Local Model for Gluing Picture

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

1.9 Local Model for Gluing Picture

Similarly, we can follow exactly the same as Chapter 1.8.3 and prove the second statement of the theorem.

from the gluing part. By Proposition 1.8.7, we know overU,(A,Φ)isC¹close to

(A1,Φ₁)thus proves thatdΘis injective.

Now we will characterize the Nahm pole solutions we found by our gluing construc- tion. Given solutions(Ai,Φ_i). Let(A^],Φ^])be the approximate solution. Letd_q^λ be the metric on the spaceB given by

d_q^λ([(A1,Φ₁)],[(A2,Φ₂)])= inf

u∈Gk(A1,Φ₁) −u(A2,Φ₂)k

y^λ+1p−1L^q. (1.105) Then, we can define an open neighborhoodU()of(A^],Φ^])by

U_(A],Φ^])()={(A,Φ) ∈ B |d_q^λ((A,Φ),(A^],Φ^]))| < , kKW(A,Φ)k

y^λ+p1−1L^p < }.

(1.106) Then we have the following theorem

Theorem 1.9.2. For? = 0,1,2, ifH₍^?_A

i,Φi) = 0, then for small enough, any point (A,Φ) ∈ U() can be represented by the following form (A,Φ) = (A^],Φ^])+Qφ, where kφk

y^λ+1p−1L^p ≤C andQis the right inverse operator defined in Proposition 1.8.4.

We prove Theorem 1.9.2 by the method of continuation. We need a new interpreta- tion of the operator.

Given (Ai,Φ) satisfying the assumption of Theorem 1.9.2, let (A^],Φ^]) be the ap- proximate solution over X^]T. In this section, for simplification, we denote L the linearization operator of (A^],Φ^])and let Q be the right inverse of L. Combining this with the embeddingy^λ+1pH₀^1,p,→ y^λ+1p−1L^q, we have

L : y^λ+1pH₀^1,p(Ω¹(g_P) ×Ω¹(g_P)) → y^λ+1p−1L^p(Ω²(g_P) ×Ω⁰(g_P)),

Q: y^λ+1p−1L^p(Ω²(g_P) ×Ω⁰(g_P)) → y^λ+p1−1L^q(Ω¹(g_P) ×Ω¹(g_P)). (1.107) Let B ∈U_(A],Φ^])(), then WLOG, we assumeB = (A^],Φ^])+(a,b)and considerBt

which is a path of connection pairs defined as follows:

Bt :=(A^],Φ^])+t(a,b) and we can define the following setS:

Definition 1.9.3. Given δ small enough, define S ⊂ [0,1]to be the interval of all t ∈ [0,1] such that there exists gauge transform u : [0,t] → G and φ : [0,t] → Ω²(g_P) ×Ω⁰(g_P)such that

(1)φ(0)= 0, u(0)= 1,

(2)ut(Bt)= (A^],Φ^])+Q(φt)withkφtk

y^λ+1p−1L^p < δ.

Our target is to proveS =[0,1]. By definition ofS, we have the following Proposi- tion:

Proposition 1.9.4. Sis non empty.

Proof. AsB₀ =(A^],Φ^]), take φ₀= 0 andu(0)= 1, we know 0 ∈S.

Now, we are going to prove S is an open set and before the proving, we will need some preparations.

Letd⁰to bed⁰

(A^],Φ^])in the Kuranishi complex (1.7), where forξ ∈ Ω⁰(g_P), d⁰(ξ) = (−d_A]ξ,[ξ,Φ^]]). For anyξ ∈Ω⁰(g_P)andφ ∈Ω²(g_P) ×Ω⁰(g_P), define the operator

Π:Ω⁰(g_P) ×Ω²(g_P) ×Ω⁰(g_P) →Ω¹(g_P) ×Ω¹(g_P),

(ξ , φ) → d⁰(ξ)+Q(φ). (1.108)

LetV₁be a norm overΩ⁰(g_P) ×Ω²(g_P) ×Ω⁰(g_P)defined as follows:

k(ξ, φ)k_V1 = kd⁰(ξ)k

y^λ+1p−1L^q +kφk

y^λ+1p−1L^p. For(a,b) ∈ Ω¹(g_P) ×Ω¹(g_P), we define another normV2as

k(a,b)kV₂ = k(a,b)k

y^λ+1p−1L^q +kL(a,b)k

y^λ+p1−1L^p. Then we have the following Proposition:

Proposition 1.9.5. ConsideringΠas operator fromV₁toV₂: Π:V1→V2,

we have

(1)Πis a bounded operator fromV₁toV₂,

(2) There exists a constantCindependent ofT such that k(ξ, φ)k_V1 ≤ CkΠ(ξ, φ)k_V2.

Proof. (1) We have the following computation for the operatorΠ:

kΠ(ξ, φ)k_B2

≤kd⁰(ξ)+Q(φ)k

y^λ+1p−1L^q+kL ◦d⁰(ξ)+L ◦Q(φ)k

y^λ+1p−1L^p

(Here we useL ◦d⁰(ξ)=[KW(A^],Φ^]), ξ]andL ◦Q= Id)

≤kd⁰(ξ)k

y^λ+1p−1L^q+kφk

y^λ+1p−1L^p +k[KW(A^],Φ^]), ξ]k

y^λ+1p−1L^p +kφk

y^λ+p1−1L^p

≤kd⁰(ξ)k

y^λ+1p−1L^q+kφk

y^λ+1p−1L^p.

(1.109) (2) Takeα = d⁰(ξ)+Q(φ), then we have Lα = [KW(A^],Φ^]), ξ]+φ. We have the following estimate:

kd⁰(ξ)k

y^λ+1p−1L^q ≤ kα−Qφk

y^λ+1p−1L^q

≤ kαk

y^λ+1p−1L^q +kQφk

y^λ+1p−1L^q

≤ kαk

y^λ+1p−1L^q +kφk

y^λ+p1−1L^p.

(1.110)

In addition, by the relationLα=[KW(A^],Φ^]), ξ]+φ, we have kφk

y^λ+p1−1L^p ≤ kLαk

y^λ+1p−1L^p +k[KW(A^],Φ^]), ξ]k

y^λ+1p−1L^p

≤ kαkV₂+kξk_C0(Her ewe use Pr oposition1.8.2)

≤ kαk_V2+kd⁰(ξ)k

y^λ+1p−1L^q

= kαkV₂+kα−Qφk

y^λ+1p−1L^q

≤ kαk_V2+kαk

y^λ+1p−1L^q+kφk

y^λ+1p−1L^p.

(1.111)

By taking small enough, we get kφk

y^λ+1p−1L^p ≤Ckαk_V2. By definition, k(ξ, φ)kV₁ = kd⁰(ξ)k

y^λ+p1−1L^q + kφk

y^λ+p1−1L^p. Combining equations (1.110) and (1.111), we obtain

k(ξ, φ)kV₁ ≤ CkαkV₂ =CkΠ(ξ, φ)kV₂.

. By this estimate, we get an immediate corollary:

Corollary 1.9.6. Πis an injective operator.

Proposition 1.9.7. For ? = 0,1,2, if H_(A^?

i,Φ_i) = 0, the operator Π is a surjective operator fromV1toV2

Proof. AsQis the inverse of L, by the assumptionH₍^?_A

i,Φ_i) =0, we know Ind Π =

−Ind D_(A,Φ) =0.By Proposition 1.9.5, we knowΠis injective, thusΠis surjective.

Proposition 1.9.8. Sis an open set in[0,1].

Proof. By Proposition 1.9.7,Πis surjective. By the implicit function theorem, we

get the result immediately.

Now, we hope to prove that the set S is a closed set. To begin with, we prove that the condition (2) in Definition 1.9.3 is a closed condition:

Lemma 1.9.9. For suitableδand, we havekφtk

y^λ+1p−1L^p ≤ ¹

2δ.

Proof. By the relationut(Bt)= (A^],Φ^])+Q(φt), we have:

KW(ut(Bt))= KW(A^],Φ^])+φt +{Q(φt),Q(φt)}. (1.112) Therefore, we have

kφtk

y^λ+1p−1L^p ≤ kKW(A^],Φ^])k

y^λ+1p−1L^p+ kKW(Bt)k

y^λ+p1−1L^p +kQ(φt)k²

y^λ+p1−1L^q

(Her e we use Pr oposition1.8.2and de f inition (1.106))

≤ (T)++C²kφtk²

y^λ+1p−1L^q

.

(1.113) Forδ < ¹

2C², (T) ≤ ¹₄δ and ≤ ¹

4δ, we have kφtk ≤ ¹

2δ, so the open condition is

also closed.

Proposition 1.9.10. For δ small enough and suitable parameter T and , S is a closed set in[0,1].

Proof. Now is routine to prove the setS is closed. Let assume a sequenceti ∈ S withti →t₀. For simplification, we denoteBi := Bt_i andφi :=φt_i. By the definition ofS, we have the relationshiput(Bi)=(A^],Φ^])+Q(φi).

By Lemma 1.9.9, we have the closed conditionkφtk

y^λ+1p−1L^p ≤ ¹

2δ. By definition of Bi, we haveBi =(A^],Φ^])+ti(a,b)and(a,b) ∈ y^λ+1pH₀^1,p ⊂ y^λ+1p−1L₁^p. We knowBi

strongly converges iny^λ+1p−1L₁^p.

By the uniform bound on theφi, theφiconverges to a limitφ₀weakly iny^λ+p1−1L^p. Define Ai =(A^],Φ^])+Q(φi). Ai is uniformly bounded in y^λ+1pH₀^1,p ,→ y^λ+p1−1L₁^p. Therefore, Aiconverges weakly iny^λ+1p−1L₁^p.

Asuiis a gauge transformation, by the relationui(Bi)= Ai, we havedui= uiAi−Biui. By the boundedness ofAiandBi, we knowui weakly converges tou0in y^λ+p1−1L₂^p. Therefore, by the Sobolev embedding theorem,ui strongly converges in y^λ+1p−1L₁^p tou₀. Therefore, we have the relationshipu₀(B₀)= A₀which implyt₀ ∈S.

We get an immediate corollary from Proposition 1.9.4, Proposition 1.9.8 and Propo- sition 1.9.10:

Corollary 1.9.11. For the setSin definition 1.9.3, we haveS =[0,1]. The proof of Theorem 1.9.2 follows immediately.

Local Model for Regular Moduli Space

Now, we are able to construct a local model for the gluing picture in the acyclic case without the assumption onH¹.

Denote ni = Ind(Pi) and we don’t assume ni = 0. Denote M^?_P

i (M^?_P) to be the moduli space which only consists of solutions to the Kapustin-Witten equations overXi (X^]T), which haveH²= 0.

Fori= 1,2, given two solutions(Ai,Φi) ∈ M^?_P

i, there exists an open neighborhood Uisuch that we can find functions

χ :Ui ⊂ M^?_P

i →Rⁿⁱ

which give local coordinates around(Ai,Φ_i)in the moduli spacesM^?_P

i. Denote U_P()={(A,Φ) ∈ B |∃(A₀,Φ₀) ∈ M^?_P, d_q^λ((A,Φ),(A₀,Φ₀))< , kKW(A,Φ)k

y^λ+p1−1L^p < }.

(1.114) Then by the exponential decay result (Theorem 1.7.1), we know that by choosing suitable compact setsGi ⊂ Xi and cut-off functions, we have a natural inclusionUi

intoM^?_P

i. Chooseyi ∈ Im(χi(Ui))and define the cut-down moduli space L = χ₁⁻¹(y₁) ∩ χ₂⁻¹(y₂) ∩ M^?_P ⊂UP(),

which has virtual dimensional 0.

ForT is large enough, recall I : C_P1 × C_P2 → C_P is the operator defined in (1.74) that constructs the approximate solution. Denote by(A₀,Φ₀):= I(χ₁⁻¹(y₁), χ₂⁻¹(y₂))

the approximate solution constructed by χ₁⁻¹(y₁) and χ₂⁻¹(y₂). Then we have the following Proposition. Compare this to Theorem 1.9.2:

Proposition 1.9.12. Forsmall enough, there exists a unique solution(A⁰,Φ⁰)inL such thatU(A₀,Φ0)() ∩L = (A⁰,Φ⁰).

Now, we will define a distance to make a comparision between connection pairs (A₀,Φ₀)overX^]T and(Ai,Φ_i)overXi.

We can define the normd as

d((A^],Φ^]);(A1,Φ₁),(A2,Φ₂))=in fu∈G_Pk(A0,Φ₀) −I((A1,Φ₁),(A2,Φ₂))k_Lq(X^]T), (1.115) where the I is the operator that constructs the approximate solutions defined in (1.74).

Summarizing Proposition 1.9.1 and Theorem 1.9.2, we obtain the following state- ment:

Theorem 1.9.13. Denote by Ui the compact sets of regular points in the moduli space M^?_P

i. There existT₀, ₀ such that for T > T₀ and < ₀, there exist open neighborhoodsNiofUiand a map

Θ: N1×N2 → M^?_P, such that

(1)Θis a diffeomorphism to its image, and the image contains regular points, (2)d(Θ((A₁,Φ₁),(A₂,Φ₂));(A₁,Φ₁),(A₂,Φ₂))) ≤ for any(Ai,Φ_i) ∈ Ni,

(3)Any connection(A^],Φ^]) ∈ M^?_P withd((A^],Φ^]);(A1,Φ₁),(A2,Φ₂)) ≤ for some (Ai,Φ_i) ∈ Ni lies in the image ofΘ.

Now we will have a brief discussion of the local gluing picture in the general case theH²is non-vanishing. For(Ai,Φ_i) ∈ M_iwith H²(Ai,Φ_i)non-vanishing, we can do the trick as in Section 1.8 by adding some finite dimensional linear space as the obstruction class and have a similar obstruction type statement as in Theorem 1.9.13. We will precise by state the theorem in general in the next subsection.

Conclusions

Now, we can summarize what we have proved and state the following theorem Theorem 1.9.14. Let (Ai,Φ_i) be connections pairs over manifolds Xi with Nahm poles over Zi, for sufficiently largeT, there is a local Kuranishi model for an open set in the moduli space over X^]T:

(1)There exists a neighborhoodN of{0} ⊂ H_(A¹

1,Φ₁) ×H_(A¹

2,Φ₂) and a mapΨ from N toH_(A²

1,Φ₁)×H_(A²

2,Φ₂).

(2) There exists a mapΘ which is a homeomorphism from Ψ⁻¹(0) to an open set V ⊂ M^?

X^].