# Donaldson Invariants and Moduli of Yang-Mills Instantons

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1 Donaldson Invariants and Moduli of Yang-Mills Instantons Lincoln College Oxford University (slides posted at users.ox.ac.uk/ linc4221) The ASD Equation in Low Dimensions, 17 November 2017

2 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 grad g f = 0 is independent of g (Poincaré-Hopf).

3 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 A 2 dμ. The corresponding Euler-Lagrange equations are M d A F A = 0. Using the gauge group G one generates lots of solutions from one. But one can fix a gauge d A a = 0

4 Recall (ASD Equations) In dimension 4, the Hodge star splits the 2-forms Λ 2 = Λ 2 + Λ 2. This splits the curvature Have F A = F + A + F A. κ(p) = F + A 2 F A 2 YM(A) = F + A 2 + F A 2, where κ(p) is a topological invariant (e.g. for SU(2)-bundles, κ(p) = 8π 2 c 2 (P)). This comes from Chern-Weil theory. Key Idea. Anti-self-dual connections, e.g. ones with F + A = 0, minimise the Yang-Mills functional among connections on P.

5 Recall (ADHM Construction) Atiyah-Drinfeld-Hitchin-Manin constructions describes all finite energy instantons over R 4 and S 4 in terms of algebraic data: a k-dimensional complex vector space H, self adjoint linear operators T i : H H, n-dim. vector space E, and a map P : E H S + satisying the ADHM equations [ADHM]. Example Consider SU(2) bundle P R 4 with c 2 (E) = 1. Then the ADHM matricest i are 1 1 and P Hom(E, S + ) is rank 1. So the moduli space of YM instantons is R 4 R +. The Point. Instantons occur in moduli spaces.

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

7 Metrical Proporties B carries an L 2 -metric A B L 2 := ( descends to a distance function M A B 2 dμ) 1/2 d([a], [B]) := inf u G A u B (clear, except for d([a], [B]) = 0 [A] = [B]; uses compactness of G)

8 Local Models of the Orbit Space B Definition Fix A A and ɛ > 0. Define T A,ɛ = {a Ω 1 (g E ) d A a = 0, a L 2 l 1 < ɛ} By Uhlenbeck s gauge fixing theorem and the implicit function theorem, A + T A,ɛ meets every nearby orbit. Definition (/Lemma) Write Γ A = Iso G A. It is isomorphic C G (H A ), where H A denotes the holonomy group of the connection A. Theorem For small ɛ > 0 the projection A B induces a homeomorphism from T A,ɛ /Γ A to a neighborhood of [A] in B.

9 Reducibility Write A for the subspace of A with minimial G-isotropy groups A = {A A Γ A = C(G)}. These are called irreducible connections. Theorem The quotient B = A /G is a smooth Banach manifold (the moduli space of irreducible connections).

10 Local Models of the ASD Moduli Space Let M P (g) be the moduli space of ASD connections on P (X, g). Define a map ψ : T A,ɛ Ω + (g E ) by and write Theorem ψ(a) = F + (A + a) Z (ψ) := ψ 1 (0) T A,ɛ The previous homeomorphism induces a homeomorphism from Z (ψ)/γ A to a neighborhood of [A] in M P (g). Have a bundle E := A G/C(G) Ω + (g E ) B. Taking the ASD portion of curvature yields a section Ψ g : B E s.t. Ψ 1 (0) = MP (g) B (the the moduli space of irreducible ASD connections).

11 Deformation-Obstruction Theory Have a linear elliptic operator C (ad(p) Λ 1 T X) d + A d A C (ad(p) (Λ 2 +T X) (Λ 0 T X)) its kernel is deformations of the ASD condition along gauge fixing. ker (d + A d A ) = T AM and coker (d + A d A ) = O AM. Remark. It will turn out that for generic g, O A M = 0. This is a nice deformation-obstruction theory.

12 Deformation-Obstruction Theory The deformation-obstruction theory of the ASD moduli problem is governed by an elliptic complex (d + A d A = 0 by ASD condition). Ω 0 X (g E) d A Ω 1 X (g E) d + A Ω + X (g E) Note H 1 A = ker δ A and H 2 A = coker δ A. The virtual dimension of the moduli problem is for SU(2) and for SO(3). 8c 2 (E) 3(1 b 1 (X) + b + (X)) s = 2p 1 (E) 3(1 b 1 (X) + b + (X))

13 Analogy with other gauge theories Flat connections (Casson theory) Ω 0 d A Ω 1 d A Ω 2 d A Ω 3 Holomorphic structures (Donaldson-Thomas theory) Ω 0,0 A Ω 0,1 A Ω 0,2 A Ω 0,3 G 2 -instantons (Donaldson-Segal programme, ongoing) Ω 0 d A Ω 1 ψ d A Ω 6 d A Ω 7 Spin(7)-instantons (unsure of current state) Ω 0 d A Ω 1 π 7 d A Ω 2 7

14 Finite-Dimensional Models Theorem If A is an ASD connection over a neighborhood of X, a neighborhood of [A] in M is modelled on a quotient f 1 (0)/Γ A, where f : ker (d + A d A ) coker (d + A d A ) is a Γ A-equivariant map. Definition An ASD connection is called regular if H 2 A = 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.

15 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 then the fibre F 1 (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.

16 Transversality Write C for the space of conformal classes of Riemannian metrics on X. We obtain a parameterized bundle E B C along with section Ψ : C B E s.t. Ψ {g} B = Ψ g. 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 B C is regular. Corollary There is a dense subset C C such that for [g] C the moduli space M P (g) is a regular (as the zero set of the section Ψ g determined by [g]).

17 Compactness Definition An ideal ASD connection over X is a pair ([A], (x 1,..., x l )). A sequence of gauge equivalence classes [A α ] converges weakly to a limiting ideal connection ([A], (x 1,..., x l )) if X f F(A α) 2 dμ X f F(A) 2 dμ + 8π 2 l r=1 f (x r ) (the action densities converge as measures) there are bundle maps ρ α : P l X {x1,..,x l } P k X {x1,..,x l } s.t. ρ α(a α ) 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.

18 Orientation One can make sense of orientation data on all of B. At each connection A (instanton or not) write elliptic operator At each point form L A : ad(e) T X ad(e) (Λ 2 + Λ 0 ) Λ top (ker L A ) (Λ top (coker L A )) 1 as the fibre of a real line bundle L B (orientation line bundle). This induces an orientation on M B.

19 Invariants in dimension zero First suppose a zero-dimensional moduli space M ASD (g). Wish to show a count (with correct multiplicities) is independent of g. Example (Toy Example.) Take π : V B finite-rank vector bundle and s section of π which is transverse to the zero section. Extract a homology class [Z (s)] H d (B), d := dim B rk V. This is the Euler class of V. For each choice of metric g, define an integer q(g) = ɛ(a), A M P (g) where the sign ɛ is determined by our orientation data L.

20 Invariants in dimension zero Theorem For any two generic metrics g 0 and g 1 on X, q(g 0 ) = q(g 1 ). Following the masters, we return to our toy example before proving this theorem. Let C be a finite-dimensional manifold parameterising sections s c. Have a parameterized zero set Z B C. The number of zeros of some s c is the number of elements of the fibre π : Z C over c C the degree of a proper map.

21 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. The boundary component at 0 is π 1 (c 0 ) and the boundary component is π 1 (c 1 ). And the total oriented boundary of a compact one-dimensional manifold is zero.

22 Invariance of Donaldson Invariants Proof. Take a path γ : [0, 1] C in the space of conformal classes of metrics which joins g 0 and g 1. Using Fredholm transversality, perturb it to be transverse to the Fredholm map M C. For generic γ, the space M = {(A, t) A M(γ(t))} B [0, 1] is an oriented one-manifold with boundary components M(g 0 ) and M(g 1 ). Uhlenbeck s compactness theorem completes the proof [We2].

23 Polynomial Invariants The dimension of M ASD is not, in general, zero. Suppose b + (X) > 1 (avoid reducible solutions). Take M ASD even dimensional dim M ASD = 2d = 8c 2 (E) 3(b + (X) + 1). Let [Σ 1 ],..., [Σ d ] be classes in H 2 (X, Z). Atiyah-Bott [] shown that the cohomology ring BX is a polynomial algebra generated in degrees two and four, where the two-dimensional generators are gotten from a certain map μ : H 2 (X) H 2 (B X ). In this way, in positive dimension, we can get integers q = μ(σ 1 )... μ(σ d ), [M]

24 Polynomial Invariants This gives a symmetric mutlilinear function q : H 2 (X)... H 2 (X) Z such that ˉq([Σ 1 ],..., [Σ d ]) = q([σ 1 ],..., [Σ d ]) (reversing orientation) if f : X Y is an orientation preserving diffeomorphism then q(f ([Σ 1 ]),..., f ([Σ d ])) = q([σ 1 ],..., [Σ d ]) 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.]).

25 Wall-Crossing Phenomena When b 2 + = 1, the Donaldson invariants are not independent of the metric. The paths between g 0 and g 1 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 Φ X,g 1 Φ X,g 2 = δ X g 1,g 2, where the δ X g 1,g 2 can be computed explicitly using modular forms [Gott96].

26 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), [DK90 ] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford U.P., [DT98 ] S. K. Donaldson and R. Thomas, Gauge theory in higher dimensions, The Geometric Universe, Oxford UP, 1998, pp [Gott96 ] L. Göttsche. Modular Forms and Donaldson Invariants for 4-Manifolds with b+=1. Journal of the American Mathematical Society, Vol. 9, Number 3, July 1996.

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