → C generic γ, the space

= (A, t) A M(γ(t)) ∗ [0, 1] M { | ∈ } ⊂ B ×

is an oriented one-manifold with boundary components M(g0) and M(g1). Uhlenbeck’s compactness theorem completes the proof [We2].

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Polynomial Invariants

The dimension of MASD is not, in general, zero. Suppose + b (X) > 1 (avoid reducible solutions). Take MASD even dimensional

+ dim M = 2d = 8c2(E) 3(b (X) + 1). ASD −

Let [Σ1], ..., [Σd ] be classes in H2(X, Z). Atiyah-Bott [] shown that the cohomology ring X∗ is a polynomial algebra generated in degrees two and four, whereB the two-dimensional generators are gotten from a certain map

2 μ : H2(X) H ( ∗ ). → BX In this way, in positive dimension, we can get integers

q = μ(Σ1) ... μ(Σ ), [M] h ∪ ∪ d i

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Polynomial Invariants

This gives a symmetric mutlilinear function

q : H2(X) ... H2(X) Z × × → such that

qˉ([Σ1], ..., [Σ ]) = q([Σ1], ..., [Σ ]) (reversing orientation) d − d if f : X Y is an orientation preserving diffeomorphism then → q(f ([Σ1]), ..., f ([Σd ])) = q([Σ1], ..., [Σd ])

Proving these more general invariants exists and satisfy the above properties is analytically more difficult than the aforementioned simple (dim. zero) invariants (done in [DK, Sect. 9.2.]).

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons Wall-Crossing Phenomena

2 When b+ = 1, the Donaldson invariants are not independent of the metric.

The paths between g0 and g1 cannot always avoid reducible solutions–which occur as codimension one walls in . C Within chambers, the Donaldson invariants are constant. When crossing a wall, they change by an explicit formula

ΦX,g1 ΦX, = δX , − g1,g2 where the X can be computed explicitly using modular δg1,g2 forms [Gott96].

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons References

[AB82 ] M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London, Series A 308 (1982), 523-615. [DK90 ] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford U.P., 1990. [DT98 ] S. K. Donaldson and R. Thomas, in higher dimensions, The Geometric Universe, Oxford UP, 1998, pp. 31-47. [Gott96 ] L. Gottsche.¨ Modular Forms and Donaldson Invariants for 4-Manifolds with b+=1. Journal of the American Mathematical Society, Vol. 9, Number 3, July 1996.

Jacob Gross Donaldson Invariants and Moduli of Yang-Mills Instantons