Bubble tree compactification of instanton moduli spaces on 4-orbifolds
BBubble tree compactification of instanton moduli spaceson 4-orbifolds
Shuaige Qiao
Abstract
In the study of moduli spaces defined by the anti-self-dual (ASD) Yang-Mills equa-tions on SU (2) or SO (3) bundles over closed oriented Riemannian 4-manifolds M ,the bubble tree compactification was defined in [T88], [F95], [C02], [C10], [F14] and[F15]. The smooth orbifold structure of the bubble tree compactification away fromthe trivial stratum is defined in [C02] and [C10]. In this paper, we apply the tech-nique to 4-orbifolds of the form M/ Z α satisfying Condition 1.1 to get a bubble treecompactification of the instanton moduli space on this particular kind of 4-orbifolds. Contents P i and getting an approximate ASD connection A (cid:48) . . . . . . . . . . 113.2 Constructing an ASD connection from A (cid:48) . . . . . . . . . . . . . . . . . . . . 18 Z α -equivariant bundles and Z α -invariant moduli spaces 326 Invariant ASD connections on S S . . . . . . . . . . . . . . . . . . . . . . . . . 366.2 Γ-invariant connections in M ( S ) . . . . . . . . . . . . . . . . . . . . . . . 376.3 Z p -invariant connections in M k ( S ) . . . . . . . . . . . . . . . . . . . . . . . 386.4 Balanced Z p -invariant connections on S . . . . . . . . . . . . . . . . . . . . 39 M/ Z α a r X i v : . [ m a t h . DG ] F e b An example: instanton moduli space on SO (3) -bundle over CP and weightedcomplex projective space CP r,s,t ) M p = − ( CP ) and its bubble tree compactification . . . . . . . . . . . . . . . 488.2 M Z a ( P − ( CP )) and its bubble tree compactification . . . . . . . . . . . . . 52 From the late 1980s, gauge theoretic techniques were applied in the area of finite groupactions on 4-manifolds. [F89] showed that on S , there is no smooth finite group actionwith exactly 1 fixed point by arguing that instanton-one invariant connections form a 1-manifold whose boundary can be identified as fixed points of the group action. In [BKS90],gauge theoretic techniques were used in studying fixed points of a finite group action on3-manifolds. [FS85] studied pseudofree orbifolds using ASD moduli spaces. A 4-dimensionalpseudofree orbifold is a special kind of orbifold which can be expressed as M /S , a quotientof a pseudofree S -action on a 5-manifold M . [A90] studied the orbifold S / Z α , which isa compactification of L ( α, β ) × R where L ( α, β ) is a Lens space. Austin gave a criterionfor existence of instantons on S / Z α and calculated the dimension of the instanton modulispace. A more general kind of orbifold, orbifold with isolated fixed points, was discussed in[F92], especially when the group-action around each singular point is a cyclic group.Let ( M, g ) be a closed, connected, oriented 4-dimensional Riemannian manifold, G = SU (2) or SO (3), P be a principal- G bundle over M , k = c ( P ) or − p ( P ) / A be thespace of connections, G be the space of gauge transformations, and B = A / G . There is aYang-Mills functional on B which measures the energy of a connection A :[ A ] (cid:55)→ Y M ( A ) := (cid:90) | F A | dvol where F A ∈ Ω ( adP ) is the curvature of A . Since 2-forms on M can be decomposed intoself-dual and anti-self-dual parts, we haveΩ ( adP ) = Ω , + ( adP ) ⊕ Ω , − ( adP ) . Denote by P + : Ω ( adP ) → Ω , + ( adP ) the projection. A connection A is anti-self-dual(ASD) if P + F A = 0. Denote the moduli space of ASD connections by M k ( M ). If A is ASD,we have Y M ( A ) = 8 π k .A connection A is reducible if there exists a principal U (1)-bundle Q such that P = Q × U (1) G and A is induced by a connection on Q . Note that A is reducible ⇔ Γ A = U (1) ,A is irreducible ⇔ Γ A = (cid:40) Z in SU (2)-bundle0 in SO (3)-bundle , where Γ A ⊂ G is the stabiliser of A . The irreducible part of A , B and M k are denoted by A ∗ , B ∗ and M ∗ k respectively. 2n 4-manifolds, the bubble tree compactification of ASD moduli spaces is defined in[T88], [F95], [C02], [C10], [F14] and [F15]. The smooth orbifold structure of the bubbletree compactification away from the trivial stratum is defined in [C02] and [C10]. We aimto apply the technique in [C02] and [C10] to define the smooth orbifold structure of thecorresponding bubble tree compactification for 4-orbifolds of the form M/ Z α which satisfythe following condition: Condition 1.1.
The Z α -action on M is free away from finite points and { x , . . . , x n } ⊂ X := M/ Z α are singularities such that for each i a neighbourhood of x i in X is cL ( a i , b i ), acone over a Lens space where a i divides α and b i is coprime to a i . (cid:4) In Section 2, we prove the equivariant removable singularity theorem.In Section 3, we introduce the equivariant Taubes gluing construction by following themethods in Section 7.2 of [DK90].In Section 4, we briefly review the bubble tree compactification for 4-manifolds.In Section 5, we first characterise Z α -equivariant bundles over M satisfying Condition1.1. Recall that SU (2) (or SO (3))-bundles over a simply connected manifold M are classifiedby c (or ( p , w )). While Z α -equivariant bundles are characterised by these characteristicclasses along with isotropy representations of each singular point. In Section 5 we definea set {O i } i ∈ I and an injective map from the space of isomorphic classes of Z α -equivariantbundles to {O i } i ∈ I where I is the index set defined in (5.3). That is to say, Z α -equivariantbundles are characterised by the index set I . After that, we make use of some results in[F89], [F92] and [FS85] to describe the Z α -invariant moduli spaces on M .In Section 6, we state some results from [A90], [F92] and [FS85] to describe the Z α -invariant ASD moduli spaces on S .In Section 7, we define the bubble tree compactification on M/ Z α . Observe that abubble on singular points of an orbifold M/ Z α may be a “singular bubble” S / Z α . Thegluing parameter needs to be Z α -equivariant. After dealing with these problems properly,techniques in [C02] can be applied and we get the bubble tree compactification for M/ Z α . Theorem 1.2. (Theorem 7.9) Suppose we have the Z α -action on M satisfies Condition 1.1and P → M is an SU (2) (or SO (3) )-bundle with c (or p )=k. Let the bubble tree compacti-fication of the irreducible Z α -invariant instanton moduli space be M k, Z a := (cid:71) i ∈ I M O i , where I is the index set defined in (5.3). Then each component M O i is an orbifold away fromthe stratum with trivial connection as the background connection of the bubble tree instantons.The dimension of the component M O with O = { k, ( a i , b i , m i ) ni =1 } is, for SU (2) case, c α − b +2 ) + n (cid:48) + n (cid:88) i =1 a i a i − (cid:88) j =1 cot (cid:18) πjb i a i (cid:19) cot (cid:18) πja i (cid:19) sin (cid:18) πjm i a i (cid:19) , where n (cid:48) is the number of { m i | m i (cid:54)≡ mod a i } . For SO (3) case replace c by − p .
3n Section 8, we calculate examples for the complex projective space CP and the weightedcomplex projective space CP r,s,t ] . Acknowledgements . The paper would not be possible without my supervisor Bai-LingWang’s guidance and encouragement. I want to express to him my heartfelt appreciation. Myspecial thanks to the ANU-CSC scholarship and the MSI ‘Kick-start’ Postdoctoral Fellowshipfor the financial support.
The Uhlenbeck removable singularity theorem tells us that any ASD connection A on a trivialprincipal G -bundle over a punctured 4-dimensional ball with L -bounded curvature can beextended to the origin of the ball. This section discusses the case when there is a finite groupΓ acting on ( B \ { } ) × G and A is a Γ-invariant ASD connection. The main result showsthat the Γ-action can also be extended to B × G such that the extended connection is Γ-invariant. Moreover, we will see how the connection A determines the isotropy representation ρ : Γ → G at the origin. We first state Theorem 4.1 of [U82] here.
Theorem 2.1. (Uhlenbeck’s removable singularity theorem) Let A be an ASD connection onthe bundle P = ( B \ { } ) × G . If || F A || L < ∞ , then there exists an ASD connection ˜ A on B × G and a gauge transformation g : P → P such that g ∗ A = ˜ A | B \{ }× G . In the proof of this theorem, a trivialisation τ on P is defined under which A can beexpressed as a g -valued one-form on B \ { } A τ : T ( B \ { } ) → g satisfying | A τ ( x ) | x → −−→
0. Thus A τ can be extended to the origin.To get a better understanding of this theorem, we translate B \ { } into S × (0 , ∞ ) bythe map f : S × (0 , ∞ ) → B \ { } ( ψ, t ) (cid:55)→ ( ψ, e − t ) . Then f ∗ A is an ASD connection on S × (0 , ∞ ) × G . To simplify the notation, we denote f ∗ A also by A . When we restrict A to the slice S × { t } , we get a connection A t on S × { t } × G . Theorem 2.2. (Theorem 4.18 of [D02]) Let Y be a 3-dimensional closed oriented Rieman-nian manifold and A be an ASD connection over a half-tube Y × (0 , ∞ ) with (cid:82) Y × (0 , ∞ ) | F A | < ∞ . Then the connections [ A t ] converge (in C ∞ topology on A Y / G Y ) to a limiting flat con-nection [ A ∞ ] over Y as t → ∞ .
4y Theorem 2.2, there exists a flat connection A ∞ such that, after gauge transformationif necessary, A t → A ∞ in C ∞ topology. The removable singularity theorem means that thereexists a gauge transformation g on S × (0 , ∞ ) × G such that lim t →∞ g | S ×{ t } = g ∞ and g ∗∞ A ∞ = A trivial is the trivial connection on S × G . Thus A can be extended to the infinity point (corre-sponding to the origin in the punctured ball). Γ -equivariant removable singularity Let Γ be a finite group acting on P = B \ { } × G through ρ : Γ → Aut ( P )such that ρ comes from an element in Hom(Γ , G ), i.e., for all γ ∈ Γ , x , x ∈ B \ { } , wehave pr (cid:0) ρ ( γ )( x , g ) (cid:1) = pr (cid:0) ρ ( γ )( x , g ) (cid:1) where pr : ( B \ { } ) × G → G is the projectivemap. Assume that this Γ-action on P induces a Γ-action on B \ { } that preserves themetric. Theorem 2.3.
Suppose A is an ASD Γ -invariant connection on P = ( B \ { } ) × G with || F A || L < ∞ . Then there exists an ASD connection ˜ A on B × G , a Γ -action on B × G ˜ ρ : Γ → Aut ( B × G ) and a Γ -equivariant gauge transformation g : P → P such that ˜ A is Γ -invariant and g ∗ A =˜ A | B \{ }× G .Proof. Let ˜ A and g be defined as in Theorem 2.1 and define ˜ ρ by˜ ρ ( γ )( p ) = (cid:40) g − ◦ ρ ( γ ) ◦ g ( p ) ∀ p ∈ B \ { } × G lim q → p g − ◦ ρ ( γ ) ◦ g ( q ) ∀ p ∈ { } × G .
To show ˜ ρ is well-defined, it suffices to show the limit in the definition exists.For all γ ∈ Γ, we have ( g − ◦ ρ ( γ ) ◦ g ) ∗ ˜ A = ˜ A (2.1)on B \ { } × G . Choose a trivialization (i.e. a section) τ of B × G so that ˜ A can be writtenas ˜ A τ : T B → g and that ˜ A τ (0) = 0. Then g − ◦ ρ ( γ ) ◦ g : P → P can be seen as a map ˜ ρ τ : B \ { } → G under this trivialization: γ · τ ( b ) = τ ( γ · b ) · ˜ ρ τ ( b )where γ ∈ Γ, b ∈ B \ { } . Under this trivialization, the connection ( g − ◦ ρ ( γ ) ◦ g ) ∗ ˜ A canbe written as (cid:0) ( g − ◦ ρ ( γ ) ◦ g ) ∗ ˜ A (cid:1) τ : T ( B \ { } ) → g ( b, v ) (cid:55)→ ( ˜ ρ τ ) − d ˜ ρ τ ( b, v ) + ( ˜ ρ τ ) − ( b ) ˜ A τ ( γ · b, γ ∗ v ) ˜ ρ τ ( b ) . ρ τ ) − d ˜ ρ τ + ( ˜ ρ τ ) − ˜ A τ ˜ ρ τ = ˜ A τ . Taking the limit of the equality as x →
0, since ˜ A τ (0) = 0, we have d ˜ ρ τ →
0, which meansthat ˜ ρ τ tends to a constant in G as x → ρ induces a Γ-action on { } × G , which is called an isotropy representation and denoted as ˜ ρ : Γ → G. (2.2)Since A is Γ-invariant, we can treat A t as a connection A Γ t on S × ρ G . From the discussionof the last section, we know that [ A Γ t ] → [ A Γ ∞ ] (2.3)where A Γ ∞ is a flat connection on S × ρ G . Moreover, the following theorem shows that [ A Γ ∞ ]can be seen as an element in Hom(Γ , G ). It is not surprising that ˜ ρ and A Γ ∞ are related. Theorem 2.4.
Suppose B is connected, π : P → B is a principal G -bundle and A is a flatconnection on P , then there exists a representation π ( B ) → G unique up to conjugationsuch that(a) If p : ˜ B → B is the universal covering space of B and π ( B ) acts on ˜ B as coveringtransformation, then P is isomorphic to ˜ B × π ( B ) G .(b) There exists an isomorphism Φ : ˜ B × π ( B ) G → P such that q ∗ Φ ∗ A = Θ where q :˜ B × G → ˜ B × π ( B ) G is the quotient map and Θ is the trivial connection on ˜ B × G .(c) The holonomy induces an injective map: hol : A flat ( P ) / G ( P ) → Hom ( π ( B ) , G ) conjugation , and a bijection: hol : (cid:71) isomorphismclasses of P A flat ( P ) / G ( P ) → Hom ( π ( B ) , G ) conjugation . Proof. (a) Suppose that π : P → B is a principal G -bundle with a flat connection A .Through the parallel transport associated to A , each smooth path l in B induces an isomor-phism from P | l (0) to P | l (1) . Since A is flat, this isomorphism is invariant under homotopiesof the paths which fix the end points. Choose a base point x ∈ B and a point p ∈ P | x . If l is a closed path through x , i.e., [ l ] ∈ π ( B, x ), the corresponding isomorphism of P | x toitself is a right translation by an element of G , denoted as hol A,p ([ l ]). By Lemma 3.5.1 in[M98], hol A,p : π ( B ) → G
6s an anti-homomorphism, i.e., for any loops λ, µ in Bhol
A,p ([ λ ] − ) = hol A,p ([ λ ]) − , hol A,p ( λ ∗ µ ) = hol A,p ( µ ) · hol A,p ( λ ) , where ∗ is the multiplication operator in π ( B ). Thus hol − A,p defines a left action of π ( B )on G : π ( B ) × G → G ([ α ] , g ) (cid:55)→ hol A,p ([ α ]) − · g. (2.4)By Lemma 3.5.3 in [M98], if we change the point p on the fibre, we will get an elementgiven by conjugation of hol A,p : hol A,p · g = g − · hol A,p · g. Similarly, if we change (
A, p ) by a gauge transformation g , hol f ( A,p ) is conjugate to hol A,p .So we get a map hol : A flat ( P ) / G ( P ) → Hom ( π ( B ) , G )conjugation . Suppose p : ( (cid:101) B, ˜ x ) → ( B, x ) is the universal covering of B . Denote by (cid:101) B × π ( B ) G theassociated bundle defined through hol A,p . DefineΦ : (cid:101) B × π ( B ) G → P as follows:Given [(˜ x, g )] ∈ (cid:101) B × π ( B ) G , choose a path ˜ l from ˜ x to ˜ x , and project ˜ l onto a path l with end points x = p (˜ x ), x := p (˜ x ) in B . Then the isomorphism P | x → P | x induced by l sends p · g ∈ π − ( x ) to Φ(˜ x, g ), as shown in Figure 1.Figure 1Check Φ is well defined: 7irst, let ˜ l , ˜ l be two different paths in (cid:101) B with endpoints ˜ x and ˜ x and l , l be theirprojections. ˜ l , ˜ l are homotopic since (cid:101) B is a universal covering, therefore l and l arehomotopic, which implies that the isomorphisms induced by l and l are the same. Second,we need to show that for any [ α ] ∈ π ( B ),Φ(˜ x · [ α ] , g ) = Φ(˜ x, [ α ] − · g ) . (2.5)the right hand side of which is equal to Φ(˜ x, hol A,p ([ α ]) · g ) by (2.4).Suppose p : (cid:101) B → B lifts the loop α to a path ˜ α from ˜ x to ˜ x in (cid:101) B . Choose a path ˜ β in˜ B from ˜ x to ˜ x . Project it by p to get a path β from x to x in B . Then lift β to a path ˜ β such that one of end points of ˜ β is ˜ x . Denote the other end point of ˜ β by ˜ y . Since π ( B )acts on (cid:101) B as a covering transformation, ˜ x · [ α ] = ˜ y . Figure 2 describes the relations betweenthese points and paths. Figure 2Note that ˜ β − is a path from ˜ x to ˜ x and ˜ α ∗ ˜ β − is a path from ˜ x to ˜ y . By the definitionof Φ, the isomorphism induced by parallel transport along β − (also denoted as β − ) sends p · hol A,p ([ α ]) · g to Φ(˜ x, hol A,p ([ α ]) · g ) and the isomorphism along α ∗ β − (also denotedas α ∗ β − ) sends p · g to Φ(˜ x · [ α ] , g ). That is to say β − ( p · hol A,p ([ α ]) · g ) = Φ(˜ x, hol A,p ([ α ]) · g ) , ( α ∗ β − )( p · g ) = Φ(˜ x · [ α ] , g ) . Formula (2.5) follows from the identity:( α ∗ β − )( p · g ) = β − ( p · g · hol A,p · g ([ α ])) = β − ( p · hol A,p ([ α ]) · g ) . (b) For any ˜ x ∈ ˜ B , choose a little neighbourhood U of ˜ x so that U ∩ U · [ α ] = ∅ for anynon-zero [ α ] ∈ π ( B ). Then for any g ∈ G , { [˜ y, g ] | ˜ y ∈ U } ⊂ ˜ B × π ( B ) G is a local horizontal8ection of ˜ B × π ( B ) G with respect to the connection Φ ∗ A . Thus ˜ B × { g } is a horizontalsection in ˜ B × G with respect to the connection q ∗ Φ ∗ A , which means q ∗ Φ ∗ A is the trivialconnection.(c) It suffices to find the inverse of the map hol : (cid:71) isomorphismclasses of P A flat ( P ) / G ( P ) → Hom ( π ( B ) , G )conjugation . For any [ ρ ] ∈ Hom ( π ( B ) , G ) /conjugation , define P := ˜ B × ρ G. Check that P is well-defined up to isomorphism: For any h ∈ G , define the following map˜ B × ρ G → ˜ B × h − ρh G [˜ x, g ] ρ (cid:55)→ [˜ x, h − · g ] h − ρh , where [ · ] ρ , [ · ] h − ρh are equivalence classes in ˜ B × ρ G and ˜ B × h − ρh G respectively. This is awell-defined isomorphism between principal G -bundles since[˜ x · [ α ] , h − · ρ ([ α ]) − · g ] h − ρh = [˜ x, h − · g ] h − ρh , ∀ [ α ] ∈ π ( B ) . We define A to be the connection on P corresponding to the distribution H , which isdefined by H [˜ x,g ] = q ∗ ( T ˜ x ˜ B × { } ) , where q : ˜ B × G → ˜ B × ρ G is the quotient map and T ˜ x ˜ B × { } ⊂ T (˜ x,g ) ( ˜ B × G ).Next we check that H is well-defined and invariant under the G -action.For any ˜ x, ˜ x (cid:48) ∈ ˜ B , [ α ] ∈ π ( B ) satisfying ˜ x (cid:48) = ˜ x · [ α ], let U (cid:51) ˜ x, U (cid:48) (cid:51) ˜ x (cid:48) be smallneighbourhoods such that p : ˜ B → B maps U and U (cid:48) homeomorphically onto the same set p ( U ) = p ( U (cid:48) ). Then we have U · [ α ] = U (cid:48) . For any g ∈ G , q ( U × { g } ) = { [˜ y, g ] | ˜ y ∈ U } , q ( U (cid:48) × { g } ) = { [˜ y, g ] | ˜ y ∈ U (cid:48) } . This implies q ( U × { [ α ] − · g } ) = q ( U (cid:48) × { g } ) . Therefore H is well-defined. H is invariant under the G -action since it pulls back to the distribution˜ H (˜ x,g ) := T ˜ x ˜ B × { } ⊂ T (˜ x,g ) ( ˜ B × G ) , which is G -invariant. A is flat since ˜ H corresponds to the trivial connection Θ on ˜ B × G .It is obvious that the map ρ (cid:55)→ A constructed above is the inverse of hol .9e now show the relation between the isotropy representation ˜ ρ defined in (2.2) and A Γ ∞ defined in (2.3).Since A Γ ∞ is the connection on S × ρ G such that q ∗ A Γ ∞ = A ∞ where q : S × G → S × ρ G is the quotient map. Define the following bundle isomorphism g Γ ∞ : S × ˜ ρ G → S × ρ G [ x, g ] ˜ ρ (cid:55)→ [ x, g ∞ · g ] ρ . It is well-defined since ˜ ρ = g − ∞ ρg ∞ by definition, thus g Γ ∞ ([ x · γ, ˜ ρ ( γ ) − · g ] ˜ ρ ) = g Γ ∞ ([ x · γ, g − ∞ ρ ( γ ) − g ∞ · g ] ˜ ρ )=[ x · γ, ρ ( γ ) − g ∞ · g ] ρ = [ x, g ∞ · g ] ρ = g Γ ∞ ([ x, g ] ˜ ρ ) . Then the following two compositions S × G g ∞ −→ S × G q −→ S × ρ GS × G q −→ S × ˜ ρ G g Γ ∞ −→ S × ρ G give the same map. Therefore q ∗ ( g Γ ∞ ) ∗ A Γ ∞ = g ∗∞ q ∗ A Γ ∞ = g ∗∞ A ∞ = A trivial , which implies hol ([ A Γ ∞ ]) = [ ˜ ρ ].To sum up, the Γ-invariant connection A on S × (0 , ∞ ) × G with finite L -curvaturegives a limit Γ-invariant flat connection A ∞ on S , which determines a representation inHom(Γ , G ) through holonomy. This representation is the isotropy representation on theorigin of B after extending Γ-action to the origin. Taubes’ gluing construction tells us that given two anti-self-dual connections A , A on 4-manifolds X , X respectively, we can glue them together to get a new ASD connection onthe space X λ X . This section follows the idea from Section 7.2 of [DK90] and discusses thecase when there is a finite group Γ acting on the 4-manifolds X , X with x , x as isolatedfixed points, how to glue two ASD Γ-invariant connections over X , X together to get anASD Γ-invariant connection on X λ X .Suppose X , X are smooth, oriented, compact, Riemannian 4-manifolds, and P , P areprincipal G -bundles over X , X respectively. Let Γ be a finite group acting on P i , X i fromleft which is smooth and orientation preserving and the action on P i cover the action on X i . P i X i (cid:51) x i Γ G ∈ X Γ1 , x ∈ X Γ2 are two isolated fixed points with equivalent isotropy representations.i.e., there exists h ∈ G such that ρ ( γ ) = hρ ( γ ) h − ∀ γ ∈ Γ (3.1)where ρ , ρ are isotropy representations of Γ at x , x respectively.Now we fix two metrics g , g on X , X such that the Γ-action preserves the metrics.This can be achieved by the following proposition. Proposition 3.1.
For any Riemannian metric g on X , ˜ g := 1 | Γ | (cid:88) γ ∈ Γ γ ∗ g defines a Γ -invariant metric. The proof is straightforward. We omit it here. P i and getting an approximate ASD connection A (cid:48) The first step is to glue manifolds X and X by connecting sum.Fix a large enough constant T and a small enough constant δ . Let λ > λe δ ≤ b where b := λe T . We first glue X (cid:48) := X \ B x ( λe − δ ) and X (cid:48) := X \ B x ( λe − δ ) together as shown in Figure 3, where e ± are defined in polar coordinates by e ± : R \ { } → R × S rm (cid:55)→ ( ± log rλ , m )and f : Ω = ( − δ, δ ) × S → Ω = ( − δ, δ ) × S (3.2)is defined to be a Γ-equivariant conformal map that fixes the first component. Denote theconnected sum by X λ X or X .On the new manifold X , we define the metric g λ to be a weighted average of g , g on X and X , compared by the diffeomorphism f . If g λ = (cid:80) m i g i on X (cid:48) i , we can arrange1 ≤ m i ≤
2. This means points are further away from each other on the gluing area.We now turn to the bundles P i . Suppose A i are ASD Γ-invariant connections on P i . Wewant to glue P i | X (cid:48) i together so that A and A match on the overlapping part.The first step is to replace A i by two Γ-invariant connections which are flat on the annuliΩ i . Define the cut-off function η i on X i as following: η ( x ) = (cid:40) x ∈ [ − δ, + ∞ ] × S x ∈ X \ B x ( b ) , η ( x ) = (cid:40) x ∈ [ −∞ , δ ] × S x ∈ X \ B x ( b ) , (3.3)which are shown in Figure 4. Note that η i ( x ) depend only on | x − x i | when x ∈ B x i ( b ).Therefore η i are Γ-invariant.Now we introduce a lemma, which comes from Lemma 2.1 of [U82].11igure 3Figure 4 Lemma 3.2.
For any connection A on R , recall that an exponential gauge associated to itis a gauge under which the connection satisfies A (0) = 0 , (cid:88) j =1 x j A j ( x ) = A r = 0 . n an exponential gauge in R n , | A ( x ) | ≤ · | x | · max | y |≤| x | | F ( y ) | , where F is the curvature of the connection. By Lemma 3.2, choose exponential gauge on B x i ( b ) so that for x ∈ B x i ( b ), we have | A i ( x ) | ≤ d ( x, x i ) max d ( y,x i ) ≤ d ( x,x i ) | F A i ( y ) | . Under this trivialisation, define A (cid:48) i = η i A i . That is to say, A (cid:48) i equals to A i on X i \ B x i ( b ),and equals to η i A i on B x i ( b ) under the chosen trivialisation. Then A i − A (cid:48) i is supported on B x i ( b ). Since X i is compact, | F A i | ≤ const. Together, we have | A i − A (cid:48) i | ≤ | A i | ≤ b | F A i ( y ) | ≤ const · b. We can also get a L -bound || A i − A (cid:48) i || L ≤ (cid:18) vol ( B x i ( b )) · const · b (cid:19) = const · b . (3.4)Since | F A (cid:48) i | ≤ const and F + A (cid:48) i is supported on the annulus B x i ( b , b ) with radii b and b , we have || F + A (cid:48) i || L ≤ (cid:18) vol ( B x i ( b )) max | F + A (cid:48) i ( y ) | (cid:19) ≤ (cid:18) vol ( B x i ( b )) max | F A (cid:48) i ( y ) | (cid:19) ≤ const · b . (3.5)The next step is to glue P | X (cid:48) and P | X (cid:48) together to get a principal G -bundle P over X and glue A (cid:48) and A (cid:48) together to get a Γ-invariant connection A (cid:48) on P . Lemma 3.3.
There exists a canonical (Γ , G ) -equivariant map ϕ : G ∼ = P | x i ϕ −→ P | x ∼ = Gg (cid:55)→ hg, where h is defined in (3.1) and (Γ , G ) -equivariant means ϕ is Γ -equivariant and G -equivariant.Proof. The G -equivariance is obvious and the Γ-equivariance follows from: P | x ϕ −→ P | x γ −→ P | x , P | x γ −→ P | x ϕ −→ P | x g (cid:55)→ hg (cid:55)→ ρ ( γ ) hg g (cid:55)→ ρ ( γ ) g (cid:55)→ hρ ( γ ) g and ρ ( γ ) hg = hρ ( γ ) h − hg = hρ ( γ ) g for any γ ∈ Γ.13enote the subgroup of (Γ , G )-equivariant gluing parameters by Gl Γ := Hom (Γ ,G ) ( P | x , P | x ) . (3.6) Proposition 3.4.
The subgroup of (Γ , G ) -equivariant gluing parameters Gl Γ takes threeforms: Gl Γ ∼ = G if ρ (Γ) , ρ (Γ) ⊂ C ( G ) ,U (1) if ρ (Γ) , ρ (Γ) (cid:54)⊂ C ( G ) and are contained in some U (1) ⊂ G,C ( G ) if ρ (Γ) , ρ (Γ) are not contained in any U (1) subgroup in G, (3.7) where C ( G ) is the center of G .Proof. By formula (3.1), ρ (Γ) and ρ (Γ) are isomorphic and have isomorphic centralisers.For any element h (cid:48) in the centraliser of ρ (Γ), ϕ (cid:48) : g (cid:55)→ hh (cid:48) g is also a (Γ , G )-equivariant mapbetween P | x and P | x since for all γ ∈ Γ hh (cid:48) ρ ( γ ) g = hρ ( γ ) h (cid:48) g = ρ ( γ ) hh (cid:48) g. For any element ϕ (cid:48) ∈ Gl Γ , it can be written as g (cid:55)→ h (cid:48) g for some h (cid:48) ∈ G . Then h − h (cid:48) is inthe centraliser of ρ (Γ) since for any γ ∈ Γ, g ∈ G , we have ρ ( γ ) h (cid:48) g = h (cid:48) ρ ( γ ) g ⇒ h − ρ ( γ ) h (cid:48) g = h − h (cid:48) ρ ( γ ) g ⇒ ρ ( γ ) h − h (cid:48) g = h − h (cid:48) ρ ( γ ) g, which implies h − h (cid:48) commutes with ρ ( γ ).Therefore Gl Γ is isomorphic to the centraliser of ρ (Γ) in G . The three cases in (3.7)are the only three groups that are centraliser of some subgroup in G when G = SU (2) or SO (3).Recall that annuli Ω i are identified by f : Ω → Ω defined in (3.2). Take ϕ ∈ Gl Γ , wedefine the identification between P | Ω and P | Ω , also denoted by f : P | Ω → P | Ω , as thefollowing. For any point q ∈ P | Ω , choose a path from π ( q ) to x ; then lift it to a pathbeginning at q by parallel transport corresponding to A (cid:48) . The resulting path has an endpoint p . On the other hand, choose a path from x to f ( π ( q )) and lift it to a path beginningat ϕ ( p ) by parallel transport corresponding to A (cid:48) . The resulting path has an end point f ( q ),as shown in Figure 5.This map f is well-defined since A (cid:48) i are flat on B x i ( λe δ ), thus a choice of paths on B x i ( λe δ )does not matter. Also, f is Γ-equivariant since A (cid:48) i are Γ-invariant.Denote ( P | X (cid:48) ) ϕ ( P | X (cid:48) ) by P and the induced connection on P by A (cid:48) ( ϕ ). In general,for different gluing parameter ϕ , ϕ , A (cid:48) ( ϕ ) and A (cid:48) ( ϕ ) are not gauge equivalent. Proposition 3.5. (Proposition 7.2.9 in [DK90]) Suppose G = SU (2) , and X i are simplyconnected, then the isomorphism class of the bundle P is independent of gluing parametersand the connections A (cid:48) ( ϕ ) , A (cid:48) ( ϕ ) are gauge equivalent if and only if the parameters ϕ , ϕ are in the same orbit of the action of Γ A × Γ A on Gl . Proof. (1). Suppose A (cid:48) ( ϕ ), A (cid:48) ( ϕ ) are gauge equivalent, we show there exist ˜ σ ∈ Γ A , ˜ σ ∈ Γ A such that ϕ = ˜ σ − ◦ ϕ ◦ ˜ σ .Since A (cid:48) ( ϕ i ) are connections on the bundle ( P | X (cid:48) ) ϕ i ( P | X (cid:48) ) over X = X λ X , thenon the cylindrical ends, we have the following local trivialisations (exponential gauge) of P i P | ( − T, + ∞ ) × S ∼ = ( − T, + ∞ ) × S × G,P | ( −∞ ,T ) × S ∼ = ( −∞ , T ) × S × G. Under these trivializations, A ∈ Ω (( − T, + ∞ ) × S , g ), A ∈ Ω (( −∞ , T ) × S , g ) are Liealgebra valued 1-forms on the cylindrical ends and A (cid:48) = η A , A (cid:48) = η A . Suppose ϕ i : g (cid:55)→ h i g . Define the bundle isomorphisms α i : ( P | X (cid:48) ) id ( P | X (cid:48) ) → ( P | X (cid:48) ) ϕ i ( P | X (cid:48) )15y α i ( p ) = p for p ∈ P | X (cid:48) or p ∈ P | X (cid:48) \ ( − δ,T ) × S α i ( t, m, g ) = ( t, m, β i ( t ) g ) for ( t, m, g ) ∈ P | ( − δ,T ) × S , where β i are smooth functions from ( − δ, T ) to G satisfying β i ( t ) = h i when t ∈ ( − δ, δ ) and β i ( t ) = id when t = T . The well-definedness of the bundle isomorphism α i is obvious.Pulled back by α i , the connections A (cid:48) ( ϕ ) and A (cid:48) ( ϕ ) become α ∗ A (cid:48) ( ϕ ) and α ∗ A (cid:48) ( ϕ )on ( P | X (cid:48) ) id ( P | X (cid:48) ). Since they are equivalent, there exists a gauge transformation ˜ σ inAut(( P | X (cid:48) ) id ( P | X (cid:48) )) such that ˜ σ ∗ α ∗ A (cid:48) ( ϕ ) = α ∗ A (cid:48) ( ϕ ) . Therefore ( α ˜ σα − ) ∗ A (cid:48) ( ϕ ) = A (cid:48) ( ϕ ). Since A (cid:48) ( ϕ i ) both equal to A (cid:48) on X (cid:48) , we have that α ˜ σα − (cid:12)(cid:12) X (cid:48) stabilises A (cid:48) on X (cid:48) . In fact, α ˜ σα − (cid:12)(cid:12) X (cid:48) can be extended to X as a gaugetransformation of P that stabilises A (cid:48) . This is because under the exponential gauge over( − δ, δ ) × S ⊂ X (cid:48) , we have α ˜ σα − = h σh − : ( − δ, δ ) × S → G, ( h σ − h − ) d ( h σh − ) + ( h σ − h − ) A (cid:48) ( h σh − ) = A (cid:48) , where σ : ( − δ, δ ) × S → G is the map corresponding to the gauge transformation ˜ σ under theexponential gauge. Since A (cid:48) = 0 on ( − δ, δ ) × S , we find σ : ( − δ, δ ) × S → G is a constantfunction, thus α ˜ σα − (cid:12)(cid:12) X (cid:48) can be extended to X . We denote this gauge transformation of P as ˜ σ .Similarly, ˜ σ (cid:12)(cid:12) X (cid:48) stabilises A (cid:48) and can be extended to ˜ σ on X which, again, stabilises A (cid:48) .We also get ˜ σ − ϕ ˜ σ = ϕ . To show ϕ , ϕ are in the same orbit of the action of Γ A × Γ A on Gl , it is enough to show˜ σ stabilises A and ˜ σ stabilises A :If A (cid:48) is irreducible, then ˜ σ ∈ {± } ⊂ Ω ( X , AdP ), thus ˜ σ also stabilises A .If A (cid:48) is reducible, by Lemma 4.3.21 of [DK90], we know that A is reducible since A = A (cid:48) on X \ B x ( b ). Then P reduces to a U (1)-principal bundle Q (cid:48) so that Γ A (cid:48) consists of constantsections of Q (cid:48) × U (1) U (1) ⊂ P × SU (2) SU (2). P also reduces to another U (1)-principal bundle Q so that Γ A consists of constant sections of Q × U (1) U (1) ⊂ P × SU (2) SU (2). Since A = A (cid:48) on X \ B x ( b ), we have Q (cid:48) (cid:12)(cid:12) X \ B x ( b ) = Q (cid:12)(cid:12) X \ B x ( b ) .Since ˜ σ ∈ Γ A (cid:48) , ˜ σ is a constant section of Q (cid:48) × U (1) U (1). By definition of ˜ σ ,˜ σ (cid:12)(cid:12) X \ B x ( b ) = ˜ σ (cid:12)(cid:12) X \ B x ( b ) . Therefore ˜ σ is constant on Q (cid:48) × U (1) U (1) over X \ B x ( b ). This in turn implies that ˜ σ iscontant on Q × U (1) U (1) over X \ B x ( b ). Then ˜ σ (cid:12)(cid:12) X \ B x ( b ) extends constantly on Q × U (1) U (1)to a gauge transformation ˜ σ A on X that stabilises A .16n summary, ˜ σ | X (cid:48) extends to ˜ σ on X , which stabilises A (cid:48) ; ˜ σ | X \ B x ( b ) extends to ˜ σ A on X , which stabilises A . Next we show that ˜ σ A = ˜ σ .It is enough to show ˜ σ A = ˜ σ on B x ( b ) since they both equal to ˜ σ on X \ B x ( b ).Under the exponential gauge on B x ( b ), ˜ σ A and ˜ σ can be written as σ A , σ : B x ( b ) → G respectively. Then we have σ − A dσ A + σ − A A σ A = A ,σ − dσ + σ − A (cid:48) σ = A (cid:48) , ( A )( ∂∂r ) = ( A (cid:48) )( ∂∂r ) = 0 , which implies that ∂σ A ∂r = ∂σ ∂r = 0. This means σ A , σ are constant on B x ( b ). Moreover,they are the same constant since they have the same value on ∂B x ( b ). That is, ˜ σ A = ˜ σ ,i.e., ˜ σ ∈ Γ A .Similarly, ˜ σ stabilises A .(2). Suppose there exist ˜ σ ∈ Γ A , ˜ σ ∈ Γ A such that ϕ = ˜ σ − ◦ ϕ ◦ ˜ σ , we now showthat A (cid:48) ( ϕ ), A (cid:48) ( ϕ ) are gauge equivalent.We first show that ˜ σ i ∈ Γ A i implies ˜ σ i ∈ Γ A (cid:48) i . Since A ∈ Ω (( − T, + ∞ ) × S , g ), A ∈ Ω (( −∞ , T ) × S , g ) are given in exponential gauge, under the coordinate ( r, ψ ), wehave A ( ∂∂r ) = A ( ∂∂r ) = 0.Under the exponential gauge, ˜ σ corresponds to σ : ( − T, + ∞ ) × S → G , and ˜ σ corresponds to σ : ( −∞ , T ) × S → G . Then we have the formulas σ − dσ + σ − A σ = A x ∈ ( − T, + ∞ ) × S ,σ − dσ + σ − A σ = A x ∈ ( −∞ , T ) × S . Therefore ∂σ i ∂r = 0. Then σ i are constant functions and σ − i η i A i σ i = η i A i , which impliesthat ˜ σ i stabilises A (cid:48) i .Define a gauge transformation on ( P | X (cid:48) ) id ( P | X (cid:48) ) by˜ σ = (cid:40) ˜ σ x ∈ X (cid:48) α − ˜ σ α x ∈ X (cid:48) . The well-definedness of ˜ σ follows from ϕ = ˜ σ − ◦ ϕ ◦ ˜ σ . It is easy to see that ˜ σ ∗ ( ˜ α ∗ A (cid:48) ( ϕ )) =˜ α ∗ A (cid:48) ( ϕ ). Therefore A (cid:48) ( ϕ ) on ( P | X (cid:48) ) ϕ ( P | X (cid:48) ) and A (cid:48) ( ϕ ) on ( P | X (cid:48) ) ϕ ( P | X (cid:48) ) are gaugeequivalent.We denote A (cid:48) ( ϕ ) by A (cid:48) when the gluing parameter is contextually clear.17 .2 Constructing an ASD connection from A (cid:48) The general idea is to find a solution a ∈ Ω ( X, adP ) Γ so that A := A (cid:48) + a is anti-self-dual,i.e., F + A = F + A (cid:48) + d + A (cid:48) a + ( a ∧ a ) + = 0 . (3.8)To do so, we wish to find a right inverse R Γ of d + A (cid:48) and an element ξ ∈ Ω , + ( X, adP ) Γ satisfying F + A (cid:48) + ξ + ( R Γ ξ ∧ R Γ ξ ) + = 0 . (3.9)Then a = R Γ ξ is a solution of equation (3.8).Since A i are two ASD connections, we have the complex:0 → Ω ( X i , adP i ) d Ai −−→ Ω ( X i , adP i ) d + Ai −−→ Ω , + ( X i , adP i ) → . We assume that the second cohomology classes H A , H A are both zero. The Γ-action can beinduced on this chain complex naturally. It is worth mentioning that the Γ-action preservesthe metric, so the space Ω , + ( X i , adP i ) is Γ-invariant. Define the following two averagingmaps: ave : Ω ( X i , adP i ) → Ω ( X i , adP i ) Γ a (cid:55)→ | Γ | (cid:88) γ ∈ Γ γ ∗ aave : Ω , + ( X i , adP i ) → Ω , + ( X i , adP i ) Γ ξ (cid:55)→ | Γ | (cid:88) γ ∈ Γ γ ∗ ξ. Note that these maps are surjective since any Γ-invariant element is mapped to itself.
Proposition 3.6.
The following diagram ( X i , adP i ) Ω ( X i , adP i ) Ω , + ( X i , adP i ) 00 Ω ( X i , adP i ) Γ Ω ( X i , adP i ) Γ Ω , + ( X i , adP i ) Γ d Ai d + Ai ave aved Ai d + Ai commutes.Proof. It suffices to show that d A i : Ω ( X i , adP i ) → Ω ( X i , adP i ) and γ commute for any γ ∈ Γ. For any η ∈ Ω ( X i , adP i ), we treat η as a Lie algebra valued 1-form on P i , then( d + A i )( γ ∗ η ) = γ ∗ dη + [ A i , γ ∗ η ] γ ∗ (cid:0) ( d + A i ) η (cid:1) = γ ∗ dη + γ ∗ [ A i , η ]= γ ∗ dη + [ γ ∗ A i , γ ∗ η ]= γ ∗ dη + [ A i , γ ∗ η ] .
18y Proposition 3.6, we have ( H A i ) Γ = 0 sinceIm( d + A i ◦ ave ) = Im( ave ◦ d + A i ) = Ω , + ( X i , adP i ) Γ . Lemma 3.7.
Suppose L : H → H is a linear surjection between Hilbert spaces, then there exists a linear right inverse of L .Proof. Decompose H as H = ker( L ) ⊕ H so that L | H ∼ = −→ H is an isomorphism. Then forany linear map P from H to ker( L ), R := ( L | H ) − + P is right inverse of L since ∀ ξ ∈ H , LR ( ξ ) = ξ + LP ξ = ξ .By Lemma 3.7, there exist right inverses R Γ i : Ω , + ( X i , adP i ) Γ → Ω ( X i , adP i ) Γ to d + A . Proposition 3.8. R Γ i are bounded operators from Ω , + L ( X i , adP i ) Γ to Ω L ( X i , adP i ) Γ . The proof of Proposition 3.8 follows from Proposition 2.13 of Chapter III of [LM89] andthe fact that X i are compact.By the Sobolev embedding theorem, we have || R Γ i ξ || L ≤ const. || R Γ i ξ || L , and combined with Proposition 3.8, we have || R Γ i ξ || L ≤ const. || ξ || L . (3.10)Define two operators Q Γ i : Ω , + ( X i , adP i ) Γ → Ω ( X i , adP i ) Γ by Q Γ i ( ξ ) := β i R Γ i γ i ( ξ ) , where β i , γ i are cut-off functions defined in the Figure 6 where β varies on (1 , δ ) × S , β varies on ( − δ, − × S and γ i varies on ( − , × S . We can choose β i such that (cid:12)(cid:12)(cid:12)(cid:12) ∂β i ∂t (cid:12)(cid:12)(cid:12)(cid:12) < δ pointwise, then ||∇ β i || L ≤ π (cid:18)(cid:90) δ δ dt (cid:19) / < πδ − / . (3.11)We can choose γ i such that γ + γ = 1 on Ω f Ω where f is defined in (3.2).Now we want to extend the operators Q Γ i to X = X λ X . Firstly, extend β i , γ i to X inthe obvious way. It is worth mentioning that after the extension γ + γ = 1 on X . Secondly,for any ξ ∈ Ω , + ( X, adP ) Γ , γ i ξ is supported on X (cid:48) i , thus R Γ i γ i ξ makes sense. Finally, extend β i R Γ i γ i ( ξ ) to the whole X . Therefore Q Γ i can be treated as an operator: Q Γ i : Ω , + ( X, adP ) Γ → Ω ( X, adP ) Γ . Define Q Γ := Q Γ1 + Q Γ2 : Ω , + ( X, adP ) Γ → Ω ( X, adP ) Γ . Lemma 3.9.
With definitions above, we have ∀ ξ ∈ Ω , + ( X, adP ) Γ , || d + A (cid:48) Q Γ ( ξ ) − ξ || L ≤ const. ( b + δ − / ) || ξ || L . Proof. || d + A (cid:48) Q Γ ( ξ ) − ξ || L = || d + A (cid:48) ( Q Γ1 ( ξ ) + Q Γ2 ( ξ )) − γ ξ − γ ξ || L = || d + A (cid:48) Q Γ1 ( ξ ) + d + A (cid:48) Q Γ2 ( ξ ) − γ ξ − γ ξ || L ≤ || d + A (cid:48) Q Γ1 ( ξ ) − γ ξ || L + || d + A (cid:48) Q Γ2 ( ξ ) − γ ξ || L . Suppose A (cid:48) i = A i + a i , then d + A (cid:48) i Q Γ i ξ = d + A i β i R Γ i γ i ξ + [ a i , β i R Γ i γ i ξ ] + = β i d + A i R Γ i γ i ξ + ∇ β i R Γ i γ i ξ + [ β i a i , R Γ i γ i ξ ] + . The three terms on the right hand side have the following estimates.(i). β i d + A i R Γ i γ i ξ = β i γ i ξ = γ i ξ .(ii). ||∇ β i R Γ i γ i ξ || L ≤ ||∇ β i || L || R Γ i γ i ξ || L ≤ const. δ − / || ξ || L by the Sobolev multiplicationtheorem and (3.10) and (3.11).(iii). || [ β i a i , R Γ i γ i ξ ] + || L ≤ const. || a i || L || R Γ i γ i ξ || L ≤ const. b || ξ || L by the Sobolev multipli-cation theorem and (3.4) and (3.10).Therefore || d + A (cid:48) i Q Γ i ( ξ ) − γ i ξ || L ≤ const.( b + δ − / ) || ξ || L and the result follows.The result of Lemma 3.9 means that Q Γ is almost a right inverse of d + A (cid:48) . Next we showthere is a right inverse R Γ of d + A (cid:48) . 20y Lemma 3.9, we can choose δ large enough and b small enough so that || d + A (cid:48) Q Γ ( ξ ) − ξ || L ≤ / || ξ || L , which implies1 / || ξ || L ≤ || d + A (cid:48) Q Γ ( ξ ) || L ≤ / || ξ || L . Then d + A (cid:48) Q Γ is invertible and 1 / || ( d + A (cid:48) Q Γ ) − ( ξ ) || L ≤ || ξ || L . (3.12)Define R Γ := Q Γ ( d + A (cid:48) Q Γ ) − , then it is easy to see that R Γ is the right inverse of d + A (cid:48) . Notethat R Γ depends on the gluing parameter ϕ , so we denote the operator by R Γ ϕ when thegluing parameter is not contextually clear. R Γ has the following good estimate: || R Γ ξ || L = || ( Q Γ1 + Q Γ2 )( d + A (cid:48) Q Γ ) − ( ξ ) || L ≤ || Q Γ1 ( d + A (cid:48) Q Γ ) − ( ξ ) || L + || Q Γ2 ( d + A (cid:48) Q Γ ) − ( ξ ) || L ≤ || R Γ1 γ ( d + A (cid:48) Q Γ ) − ( ξ ) || L + || R Γ2 γ ( d + A (cid:48) Q Γ ) − ( ξ ) || L (by (3.10)) ≤ const. || γ ( d + A (cid:48) Q Γ ) − ( ξ ) || L + const. || γ ( d + A (cid:48) Q Γ ) − ( ξ ) || L ≤ const. || ( d + A (cid:48) Q Γ ) − ( ξ ) || L (by (3.12)) ≤ const. || ξ || L . (3.13)Then we have || ( R Γ ξ ∧ R Γ ξ ) + − ( R Γ ξ ∧ R Γ ξ ) + || L ≤ || R Γ ξ ∧ R Γ ξ − R Γ ξ ∧ R Γ ξ || L = 12 || ( R Γ ξ + R Γ ξ ) ∧ ( R Γ ξ − R Γ ξ ) + ( R Γ ξ − R Γ ξ ) ∧ ( R Γ ξ + R Γ ξ ) || L ≤ const. || R Γ ξ − R Γ ξ || L || R Γ ξ + R Γ ξ || L (by (3.13)) ≤ const. || ξ − ξ || L ( || ξ || L + || ξ || L ) . (3.14)Define an operator T : ξ (cid:55)→ − F + ( A (cid:48) ) − ( R Γ ξ ∧ R Γ ξ ) + , then solving equation (3.9) meansto find a fixed point of the operator T . Here we apply the contraction mapping theorem to T to show there exists a unique fixed point of T . There are two things to check:1. There is an r > b , T is a map from the ball B ( r ) ⊂ Ω , + L ( X, adP ) to itself. This follows from || ξ || L < r ⇒ || T ξ || L ≤ || F + ( A (cid:48) ) || L + || R Γ ξ ∧ R Γ ξ || L ≤ const.b + || R Γ ξ || L ≤ const.b + const. || ξ || L ≤ const. ( b + r ) < r ( f or small b, r with b << r ) . T is a contraction for sufficiently small r , i.e., there exists λ < || T ξ − T ξ || ≤ λ || ξ − ξ || ∀ ξ , ξ . This follows from (3.14). 21ow we have proved that there exists a unique solution to equation (3.9).
Theorem 3.10.
Suppose A , A are Γ -invariant ASD connections on X , X respectivelywith H A i = 0 . Let λ, T, δ be positive real numbers such that b := λe T > λe δ . Then we canmake δ large enough and b small enough so that for any (Γ , G ) -equivariant gluing parameter ϕ ∈ Hom ( G, Γ) ( P | x , P | x ) , there exists a ϕ ∈ Ω ( X, adP ) Γ with || a ϕ || L ≤ const. b such that A (cid:48) ( ϕ ) + a ϕ is a Γ -invariant ASD connection on X . Moreover, if ϕ , ϕ are in the same orbitof Γ A × Γ A on Gl , then A (cid:48) ( ϕ ) + a ϕ , A (cid:48) ( ϕ ) + a ϕ are gauge equivalent.Proof. We only need to prove the last statement.If ϕ , ϕ are in the same orbit of Γ A × Γ A on Gl , then A (cid:48) ( ϕ ) , A (cid:48) ( ϕ ) are gauge equivalent.For some gauge transformation σ we have σ ∗ A (cid:48) ( ϕ ) = A (cid:48) ( ϕ ). Applying σ ∗ on both sides ofthe following formula F + A (cid:48) ( ϕ ) + ξ ( ϕ ) + ( R Γ ϕ ξ ( ϕ ) ∧ R Γ ϕ ξ ( ϕ )) + = 0gives σ ∗ F + A (cid:48) ( ϕ ) + σ ∗ ξ ( ϕ ) + σ ∗ ( R Γ ϕ ξ ( ϕ ) ∧ R Γ ϕ ξ ( ϕ )) + = 0 . (3.15)Since σ ∗ and d + A (cid:48) commute and R Γ = Q Γ ( d + A (cid:48) Q Γ ) − , then σ ∗ and R Γ commute. Then (3.15)becomes F + A (cid:48) ( ϕ ) + σ ∗ ξ ( ϕ ) + ( R Γ ϕ σ ∗ ξ ( ϕ ) ∧ R Γ ϕ σ ∗ ξ ( ϕ )) + = 0This means σ ∗ ξ ( ϕ ), ξ ( ϕ ) are solutions to F + A (cid:48) ( ϕ ) + ξ + ( R Γ ϕ ξ ∧ R Γ ϕ ξ ) + = 0, which implies σ ∗ ξ ( ϕ )= ξ ( ϕ ). The following deduction completes the proof. σ ∗ ξ ( ϕ ) = ξ ( ϕ ) ⇒ σ ∗ R Γ ϕ ξ ( ϕ ) = R Γ ϕ σ ∗ ξ ( ϕ ) = R Γ ϕ ξ ( ϕ ) ⇒ σ ∗ a ϕ = a ϕ ⇒ σ ∗ ( A (cid:48) ( ϕ ) + a ϕ ) = A (cid:48) ( ϕ ) + a ϕ . In this section we sketch the construction of bubble tree compactification for 4-manifolds.Details can be found in [C02].
Definition 4.1. (1). A rooted tree ( T, v ) is a triple ( V T , E T , v ) where V T is a set of discretepoints (called vertices ), v ∈ V T (called the root ), and E T is a set of segments withvertices as end points (called edges ) such that any two vertices are connected by exactlyone path.(2). A vertex v is called an ancestor of a vertex v if v lies on the path from the root v to v . And v is called a descendant of v .(3). A vertex v is called the parent of a vertex v if v is an ancestor of v and there is anedge between them. v is called a child of v .224). A vertex v is called a leaf if it has no child.(5). A subtree of T (with root v for some v ∈ V T ) is a tree, denoted by ( t ( v ) , v ), such that V t ( v ) ⊂ V T is the union of { v } and the set of all descendants of v and E t ( v ) is the set ofedges connecting vertices in V t ( v ) . (cid:4) Definition 4.2. A bubble tree is a weighted-rooted-tree ( T, v , w ) (or briefly denoted by T )satisfying • w , called the weight , is a map from V T to Z ≥ . • For any non-root vertex v , either w ( v ) (cid:54) = 0 or child ( v ) ≥ W ( v i ) := (cid:88) v (cid:48) ∈ V t ( vi ) w ( v (cid:48) ) > v i ∈ child ( v ) . Here W ( v i ) is called the total weight of the vertex v i or the total weight of the tree t ( v i ).A vertex in a bubble tree is called a ghost vertex if it is non-root and has weight 0. A bubbletree is called a ghost bubble tree if it has a ghost vertex. Denote T k := { ( T, v , w ) | W ( v ) = k } . (cid:4) Note that T k is a finite set of bubble trees with total weight k . Here is a bubble treeexample and a non bubble tree example. Examples 4.3. (cid:4)
Vertices and edges in a bubble tree have geometric meaning in our situation. The rootvertex corresponds to the base manifold X , each non-root vertex corresponds to a bubble S , and each edge corresponds to a gluing point on X or an S . Consider the bubble tree inExample 4.3, the associated bubble tree space can be drawn as in Figure 7.By Definition 4.2, any subtree of a bubble tree T is again a bubble tree. For any v ∈ V T with children { v , . . . , v n } , we define the set of all its subtrees to be m v := ( t ( v ) , . . . , t ( v n )).Let S m v be the subgroup of the permutation group S n whose elements fix m v .23igure 7: An example of bubble tree Definition 4.4.
Suppose (
T, v , w T ) is a bubble tree, e is an edge of T , v , v are the twovertices connected by e such that v is the parent of v . A bubble tree ( T (cid:48) , v , w T (cid:48) ) is the contraction of T at e if • V T (cid:48) = V T \ { v } , E T (cid:48) = E T \ { e } . • w T (cid:48) ( v ) = w T ( v ) + w T ( v ), child T (cid:48) ( v ) = child T ( v ) ∪ child T ( v ). T (cid:48) is also denoted as T \ { v } . (cid:4) The following lemma is obvious.
Lemma 4.5.
There is a partial order on T k : T < T (cid:48) if T (cid:48) is the contraction of T at someedges of T . Definition 4.6.
Given a tuple m = ( k , . . . , k n ) ∈ Z n + , a generalised instanton on S associ-ated to m is an element([ A ] , ( p , . . . , p n )) ∈ M k ( S ) × ( S \ {∞} ) n \ (cid:52) =: M k , m ( S ) , where k ∈ Z ≥ , (cid:52) is the big diagonal of ( S \ {∞} ) n , i.e. the complement of the subset of( S \ {∞} ) n whose elements are distinct n -tuples. p i are called mass points and k i are calledthe weight at p i . The space of generalised instantons on S is denoted by M k , m . The spaceof balanced generalised instantons is defined to be M bk , m := M k , m /H, where H ⊂ Aut ( R ) is the subgroup generated by translations and dilations in R . Anotherequivalent way to define M bk , m is M bk , m = (cid:26) ([ A ] , ( p , . . . , p n )) ∈ M k , m (cid:12)(cid:12)(cid:12) m ( A ) · k + p k + · · · + p n k n = 0One of the following (a) and (b) holds. (cid:27) where m ( A ) = 1 || F A || (cid:90) R y | F A | dy is the mass center of A and the ( a ) , ( b ) conditions are24a). The mass of a subdomain is fixed and all the mass points p i are on the complement ofthe subdomain, i.e., (cid:90) R \ B (1) | F A | = (cid:126) , p i ∈ B (1) ∀ i = 1 , . . . , n where (cid:126) is a fixed constant less than 4 π .(b). The mass of a subdomain is less than (cid:126) and all the mass points p i are on the complementof the subdomain. Moreover, at least one mass point lies on the boundary of thesubdomain, i.e., (cid:90) R \ B (1) | F A | < (cid:126) , p i ∈ B (1) , { p i ∈ ∂B (1) } > . We will mainly use the second description of M bk , m . (cid:4) Definition 4.7.
Given a bubble tree (
T, v , w ), the space of bubble tree instantons associatedto T is defined to be S T ( X ) := M w ( v ) ( X ) × fiber product (cid:89) v i ∈ child ( v ) P v i ( X ) (cid:46) S m v , (4.1)where • P v i ( X ) is the pull-back bundle of (cid:16) F r ( X ) × SO (4) S bt ( v i ) (cid:17) → X by X n ( v ) \ (cid:52) i th proj −−−−→ X ,where (cid:52) is the big diagonal and n(v)= • S bt ( v ) = M bw ( v ) ( S ) if t ( v ) is the tree with only one vertex v . • S bt ( v ) is a subset of S t ( v ) ( S \ {∞} ) if t ( v ) is not the tree with only one vertex. S bt ( v ) consists of those elements in S t ( v ) ( S \ {∞} ) such that on each bubble the inducedgeneralised instanton is balanced.Elements in S bt ( v ) are called balanced bubble tree instantons . In the definition of a bubble treeinstanton (4.1), if we use S t ( v ) instead of S bt ( v ) (i.e., the data on each bubble need not to bebalanced), then we get a larger space, denoted by (cid:101) S T ( X ). Given a bubble tree instanton,the underlying connection on X in it is called the background connection . (cid:4) Lemma 4.8. S T ( X ) is a smooth manifold. roof. By definition, fiber product (cid:89) v i ∈ child ( v ) P v i ( X ) is a fibre bundle over X n ( v ) \(cid:52) with fibre (cid:89) v i ∈ child ( v ) S bt ( v i ) ,which is a smooth manifold by Lemma 3.7 of [C02]. Since the S m v -action on X n ( v ) \ (cid:52) isfree, the action on fiber product (cid:89) v i ∈ child ( v ) P v i ( X ) is also free, therefore S T ( X ) is a smooth manifold. Remark 4.9.
In Definition 4.6, if we remove the “ASD” condition, that is, consider B ratherthan M , then we get a space of generalised connections and balanced generalised connections : B k , m := B k × (cid:16) ( S \ {∞} ) n \ (cid:52) (cid:17) , B bk , m := (cid:26)(cid:0) [ A ] , ( x , . . . , x n ) (cid:1) ∈ B k , m (cid:12)(cid:12)(cid:12) m ([ A ]) e ([ A ]) + k x + · · · + k n x n = 0,one of (a),(b) holds. (cid:27) , where B k ⊂ B is an open neighbourhood of M ⊂ B with elements having energy in aninterval ( k − (cid:15), k + (cid:15) ) for some fixed constant (cid:15) (without loss of generality we take theinterval to be ( k / , k / e ([ A ]) = 18 π (cid:90) R | F A | dvol is the energy of A . Since the H -action preserves e ([ A ]), B bk , m is well-defined. Here we use B k rather than the whole B , since if a connection A has energy less than (cid:126) and m is empty,then both conditions ( a ) and ( b ) cannot be satisfied. We shall see that B k is enough forlater applications.Correspondingly, if we remove the “ASD” condition in Definition 4.7, then we get a spaceof bubble tree connections B T ( X ). Moreover, if the data on each bubble is not balanced, weget a space (cid:101) B T ( X ).In summary, for elements in S T ( X ), a connection on each bubble needs to be “ASD”and “balanced”; for elements in (cid:101) S T ( X ), a connection on each bubble needs to be “ASD”,not necessarily “balanced”; for elements in B T ( X ), a connection on each bubble needs tobe “balanced”, not necessarily “ASD”; for elements in (cid:101) B T ( X ), a connection on each bubbleneed not be “ASD” or “balanced”. (cid:4) Bubble tree instantons play a similar role in bubble tree compactification as ideal connec-tions in Uhlenbeck compactification but with more information on the description of ASDconnections at each bubbling point: as the limit of a sequence of ASD connection with en-ergy blow-up, ideal connections only tell us where the energy blows up, while bubble treeinstantons tell us where and how the energy blows up. A bubble tree instanton will bewritten as (cid:8)(cid:2) [ A v ] , [ x , . . . , x n ( v ) ] (cid:3)(cid:9) v ∈ V T , x , . . . , x n ( v ) ] = ( Y n ( v ) \ (cid:52) ) / ∼ , Y = (cid:40) X v = v S \ {∞} v (cid:54) = v , such that when v (cid:54) = v , (cid:2) [ A v ] , [ x , . . . , x n ( v ) ] (cid:3) is balanced. Consider the bubble tree in Exam-ple 4.3, Figure 8 gives an associated bubble tree instanton.Figure 8: An example of a bubble tree instantonWe denote M k ( X ) := (cid:91) T ∈T k S T ( X ) ,S k ( X ) := (cid:91) T ghost bubbletree in T k S T ( X ) . Given an element in S T ( X ), to glue the background connection with all connections oneach bubble sphere together, we need gluing data ( ρ, λ ) ∈ Gl × R + for each gluing point,where Gl is the space of gluing parameters defined by taking Γ to be the trivial group in Gl Γ . ( Gl is also defined in page 286 in [DK90]). Motivated by this, we define the gluingbundle over S T ( X ) with fibre Gl T = (cid:89) e ∈ edge ( T ) Gl × R + , denoted by: GL T ( X ) → S T ( X ) . Here we give a description of GL T ( X ), see page 32 of [C02] for details. Points in GL T ( X )are of the following inductive form: (here we use notations from page 335 of [T88]) e = [ A , { q α , e α } α ] , e α = ( λ α , [ p α , A α , { q αβ , e αβ } β ]) , e αβ = . . . where • A ∈ A ASD ( P ), P → X is a G -bundle with c equal to w ( v ). • q α ∈ P ˜ × F r ( X ), the index α runs over child ( v ), and for different α, α (cid:48) , q α , q α (cid:48) are ondifferent fibres of P ˜ × F r ( X ) → X . 27 λ α ∈ R + , p α ∈ P α | ∞ , P α → S is a G -bundle with c equals to w ( α ), A α ∈ A ASD ( P α ), q αβ ∈ P α | S \∞ , the index β runs over child ( α ). Example 4.10.
Consider the bubble tree in Example 4.3, an element in its gluing bundlecan be written as in Figure 9. Given such an element, we have all the data we need to do theFigure 9: A point in GL T ( X )Taubes’ gluing construction. For λ i small enough, define the pre-glued approximated ASDconnection Ψ (cid:48) T ( e ) := A ( ρ ,λ ) A . . . ( ρ ,λ ) A , and the corresponding unique ASD connectionΨ T ( e ) := Ψ (cid:48) T ( e ) + a (Ψ (cid:48) T ( e )) . The image of Ψ T is in M k where k is the total weight of T . Note that when T is a treewhose root has weight 0 and X is a space with b +2 >
0, the Atiyah-Hitchin-Singer complexof the trivial connection on X has nonzero obstruction space H , so the right inverse of themap d +Ψ (cid:48) T ( e ) is not defined in general. Therefore we exclude this case. (cid:4) Since G = SU (2) or SO (3), Gl × R + ∼ = R \ { } or ( R \ { } ) / Z , therefore GL T ( X ) hasfibre (cid:89) e ∈ E T ( R \ { } ) or (cid:89) e ∈ E T ( R \ { } ) / Z . Therefore GL T ( X ) can be identified as a vector bundle (up to Z actions) with the zerogluing data removed. We add zero gluing data to GL T ( X ) to get the whole vector bundle,which is also denoted by GL T ( X ). Fibre of GL T ( X ) is Gl T := (cid:89) e ∈ E T R or (cid:89) e ∈ E T R / Z . It is obvious that the Taubes’ gluing construction can be applied to elements in GL T ( X )with zero gluing parameters. So we can define Ψ (cid:48) T , Ψ T on GL T ( X ), but the image of elements28ith zero gluing parameters may not be in M k ( X ) since after gluing a balanced bubble treeinstanton on an S we get a non-balanced connection in general. To fix this, define a newmap by composing Ψ T with the balanced map ‘ b ’:Ψ bT ( e ) := b ◦ Ψ T ( e ) ∈ M k ( X )where b is the map making all generalised connections (defined in Remark 4.9) on all bubblesbalanced by translations and dilations (note that if all gluing parameters in e are nonzero,then b is the identity).When gluing two connections using different gluing parameters ρ , ρ , as long as theyare in the same orbit of the Γ A × Γ A -action on Gl , they yield the same element in M ( X ).Because of this, Ψ T and Ψ bT may fail to be injective. To fix this, we define a Γ T -action on GL T ( X ).For an ASD connection on S , the stabiliser is C ( G ) (centre of G ) if it has non-zeroenergy, and is G if it has zero energy. Take G = SO (3), then C ( G ) = 1. DefineΓ T := (cid:89) v is a ghost vertex SO (3) v . For each ghost vertex v , suppose the edge connecting v and its parent v − is e − , the edgesconnecting v and its children v , . . . , v n are e , . . . , e n , then SO (3) v acts on GL T ( X ) byacting on gluing parameters associated to e − , e , . . . , e n : ∀ g ∈ SO (3) , ( ρ − , ρ , . . . , ρ n ) · g = ( g − ρ − , ρ g, . . . , ρ n g ) . Here ρ i is the gluing parameter associated to edge e i . Note that the way SO (3) v acts on ρ − is different from the way it acts on ρ i for i = 1 , . . . , n . This is because ρ − ∈ Hom G ( P ( S v − ) | x − , P ( S v ) | ∞ ) ,ρ i ∈ Hom G ( P ( S v ) | x i , P ( S v i ) | ∞ ) , i = 1 , . . . , n, where S v − , S v , S v i are bubble spheres corresponds to the vertices v − , v, v i respectively, x − ∈ S v − \ {∞} , x i ∈ S v \ {∞} are gluing points associated to e − , e i respectively. And SO (3) v acts on ρ − , ρ i through acting on the trivial bundle P ( S v ).This provides a fibre bundle GL T ( X ) / Γ T → S T ( X )with a particular section defined by the zero section of GL T ( X ). The map Ψ T on a smallneighbourhood of the zero section of GL T ( X ) induces a well-defined map on a small neigh-bourhood of the zero section of GL T ( X ) / Γ T , which is also denoted by Ψ T .Suppose T (cid:48) is the contraction of T at e , . . . , e n , define Gl T,T (cid:48) := (cid:32) (cid:89) e ,...,e n R \ { } (cid:33) × { } ⊂ Gl T and define a sub-bundle GL T,T (cid:48) of GL T ( X ) with fibre Gl T,T (cid:48) . Denote by GL T ( (cid:15) ) and GL T,T (cid:48) ( (cid:15) ) the (cid:15) -neighbourhoods of the zero-sections.29 heorem 4.11. (Theorem 3.32 of [C02]) For any T ∈ T k and any precompact subset U of S T ( X ) , there exists (cid:15) such that Ψ T : GL T ( (cid:15) ) | U → (cid:91) T ∈T k (cid:101) S T ( X ) maps GL T,T (cid:48) ( (cid:15) ) | U / Γ T to (cid:101) S T (cid:48) ( X ) diffeomorphically onto its image for any T (cid:48) > T . Proposition 4.12.
For any T ∈ T k and T (cid:48) > T , let U, (cid:15) be as in Theorem 4.11, then Ψ bT = b ◦ Ψ T : GL T,T (cid:48) ( (cid:15) ) | U / Γ T Ψ T −−→ (cid:101) S T (cid:48) ( X ) b −→ S T (cid:48) ( X ) maps GL T,T (cid:48) ( (cid:15) ) | U / Γ T to S T (cid:48) ( X ) diffeomorphically onto its image for any T (cid:48) > T .Proof. It suffices to find a smooth inverse map of Ψ bT from the image of Ψ bT to GL T,T (cid:48) ( (cid:15) ) | U / Γ T .Recall that Ψ bT is the composition of the following three maps: GL T,T (cid:48) ( (cid:15) ) | U / Γ T Ψ (cid:48) T −−→ im (Ψ (cid:48) T ) a −→ (cid:101) S T (cid:48) ( X ) b −→ S T (cid:48) ( X ) . (cid:84)(cid:101) B T (cid:48) ( X )Without loss of generality, assumei.e., T (cid:48) is the contraction of T at the middle vertex. Then GL T,T (cid:48) ( (cid:15) ) = M k ( X ) × F r ( X ) × SO (4) M bk , m × M bk × B (0 , (cid:15) )where m is the tuple consists of total weight of children of v in T , i.e., m = ( k ), and B (0 , (cid:15) )is the 4-dimensional (cid:15) -ball in R with the origin removed. So it suffices to find a smoothinverse map of M bk , m × M bk × B (0 , (cid:15) ) / Γ ( A , A , ρ, λ ) im (Ψ (cid:48) ) A ( ρ,λ ) A M k + k A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) M bk + k ( S ) h (cid:0) A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) (cid:1) . Ψ (cid:48) Ψ ab h ∈ H = Aut ( R ) pulls back A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) to a balanced connection,hence h depends on A , A , ρ, λ . It is obvious that h (cid:0) A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) (cid:1) = h (cid:0) A ( ρ,λ ) A ) (cid:1) + h (cid:0) a ( A ( ρ,λ ) A ) (cid:1) . Recall that in Section 3, given two points x ∈ X , x ∈ X and a small positive parameter λ , we denote the glued space by X λ X . Let X = X = S , then we denote by X ( x,λ ) X or S ( x,λ ) S the space X λ X with x = x ∈ X \ {∞} and x = ∞ ∈ X . Note that thecanonical conformal diffeomorphism X ∼ = X ( x,λ ) X maps B x ( λe δ ) onto X \ B ∞ ( λe − δ )where δ is the constant fixed in Taubes’ gluing construction. We denote this diffeomorphismby i : B x ( λe δ ) → X \ B ∞ ( λe − δ ).The H -action on R induces an H -action on X \ {∞} × R > in the following way: sincethe image of a ball in R under the map h ∈ H is a ball, we can define h ( x, λ ) := ( h ( x ) , h ( λ )) , s.t. B h ( x ) ( h ( λ ) e T ) = h ( B x ( λe T )) , where B x ( λe T ) ⊂ X \ {∞} ∼ = R , T > H -action on X \ {∞} × R > lifts to an H -action on P ( X ) | X \{∞} × R > in the obviousway. Moreover, the H -action on P ( X ) | X \{∞} × R > induces an H -action on (cid:91) x ∈ X \{∞} Hom G ( P ( X ) | x , P ( X ) | ∞ ) . To construct the inverse of b ◦ a ◦ Ψ (cid:48) , first observe that h (cid:0) A ( ρ,λ ) A ) (cid:1) lies in the imageof Ψ (cid:48) since h (cid:0) A ( ρ,λ ) A ) (cid:1) = h ( A ) ( h ( ρ ) ,h ( λ )) (cid:18) h ( λ ) (cid:19) ( A ) . (4.2)To see this, note that both left hand side and right hand side of (4.2) equal to (cid:40) η ( h − ) · h ( A ) outside the ball B h ( x ) ( h ( λ ) e − δ ) ,i ∗ ( η · A ) on B h ( x ) ( h ( λ ) e δ ) , where x ∈ X is the point such that ρ ∈ Hom G ( P ( X ) | x , P ( X ) | ∞ ), η , η are defined in(3.3).We claim that h commutes with a : h ( a ( A ( ρ,λ ) A )) = a ( h ( A ( ρ,λ ) A )) . (4.3) a is a map defined on the image of Ψ (cid:48) and h ( A ( ρ,λ ) A ) lies in the image of Ψ (cid:48) , thereforethe right hand side of (4.3) makes sense. (4.3) holds since the two metrics on X ( ρ,λ ) X and X ( h ( ρ ) ,h ( λ )) X are conformally equivalent.Therefore h (cid:0) A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) (cid:1) = h ( A ( ρ,λ ) A ) + a ( h ( A ( ρ,λ ) A ))31s an element in the image of Ψ andΨ − (cid:0) h (cid:0) A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) (cid:1)(cid:1) = (cid:18) h ( A ) , (cid:18) h ( λ ) (cid:19) ( A ) , h ( ρ ) , h ( λ ) (cid:19) ∈ M k , m × M k × B (0 , (cid:15) ) , which is in the same H -orbit as the element ( A , A , ρ, λ ). Since ( A , A , ρ, λ ) is balanced,after applying the balanced map b we have b ◦ Ψ − (cid:0) h (cid:0) A ( ρ,λ ) A + a ( A ( ρ,λ ) A ) (cid:1)(cid:1) = ( A , A , ρ, λ ) ∈ M bk , m × M bk × B (0 , (cid:15) ) . So the inverse of b ◦ Ψ is b ◦ Ψ − .Proposition 4.12 gives a set of maps on an open cover of M k ( X ) D ( X, k ) := (cid:8)(cid:0) Ψ bT ( GT T ( (cid:15) ) | U ) / Γ T , (Ψ bT ) − (cid:1)(cid:9) T ∈T k . (4.4)If T contains no ghost vertex, Γ T -action is free up to a finite group on Ψ bT ( GT T ( (cid:15) ) | U ). Soaway from the ghost strata, (4.4) give an orbifold atlas on M k ( X ) \ S k ( X ). In [C02], itis proved that after perturbing the atlas (4.4) and applying so-called “flip resolutions” toresolve ghost strata, we get a smooth orbifold M k ( X ). Z α -equivariant bundles and Z α -invariant moduli spaces Let Γ be a finite group acting on M from left and preserve metric g on M . The Γ-actionon M may not lift to P in general. Let H be the bundle automorphisms of P covering anelement in Γ ⊂ Diff( M ). Then G is a subset of H covering identity in Diff( M ). There is anexact sequence: 1 → G → H → Γ → . Note that the Γ-action on M induces a well-defined Γ-action on B : given γ ∈ Γ, [ A ] ∈ B ,define γ · [ A ] := [ h ∗ γ A ] where h γ ∈ H covers γ ∈ Diff( M ). The well-definedness of thisaction follows from the fact that two elements in H covering the same γ differ by a gaugetransformation. The metric g on M is Γ-invariant, therefore M k ⊂ B is Γ-invariant. Denote B Γ := { [ A ] ∈ B | γ · [ A ] = [ A ] } , B ∗ Γ := { [ A ] ∈ B ∗ | γ · [ A ] = [ A ] } , M Γ := { [ A ] ∈ M | γ · [ A ] = [ A ] } . Suppose [ A ] ∈ B Γ , we get the following short exact sequence:1 → Γ A → H A → Γ → , (5.1)where Γ A ⊂ G , H A ⊂ H are stabilisers of A . If A is irreducible, Γ A = Z (or 0 in SO (3)cases) and H A → Γ is a double cover (or an isomorphism in SO (3) cases). In summary, any32rreducible Γ-invariant connection in B induces a group action on P that double covers (orcovers) the Γ-action on M . Denote A H A := { A | h ∗ γ A = A, ∀ h γ ∈ H A } , G H A := { g | gh γ = h γ g, ∀ h γ ∈ H A } , B H A := A H A / G H A . Now there are two kinds of “invariant moduli space”: B Γ and A H A / G H A . A prerequisite of the former is a Γ-action on X , of the latter is a H A -action on P .Let Γ be a cyclic group Z α . Suppose the Z α -action on M satisfies Condition 1.1. Asstated in [F92], the irreducible ASD part of A H A / G H A (denoted by A ∗ ,ASD, H A / G H A ) is, aftera perturbation, a smooth manifold. While the irreducible ASD part of B Z α , denoted by M ∗ Z α , is not a manifold in general. It might contain components with different dimension aswe will see in the example of the weighted complex projective space in Section 8. Proposition 5.1. (Proposition 2.4 of [F89]) The natural map A ∗ , H A / G H A → B ∗ Z α is a homeomorphism onto some components. Restricting to the ASD part, we get A ∗ ,ASD, H A / G H A → M ∗ Z α , which is also a homeomorphism onto some components. From the discussion above, we have B ∗ Z α = (cid:91) [ A ] ∈B ∗ Z α A ∗ , H A / G H A , where each A ∗ , H A / G H A is viewed as a subset in B ∗ Z α through the map in Proposition 5.1.The union is not disjoint. Furthermore, each H A can be seen as an element in { φ ∈ Hom ( Z α , H ) | ˜ φ ∈ Hom ( Z α , Diff( M )) is a double cover of the Z α -action on M } (or { φ ∈ Hom ( Z α , H ) | ˜ φ ∈ Hom ( Z α , Diff( M )) is the Z α -action on M } ) (5.2)where ˜ φ is the group action on M induced by φ . Denote by J the quotient of (5.2) byconjugation of gauge transformations. Then we have Theorem 5.2.
The map (cid:71) [ φ ] ∈ J A ∗ ,φ / G φ → B ∗ Z α is bijective. roof. We already have surjectivity. Proposition 5.1 implies injectivity on each disjoint set.Hence it suffices to show that if A ∈ A ∗ ,φ , A ∈ A ∗ ,φ and A = g · A for some g ∈ G , then[ φ ] = [ φ ] . This follows from g · φ · g − · A = g · φ · A = g · A = A = φ · A . Next we classify the index set J . Proposition 5.3.
The map isomorphism classes of Z α -equivariant SU (2) bundle P → M double covering Z α -action on M → Z × Z a × · · · × Z a n P (cid:55)→ ( c ( P ) , m , . . . , m n ) is injective where c is the second Chern number and m i is defined as follows: let f i : Z a i → S ⊂ SU (2) be the isotropy reprensentation at points π − ( x i ) = { x i , . . . , x α/a i i } , then m i ∈ Z a i ∼ = H ( L ( a i , b i )) is the Euler class of the S -bundle S × S f i → S / Z a i = L ( a i , b i ) . Proof. (Sketch) The proof is basically the same as that of Proposition 4.1 in [FS85].If f i is the trivial map, then P | ∂B ( x i ) /f i is the trivial bundle over L ( a i , b i ) and m i = 0. If f i is non-trivial, then P | ∂B ( x i ) reduces to an S -bundle and the Euler class of this S -bundlequotient by f i is the weight of f i . Therefore any two isomorphic Z α -equivariant SU (2)bundles have the same m , . . . , m n . This gives the well-definedness.Given ( c , m , . . . , m n ), we use m , . . . , m n to get the unique Z α -equivariant SU (2)-bundle over a neighbourhood N of singularities in M , N := n (cid:91) i =1 α/a i (cid:91) j =1 B ( x ji ) , and hence get an SU (2)-bundle over ∂N/ Z α . If ( c , m , . . . , m n ) lies in the image of the mapin this proposition, then the SU (2)-bundle over ∂N/ Z α extends uniquely to M \ N/ Z α since c is fixed. This shows the injectivity.For the SO (3) case, there is a similar injective map: isomorphism classes of Z α -equivariant SO (3) bundle P → M covering Z α -action on M → Z × H ( M ; Z ) × Z a × · · · × Z a n P (cid:55)→ ( p ( P ) , w ( P ) , m , . . . , m n ) . SU (2) case, since − ∈ Z α acts on P either as id or as − id , so m , . . . , m n all are either even or odd. For the SO (3) case, not allpairs ( p , w ) occur as ( p ( P ) , w ( P )) for some bundle P . In this chapter we do not describewhat the image subset is. If an element ( c , m , . . . , m n ) is not in the image, then we saythe set of bundles and connections corresponding to ( c , m , . . . , m n ) is empty. Denote I := { c ( P ) } × Z a × · · · × Z a n ( or { ( p ( P ) , w ( P )) } × Z a × · · · × Z a n ) . (5.3)We have B ∗ Z α = (cid:71) i ∈ I A ∗ ,φ i / G φ i , M ∗ Z α = (cid:71) i ∈ I A ∗ ,ASD,φ i / G φ i , where φ i is the Z α (or Z α )-action on P corresponds to i ∈ I . Definition 5.4.
Let M , the Z α -action on M and { a i , b i } ni =1 be defined as above. The bundle P → M characterised by ( c , m , . . . , m n ) (or ( p , w , m , . . . , m n )) is called a bundle of type O where O = { c , ( a i , b i , m i ) ni =1 } (or ( p , w , ( a i , b i , m i ) ni =1 )). In the SO (3) case, sometimes w in O is omitted if w is fixed and contextually clear. (cid:4) So far we have discussed irreducible connections. For reducible connections, the onlydifference between the ordinary case and the Z α (or Z α )-equivariant case is that the U (1)-reduction of P needs to be Z α (or Z α )-equivariant. Proposition 4.1, 5.3 and 5.4 in [FS85]give a full description of equivariant U (1)-reduction and invariant reducible connections.To summarise, Table 1 shows the similarity and difference in the study of instantonmoduli spaces on manifolds and this special kind of orbifolds.Table 1 Objects manifold M orbifold X := M/ Z α Bundles
P SU (2)-bundleor SO (3)-bundle Z α -equivariant SU (2)-bundle on M or Z α -equivariant SO (3)-bundle on M Classification of
P c or ( p , w ) ( c , m , . . . , m n )or ( p , w , ( a i , b i , m i ) ni =1 )Classification of U (1)-reductions Q ± e ∈ H ( M ; Z ) with e = − c or w = e mod 2 and e = p ± e ∈ H ( X \ { x , . . . , x n } ; Z ) with e | L ( a i ,b i ) = m i and i ∗ π ∗ ( e ) = − c or w = i ∗ π ∗ ( e ) mod 2 and i ∗ π ∗ ( e ) = p Elliptic complex Ω ( adP ) d A −−→ Ω ( adP ) d + A −−→ Ω , + ( adP ) Ω ( adP ) Z α d A −−→ Ω ( adP ) Z α d + A −−→ Ω , + ( adP ) Z α or Ω ( adP ) Z α d A −−→ Ω ( adP ) Z α d + A −−→ Ω , + ( adP ) Z α Dimension ofASD moduli space 8 c − b + )or − p − b + ) index of the complex above,shown in formula (5.4) In Table 1, π : M → X is the projection and i is the inclusion map from punctured M to M . Remark 5.5.
The calculation of the index of the invariant elliptic complex in Table 1 is inSection 6 of [FS85]. Two points need to be mentioned in our case.35irst, [FS85] considered pseudofree orbifolds, which can also be expressed as M/ Z α .Although M/ Z α has isolated singular points { x , . . . , x n } , the branched set of the branchedcover π : M → M/ Z α is { x , . . . , x n } union with a 2-dimensional surface F in M/ Z α . In theset-up of [FS85], the Z α -action on the SO (3)-bundle P → M is trivial when restricted to P | π − ( F ) , hence the Lefschetz number restricted to π − ( F ) is easy to calculate and the indexis the same as the case when the branched set consists only of isolated points.Second, by the construction of P in [FS85], m i is relatively prime to a i , (i.e. m i is agenerator in H ( L ( a i , b i ); Z )) but the method used in [FS85] also works for the case when Z a i acts by any weight on the fibre over the singular point. ([A90] used the same approachto calculate this index for M = S .) In this more general case the result of Theorem 6.1 in[FS85] becomes − p ( P ) α − b +2 ) + n (cid:48) + n (cid:88) i =1 a i a i − (cid:88) j =1 cot (cid:18) πjb i a i (cid:19) cot (cid:18) πja i (cid:19) sin (cid:18) πjm i a i (cid:19) , (5.4)where n (cid:48) is the number of { m i | m i (cid:54)≡ a i } . For the SU (2) case, we replace − p ( P )in this formula by 8 c ( P ). (cid:4) S S Let P → S be the SU (2)-bundle with c ( P ) = 1. The ASD moduli space M ( S ) is a5-dimensional ball. In this section we describe this moduli space explicitly.The 4-sphere S can be covered by two open sets S ∼ = HP = U ∪ U with coordinatecharts and transition map as ϕ : U := { [ z , z ] | z (cid:54) = 0 } → H [ z , z ] (cid:55)→ z /z ϕ : U := { [ z , z ] | z (cid:54) = 0 } → H [ z , z ] (cid:55)→ z /z ϕ ◦ ϕ − : H \ { } → H \ { } z (cid:55)→ z − . The bundle P over S can be reconstructed as U × Sp (1) (cid:71) U × Sp (1) / ∼ , ( z, g ) ∼ (cid:18) z − , z | z | g (cid:19) ∀ z ∈ U \ { } . According to Section 3.4 of [DK90] and Chapter 6 of [FU84], there is a standard ASDconnection θ defined by A = Im ¯ zdz | z | ∈ Ω ( U , su (2)) , SO (5)-action on S . The moduli space M ( S ) is a5-dimensional ball: M ( S ) = (cid:8) T ∗ λ,b θ | λ ∈ (0 , , b ∈ S (cid:9) , where T λ,b is a conformal transformation on S that is induced by the automorphism x (cid:55)→ λ ( x − b ) on R . In particular, for b = 0, the connection T ∗ λ, θ can be represented under thetrivialisation over U as A ( λ ) := Im ¯ zdzλ + | z | ∈ Ω ( U , su (2)) . (6.1) Γ -invariant connections in M ( S ) In this section we describe the Γ-invariant part of M ( S ) for some finite group Γ.Let SO (4) ∼ = Sp (1) × ± Sp (1) act on S by the extension of the action on H :( Sp (1) × ± Sp (1)) × H → H ([ e + , e − ] , z ) (cid:55)→ e + ze − − . This action has no lift to the bundle P , hence the action can not be defined on A ( P ) in thestandard way. But this SO (4)-action lifts to a Spin (4) ∼ = Sp (1) × Sp (1) action on P in thefollowing way: ( e + , e − ) · ( z, g ) = ( e + ze − − , e − g ) ∀ ( z, g ) ∈ U × Sp (1) , ( e + , e − ) · ( z, g ) = ( e − ze − , e + g ) ∀ ( z, g ) ∈ U × Sp (1) . This lifting defines an SO (4)-action on A and G : For any element γ in SO (4) acting on S ,there are two elements ˜ γ, − ˜ γ in Spin (4) that act on P covering γ . For any A ∈ A , we define γ · A := ˜ γ · A . Since {± } ⊂ Γ A for any A ∈ A , this action is well-defined. γ acts on G byconjugation of ± ˜ γ .From Section 6.1, we know M ( S ) is a 5-dimensional ball parametrised by b , the masscentre in S , and λ , the scale of concentration of mass around the mass centre. Then wehave the following proposition. Proposition 6.1.
For any non-trivial finite subgroup Γ ⊂ SO (4) such that Γ acts freely on R \ { } , the Γ -invariant ASD moduli space is M Γ1 ( S ) := A ASD, Γ ( P ) / G Γ ( P ) ∼ = { A ( λ ) | λ ∈ (0 , ∞ ) } where A ( λ ) is defined in (6.1). Moreover, this space is a smooth 1-dimensional manifolddiffeomorphic to R . The proof can be found in Lemma 5.2 of [F92]. If Γ is a cyclic group, there is anotherway to calculate the dimension of M Γ1 ( S ), which will be shown in Example 6.4.37 .3 Z p -invariant connections in M k ( S ) In this section we describe the Z p -invariant part of M k ( S ).Suppose Z p acts on S as the extension of the following action: e πip · ( z , z ) = ( e πip z , e πiqp z ) , where e πip is the generator of Z p , ( z , z ) ∈ C ∼ = R , q is an integer relatively prime to p .Identify C with H by ( z , z ) (cid:55)→ z + z j , then this action is ∀ z ∈ H , e πip · z = e πi (1+ q ) p ze πi (1 − q ) p . From the discussion in Section 5, H A is Z p or Z p × Z (depending on whether (5.1) splits ornot). This means there always exists a Z p -action on P that is a double covering Z p -actionon S .According to Section 2.3 of [A90] or the Proposition 5.3 in this paper, a Z p -equivariant SU (2)-bundle over S can be described by a triple ( k, m, m (cid:48) ): where k is the second Chernnumber of P , m and m (cid:48) are the weights of the Z p -action on the fibres over { } and {∞} respectively. By ‘ Z p -action on a fibre has weight m ’ we mean there is a trivialisation nearthis fibre such that the generator e πip of Z p acts on this fibre by e πip · g = e mπip g where g ∈ Sp (1). Example 6.2.
Consider the bundle P → S defined in Section 6.2. In the trivialisationsover U , U , the Z p -action is defined by e πip · ( z, g ) = ( e πi (1+ q ) p ze πi (1 − q ) p , e πi ( q − p g ) ∀ ( z, g ) ∈ U × Sp (1) ,e πip · ( z, g ) = ( e πi ( q − p ze πi ( − q − p , e πi ( q +1) p g ) ∀ ( z, g ) ∈ U × Sp (1) . Theorem 6.3. (Lemma 5.1, Proposition 4.3, Theorem 5.2 and Section 4.4 of [A90]) Supposean SU (2) -bundle P → S with Z p -action on P that is a double covering of the Z p -actionon S has second Chern number and action-weights ( k, m, m (cid:48) ) .(1). If ( k, m, m (cid:48) ) satisfies the condition that there exists a, b ∈ Z s.t. (cid:40) aq ≡ m (cid:48) + m mod p, b ≡ m (cid:48) − m mod p. (6.2) Then the space of Z p -invariant ASD connections in M k ( S ) , denoted by M Z p ( k,m,m (cid:48) ) ( S ) is non-empty. Furthermore, ab ≡ k mod p .(2). M Z p ( k,m,m (cid:48) ) ( S ) (cid:54) = ∅ if and only if there is a sequence of finite triples { ( k i , m i , m (cid:48) i ) } ni =1 satisfying formula (6.2) and k i > , k = n (cid:88) i =1 k i , m ≡ m mod p , m (cid:48) i ≡ m i +1 mod p , m ≡ m (cid:48) n mod p. ( k, m, m (cid:48) ) satisfies the conditions in (2), the dimension of the invariant moduli spaceisdim M Z p ( k,m,m (cid:48) ) ( S ) = 8 kp − n + 2 p p − (cid:88) j =1 cot πjp cot πjqp (cid:18) sin πjm (cid:48) p − sin πjmp (cid:19) , (6.3) where n ∈ { , , } is the number of elements of the set { x ∈ { m, m (cid:48) } | x (cid:54)≡ , p mod p } . Example 6.4.
Consider the bundle P in Example 6.2, the second Chern number and actionweights are (1 , q − , q + 1), which satisfies (6.2) for a = b = 1. M Z p (1 ,q − ,q +1) ( S ) is non-emptyand by formula (6.3) the dimension is given bydim M Z p (1 ,q − ,q +1) ( S )= 8 p − n + 8 p p − (cid:88) j =1 cos (cid:18) πjp (cid:19) cos (cid:18) πjqp (cid:19) = 8 p − n + 8 p p − (cid:88) j =1 (cid:16) πjp (cid:17) (cid:16) πjqp (cid:17)
2= 8 p − n + 2 p p − (cid:88) j =1 (cid:18) πjp (cid:19) + cos (cid:18) πjqp (cid:19) + cos (cid:16) πj ( q +1) p (cid:17) + cos (cid:16) πj ( q − p (cid:17) = 1 . where n ∈ { , , } be the number of q − , q + 1 (cid:54)≡ p . The last equality follows fromthe fact that (cid:80) p − j =1 cos (cid:16) πjmp (cid:17) equals to − m (cid:54)≡ p , or ( p −
1) if m ≡ p . Thismatches the result in Proposition 6.1. (cid:4) Z p -invariant connections on S Recall that the space of balanced connections on S is defined to be A bk := A k /H, where H ⊂ Aut ( R ) is the subgroup generated by translations and dilations. Correspond-ingly, the balanced moduli space and the balanced ASD moduli space are B b := B /H , M bk := M k /H. When k (cid:54) = 0, an equivalent definition of M bk is M bk = (cid:26) [ A ] | (cid:90) R x | F A | = 0 , (cid:90) B (1) | F A | = (cid:126) (cid:27) , B (1) is the unit ball in R and (cid:126) is a positive constant less than 4 kπ .Given Z p -action on P → S whose second Chern number and action weights on { } , {∞} are ( k, m, m (cid:48) ), there is an R + -action on M Z p ( k,m,m (cid:48) ) induced by the dilations on R . Denotethe balanced Z p -invariant moduli space by M Z p ,b ( k,m,m (cid:48) ) := M Z p ( k,m,m (cid:48) ) / R + . Note that in the Z p -invariant case we only quotient by the dilation action since translationdoes not preserve Z p -invariance of a connection in general. M/ Z α Recall that in the bubble tree compactification for a 4-manifold M , there are 3 importantconcepts:(1) Bubble tree T ,(2) Space of bubble tree instantons S T ( M ),(3) Gluing bundle GL T ( M ) → S T ( M ).Now given a Z α (or Z α )-equivariant bundle over M characterised by O defined in Definition5.4, we need to define the following corresponding version:(1) O -bubble-tree T ,(2) O -invariant bubble tree instanton space S O T ( M ),(3) O -invariant gluing bundle GL O T ( M ) over S O T ( M ). Definition 7.1.
Let Z a act on S ∼ = C ∪ {∞} by e πi/a · ( z , z ) = ( e πi/a z , e πib/a z )for some coprime integers a, b . Given O = { k, ( a, b ) , ( m, m (cid:48) ) } where k ∈ Z ≥ , m, m (cid:48) ∈ Z a .An O -bubble-tree on S / Z a is a triple ( T, v , w ), where T is a tree with root v and w is amap on the vertices of T , such that(1). ( T, v , w ) has a path of the formfor some j ≥ k i are called the weight of v i .402). On the complement of this path, w assigns to each vertex a non-negative integer, whichis also called the weight of the vertex.(3). Assign • each vertex in { v , . . . , v j } a singular bubble S / Z a , • each vertex in V T \ { v , . . . , v j } a bubble S , • each edge e i ∈ { e , . . . , e j − } the north pole of the singular bubble corresponds to v i , • each edge e ∈ E T \ { e , . . . , e j − } a point on the bubble corresponds to v or apoint away from the north pole on the singular bubble corresponds to v , where v is the elder vertex among the two vertices connected by e .Define the pullback of ( T, v , w ) to be the triple ( ˜ T , v , ˜ w ) with a Z a -action such that T = ˜ T / Z a . Take a = 3, j = 1, then ( T, v , w ) and ( ˜ T , v , ˜ w ) defined in Figure 10 givesan example of the pullback of T .Figure 10: The pullback of T (4). Let ( ˜ T , v , ˜ w ) be the pullback of ( T, v , w ). By composing ˜ w with pr , the projectionon to the first coordinate, ( ˜ T , v , pr ◦ ˜ w ) defines a bubble tree with total weight k . (cid:4) A singular bubble in an O -bubble-tree is called ghost bubble if it has weight 0. ByDefinition 7.1, an O -bubble-tree could have a ghost singular bubble with only one child.Figure 10 gives such an example if we take k = 0, k (cid:54) = 0.41 efinition 7.2. Let M/ Z α be the orbifold defined as before. Given O = { k, ( a i , b i , m i ) ni =1 } ,an O -bubble-tree on M/ Z α is a triple ( T, v , w ), where T is a tree with root v and w is amap on vertices of T such that(1). w maps v to O := { m ( v ) , ( a i , b i , m i ) ni =1 } for some m ( v ) , m i .(2). Among children of v , for each i ∈ { , . . . , n } , there is at most one vertex v such that w ( v ) = ( m ( v ) , ( a i , b i ) , ( m, m (cid:48) )) , for some m ( v ) , m, m (cid:48) . If such v exists, we assign the edge connecting v and v the singular point of the cone cL ( a i , b i ) ⊂ M/ Z α . The subtree ( t ( v ) , v, w ) is a { k i , ( a i , b i ) , ( m i , m i ) } -bubble-tree on S / Z a i for some k i > v does not exists, m i = m i .(3). For all other vertices of T , w assigns each a non-negative integer.(4). Define the pullback of ( T, v , w ) to be the triple ( ˜ T , v , ˜ w ) with a Z α -action such that T = ˜ T / Z α . By composing ˜ w with pr , the projection to the first coordinate, ( ˜ T , v , pr ◦ ˜ w ) defines a bubble tree with total weight k . (cid:4) Denote the space of O -bubble-trees by T O . The pullback of an O -bubble-tree on M/ Z α is a bubble tree on M . The difference between O -bubble-trees and bubble trees is thatthe w map in O -bubble-trees gives more information. When there is no group action, theweight-map w in a bubble tree assigns each vertex simply an integer since SU (2) or SO (3)-bundles are characterised by c or p when w is fixed. For an O -bubble-tree ( T, v , w ), tocharacterise Z α (or Z α )-equivariant bundles over M and S ’s, the weight-map w containsmore information, as defined in Definition 7.2.In the following discussion, we also treat O -bubble-trees as the trees pulled back to M ,then there is an obvious Z α -action on each O -bubble-tree on M . Examples 7.3.
Suppose α = 6 , a = 2 , a = 3 , b = b = 1 and the space X = M/ Z α hastwo singularities whose neighbourhoods are cL ( a , b ) and cL ( a , b ). The branched coveringspace M has a Z -action on it such that the action is free away from 5 points x , . . . , x ( x , x , x are in the same Z -orbit with stabiliser Z and x , x are in the same Z -orbitwith stabiliser Z ), as shown in Figure 11.Figure 12 is an O -bubble-tree for O = ( k, ( a i , b i , m i ) i =1 )where k = m ( v ) + 6 m ( v ) + 2 m ( v ) + 6 m ( v ). The right diagram in Figure 12 is the tree on X and the left one is the tree pulled back to M . Figure 13 is another way to draw this tree.In Figure 13, x , x are defined in Example 7.3; the points y , . . . , y ∈ M \ { x , . . . , x } arein the same Z -orbit (these six points are the preimage of some non-singular point y ∈ X under the branched covering map); e , e , e are in the same orbit of the induced Z -actionon the bubble. Note that given the definition of w ( v ) and w ( v ), the vertex v correspondsto a bubble S and has to be attached to non-singular points in X . v corresponds to asingular bubble S / Z a and has to be attached to the singular point of the cone cL ( a , b ).42igure 11Figure 12Figure 13 (cid:4) Definition 7.4.
Given an O -bubble-tree T on M , it induces a unique (if it exists) Z α (or Z α )-action on bundles over M and S ’s, therefore also on A , G and S T ( M ). The space of O -invariant bubble tree instantons , denoted as S O T ( M ), is the Z α (or Z α )-invariant part of S T ( M ). In particular, if T is the trivial tree, i.e., T has only one vertex, we denote thecorresponding space of O -invariant bubble tree instantons by M O . (cid:4) xamples 7.5. The following is an example of an O -invariant bubble tree instanton associ-ated to the tree in Example 7.3, where A is Z (or Z )-invariant, A is Z (or Z )-invariant. (cid:4) Denote the compactified O -invariant moduli space and Z α -invariant moduli space by M O := (cid:91) T ∈T O S O T ( M ) , M k, Z α := (cid:91) i ∈ I M O i , where I is the index set defined in (5.3). Definition 7.6.
Given an O -bubble-tree T , the O -invariant gluing bundle GL O T ( M ) over S O T ( M ) is the Z α (or Z α )-invariant part of GL T ( M ). (cid:4) Recall that the fibre of GL T ( M ) is Gl T = (cid:89) e ∈ E ( T ) R / Z where E ( T ) is the set of edges in T . Since Z α ⊂ U (1), by Proposition 3.4, the fibre of GL O T ( M ) is Gl O T = (cid:89) e ∈ E ( T ) R / Z × (cid:89) e ∈ E ( T ) R / Z where E ( T ) , E ( T ) are the subsets of edges in T on X whose corresponding isotropy repre-sentation is trivial and non-trivial respectively. Therefore GL O T ( M ) is a vector bundle (upto a finite group action) over S O T ( M ). Similarly to the no-group-action version, supposing T (cid:48) ∈ T O is the contraction of T at edges e , . . . , e n , we define GL O T,T (cid:48) ( M ) to be the sub-bundleof GL O T ( M ) whose fibres consist of points having non-zero gluing parameters correspondingto e , . . . , e n and zero gluing parameters corresponding to other edges. Sometimes we omit M in S O T ( M ), GL O T ( M ), GL O T,T (cid:48) ( M ) when it is clear from the context.44efine Γ O T := (cid:89) v ∈ V ( T ) SO (3) × (cid:89) v ∈ V ( T ) U (1)where V ( T ) is the subset of ghost vertices in T on X whose corresponding bubble is S or S / Z a i with trivial isotropy representations on the fibre over its south and north poles. V ( T ) is the complement of V ( T ) in the set of ghost vertices.Given the definition of the space of O -invariant bubble tree instantons S O T , and the O -equivariant gluing bundle GL O T , we define the gluing map and orbifold structure on M O ( M )away from the ghost strata.First consider the simplest case: consider the tree T on X with two vertices v , v andone edge. Fix an l ∈ { , . . . , n } , let w ( v ) = ( k , ( a i , b i , m i ) ni =1 ) , m i = m i for i (cid:54) = l,w ( v ) = ( k , ( a l , b l ) , ( m l , m l )) . In this case the point assigned to the edge is the singular point of the cone cL ( a l , b l ) ⊂ X .For the gluing map associated to this tree we have the following theorem. Theorem 7.7.
Let O = { k , ( a i , b i , m i ) ni =1 } , l ∈ { , . . . , n } and U ⊂ M O ( M ) , U b ⊂M Z al ,b ( k ,m l ,m l ) ( S ) be open neighbourhoods. Assume I ∼ = U (1) if m l (cid:54)≡ mod a l and I ∼ = SO (3) if m l ≡ mod a l , then by the equivariant Taubes gluing construction in Section 3, we havethe following map gluing connections at the singular point of the cone cL ( a l , b l ) ⊂ X Ψ T : U × U b × I × (0 , (cid:15) ) → M O ( M ) , where O = { k + αk a l , ( a i , b i , m i ) ni =1 } and m i = m i for i (cid:54) = l . Ψ T is a diffeomorphism onto itsimage.Proof. Since Ψ T is the restriction of the map in Theorem 3.8 in [C02] to the Z α -invariantpart, it suffices to check the dimensions of U × U b × I × (0 , (cid:15) ) and M O ( M ) are equal. Takean SU (2)-bundle as an example, with k its second Chern number. dim ( U ) = 8 k α − b +2 ) + n (cid:48) + n (cid:88) i =1 a i a i − (cid:88) j =1 cot (cid:18) πjb i a i (cid:19) cot (cid:18) πja i (cid:19) sin (cid:18) πjm i a i (cid:19) dim ( U b ) = 8 ka l − n (cid:48)(cid:48) + 2 a l a l − (cid:88) j =1 cot (cid:18) πjb l a l (cid:19) cot (cid:18) πja l (cid:19) (cid:18) sin (cid:18) πjm l a l (cid:19) − sin (cid:18) πjm l a l (cid:19)(cid:19) dim ( M O ( M )) = (cid:18) k α + 8 ka l (cid:19) − b +2 ) + n (cid:48)(cid:48)(cid:48) + (cid:88) i (cid:54) = l a i a i − (cid:88) j =1 cot (cid:18) πjb i a i (cid:19) cot (cid:18) πja i (cid:19) sin (cid:18) πjm i a i (cid:19) + 2 a l a l − (cid:88) j =1 cot (cid:18) πjb l a l (cid:19) cot (cid:18) πja l (cid:19) sin (cid:18) πjm l a l (cid:19) n (cid:48) is the number of { m i | m i (cid:54)≡ a i } , n (cid:48)(cid:48) is the number of { m l , m l (cid:54)≡ a l } , n (cid:48)(cid:48)(cid:48) is the number of { m i | m i (cid:54)≡ a i } .We check n (cid:48)(cid:48)(cid:48) = − n (cid:48) + n (cid:48)(cid:48) + dim ( I ) + 1.If m l (cid:54)≡ m l (cid:54)≡ n (cid:48)(cid:48)(cid:48) − n (cid:48) = 0 , n (cid:48)(cid:48) = 2 , dim ( I ) = 1 . If m l ≡ m l (cid:54)≡ n (cid:48)(cid:48)(cid:48) − n (cid:48) = 1 , n (cid:48)(cid:48) = 1 , dim ( I ) = 3 . If m l (cid:54)≡ m l ≡ n (cid:48)(cid:48)(cid:48) − n (cid:48) = − , n (cid:48)(cid:48) = 1 , dim ( I ) = 1 . If m l ≡ m l ≡ n (cid:48)(cid:48)(cid:48) − n (cid:48) = 0 , n (cid:48)(cid:48) = 0 , dim ( I ) = 3 . Now we state the result for general O -bubble tree T . Theorem 7.8.
For any O -bubble tree T and precompact subset U ⊂ S O T ( M ) , there exists (cid:15) > such that Ψ bT : ( GL O T ( (cid:15) ) | U ) / Γ O T → M O is a local diffeomorphism from ( GL O T,T (cid:48) ( (cid:15) ) | U ) / Γ T to S O T (cid:48) for any O -bubble tree T (cid:48) satisfying T (cid:48) > T . The proof is the same as the proof of the corresponding theorem of manifold withoutgroup action. Then we get a set D ( X, O ) = { Ψ bT ( GL O T ( (cid:15) ) | U ) / Γ O T , (Ψ bT ) − } T ∈T O , which gives an orbifold structure away from the ghost strata. Just like the case with no Z α -action, we can perturb the map Ψ bT to be Ψ bT so that this atlas is smooth, apply flipresolutions to resolve ghost strata and get M O . Theorem 7.9.
Suppose we have the Z α -action on M satisfies Condition 1.1 and P → M is an SU (2) (or SO (3) )-bundle with c (or p )=k. Let the bubble tree compactification of theirreducible Z α -invariant instanton moduli space be M k, Z a := (cid:71) i ∈ I M O i , where I is the index set defined in (5.3). Then each component M O i is an orbifold away fromthe stratum with trivial connection as the background connection of the bubble tree instantons.The dimension of the component M O with O = { k, ( a i , b i , m i ) ni =1 } is, for SU (2) case, c α − b +2 ) + n (cid:48) + n (cid:88) i =1 a i a i − (cid:88) j =1 cot (cid:18) πjb i a i (cid:19) cot (cid:18) πja i (cid:19) sin (cid:18) πjm i a i (cid:19) , where n (cid:48) is the number of { m i | m i (cid:54)≡ mod a i } . For SO (3) case replace c by − p . An example: instanton moduli space on SO (3) -bundleover CP and weighted complex pro jective space CP r,s,t )For any SO (3)-bundle P on CP , the second Stiefel-Witney class w ( P ) is either 0 or 1. Inthis chapter we describe the ASD moduli space of SO (3)-bundles over CP with w = 1.Choose a Spin C (3) structure on P , i.e. a U (2)-bundle E lifting P , we have w ( P ) = c ( E ) mod 2 , p ( P ) = c ( E ) − c ( E ) . Since w = p mod 2, the SO (3)-bundles on CP are classified by p . If P admits some ASDconnection, we have p ( P ) = − , − , − , . . . , then along with the fact that b − ( CP ) = 0,the bundle P is not reducible. Hence there is no reducible connection on P if it admits anASD connection.The bundle with p = − ( CP ) consisting ofall anti-self-dual 2-forms, denoted as Λ , − ( CP ). By Section 4.1.4 of [DK90], the ASD modulispace of this bundle is a single point: the standard connection induced by the Fubini-Studymetric.The first subsection calculates M p = − ( CP ) and its compactification, in which case theUhlenbeck compactification and bubble tree compactification coincide. The second subsec-tion introduces a cyclic group action on CP , and calculates the Z a -invariant moduli space M Z a p = − ( CP ) and its compactification.Theorem 8.1 and 8.2 are the main tools used in this section. Theorem 8.1. (Proposition 6.1.13 in [DK90]) If V is an SO (3) bundle over a compact,simply connected, K¨ahler surface X with w ( V ) the reduction of a (1,1) class c , there isa natural one-to-one correspondence between the moduli space M ∗ ( V ) of irreducible ASDconnections on V and isomorphism classes of stable holomorphic rank-two bundles E with c ( E ) = c and c ( E ) = ( c − p ( V )) . Since CP is a K¨ahler surface and the generator of H ( CP ; Z ) ∼ = Z is a type (1,1) classwhich is also an integral lift of the generator of H ( CP ; Z ) = Z , the theorem above impliesthe following correspondence M p =1 − k ( CP ) ↔ isomorphism classes ofstable holomorphic rank-2 bundlesover CP with c = − , c = k . Here we choose a generator of the cohomology ring H ∗ ( CP ; Z ) so that the Chern classes ofbundles over CP can be expressed as integers. Theorem 8.2. (Theorem 4.1.15 of Chapter 2 in [OSS88]) Let
V, H, K be complex vectorspaces with dimension , k − , k respectively. There is a one-to-one correspondence isomorphism classes ofstable holomorphic rank-2 bundleover CP with c = − , c = k ↔ P/G here G = GL ( H ) × O ( K ) and P = α ∈ Hom ( V ∗ , Hom ( H, K )) (cid:12)(cid:12)(cid:12)(cid:12) ( F ) ∀ h (cid:54) = 0 , the map α h : V ∗ → K defined by α h ( z ) := α ( z )( h ) has rank ≥ .(F2) α ( z (cid:48) ) t α ( z (cid:48)(cid:48) ) = α ( z (cid:48)(cid:48) ) t α ( z (cid:48) ) for all z (cid:48) , z (cid:48)(cid:48) ∈ V ∗ . and G acts on P by ( g, φ ) · α = φ ◦ α ( z ) ◦ g − . M p = − ( CP ) and its bubble tree compactification In Theorem 8.2, P is a subspace of Hom ( V ∗ , Hom ( H, K )) satisfying conditions (F1) and(F2). (F2) is a closed condition while (F1) is an open condition, hence
P/G fails to becompact. In this section we compactify
P/G for k = 2.The space P/G for k = 2 is described in Section 4.3 of Chapter 2 of [OSS88]. In thiscase, dim C H = 1 , dim C K = 2, and the condition (F2) is automatically true, so P = { α ∈ M , ( C ) | α has rank ≥ } ,G = O (2 , C ) × C × , where M , ( C ) is space of 2 × C × is space of non-zero complex numbers.Let S be the space of 3 × P/G → S / C × =: P ( S )[ α ] (cid:55)→ [ α t α ] . There is an obvious way to compactify P ( S ): By adding the projective space of S , the3 × P ( S \ { } ). However, this is different fromthe bubble tree compactification.To compactify P/G as the bubble tree compactification, we need to figure out whichpoints in
P/G correspond to connections near lower strata in M p = − ( CP ) via the one-to-one correspondence M p = − ( CP ) ↔ P/G.
To do this we first introduce some concepts in algebraic geometry.Recall that any topological rank r complex vector bundle E on CP with first Chern class c has the form E topo ∼ = O CP ( c ) ⊕ C r − , where O CP n ( c ) = O CP n (1) ⊗ c , O CP n (1) is dual to the canonical bundle O CP n ( −
1) on CP n , C r − is the trivial rank r − CP . By the theorem of Grothendieck(ref. Theorem 2.1.1 in Chapter 1 of [OSS88]), given any holomorphic structure on E , it hasa unique form E holo ∼ = O ( a ) ⊕ · · · ⊕ O ( a r ) , where a , . . . , a r ∈ Z , a ≥ · · · ≥ a r , a + · · · + a r = c . E → CP be any rank- r holomorphic complex vector bundle, L be any complexprojective line in CP , i.e., L ∼ = CP and L ⊂ CP , then E | L holo ∼ = O ( a ( L )) ⊕ · · · ⊕ O ( a r ( L )) . We get a map a E : { complex prjective lines in CP } → Z r L (cid:55)→ a E ( L ) := ( a ( L ) , . . . , a r ( L )) .a E ( L ) is called the splitting type of E on L . Define a total order on Z r : ( a , . . . , a r ) < ( b , . . . , b r ) if the first non-zero b i − a i is positive. Definition 8.3.
The generic splitting type of E is a E := inf L ⊂ CP ,L ∼ = CP a E ( L ) . The set of jump lines of E is S E := { L | L ⊂ CP , L ∼ = CP , a E ( L ) > a E } . (cid:4) Examples 8.4. (i). The stable holomorphic bundle corresponding to the point in M p = − ( CP )is E := T ∗ CP ⊗ O CP (1). The stability of E is proved in Theorem 1.3.2 in Chapter 2of [OSS88]. Its Chern classes are c ( E ) = c ( T ∗ CP ) + 2 c ( O (1)) = − ,c ( E ) = c ( T ∗ CP ) + c ( T ∗ CP ) c ( O (1)) + c ( O (1)) = 1 . The GL (3 , C )-action on C induces an action on CP and E such that E holo ∼ = t ∗ E forany t ∈ GL (3 , C ). Therefore the splitting type of E on any projective line L in CP isthe same, i.e., the set of jump lines S E = ∅ . The splitting type of E is (0 , − E to be the stable holomorphic rank 2 bundle corresponding to [ α ] ∈ P/G . Then α can be expressed as a map from V ∗ to C with rank 2, where V is a 3-dimensionalcomplex vector space. For any element z ∈ V ∗ , it defines a complex plane ker ( z ) in V and thus a projective line L z in P ( V ). From Section 4.3 in Chapter 2 of [OSS88], L z is a jump line if and only if α ( z ) = 0. There is only one such L z since α has rank 2.For example, if α = (cid:20) (cid:21) we have z = (0 , , t and L z = { [ z , z , } is the projective line defined by z . (cid:4) P − ( CP ) : the SO (3)-bundle Λ , − ( CP ) on CP P ( S ) : the SO (3) bundle on S with p = − A : the point in M ( P − ( CP )) A : the point in M b ( P ( S )) E − , : the U (2)-bundle on CP with c = − , c = 2 . For any z ∈ CP and gluing parameter ρ , the glued ASD connection Ψ( A , A , z, ρ ) inducesa holomorphic structure E on E − , . From the example above we know that there is a uniquejump line of E . Proposition 8.5.
Using the notations above, the jump line of E is the projective line L z defined by z .Proof. Without loss of generality, assume z = [1 , , ∈ CP . There is an SU (2) action on CP induced by the action of the group (cid:20) SU (2) (cid:21) ⊂ SU (3)on C . This action fixes z = [1 , , { [1 , z , z ] } ⊂ CP , the Fubini-Studymetric is ω F S = √− ∂ ¯ ∂ log(1 + | z | + | z | ) . It is U (3)-invariant and thus SU (2)-invariant. The group SU (2) acts on S ∼ = C ∪ {∞} inthe obvious way.Let A be the glued connection Ψ( A , A , z, ρ ) on P − ( CP ), the SO (3)-bundle on CP with p = −
7. Given any element g ∈ SU (2), lift g to ˜ g , ˜ g , automorphisms on the bundle P − ( CP ) and P ( S ) covering g respectively. By acting some element in gauge transforma-tion on ˜ g , we can make ˜ g , ˜ g to be ρ -equivariant when restricting ˜ g , ˜ g to P − ( CP ) | z and P ( S ) | ∞ respectively. Therefore ˜ g , ˜ g induce an automorphism ˜ g on P − ( CP ) covering g .Since A , A are SU (2)-invariant, the glued connection A is gauge equivalent to ˜ g · A . Sothe holomorphic structure E on E − , induced by A satisfies E |
L holo ∼ = E | g · L for any g ∈ SU (2) and projective line L in CP . Since there is only one jump line on E , thisjump line has to be invariant under SU (2)-action, i.e., the projective line { [0 , z , z ] } .To sum up, when gluing A on S to A on CP at z ∈ CP , the holomorphic structureon L z ⊂ CP is changed and L z becomes a jump line of the holomorphic structure on E − , induced by A . 50o compactify M p = − ( CP ), we first define a fibre bundle π : P/G → CP [ α ] (cid:55)→ π ( α ) = z, s.t. α ( z ) = 0 , i.e., π maps α to the point in CP that defines the jump line of the holomorphic struc-ture induced by α . Then the bubble tree compactification (which is same as Uhlenbeckcompactificaition in this case) of M p = − ( CP ) is by compactifying P/G through the one-point-compactification of each fibre of π : P/G → CP : M ( P − ( CP )) = M ( P − ( CP )) ∪ M ( P − ( CP )) × CP = P/G ∪ CP . Next we describe the neighbourhood of points in the lower stratum of the compactifiedspace. To do this we first introduce a notion called “jump line of the second kind”, which wasdeveloped by K.Hulek in [H79]. The set of jump lines only contains part of the informationof a stable holomophic structure of E − , since points on the same fibre of π : P/G → CP have the same jump line. We shall see in the case c = − , c = 2, the set of jump lines ofsecond kind determines the holomorphic structure completely. Definition 8.6. (Definition 3.2.1 of [H79]) Given a holomorphic complex vector bundle E → CP , a jump line of the second kind is a projective line L in CP defined by theequation z = 0 such that h ( E | L ) (cid:54) = 0, where L is the subvariety in CP defined by z = 0.Denote the space of jump lines of the second kind by C E . (cid:4) As claimed in page 242 of [H79], suppose that E corresponds to the element α ∈ P/G ,then this space is given by the equation C E = { z ∈ CP | det ( α ( z ) t α ( z )) = 0 } . The relation between the space of jump lines and that of the second kind is given by Propo-sition 9.1 of [H79]: S E ⊂ Singular locus of C E . In the case k = 2, if α ∈ P/G is represented by a matrix (cid:20) a b cd e f (cid:21) , then C E = (cid:8) z = [ z , z , z ] ∈ CP | az + bz + cz = ± i ( dz + ez + f z ) (cid:9) = L z (cid:48) ∪ L z (cid:48)(cid:48) ,S E = L z (cid:48) ∩ L z (cid:48)(cid:48) = { z | α ( z ) = 0 } , where L z is the projective line defined by z and z (cid:48) = [ a + id, b + ie, c + if ], z (cid:48)(cid:48) = [ a − id, b − ie, c − if ].In summary, C E is the union of two different projective lines whose intersection is asingle point that defines the only jump line of E . Therefore C E can be seen as an element in51 P × CP \(cid:52) Σ with (cid:52) being the diagonal and Σ being the permutation group. Moreover, C E determines E completely through the following map CP × CP \ (cid:52) Σ φ −→ P/G { [ a, b, c ] , [ d, e, f ] } (cid:55)→ (cid:20) a + d b + e c + f a − d i b − e i c − f i (cid:21) . One can check that φ is well-defined and a one-to-one correspondence. Under this corre-spondence, the fibre bundle π : P/G → CP is the composition P/G → CP × CP \ (cid:52) Σ → CP (cid:20) u t v t (cid:21) (cid:55)→ { u + iv, u − iv } (cid:55)→ z. Here z satisfies ( u + iv ) t z = ( u − iv ) t z = 0. The fibre of π is CP × CP \(cid:52) Σ .There is another way to interpret the fibre of π . Note that α , α are in the same fibre ifand only if there exists A ∈ GL (2 , C ) such that A · α = α . Therefore π : P/G → CP is a fibre bundle with fibre GL (2 , C ) /G , where G = O (2 , C ) × C × acts on GL (2 , C ) by ( φ, λ ) · A = φ · A · λ − . There are homeomorphisms GL (2 , C ) /G → S × / C × = S × \ { } C × − S × / C × ∼ = CP − CP A (cid:55)→ A t A where S i × is the space of symmetric 2 × i and S × is the space ofsymmetric 2 × M Z a ( P − ( CP )) and its bubble tree compactification Definition 8.7.
Suppose r, s, t are pairwise coprime positive integers, then the weightedcomplex projective space CP r,s,t ) is defined to be CP r,s,t ) = S /S , where S acts on S by g · ( z , z , z ) = ( g r z , g s z , g t z ) for g ∈ S .Let a = rst , then Z a = Z r ⊕ Z s ⊕ Z t . Define a Z a -action on CP by( e πi/r , e πi/s , e πi/t ) · [ z , z , z ] = [ e πi/r z , e πi/s z , e πi/t z ] , CP / Z a is diffeomorphic to CP r,s,t ) through[ z , z , z ] (cid:55)→ [ z r , z s , z t ] . Hence we can investigate connections on CP r,s,t ) through Z a -invariant connections on CP .The singularities of the Z a -action and stabilisers of these singularities on CP are:Γ [1 , , = Γ [0 , , = Γ [0 , , = Z a Γ [0 ,z ,z ] = Z r , z z (cid:54) = 0 , Γ [ z , ,z ] = Z s , z z (cid:54) = 0 , Γ [ z ,z , = Z t , z z (cid:54) = 0 , where Γ z is the stabiliser of z . The Z a -action on CP naturally induces an action on thebundle P − ( CP ) = Λ , − ( CP ).For simplicity, assume s = t = 1 in the following calculation. Then g ∈ Z a acts on P − ( CP ) | [1 , , and fixes this fibre. To see this, under the local chart [1 , z , z ] around[1 , , , ,
0] have the following basis (cid:26) i dz ∧ d ¯ z − dz ∧ d ¯ z ) ,
12 ( dz ∧ d ¯ z − d ¯ z ∧ dz ) , i dz ∧ d ¯ z + d ¯ z ∧ dz ) (cid:27) . Hence the Z a -action fixes the fibre P − ( CP ) | [1 , , . Recall the Z a -invariant instanton-1 mod-uli space of S introduced in Section 6.2. Consider the Z a -action defined on S , which isinduced by the action e πi/a · ( z , z ) = ( e πi/a z , e πi/a z ) on C , or equivalently, by the action e πi/a · z = e πi/a z on H . Then Example 6.2 lifts this action to P ( S ) such that Z a fixesthe fibre over the south pole P ( S ) | ∞ . That is to say, the Z a -action has weight zero on P − ( CP ) | [1 , , and P ( S ) | ∞ .Any gluing parameter ρ : P − ( CP ) | [1 , , → P ( S ) | ∞ at the point [1 , ,
0] is Z a -equivariant,which implies that any connection obtained by gluing A ∈ M ( P − ( CP )) and A ∈ M b ( S )at [1 , ,
0] is Z a -invariant.The Z a -action on P/G induced from Z a -action on CP is e πi/a · (cid:20) x x x x x x (cid:21) = (cid:20) e πi/a x x x e πi/a x x x (cid:21) . Hence the Z a -invariant part of P/G ∼ = CP × CP \(cid:52) Σ is (cid:18) CP × CP \ (cid:52) Σ (cid:19) Z a = CP × CP \ (cid:52) Σ ∪ (cid:0) { [1 , , } × CP (cid:1) , where CP ⊂ CP is { [0 , z , z ] } . The first part of this union is the fibre over [1 , ,
0] in thefibre bundle π : P/G → CP , which corresponds to the space of holomorphic structures on E − , with jump line { [0 , z , z ] } . This fibre π − ([1 , , M ( P − ( CP )) obtained by gluing A ∈ M ( P − ( CP )) and A ∈ M b ( S ) at [1 , , Z a -invariant ASD moduli space is M Z a ( P − ( CP )) = π − ([1 , , ∪ CP , where π − ([1 , , is the one point compactificaiton of π − ([1 , , CP ⊂ CP isthe projective line { [0 , z , z ] } . Remark 8.8.
In this particular case, we have described gluing at the isolated fixed pointin CP . Now consider those fixed points in the fixed sphere { [0 , z , z ] } ⊂ CP . Supposethere exists a gluing parameter ρ : P − ( CP ) | [0 ,z ,z ] → P ( S ) | ∞ that is Z a -equivariant, wecan glue the Z a -invariant A ∈ M ( P − ( CP )) and Z a -invariant A ∈ M ( S ) at [0 , z , z ]using the gluing data ( ρ, λ ) and get a set of Z a -invariant connections in M ( P − ( CP )) for allsmall enough λ . This set of connections is contained in ( P/G ) Z a ∩ π − ([0 , z , z ]). Therefore( P/G ) Z a ∩ π − ([0 , z , z ]) is non-compact since λ can be arbitrarily small. This contradictsthe fact that ( P/G ) Z a ∩ π − ([0 , z , z ]) is compact. Thus Z a -equivariant gluing parameters donot exist, that is, the isotropy representation at [0 , z , z ] induced by [ A ] is not equivalentto the isotropy representation at the south pole of S induced by [ A ]. (cid:4) References [A90] D.M.Austin. SO (3) -invariants on L ( p, q ) × R . J. Differential Geometry, 32: 383-413,1990. [BKS90] N.P.Buchdahl, S.Kwasik, R.Schultz. One Fixed Point Actions on Low-dimensionalSphere. Inventiones Mathematicae, 102: 633-662, 1990. [C02] B.Chen.
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