Donaldson Invariants and Moduli of Yang-Mills Instantons 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 space , 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 topology) 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 (instanton 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).