Moduli Theory

Chapter I: A Gluing Theorem for the Kapustin-Witten Equations with a Nahm

1.6 Moduli Theory

Corollary 1.5.21. If the cylindrical end of X has the formS³× [0,+∞), we have Ind⁺_X(P)=−3χ(X) −3, Ind⁻_X(P)=−3χ(X)+3. (1.48) Proof. Denote by ¯X a smooth compactification ofXover the tubeS³× [0,+∞)and denote by ¯Pthe extension of the bundle P.

By Proposition 1.5.18, we have

Ind⁺_X(P)+Ind⁻_W = IndX¯(P),¯ Ind⁻_X(P)+Ind⁺_W =IndX¯(P¯). In addition, by Proposition 1.5.16, we know that

IndX¯(P)¯ =−3χ(X)¯ = −3χ(X) −3.

We get the result we want.

Definition 1.6.1. Given a smooth Nahm pole solution(A₀,Φ₀), we define the framed configuration spaceC_p,λ^{f r} as follows:

C_p,λ^{f r} :={(A0,Φ₀)+(a,b) | (a,b) ∈ y^λ+p1H₀^1,p(Ω¹(g_P) ×Ω¹(g_P))}. (1.49) Here y^λ+1pH₀^1,p(Ω¹(g_P) ×Ω¹(g_P))is the completion of smooth 1-forms with respect to the normy^λ+1pH₀^1,p.

We have some basic properties of the framed configuration space:

Proposition 1.6.2. (1) Any(A,Φ) ∈ C_p,λ^{f r}satisfies the Nahm pole boundary condition.

(2) AssumeXhas non-vanishing boundary and cylindrical end which identified with Y × (0,+∞). Let P be anSU(2) bundle over it, let (A₁,Φ₁)be a connection pair satisfying the Nahm pole boundary condition, denote

Í3 j=1e_jt¹_j

y is the leading part of Φ₁. If(A1,Φ₁−

Í3 j=1e_jt¹_j

y ) ∈ H₀^1,p(X),then there exists a global gauge transformation g ∈ Gsuch thatg(A₁,Φ₁) ∈ C_p,λ^{f r}.

Proof. For (1), as forλ∈ [1− ¹_p,1),p >2, any differential form which blows-up as y⁻¹is not contained iny^λ+1pH₀^1,p, the result follows immediately.

For (2), for i = 0,1, (Ai,Φ_i) both satisfies the Nahm pole boundary condition (Definition 1.2.1). Then there exists orthogonal basis ej ∈ T^?Y and tⁱ_j ∈ g_P for

j = 1,2,3 such that [tⁱ_j

1,tⁱ_j

2] = j₁j₂j₃tⁱ_j

3 where j₁j₂j₃ is the Kronecker symbol of j₁, j₂, j₃. In addition, let the asymptotic expansion of Φ_i at y = 0 to be

Í3 j=1e_jtⁱ_j

y +

O(y). By the commutation relation of tⁱ_j, there exists a ˆg : Z → SU(2) such that ˆg(

Í3 j=1e_jt¹_j

y )gˆ⁻¹ = ^Í3j=1_y^ejt0j. By the Hopf theorem, the homotopy type of maps from Y³ to SU(2) is totally determined by the degree. Given ˆg : Z → SU(2), considerYT₀ =Y × {T₀} ⊂ X, we can choose a band to connect Z andY which is homeomorphic to Z]YT₀. We can choose a map ˆg⁰ : YT₀ → SU(2) whose degree equals minus degree ofgand extend these two maps toZ]YT₀ and denote as ˜g. Then g˜has degree zero and can be extended to whole X, which we denote asg.

By the assumption that(Ai,Φi)are smooth, we haveg(A1,Φ₁)−(A0,Φ₀) ∈ y^λ+1pH₀^1,p. Remark. For a general compact 4-dimensional manifold with boundary, Proposi- tion 1.6.2 is not true. ForD⁴, the 4-dimensional unit disc, a choice of frame gives a

map fromS³ → SO(3)which isπ₃(SO(3)). Two frames corresponding to different elements inπ3(SO(3))can not be globally gauge equivalent.

As we fixed a base connection(A₀,Φ₀)to define the framed configuration space, we can also consider the gauge group that preserves the frame.

Definition 1.6.3. The framed gauge groupG^{f r} is defined as follows:

G^{f r} := {g ∈ Aut(P) | g|Y = 1}. (1.50) Giveng ∈ G^{f r}, the action ofgon(A₀,Φ₀)will be

g(A₀,Φ₀)= (A₀−dA₀g g⁻¹,gΦ₀g⁻¹). (1.51) Then, we have

g(A₀,Φ₀) − (A₀,Φ₀)=(−dA₀g g⁻¹, [g,Φ₀]g⁻¹). (1.52) We consider the following weighted frame gauge groupG_p^{f r}_,λ:

G_p,λ^{f r} = {g ∈ G^{f r} | dA₀g g⁻¹∈ y^λ+1pH₀^1,p(Ω¹), [g,Φ₀]g⁻¹∈ y^λ+1pH₀^1,p(Ω¹)}. (1.53) For convenience, we denote d⁰(ξ) := d₍⁰_A

0,Φ0)(ξ) = (dA₀ξ,[Φ₀, ξ])and we have the following lemma on the weighted frame gauge groupG_p,λ^{f r}.

Lemma 1.6.4. G_p,λ^{f r} = {g ∈ G^{f r} | ∇₀g ∈ y^λ+p1L^p, ∇²

0g ∈ y^λ+1p−1L^p, [Φ₀,g] ∈ y^λ+1pL^p, ∇₀[Φ₀,g] ∈ y^λ+p1−1L^p}.

Proof. Obviously{g ∈ G^{f r} | d⁰g ∈ y^λ+p1L^p, ∇₀(d⁰)g ∈ y^λ+1p−1L^p} ⊂ G_p,λ^{f r}. For the other side, we argue as follows: take α := dA₀gg⁻¹, then α ∈ y^λ+1pH₀^1,p ,→ y^λ+1p−1L^2p.BydA₀g = αg, we have∇₀g ∈ y^λ+1p−1L^2psince the pointwise norm of gis 1. In addition,∇₀dA₀g =(∇₀α)g+α∇₀g.As∇₀α∈ y^λ+1p−1L^pand the pointwise norm ofgis 1, we have∇₀αg ∈ y^λ+1p−1L^p. Asα ∈ y^λ+1p−1L^2p,∇₀g ∈ y^λ+1p−1L^2p, we haveα∇₀g ∈ y^λ+1p−1L^p. For[g,Φ₀]g⁻¹ ∈ y^λ+p1H₀^1,p(Ω¹), we have[g,Φ₀]g⁻¹ ∈ y^λ+1pL^p. Lettingβ =[g,Φ₀]g⁻¹, then β ∈ y^λ+p1L^pimplies βg =[g,Φ₀] ∈ y^λ+1pL^p. In addition, β ∈ y^λ+1pH₀^1,pimplies∇₀β ∈ y^λ+p1L^pand β ∈ y^λ+1p−1L^2p. In addition, we have∇₀g ∈ y^λ+1p−1L^p, thus∇₀[g,Φ₀]=∇₀(βg)=(∇₀β)g+β∇₀g ∈ y^λ+1p−1L^p. Therefore,G_p,λ^{f r} ⊂ {g ∈ G^{f r} | ∇₀g ∈ y^λ+1pL^p, ∇²

0g ∈ y^λ+1p−1L^p}.

Thus we can rewriteG_p,λ^{f r} asG_p,λ^{f r} ={g ∈ G^{f r} |d⁰g ∈y^λ+1pL^p, ∇₀d⁰g ∈ y^λ+1p−1L^p}.

Lemma 1.6.5. The space y^λ+p1+1H₀^2,p(g_P) is an algebra and y^λ+p1H₀^1,p(g_P) is a module over this algebra.

Proof. For the algebra statement, we only need to proveu₁u₂ ∈ y^λ+1p+1H₀^2,p(g_P), or equivalently,u₁u₂∈ y^λ+1p+1L^p,∇₀(u₁u₂) ∈ y^λ+1pL^p,∇²

0(u₁u₂) ∈ y^λ+1p−1L^p. Since ui ∈ y^λ+1p+1H₀^2,p, we have ui ∈ y^λ+p1+1L^p, ∇₀ui ∈ y^λ+p1L^p and ∇²

0ui ∈ y^λ+1p−1L^p.

By Proposition 1.5.6,ui ∈ y^λ+p1+1H₀^1,p,→ y^λ+p1L₁^p,→ y^λ+p1L^2p.By Corollary 1.5.7, we have u₁u₂ ∈ y^λ+1p+1L^p. In addition, we know ∇₀ui ∈ y^λ+1pH₀^1,p ,→ y^λ+p1−1L^2p andui ∈ y^λ+1pL^2p. Asλ ≥ 1−¹_p, we haveλ+¹_p+λ+¹_p−1≥ λ+ ¹_p. By the Hölder inequality in Proposition 1.5.6, we have∇₀u1u2 ∈ y^λ+p1L^p.

Forui ∈ y^λ+1p+1H₀^2,p, as p > 2 and λ ≥ 1− ¹_p, we haveui ∈ y^λ+1p−1L₂^p ,→ C⁰and

∇²

0u∈ y^λ+1p−1L^p. Therefore, we have∇²

0u₁u₂∈ y^λ+1p−1L^p.

The module statement can also be proved in a similar way.

Fix a base pointp0 ∈ Xand define a system of neighborhoods of the identity inG_P as

U = {g ∈ G^{f r} | kd⁰gk

y^λ+p1L^p ≤ , k∇₀d⁰gk

y^λ+1p−1L^p ≤ , |g(p₀) −1| ≤ }.

(1.54) This topology is independent of the base pointp₀.

With the previous lemma, we can establish a Lie group structure onG^{f r}: Corollary 1.6.6. (1)G_p,λ^{f r} is a Lie group with Lie algebra

Lie(G_p,λ^{f r})= y^λ+1p+1H₀^2,p(g_P).

(2)G_p,λ^{f r} acts smoothly onC_p,λ^{f r}.

Proof. This result follows immediately from Lemma 1.6.4 and Lemma 1.6.5.

Now, we have the following proposition of the framed configuration spaceC_p,λ^{f r} and the framed gauge groupG^{f r}

p,λ:

Proposition 1.6.7. For any(A,Φ) ∈ C_p,λ^{f r}, we haveKW(A,Φ) ∈ y^λ+p1−1L^p(Ω²⊕Ω⁰),

Proof. By the definition of C_p^{f r}_,λ, there exists (a,b) ∈ y^λ+1pH₀^1,p(Ω¹(g_P) ×Ω¹(g_P)) such that(A,Φ)=(A₀,Φ₀)+(a,b).

By Proposition 1.2.9, we haveKW(A,Φ)= KW(A₀,Φ₀)+L¹(a,b)+{(a,b), (a,b)}. Here {(a,b), (a,b)} is a quadratic term. By theorem 1.5.4, we have L¹(a,b) ∈ y^λ+1p−1L^p(Ω²×Ω⁰).By the embeddingy^λ+1pH₀^1,p,→ y^λ+1p−1L₁^p,→ y^λ+1p−1L^2p, we have{(a,b),(a,b)} ∈ y^λ+1p−1L^p. Now we will study the behavior of the gauge group (1.53) over the cylindrical end.

We have the following proposition which describes the limit behavior of the group G_p,λ^{f r}. We need the hypothesis that ρ is acyclic. Let (Aρ,Φ_ρ) be the flat SL(2;C) connection associate to ρ.

Recall that d_(A,Φ)⁰ (ξ) = (dAξ,[Φ, ξ])and ρacyclic implies Ker d₍⁰_A

ρ,Φρ) = 0 and the connectiondAρ itself may still be a reducibleSU(2)connection.

We have the following lemma over the cylindrical end:

Lemma 1.6.8. SupposeXis a manifold with boundary and cylindrical end which is identified withY × (0,+∞), for(A₀,Φ₀)a reference connection whichL₁^pconverges to(A_ρ,Φ_ρ)over the cylindrical end forp >2, then forT is large enough, we have (1)d_(A⁰

0,Φ₀) :L₂^p(Y × (T−1,T+1)) → L₁^p(Y× (T −1,T +1))is injective, (2)kξk_L^p

2(Y×(T−1,T+1)) ≤ Ckd_(A⁰

0,Φ₀)ξk_L^p

1(Y×(T−1,T+1)).

Proof. For convenience, during the proof, we writeL_k^pshort forL_k^p(Y×(T−1,T+1)).

(1) Denote(a,b)=(Aρ,Φ_ρ) − (A₀,Φ₀), then for anyξ ∈ L₂^p, we have kd₍⁰_Aρ,Φρ)ξ−d_(A⁰

0,Φ₀)ξk_L^p

1 ≤ C(k[a, ξ]k_L^p

1 +k[b, ξ]k_L^p

1) ≤C(kak_L^p

1 +kbk_L^p

1)kξk_L^p

2. Ifξ ∈Kerd_(A⁰

0,Φ₀), we obtain kd_(A⁰_ρ,Φρ)ξk_L^p

k

≤ C(kak_L^p

1 +kbk_L^p

1

)kξk_L^p

2. ForTis large enough,C(kak_L^p

1+kbk_L^p

1)is smaller than the operator norm ofd_(A⁰

0,Φ0), which impliesξ =0.

(2) If the inequality is not true, then there exists a sequence{ξn}with kξnk_L^p

1 = 1, lim

n→∞kd_(A⁰

0,Φ0)ξnk_L^p

1 =0, which implieskξnk_L^p

2 is bounded.

Then, ξn weak converges to ξ∞ in L₂^p and strongly converges to ξ∞ in L₁^p, which implies kd_(A⁰

0,Φ₀)ξ∞k_L^p

1 = 0 andkξ∞k_L^p

1 = 1. As Ker d_(A⁰

0,Φ₀) = 0, we haveξ∞ = 0, contradictingkξ∞k_L^p

1 =1.

We have the following corollary:

Corollary 1.6.9. If over the cylindrical end, (A₀,Φ₀) converges in L₁^p norm to (Aρ,Φ_ρ)and(Aρ,Φ_ρ)is an irreducible flatSL(2;C)connection, thenKerd_(A⁰

0,Φ₀) = 0.

Proof. As before, we denoted⁰ := d₍⁰_A

0,Φ₀). By Kato inequality, we know forξ ∈Ω⁰, we have the pointwise estimate

d|ξ| ≤ |dA₀ξ| ≤ |d⁰(ξ)|.

Therefore, ξ ∈ Kerd⁰ implies |ξ| is a constant. In addition, by Lemma 1.6.8, we knowξ = 0 overY × (T −1,T +1)whenT is large enough, therefore, we haveξ is

identically zero.

Proposition 1.6.10. If the limiting connection ρis irreducible, then forT is large enough, there is a constant C such that for any section ξ ∈g_P, with d⁰ξ ∈ L^p(Y × [T,+∞))and∇₀d⁰ξ ∈ L^p(Y × [T,+∞)), we have

(1)|ξ| →0at the cylindrical end andsup|ξ| ≤C(kd⁰ξk_L^p

1(Y×[T,+∞))), (2) Either|g(x) −1| →0or|g(x)+1| →0asx tends to infinity in X.

Proof. (1) For an integerk >T+1, over a bandBk :=Y × (k −1,k +1), we have kξk_C0(B_k) ≤ kξk_L^p

2(B_k) ≤Ckd⁰ξk_L^p

1(B_k). The statement follows immediately.

(2) Denote by g_k the restriction of g to the band Bk. After identifying different bands withY³× (−1,1), we can consider {gk} a sequence of gauge transformation overY³× (−1,1)withk∇₀d⁰gkk_Lp, k∇₀gkk_Lpconverging to zero. As the pointwise norm of g is always 1, by Rellich lemma,gk strongly converges tog∞ in L₁^pwhich impliesd⁰g∞ =0. By our assumption, we haveg∞ = ±1.

Now, we can define a framed quotient space as follows:

Definition 1.6.11. We defineB_p,λ^{f r} as the following weighted framed moduli space:

B_p,λ^{f r} = C_p,λ^{f r}/G_p,λ^{f r}.

In addition, we have the following definition of the moduli space:

Definition 1.6.12. The framed moduli spaceM_p,λ^{f r,ρ}(X)is defined as follows:

M^{f r,ρ}

p,λ (X)= {(A,Φ) ∈ C_p,λ^{f r}|KW(A,Φ)=0}/G_p,λ^{f r}. (1.55) We have the following basic properties of the framed moduli space:

Proposition 1.6.13. (1) For any(A,Φ) ∈ M_p,λ^{f r,ρ} satisfies the Nahm pole boundary condition.

(2) Any(A,Φ) ∈ M_p,λ^{f r,ρ} converges toρinL₁^pnorm.

Proof. (1) is an immediate consequences of Proposition 1.6.2. (2) is a consequence

of the definition ofC_p,λ^{f r}.

Slicing Theorem

Now we study the local properties of the moduli space and we will assign a suitable norm to the Kuranishi complex (1.7).

Define(Ω⁰, λ,k,p)as follows:

(Ω⁰, λ,k,p):= y^λH₀^k,p(Ω⁰(g_P)).

Here the notation y^λH₀^k,p(Ω⁰(g_P))means the completion of the smooth sections of Ω⁰(g_P)in the norm y^λH₀^k,p. Similarly, we define

(Ω¹×Ω¹, λ,k,p):= y^λH₀^k,p(Ω¹(g_P) ×Ω¹(g_P)) and

(Ω²×Ω⁰, λ,k,p):= y^λH₀^k,p(Ω²(g_P) ×Ω⁰(g_P)).

Now we rewrite the Kuranishi complex (1.7) at the point (A₀,Φ₀) with respect to the new norm as follows:

0→ (Ω⁰, λ+1+1 p,2,p)

d_(A⁰

0,Φ0)

−−−−−−→ (Ω¹×Ω¹, λ+1 p,1,p)

L_(A

0,Φ0)

−−−−−−→ (Ω²×Ω⁰, λ+1

p−1,0,p) → 0, (1.56)

Here we only consideredλ ∈ [1− ¹_p,1).

We have the following Proposition for the operatord₍⁰_A

0,Φ₀): Proposition 1.6.14. The operatord_(A⁰

0,Φ₀): d₍⁰_A

0,Φ₀) : y^λ+1+1pH₀^2,p(Ω⁰(g_P)) → y^λ+1pH₀^1,p(Ω¹(g_P) ×Ω¹(g_P)) is a closed operator.

Proof. see Appendix 2.

Corollary 1.6.15. y^λ+1pH₀^2,p(Ω¹(g_P)×Ω¹(g_P))= Imd_(A⁰

0,Φ₀)⊕(Kerd_(A^0,?

0,Φ₀)∩y^λ+1pH₀^1,p)

Proof. Letx =(x1,x2) ∈ Ω¹(g_P) ×Ω¹(g_P), by definition ofd₍⁰_A

0,Φ₀), we have hd₍⁰_A

0,Φ₀)ξ,xi= hdA₀ξ,x1i+h[Φ₀, ξ],x2i whereh, imeans theL²inner product.

Integrating by parts, we have

hdA₀ξ,x₁i= hξ,d^?_A

0x₁i −

∫

∂Xtr(ξ∧?x₁).

Asξ ∈ y^λ+1pH₀^2,p(Ω⁰(g_P))andx₁ ∈ y^λ+1pH₀^1,p(Ω¹(g_P) ×Ω¹(g_P)), we havex₁|_∂X =0 andξ|∂X =0. Therefore,

hd_(A⁰

0,Φ₀)ξ,xi= hξ,d₍⁰_A^,?

0,Φ₀)xi.

Supposex ∈Coker d_(A⁰

0,Φ₀), then for∀ξ ∈ y^λ+1p+1H₀^2,p, we obtainhd₍⁰_A

0,Φ₀)ξ,xi= 0.

As λ > −1, integrating by parts, we have hξ,d₍⁰_A^,?

0,Φ₀)xi = 0. Thus d_(A^0,?

0,Φ₀)x = 0.

Combining this with Proposition 1.6.14, we finish the proof.

Fixe a reference connection pair(A0,Φ₀) ∈ C_p,λ^{f r} and > 0. We set:

T(A,Φ), :={(a,b) ∈Ω¹(g_P) ×Ω¹(g_P) | d₍^0,?_A,Φ)(a,b)=0,k(a,b)k

y^λ+1pH₀^1,p < }. (1.57) Thus we have a natural map p :T_(A,Φ), → B_p,λ^{f r}, which is induced by the inclusion ofT(A,Φ), intoC_p,λ^{f r} composed with quotienting by the gauge groupG_p^{f r}_,λ.

We have the following slicing theorem for the moduli spaceB_p,λ^{f r}.

Theorem 1.6.16. Given a point(A,Φ) ∈ C_p,λ^{f r}, denote by[(A,Φ)] ∈ B_p,λ^{f r} the equiv- alence class under the projection map. For small > 0,

(1) if(A,Φ)is irreducible, thenT(A,Φ), is a homeomorphism to a neighborhood of [(A,Φ)]inB_p,λ^{f r}.

(2) if(A,Φ)is reducible, thenT(A,Φ),/Γ_(A,Φ)is a homeomorphism to a neighborhood of[(A,Φ)]inB_p,λ^{f r}.

Proof. Consider the map

S :T(A,Φ), × G_P/{±1} → C_P,

S(A+a,Φ+b,g)= g(A+a,Φ+b). (1.58) The map has derivative ata =0,b=0,g= 1 as:

DS:Kerd_(A,Φ)^0,? ×Ω⁰(g_P) →Ω¹(g_P) ×Ω¹(g_P),

(a,b, ξ) → (a,b)+d₍⁰_A,Φ)(ξ). (1.59) By Corollary 1.6.15, we knowDSis always surjective.

(1) If (A,Φ)is irreducible, then DS is injective, by the implicit function theorem, we know for small enough,Sis a homeomorphism.

(2) If(A,Φ)is reducible, thenDShas kernelH_(A,Φ)⁰ .LetH_(A,Φ)^0⊥ be the orthogonal of H_(A,Φ)⁰ with respect to theL²inner product. Then this time the restriction map

S:T(A,Φ),×exp(H_(A,Φ)^0⊥ )/{±1} → C_P

is a local diffeomorphism. In addition, the multiplication mapΓ_(A,Φ)×exp(H₍⁰_A,Φ)^⊥ ) → G_P at the identity will have derivative 1. Thus for g ∈ G_P close to 1, there exist l ∈ Γ_(A,Φ) and m ∈ exp(H₍^0⊥_A,Φ)) such that g = ml and the splitting is unique.

Therefore, we get a homeomorphism from T(A,Φ),/Γ_(A,Φ) to a neighborhood of [(A,Φ)]inB_p,λ^{f r}.

Kuranishi Model

Given(A0,Φ₀)a solution to the Kapustin-Witten equations, all the other solutions within the sliceT(A,Φ), are given by the setZ(Ψ)of zeros of the map

T(A,Φ), Ψ

−→ y^λ+1p−1L^p(Ω²(g_P) ×Ω⁰(g_P)),

Ψ(a,b)=KW(A₀+a,Φ₀+b)=L¹(a,b)+{(a,b),(a,b)}. (1.60)

By Theorem 1.5.9,DΨis a Fredholm operator and for the homology associated to the Kuranishi complex, we have the following identification:

H_(A,Φ)² K erL^?∩y^λ+1p−1L^p, H_(A,Φ)¹ K erL ∩ (K er d_(A^0,?

0,Φ0)∩y^λ+1pH₀^1,p). (1.61) Therefore, we have the following Kuranishi picture of the moduli space:

Proposition 1.6.17. [18, Proposition 8] For any solution(A,Φ)withKW(A,Φ)= 0, for sufficiently small, there is a map ρ from a neighborhood of the origin in the harmonic spaceH₍¹_A,Φ)to the harmonic spaceH_(A,Φ)² such that if(A,Φ)is irreducible, a neighborhood of[(A,Φ)] ∈ M_p,λ^{f r} is carried by a diffeomorphism onto

Z(ρ)= ρ⁻¹(0) ⊂ H_(A,Φ)¹

and if(A,Φ)is reducible, then a neighborhood of[(A,Φ)]is modelled on Z(ρ)/Γ_(A,Φ).