On a topology property for the moduli space of Kapustin-Witten equations

TL;DR

This paper investigates the topology of the moduli space of Kapustin-Witten equations on four-manifolds, showing triviality of solutions near generic ASD connections and non-connectedness of the moduli space, with extensions to related gauge theories.

Contribution

It proves that solutions close to generic ASD connections are trivial and that the moduli space is non-connected, extending results to Hitchin-Simpson and Vafa-Witten equations.

Findings

Solutions near generic ASD connections are trivial.

The moduli space of solutions is non-connected.

Results extend to other gauge theory equations.

Abstract

In this article, we study the Kapustin-Witten equations on a closed, simply-connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use the Taubes' compactness theorem in arXiv:1307.6447v4 to prove that if $(A,\phi)$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then $(A,\phi)$ must be a trivial solution. We also prove that the moduli space of the solutions of Kapustin-Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all $generic$. At last, we extend the results for the Kapustin-Witten equations to other equations on gauge theory such as the Hitchin-Simpson equations and Vafa-Witten on a compact K\"{a}hler surface.

Equations

$$\begin{array}{rl}& F_A^{+}-(\varphi \wedge \varphi {)}^{+}=0,\\ & d_A^{\ast }\varphi =0,(d_A\varphi {)}^{-}=0,\end{array}$$

$$\begin{array}{rl}& d_{A_{\infty }}^{+}a+(a\wedge a{)}^{+}-(\varphi \wedge \varphi {)}^{+}=0,\\ & (d_{A_{\infty }}\varphi +[a,\varphi ]{)}^{-}=0.\end{array}$$

$$dist(A,A_{\infty })\ge \delta ,$$

$$dist(A,B):=\underset{g\in {\mathcal{G}}_P}{\inf }\Vert g^{\ast }(A)-B{\Vert }_{L_1^2(X)},\forall A,B\in {\mathcal{A}}_P.$$

$$\underset{g\in {\mathcal{G}}_P}{\inf }\Vert g^{\ast }(A)-A_{\infty }{\Vert }_{L_1^2}+\Vert \varphi {\Vert }_{L_1^2}\le \delta ,$$

$$KW:{\mathcal{A}}_P\times {\Omega }^1(X,{\mathfrak{g}}_P)\to {\Omega }^{2,+}(X,{\mathfrak{g}}_P)\times {\Omega }^{2,-}(X,{\mathfrak{g}}_P),$$

$$KW(A,\phi)=\big{(}F_{A}^{+}-(\phi\wedge\phi)^{+},(d_{A}\phi)^{-}\big{)}.$$

$$M_{KW}(P,g):=\{(A,\varphi )\in {\mathcal{A}}_P\times {\Omega }^1(X,{\mathfrak{g}}_P\mid KW(A,\varphi )=0\}/{\mathcal{G}}_P$$

$$\Vert (A_1,{\varphi }_1)-(A_2,{\varphi }_2){\Vert }^2=\Vert A_1-A_2{\Vert }_{L_1^2(X)}^2+\Vert {\varphi }_1-{\varphi }_2{\Vert }_{L_1^2(X)}^2,$$

$$dist\big{(}[A_{1},\phi_{1}],[A_{2},\phi_{2}]\big{)}:=\inf_{g\in\mathcal{G}_{P}}\|(A_{1},\phi_{1})-g^{\ast}(A_{2},\phi_{2})\|.$$

$$\begin{split}dist([A,\phi],M_{ASD}):&=\inf_{g\in\mathcal{G}_{P},A_{\infty}\in M_{ASD}}\|g^{\ast}(A,\phi)-(A_{\infty},0)\|\\ &=\inf_{g\in\mathcal{G}_{P},A_{\infty}\in M_{ASD}}\big{(}\|g^{\ast}(A)-A_{\infty}\|^{2}_{L^{2}_{1}(X)}+\|\phi\|^{2}_{L^{2}_{1}(X)}\big{)}^{\frac{1}{2}},\\ \end{split}$$

$$2\Vert \varphi {\Vert }_{L^2}^2\ge \Vert F_A^{+}{\Vert }_{L^2(X)}\ge \delta .$$

$$dist(A,M_{ASD}):=\underset{g\in {\mathcal{G}}_P,A_{\infty }\in M_{ASD}}{\inf }\Vert g^{\ast }(A)-A_{\infty }{\Vert }_{L_1^2(X)}\ge \tilde{\delta },$$

$$\ker d_A^{+}{d}_A^{+,\ast }{|}_{{\Omega }^{2,+}(X,adE)}\cong \ker ({\bar{\partial }}_A{\bar{\partial }}_A^{\ast }+{\bar{\partial }}_A^{\ast }{\bar{\partial }}_A){|}_{{\Omega }^{0,2}(X,adE)}\oplus \ker d_A{|}_{{\Omega }^0(X,adE)},$$

$$\Vert \theta {\Vert }_{L^2(X)}\ge C.$$

$$\Vert {\Lambda }_{\omega }{F}_A{\Vert }_{L^2(X)}\ge \tilde{C}$$

$$M_{HS}:=\{(A,\varphi )\in {\mathcal{A}}_E^{1,1}\times {\Omega }^{1,0}(X,adE)\mid {\Lambda }_{\omega }(F_A+[\theta ,{\theta }^{\ast }])=0,{\bar{\partial }}_A\theta =\theta \wedge \theta =0\}/{\mathcal{G}}_E$$

$$\begin{array}{rl}& F_A-\varphi \wedge \varphi =0,\\ & d_A^{\ast }\varphi =0,d_A\varphi =0.\end{array}$$

$${\Omega }^0(X,{\mathfrak{g}}_P)\stackrel{d_A}{\to }{\Omega }^1(X,{\mathfrak{g}}_P)\stackrel{d_A^{+}}{\to }{\Omega }^{2,+}(X,{\mathfrak{g}}_P).$$

$$M_{ASD}:=\{A\in {\mathcal{A}}_P:F_A+\ast F_A=0\}/{\mathcal{G}}_P$$

$$0\to {\Omega }^1(X,{\mathfrak{g}}_P)\stackrel{d_{(A,\varphi )}^0}{\to }{\Omega }^1(X,{\mathfrak{g}}_P)\oplus {\Omega }^1(X,{\mathfrak{g}}_P)\stackrel{d_{(A,\varphi )}^1}{\to }{\Omega }^{2,-}(X,{\mathfrak{g}}_P)\oplus {\Omega }^{2,+}(X,{\mathfrak{g}}_P)\to 0,$$

$$d_{(A,\varphi )}^0(\xi )=(-d_A\xi ,[\xi ,\varphi ]),$$

$$d^{1}_{(A,\phi)}(a,b)=\big{(}(d_{A}b+[a,\phi])^{-},(d_{A}a+[b,\phi])^{+}\big{)}.$$

$$d^{1}_{(A,\phi)}\circ d^{0}_{(A,\phi)}(\xi)=\big{(}[\xi,(d_{A}\phi)^{-}],[\xi,(F_{A}+\phi\wedge\phi)^{+}]\big{)}.$$

$$0\to {\Omega }^{2,-}(X,{\mathfrak{g}}_P)\oplus {\Omega }^{2,+}(X,{\mathfrak{g}}_P)\stackrel{d_{(A,\varphi )}^{1,\ast }}{\to }{\Omega }^1(X,{\mathfrak{g}}_P)\oplus {\Omega }^1(X,{\mathfrak{g}}_P)\stackrel{d_{(A,\varphi )}^{0,\ast }}{\to }{\Omega }^0(X,{\mathfrak{g}}_P)\to 0.$$

$$d_{(A,\varphi )}^{1,\ast }(a^{\prime },b^{\prime })=(d_A^{\ast }{b}^{\prime }-\ast [\varphi ,a^{\prime }],d_A^{\ast }{a}^{\prime }+\ast [\varphi ,b^{\prime }]),$$

$$d_{(A,\varphi )}^{0,\ast }(a,b)=(-d_A^{\ast }a+\ast [\varphi ,\ast b]).$$

$$KW(A+a,\varphi +b)=KW(A,\varphi )+d_{A,\varphi }^1(a,b)+\{(a,b),(a,b)\},$$

$$\{(a,b),(a,b)\}:=\big{(}[a,b]^{-},(a\wedge a+[b,b])^{+}\big{)}.$$

$$d_{A_0,{\varphi }_0}^1+\{(a,b),(a,b)\}={\psi }_0-KW(A_0,{\varphi }_0).$$

Full text

††T. Huang: School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P. R. China; e-mail: htmath@ustc.edu.cn; htustc@gmail.com††Mathematics Subject Classification (2010): 58E15;81T13

Keywords. Kapustin$\textendash$Witten equations, Vafa$\textendash$Witten equations, stable Higgs bundle, gauge theory

1 Introduction

Let $X$ be an closed, oriented, smooth, four-dimensional manifold with a given Riemannian metric $g$, and let $P\rightarrow X$ be a principal $G$-bundle with $G$ being a compact Lie group. We denote by $\mathfrak{g}_{P}$ the adjoint bundle of $P$ and by $\mathcal{A}_{P}$ the space of smooth connections on $P$. On a $4$-manifold $X$ the Hodge star operator $\ast$ takes $2$-forms to $2$-forms and we have $\ast^{2}=Id_{\Omega^{2}}$. The self-dual and anti-self-dual forms, we denoted $\Omega^{+}$ and $\Omega^{-}$ are defined to be the $\pm$ eigenspace of $\ast$: $\Omega^{2}T^{\ast}X=\Omega^{+}\oplus\Omega^{-}$.

The Kapustin$\textendash$Witten equations are defined on a Riemannian $4$-manifold given a principle bundle $P$. For most present considerations, $G$ can be taken to be $SU(2)$ or $SO(3)$. The equations require a pair $(A,\phi)\in\mathcal{A}_{P}\times\Omega^{1}(X,\mathfrak{g}_{P})$ satisfies

[TABLE]

where $F_{A}$ is the curvature two-form of the connection $A$, and superscripts “-” and “+” indicate taking the anti-self-dual part or self-dual part respectively. These equations were introduced by Kapustin and Witten [15]. The motivation is from the viewpoint of $\mathcal{N}=4$ super Yang$\textendash$Mills theory in four dimensions to study the geometric Langlands program [9, 11, 15] and [28, 29, 30, 31]. One also can see Gagliardo-Uhlenbeck’s article[8]. Note that the equation $d_{A}^{\ast}\phi=0$ makes equations (1.1) an elliptic system after gauge fixing equation. Note also there are no solutions to Kapustin$\textendash$Witten equations in the case of $p_{1}(P)$ is positive.

In mathematics, the analytic properties of solutions of Kapustin$\textendash$Witten equations were discussed by Taubes [24, 25, 26] and Tanaka [21]. In [24], Taubes studied the Uhlenbeck style compactness problem for $SL(2,\mathbb{C})$ connections, including solutions to the above equations, on four-manifolds, see [25, 26]. In [21], Tanaka observed that equations on a compact Kähler surface are the same as Hitchin$\textendash$Simpson’s equations [12, 18] and proved that the singular set introduced by Taubes for the case of Simpson’s equations has a structure of a holomorphic subvariety. In [13], the author proved that there exist a lower bounded for the $L^{2}$-norm of extra fields unless the connection is anti-self-dual under some mild conditions on $X,P,g,G$.

One always uses continuous method to construct the solutions of some gauge-theoretic equations. For example, Freed-Uhlenbeck and Taubes [6, 22] used this technical to constructed the ASD connections over a closed four-manifold. In this article, we suppose that the pair $(A_{\infty}+a,\phi)$ is a smooth solution of the Kapusitin$\textendash$Witten equations, where $A_{\infty}$ is an anti-self-dual connection on $P$. Then, the pair $(A_{\infty}+a,\phi)$ satisfies the equations:

[TABLE]

We will show that if the connections are in a neighborhood of a $generic$ anti-self-dual connection $A_{\infty}$, then the smooth solutions of Kapustin$\textendash$Witten equations on a closed four-manifold must be trivial solutions, i.e., $F_{A_{\infty}+a}^{+}=0$ and $\phi=0$.

Theorem 1.1**.**

*Let $X$ be a closed, oriented, simply-connected, four-dimensional manifold with a smooth Riemannain metric $g$, $P\rightarrow X$ be a principal $SU(2)$ or $SO(3)$-bundle with $p_{1}(P)$ negative. Suppose that there exist a $generic$ ASD connection $A_{\infty}$ on $P$, i.e., $H^{0}_{A_{\infty}}=\ker d_{A_{\infty}}|_{\Omega^{0}(X,\mathfrak{g}_{P})}=0$ and $H^{2}_{A_{\infty}}=\ker d_{A_{\infty}}^{+}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}=0$, then there is a positive constant $\delta=\delta(g,P,A_{\infty})$ with following significance. If $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations over $X$, then one of following must hold:

(1) $F^{+}_{A}=0$ and $\phi=0$;

(2) the pair $(A,\phi)$ satisfies*

[TABLE]

where $dist(\cdot,\cdot)$ defined by

[TABLE]

Remark 1.2**.**

We describe another result as follows which is a small variation of the result in [13]. Let the pair $(A,\phi)$ be a smooth solution of Kapustin$\textendash$Witten equations. If $(A,\phi)$ is closed to $(A_{\infty},0)$ in $L^{2}_{1}$-topology, i.e, there exist a small enough constant $\delta\in(0,1)$ such that

[TABLE]

where $A_{\infty}$ is a $generic$ ASD connection, then $A$ must be ASD and $\phi$ identically zero. We should point out that the result is different to Theorem 1.1. We don’t make any assumptions on extra fields in theorem 1.1. The following case may happen: if the connection $A$ tends to a $generic$ ASD connection $A_{\infty}$ in $L^{2}_{1}$-topology (under moduli gauge transformations), then the $L^{2}$-norm of extra field $\|\phi\|_{L^{2}(X)}$ may tends to infinite. But thanks to a compactness theorem of Taubes, we observed that if $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ is a sequence of smooth solutions of Kapustin$\textendash$Witten equations, then there exist a subsequence $\Xi\subset\mathbb{N}$ such that the sequence $\{\|\phi_{i}\|_{L^{2}(X)}\}_{i\in\Xi}$ is a bounded subsequence under some conditions, see Lemma 3.10. Using this useful observation, we could prove Theorem 1.1.

We define the gauge-equivariant map

[TABLE]

[TABLE]

We denote by

[TABLE]

the moduli space of solutions of Kapustin$\textendash$Witten equations. The moduli space $M_{ASD}$ of all ASD connections can be embedded into $M_{KW}$ via $A_{\infty}\mapsto(A_{\infty},0)$, $A_{\infty}$ is an ASD connection on $P$. We also denote $\bar{M}_{ASD}$ by the compactification of moduli space of ASD connection.

Following the idea of Donaldson on [2, Section 4.2.1], we write $[A,\phi]$ for the equivalence class of a pair $(A,\phi)$, which is a point in the moduli space $M_{KW}$. We set,

[TABLE]

is preserved by the action of $\mathcal{G}_{P}$, so descends to define a distance function on $M_{KW}$:

[TABLE]

Let the pair $(A,\phi)$ be a smooth solution of Decoupled Kapustin$\textendash$Witten equations over a closed, simply-connected, four-manifold. We observe that if the connection $A$ is irreducible, then the extra field $\phi$ vanishes, see Theorem 3.4. Hence, we can denote by

[TABLE]

the distance between $M_{ASD}$ and $M_{KW}\backslash M_{ASD}$.

Theorem 1.3**.**

*Let $X$ be a closed, oriented, simply-connected, four-dimensional manifold with a smooth Riemannain metric $g$, $P\rightarrow X$ be a principal $SU(2)$ or $SO(3)$-bundle with $p_{1}(P)$ negative. Suppose that the connections in $\bar{M}_{ASD}(P,g)$ are all $generic$, i.e., $H^{0}_{A_{\infty}}=\ker d_{A_{\infty}}|_{\Omega^{0}(X,\mathfrak{g}_{P})}=0$ and $H^{2}_{A_{\infty}}=\ker d_{A_{\infty}}^{+}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}=0$ for any connection $[A_{\infty}]\in\bar{M}_{ASD}(P,g)$, then there is a positive constant $\delta=\delta(g,P)$ with following significance. If $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations, then one of following must hold:

(1) $F^{+}_{A}=0$ and $\phi=0$;

(2) the pair $(A,\phi)$ satisfies*

[TABLE]

In particular, there is a positive constant $\tilde{\delta}=\tilde{\delta}(g,P)$ such that

[TABLE]

unless $A$ is anti-self-dual with respect to $g$.

We now take $X$ to be a compact Kähler surface with Kähler form $\omega$, and $E$ to be a principal $G$-bundle over $X$. Tanaka observed that the Kapustin$\textendash$Witten equations are the same as Hitchin$\textendash$Simpson equations. Furthermore, if $A$ is an $SU(N)$-ASD connection on $E$, then

[TABLE]

see [14, Proposition 2.3] or [7, Chapter IV] . It’s difficult to addition certain mild conditions to ensure $\ker(\bar{\partial}_{A}\bar{\partial}_{A}^{\ast}+\bar{\partial}_{A}^{\ast}\bar{\partial}_{A})|_{\Omega^{0,2}(X,adE)}$ and $\ker d_{A}|_{\Omega^{0}(X,adE)}$ vanish at some time. But one can see that $\ker d_{A}|_{\Omega^{0}(X,adE)}=0$ is equivalent to the connection $A$ is irreducible. It is convenient to introduce the $c$-$generic$ metric which ensures the connections on the compactification of moduli space of ASD connections on $E$, $\bar{M}(E,g)$, are irreducible, one also can see [7, Chapter IV].

Definition 1.4**.**

Let $X$ be a compact, connected, Kähler surface, $c$ be a positive integer. We say that a Kähler metric $g$ on $X$ is $c$-$generic$ if for every $SU(2)$-bundle $E$ over $X$ with $c_{2}(E)\leq c$, there are no reducible ASD connections on $E$.

If we suppose that the Kähler metric $g$ is $c$-$generic$, we could deform the connection $A\in\mathcal{A}^{1,1}$ which obeys $\|\Lambda_{\omega}F_{A}\|_{L^{2}(X)}\leq\varepsilon$ to an other connection $A_{\infty}$ satisfies $\Lambda_{\omega}F_{A_{\infty}}=0$, where $\varepsilon$ is a suitable sufficiently small positive constant, see Theorem 4.8. Even though, the connection $A_{\infty}$ may be not an ASD connection, but the $(0,2)$-part $F_{A_{\infty}}^{0,2}$ of the curvature $F_{A_{\infty}}$ would estimated by $\Lambda_{\omega}F_{A}$. We then extend the results of Kapustin$\textendash$Witten equations to the Hitchin$\textendash$Simpson equations case.

Theorem 1.5**.**

*Let $X$ be a compact, Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle with $c_{2}(E)=c$ positive. Suppose that $g$ is a $c$-$generic$ metric in the sense of Definition 1.4, then there is a positive constant $C=C(g,E)$ with following significance. If the Higgs pair $(A,\theta)\in\mathcal{A}_{E}^{1,1}\times\Omega^{1,0}(X,adE)$ is a smooth solution of Hitchin-Simpson equations, then one of following must hold:

(1) $\Lambda_{\omega}F_{A}=0$, or

(2) the Higgs field $\theta$ satisfies*

[TABLE]

Furthermore, if $X$ is also simply-connected, then the curvature $F_{A}$ obeys

[TABLE]

for a positive constant $\tilde{C}=\tilde{C}(g,P)$ unless $\Lambda_{\omega}F_{A}=0$.

We denote by

[TABLE]

the moduli space of solutions of Hitchin-Simpson equations. In [12], Hitchin proved that if the bundle $E$ is a rank-2 bundle of odd degree over a Riemannian surface of genus $g>1$, then the moduli space of stable Higgs bundle is connected and simply connected, see [12, Theorem 7.6]. Following Theorem 1.5, it implies that the moduli space of stable Higgs bundle on a simply-connected Kähler surface with a generic Kähler metric is not connected. It is different from the Riemannian surface case, see Corollary 4.13.

Remark 1.6**.**

If the principal $G$-bundle $P\rightarrow X$ over the closed manifold $X$ with $p_{1}(P)=0$, the solutions $(A,\phi)$ of the Kapustin$\textendash$Witten equations are flat $G_{\mathbb{C}}$-connections with moment map condition:

[TABLE]

The [2, Proposition 2.2.3] shows that the gauge-equivalence classes of flat $G$-connections over a connected manifold, $X$, are in one-to-one correspondence with the conjugacy classes of representations $\rho:\pi_{1}(X)\rightarrow G$. If $X$ is also a simply-connected manifold, i.e., $\pi_{1}(X)$ is trivial, then the representations $\rho$ must be a trivial representation. It is no sense to consider Kapustin$\textendash$Witten equations on a principal $G$-bundle with $p_{1}(P)=0$ over a simply-connected four-manifolds.

The organization of this paper is as follows. In section 2, we first recall gauge theory in $4$-dimensional manifolds. We also discuss the elliptic deformation complex and Kuranishi model for the Kapustin$\textendash$Witten equations. In section 3, we recall a vanishing theorem for the extra fields of the decoupled Kapustin$\textendash$Witten equations. Using an optimal inequality proved by Donaldson, we observe that the $L^{2}$-norm of extra fields of non-trivial solutions of Kapustin$\textendash$Witten equations have a positive lower bounded. Thanks to Taubes’ compactness theorem [24], we observe that if $(A_{i},\phi_{i})$ is a sequence solutions of Kapustin$\textendash$Witten equations and the connection $A_{i}$ converges to an irreducible ASD connection $A_{\infty}$ in $L^{2}_{1}$ (under moduli gauge transformations), then the sequence $\{\|\phi_{i}\|_{L^{2}(X)}\}$ has a bounded subsequence. At last, we obtain our main result: if the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then $(A,\phi)$ must be a trivial solution. In section 4, we extend the above results to the global case. We will prove that if the connections on the compactification of moduli space of ASD connections, $\bar{M}_{ASD}$, are $generic$, then $\|F^{+}_{A}\|_{L^{2}(X)}$ must has a uniform positive lower bounded. In particular, the subspace of anti-self-dual solutions is not connected to the space of Kapustin$\textendash$Witten equations. We also given some four-manifolds $X$ with Riemannian metric $g$ and principle $SO(3)$-bundles $P\rightarrow X$ ensure that the connections on $\bar{M}_{ASD}$ are $generic$. At last, we construct a gluing theorem for the connection $A$ which obeys $\|\Lambda_{\omega}F_{A}\|_{L^{2}(X)}\leq\varepsilon$, see Theorem 4.8. We then extend the results for the Kapustin$\textendash$Witten equations to other coupled equations for pairs such as the Hitchin$\textendash$Simpson equations and Vafa$\textendash$Witten on compact Kähler surface with a smooth $c$-$generic$ Kähler metric $g$.

2 A neighborhood of an ASD connection

2.1 Yang-Mills theory on 4-manifolds

Let $X$ be an oriented, closed, smooth, Riemannian $4$-manifold, $P\rightarrow X$ be a principal $G$-bundle with $G$ being a compact Lie group. The Hodge start operator gives an endomorphism of $\Omega^{2}$ with property $\ast^{2}=Id_{\Omega^{2}}$. We denote by $\Omega^{2,+}$ and $\Omega^{2,-}$ the eigenvalues of $+1$ and $-1$. A $2$-form in $\Omega^{2,+}$ (or in $\Omega^{2,-}$) is called self-dual (or anti-self-dual). Decomposing the curvature $F_{A}$ of a connection $A$ according to the decomposition $\Omega^{2}=\Omega^{2,+}\oplus\Omega^{2,-}$ of the 2-forms into self-dual and anti-self-dual parts. An ASD connection $A$ on $P$ naturally induces the Yang$\textendash$Mills complex

[TABLE]

The $i$-th cohomology group $H^{i}_{A}$ of this complex if finite dimensional and the index $d=h^{0}-h^{1}+h^{2}$ ($h^{i}=dimH^{i}_{A}$ )is given by $c(G)\kappa(P)-dimG(1-b_{1}+b^{+})$. $H^{0}_{A}$ is the Lie algebra of the stabilizer $\Gamma_{A}$, the group of gauge transformation of $P$ fixing by $A$. We call a connection is $generic$ when $H^{0}_{A}=0$ and $H^{2}_{A}=0$. We denote $M_{gen}$ the subset of

[TABLE]

of $generic$ ASD connections on $P$. $M_{gen}$ becomes a smooth manifold whose tangent space is $H^{1}_{A}$. $M_{gen}$ consists exactly of all singular points and we have two types according to either case (1) in which $A$ is irreducible $H_{A}^{0}=0$ but $H^{2}_{A}\neq 0$ or case (2) in which $A$ is reducible $H_{A}^{0}\neq 0$. So if the anti-self-dual connections $[A]\in M_{ASD}$ are all $generic$, the moduli space $M_{ASD}$ is a smooth manifold. Furthermore, We call an ASD connection is $regular$ when $H^{2}_{A}=0$.

Now we will present the Kuranishi complex associated to the Kapustin$\textendash$Witten equations, see [13, section 3.1]. Tanaka obtained an analogous Kuranishi complex to the Vafa$\textendash$Witten equations [16, 19]. Let $(A,\phi)\in\mathcal{A}_{P}\times\Omega^{1}(X,\mathfrak{g}_{P})$ be a smooth solution of the Kapustin$\textendash$Witten equations. Then, the infinitesimal deformation at $(A,\phi)$ is given by the following:

[TABLE]

where the map $d^{0}_{(A,\phi)}$ and $d^{1}_{(A,\phi)}$ are defined for $\xi\in\Omega^{0}(X,\mathfrak{g}_{P})$ and $(a,b)\in\Omega^{1}(X,\mathfrak{g}_{P})\oplus\Omega^{1}(X,\mathfrak{g}_{P})$ by

[TABLE]

[TABLE]

We compute

[TABLE]

Thus the pair $(A,\phi)$ satisfies Kapustin$\textendash$Witten equations if and only if $d^{1}_{(A,\phi)}\circ d^{0}_{(A,\phi)}=0$. We can define the homology groups: $H^{0}_{(A,\phi)}=\ker d^{0}_{(A,\phi)}$, $H^{1}_{(A,\phi)}=\frac{\ker d^{1}_{(A,\phi)}}{\rm{Im}d^{0}_{(A,\phi)}}$ and $H^{2}_{(A,\phi)}=\rm{Coker}d^{1}_{(A,\phi)}$. We denote the isotropy group of the pair $(A,\phi)$ as $\Gamma(A,\phi)=\{u\in\mathcal{G}_{P}|u^{\ast}(A,\phi)=(A,\phi)\}$. Recall that $H^{0}_{(A,\phi)}$ is the Lie algebra of the stabilizer of $(A,\phi)$ and $H^{1}_{(A,\phi)}$ is the formal tangent space. The dual complex is:

[TABLE]

where

[TABLE]

[TABLE]

Proposition 2.1**.**

The map $KW(A,\phi)$ has an exact quadratic expansion given by

[TABLE]

where $\{(a,b),(a,b)\}$ is the symmetric quadratic form given by

[TABLE]

Given fixed $(A_{0},\phi_{0})$, we look for solutions to the inhomogeneous equation $KW(A_{0}+a,\phi_{0}+b)=\psi_{0}$. By Proposition 2.1, this equation is equivalent to

[TABLE]

To make this equation elliptic, it’s nature to impose the gauge-fixing condition

[TABLE]

If we define

[TABLE]

then the elliptic system can be rewritten as

[TABLE]

Local interior estimates for the elliptic system (2.2) are consider in [5] equation 3.2 in the context of $PU(2)$ monopoles. This is also considered similarly for the Vafa-Witten equations in [16].

2.2 An inequality for the connections near an ASD connection

Let $A_{\infty}$ be a fixing ASD connection on $P$, any connection $A$ can be written uniquely as

[TABLE]

In [1], Donaldson proved that the connection $A$ can be written as

[TABLE]

where $\tilde{A}_{\infty}$ is also an ASD connection and $u\in\Omega^{2,+}(X,\mathfrak{g}_{P})$, i.e., the connection $A$ satisfies

[TABLE]

when the connection $A_{\infty}$ is $regular$ and $a$ is small enough in $L^{2}_{1}$-norm. The operator $d_{A}^{+}d^{+,\ast}_{A}$ is an elliptic self-adjoint operator on the space of $L^{2}$ sections of $\Omega^{2,+}TX\otimes\mathfrak{g}_{P}$. It is a standard result that the spectrum of $d_{A}^{+}d_{A}^{+,\ast}$ is discrete, and the lowest eigenvalue is nonnegative.

Definition 2.2**.**

For $A_{\infty}\in M_{ASD}$, define

[TABLE]

is the lowest eigenvalue of $d^{+}_{A_{\infty}}d^{+,\ast}_{A_{\infty}}$.

One can see that an ASD connection $A_{\infty}$ is $regular$, i.e., $\mu(A_{\infty})>0$. The Sobolev norms $L^{p}_{k,A}$, where $1\leq p<\infty$ and $k$ is an integer, with respect to the connections defined as:

[TABLE]

where $\nabla^{j}_{A}:=\nabla_{A}\circ\ldots\circ\nabla_{A}$ (repeated $j$ times for $j\geq 0$).

Theorem 2.3**.**

([1, Proposition 22]) Let $X$ be a closed, four-dimensional, smooth Riemannian manifold with a smooth Riemannian metric, $G$ be a compact Lie group, $P$ be a smooth principal $G$-bundle over $X$. If there is a $C^{\infty}$ ASD connection $A_{\infty}$ on $P$ that is $regular$, then there is constant $\sigma=\sigma(\mu(A_{\infty}),g,G)\in(0,1]$ with the following significance. If $A:=A_{\infty}+b$ is a smooth connection on $P$obeying

[TABLE]

then there exist a solution $a:=d^{+,\ast}_{A_{\infty}}u\in\Omega^{1}(X,\mathfrak{g}_{P})$ where $u\in\Omega^{2,+}(X,\mathfrak{g}_{P})$ to Equation (2.3). In fact, the connection $\tilde{A}_{\infty}:=A-a$ is an anti-self-dual connection on $P$. Further, there exist a constant $C=C(\mu(A_{\infty}),g,G)\in(0,\infty)$ such that

[TABLE]

[TABLE]

This theorem 2.3 by follows the method of proof of [22, Theorem 2.2 ] applied to equation $F^{+}(A+d^{+,\ast}_{A_{\infty}}u)=0$.

3 Non-existence solutions on a neighborhood of a $generic$ ASD connection

3.1 Decoupled Kapustin$\textendash$Witten equations

Definition 3.1**.**

Let $G$ be a compact Lie group, $P$ be a principal $G$-bundle over a closed, smooth four-manifold $X$ with a smooth Riemannian metric $g$. The $decoupled$ Kapustin$\textendash$Witten equations on $P$ over $X$ are the equations require a pair $(A,\phi)\in\mathcal{A}_{P}\times\Omega^{1}(X,\mathfrak{g}_{P})$ satisfies

[TABLE]

Lemma 3.2**.**

If a pair $(A,\phi)$ is a smooth solution of $decoupled$ Kapustin$\textendash$Witten equations, then the extra field $\phi$ also obeys

[TABLE]

Proof.

At first, we observe that $\int_{X}tr(\phi\wedge\phi\wedge\phi\wedge\phi)=0$, hence

[TABLE]

i.e., $\phi\wedge\phi=0$. Following the identity $(d_{A}\phi)^{-}=0$, i.e., $d_{A}\phi=\ast d_{A}\phi$, we obtain that

[TABLE]

Hence take the $L^{2}$ inner of above identity with $\phi$ and integrable by parts, we obtain

[TABLE]

i.e., $d_{A}\phi=0$. We complete the proof of this lemma. ∎

We recall a vanishing theorem on the extra fields of $decoupled$ Kapustin$\textendash$Witten equations. The prove is similar to Vafa$\textendash$Witten equations [16, Theorem 4.2.1]. At first, we recall a useful lemma proved by Donaldson [2, Lemma 4.3.21].

Lemma 3.3**.**

If $A$ is an irreducible $SU(2)$ or $SO(3)$ ASD connection on a bundle $P$ over a simply connected four-manifold $X$, then the restriction of $A$ to any non-empty open set in $X$ is also irreducible.

Theorem 3.4**.**

([13, Theorem 2.9] ) Let $X$ be a closed, simply-connected, smooth, oriented, Riemannian four-manifold, $P\rightarrow X$ be an $SU(2)$ or $SO(3)$ principal bundle. Suppose that $(A,\phi)$ is a solution of the $decoupled$ Kapustin$\textendash$Witten equations. If $A$ is irreducible, then the extra fields $\phi$ vanishes.

Proof.

We denote $Z^{c}$ by the complement of the zero of $\phi$. By unique continuation of the elliptic equation $(d_{A}+d^{\ast}_{A})\phi=0$, $Z^{c}$ is either empty or dense. Since $\phi\wedge\phi=0$, $\phi$ has at most rank one. The Lie algebra of $SU(2)$ or $SO(3)$ is three-dimensional, with basis $\{\sigma^{i}\}_{i=1,2,3}$ and Lie brackets $\{\sigma^{i},\sigma^{j}\}=2\varepsilon_{ijk}\sigma^{k}$. In a local coordinate, we can set $\phi=\sum_{i=1}^{3}\phi_{i}\sigma^{i}$, where $\phi_{i}\in\Omega^{1}(X)$. We then have

[TABLE]

i.e.,

[TABLE]

On $Z^{c}$, $\phi$ is non-zero, then without loss of generality we can assume that $\phi_{1}$ is non-zero. From (3.1), there exist functions $\mu$ and $\nu$ such that

[TABLE]

Hence we would re-written $\phi$ to

[TABLE]

Then on $Z^{c}$ write $\phi=\xi\otimes\omega$ for $\xi\in\Omega^{0}(Z^{c},\mathfrak{g}_{P})$ with $\langle\xi,\xi\rangle=1$, and $\omega\in\Omega^{1}(Z^{c})$. We compute

[TABLE]

[TABLE]

Taking the inner product with $\xi$ and using the consequence of $\langle\xi,\xi\rangle=1$ that $\langle\xi,d_{A}\xi\rangle=0$, we get $d\omega=d^{\ast}\omega=0$. It follows that $d_{A}\xi\wedge\omega=0$ and $d_{A}\xi\wedge\ast\omega=0$. Since $\omega$ is nowhere zero along $Z^{c}$, we must have $d_{A}\xi=0$ along $Z^{c}$. Therefore, $A$ is reducible along $Z^{c}$. However according to Lemma 3.3, $A$ is irreducible along $Z^{c}$. This is a contradiction unless $Z^{c}$ is empty. Therefore $Z=X$, so $\phi$ identically zero. ∎

3.2 A bounded property of extra fields

In this section, we will proved that the extra fields have a lower positive bounded if the connections on the neighborhood of a $regular$ ASD connection. The result has a direct proof by using the Kuranshi model for the moduli space of Kapustin$\textendash$Witten solutions around the pair $(A_{\infty},0)$, where $A_{\infty}$ is an ASD connection. Our proof employs a Weizenböck formula, the idea could extends to the cases of Kapustin$\textendash$Witten equations and Vafa$\textendash$Witten equations on Kähler surface. We need recall a bounded on $\|\phi\|_{L^{\infty}}$ in terms of $\|\phi\|_{L^{2}}$. The technique is analogy to Vafa$\textendash$Witten equations case [16].

Theorem 3.5**.**

([13, Theorem 2.4] ) If the pair $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations over a closed, smooth, Riemannian four-manifold, then

[TABLE]

where $C=C(g)$ is a positive constant only depends on metric $g$.

If we suppose that the connection $[A]$ is in a neighborhood of a $regular$ ASD connection $[A_{\infty}]$, then the Theorem 2.3 which provides existence of an other ASD connection $\tilde{A}_{\infty}$ on $P$ and a Sobolev norm estimate for the distance between $A$ and $\tilde{A}_{\infty}$. Following the idea in [13], we then have

Proposition 3.6**.**

Let $X$ be a closed, oriented, smooth, four-dimensional Riemannian manifold with Riemannian metric $g$, $P\rightarrow X$ be a principal $G$-bundle with $G$ being a compact Lie group with $p_{1}(P)$ negative. Suppose that $A_{\infty}$ is an $regular$ ASD connection on $P$, i.e., $\ker{d^{+}_{A_{\infty}}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}}=0$, then there are positive constants $\delta=\delta(g,A_{\infty})\in(0,1)$, $C=C(g,A_{0})\in[1,\infty)$ with following significance. If $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations and the connection $A$ obeys

[TABLE]

then either the extra field obeying

[TABLE]

or $A$ is anti-self-dual with respect to $g$.

Proof.

Following Theorem 2.3, it implies that for a suitable constant $\delta$, the connection $A$ can be written as $A=\tilde{A}_{\infty}+a$, where $\tilde{A}_{\infty}$ is also an ASD connection. Following the idea in [13], we have two integrable inequalities:

[TABLE]

[TABLE]

Combining the preceding inequalities gives

[TABLE]

If $\|F^{+}_{A}\|_{L^{2}(X)}$ is non-zero, thus $\|\phi\|^{2}_{L^{2}(X)}\geq 2/c$. We complete the proof of this proposition. ∎

3.3 Uhlenbeck type compactness of Kapustin$\textendash$Witten equations

We now recall the work by Taubes on Uhlenbeck style compactness theorem of Kapustin$\textendash$Witten equations on closed four-dimensional manifolds [24]. In [21], Tanaka also briefly describe the compactness theorem. Let $X$ be a closed, oriented, smooth Riemannian four-manifold with Riemannian metric $g$, $P\rightarrow X$ be a principal $G$-bundle over $X$ with $G$ being $SU(2)$ or $SO(3)$. We consider a sequence solutions $\{(A_{i},\phi_{i})\}$ of the Kapustin$\textendash$Witten equations. We put $r_{i}:=\|\phi_{i}\|_{L^{2}(X)}$. In the case that $\{r_{i}\}$ has a bounded subsequence, the Uhlenbeck compactness theorem with the Weizenböck formula deduces the following.

Theorem 3.7**.**

*([21, Proposition 2.1] and [24]) If $\{r_{i}\}$ has a bounded subsequence, then

(1) there exist a principal bundle $P_{\Delta}\rightarrow X$ and a pair $(A_{\Delta},\phi_{\Delta})\in\mathcal{A}_{P_{\Delta}}\times\Omega^{1}(X,\mathfrak{g}_{P_{\Delta}})$ obeys the Kapustin-Witten equations;

(2) a finite set $\Sigma\subset X$ of points, a subsequence $\Xi\in\mathbb{N}$ and a sequence $\{g_{i}\}_{i\in\Xi}$ of automorphisms of $P_{\Delta}|_{X-\Sigma}$ such that $\{(g_{i}^{\ast}(A_{i}),g^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi}$ converges to $(A_{\Delta},\phi_{\Delta})$ in the $C^{\infty}$ topology on compact subsets in $X-\Sigma$.*

Analysis for the case that $\{r_{i}\}$ has no bounded subsequences was the bulk of [24, Theorem 1.1] .

Theorem 3.8**.**

*([21, Proposition 2.2] and Theorem 2.3) If $\{r_{i}\}$ has no bounded subsequence. There exists in this case the following data:

(1) A finite set $\Theta\subset X$ and a closed, nowhere dense set $Z\subset X-\Theta$,

(2) a real line bundle $\mathcal{I}\rightarrow X-(Z\cup\Theta)$,

(3) a section $v$ of $\mathcal{I}\otimes T^{\ast}X|_{X-(Z\cup\Theta})$ with $dv=d^{\ast}v=0$, the norm of $v$ extends over the whole of $X$ as a bounded $L^{2}_{1}$ function. In addition,

a) The extension of $|v|$ is continuous on $X-\Theta$ and its zero locus is the set $Z$.

b) Let $U$ denote an open set in $X-\Theta$ with compact closure. The function $|v|$ is Hölder continuous on $U$ with Hölder exponent that is independent of $U$ and of the original sequence $\{(A_{i},\phi_{i})\}_{i=1,2,\ldots}$.

c) If $p$ is any given point in $X$, then the function $dist(\cdot,p)^{-1}|\nabla v|$ extends to the whole of $X$ as an $L^{2}_{1}$-function.

(4) A principal $G$-bundle $P_{\Delta}$ over $X-(Z\cup\Theta)$ and a connection $A_{\Delta}$ of $P_{\Delta}$ with $d_{A_{\Delta}}^{\ast}F_{A_{\Delta}}=0$.

(5) An isometric $A_{\Delta}$ covariantly constant homorphism $\sigma_{\Delta}:\mathcal{I}\rightarrow\mathfrak{g}_{P}$.

In addition, there exist a subsequence $\Xi\subset\mathbb{N}$ and a sequence of automorphisms $g_{i}:P_{\Delta}\rightarrow P|_{X-(Z\cup\Theta)}$ such that

(i) $\{g_{i}^{\ast}(A_{i})\}_{i\in\Xi}$ converges to $A_{\Delta}$ in the $L^{2}_{1}$ topology on compact subset in $X-(Z\cup\Theta)$ and

(ii) The sequence $\{r^{-1}g^{\ast}_{i}(\phi_{i})\}$ converges to $v\otimes\sigma_{\Delta}$ in $L^{2}_{1}$ topology on compact subset in $X-(Z\cup\Theta)$ and $C^{0}$-topology on $X-\Theta$. Moreover, the sequence $\{r_{i}^{-1}\phi_{i}\}_{i\in\Xi}$ is pointwise bounded and it converges in the $L^{2}_{1}$ topology on $X$ to $|v|$.*

As was the case for Uhlenbeck theorem, the number of elements in $\Theta$ only dependent on the first Pontrjagin class $p_{1}(P)$. In particular, $\Theta$ is empty if the first Pontrjagin class of $p_{1}(P)=0$. As for the structure of the above $Z$, Taubes proved that $Z$ has the Hausdorff dimension at most 2, see [24].

For a sequence of connections $\{A_{i}\}$ on $P$, Tanaka defined a set $S(\{A_{i}\})$ as follows:

[TABLE]

where $\varepsilon_{0}$ is a positive constant is defined as in [20]. This set $S(\{A_{i}\})$ describes the singular set of a sequence of connections $\{A_{i}\}$. In [20], Tanaka observed that under the particular circumstance where the connections are non-concentrating and the limiting connection is non-locally reducible, one obtains an $L^{2}$-bounded on the extra fields. We extend the idea used by Tanaka for Vafa$\textendash$Witten equations (see [20, Theorem 1.3]) to the Kapustin$\textendash$Witten equations case.

Theorem 3.9**.**

Let $X$ be a closed, oriented, 4-dimensional manifold with a smooth Riemannian metric $g$, $P\rightarrow X$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. If $\{(A_{i},\phi_{i})\}$ is a sequence of smooth solutions of Kapustin$\textendash$Witten equations with $S(\{A_{i}\})$ being empty, then there exists subsequence $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{g_{i}\}_{i\in\Xi}$ such that $\{g_{i}(A_{i})\}_{i\in\Xi}$ converges weakly in the $L^{2}_{1}$-topology. If the limit is not locally reducible, then there exists a positive number $C$ such that $\int_{X}|\phi_{i}|^{2}\leq C$ for all $i\in\Xi$, and $\{(g_{i}^{\ast}(A_{i}),g_{i}^{\ast}(\phi_{i}))\}_{i\in\Xi}$ converges in the $C^{\infty}$-topology to a pair that obeys the Kapustin$\textendash$Witten equations.

We then have a useful lemma as follows.

Lemma 3.10**.**

Let $X$ be a closed, oriented, simply-connected, 4-dimensional manifold with a smooth Riemannian metric $g$, $P\rightarrow X$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. If $\{(A_{i},\phi_{i})\}$ is a sequence of smooth solutions of Kapustin$\textendash$Witten equations such that

[TABLE]

where $A_{\infty}$ is an irreducible ASD connection on $P$, then there exist a subsequence $\Xi\subset\mathbb{N}$ such that the sequence $\{r_{i}:=\|\phi_{i}\|_{L^{2}(X)}\}_{i\in\Xi}$ is a bounded subsequence.

Proof.

For the sequence of $\{A_{i}\}_{i\in\mathbb{N}}$ on $P$, one can see $S(\{A_{i}\})=\varnothing$, where $S(A_{i})$ defined as in equation (3.2). Following [19, Theorem 1.3] , there exist a subsequence $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $g^{\ast}_{i}(A_{i})_{i\in\Xi}$ converges weakly in the $L_{1}^{2}$-topology, we denote the limit connection by $\tilde{A}_{\infty}$. Under our assumption on the sequence $\{A_{i}\}_{i\in\mathbb{N}}$, there exist a gauge transformation $g_{\infty}$ such that $g_{\infty}^{\ast}(A_{\infty})=\tilde{A}_{\infty}$. Hence $\tilde{A}_{\infty}$ is also not locally reducible, since the locally reducible connection on simply-connected manifold is also reducible, see [20, Appendix B] . At last, following Theorem 3.9, it implies that there eixst a positive number $C$ such that $\|\phi_{i}\|_{L^{2}(X)}\leq C$ for all $i\in\Xi$. ∎

Proof of Theorem 1.1..

We suppose that the pair $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations and $A$ is not ASD with respect to metric $g$. If we suppose that the constant $\delta$ does not exist, we may choose a sequence $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ of solutions of Kapustin$\textendash$Witten equations on $P$ such that $dist(A_{i},A_{\infty})\rightarrow 0$ as $i\rightarrow\infty$, then following Proposition 3.6 and Lemma 3.10, there exists a subsequence $\Xi\subset\mathbb{N}$ and two positive constants $C,c$, such that

[TABLE]

Following the compactness Theorem 3.7, there exist a pair $(A_{\Delta},\phi_{\Delta})\in\mathcal{A}_{P_{\Delta}}\times\Omega^{1}(X,\mathfrak{g}_{P_{\Delta}})$ that obeys the Kapustin$\textendash$Witten equations and there has a subsequence $\Xi^{\prime}\subset\Xi$ and a sequence $\{g_{i}\}_{i\in\Xi^{\prime}}$ of automorphisms of $P_{\Delta}$ such that $\{(g_{i}^{\ast}(A_{i}),g^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi^{\prime}}$ converges to $(A_{\Delta},\phi_{\Delta})$ in the $C^{\infty}$ topology on $X$. Thus the extra field $\phi_{\Delta}$ also has a lower positive bounded:

[TABLE]

But on the other hand, under our initial assumption regarding the sequence $\{A_{i}\}_{i\in\mathbb{N}}$, we have $[A_{\Delta}]\equiv[A_{\infty}]$. Thus the connection $A_{\Delta}$ is also irreducible, following the vanishing Theorem 3.4, it implies that the extra field $\phi_{\Delta}=0$. This contradicts $\|\phi_{\Delta}\|_{L^{2}(X)}$ has a uniform positive lower bound. The preceding argument shows that the desired constant $\delta$ exists. ∎

3.4 A bounded property of curvatures

In this section, we extend the proof method of Theorem 1.1 to global case. We suppose that the connections $[A_{\infty}]\in\bar{M}_{ASD}$ are all $regular$, i.e., $\ker{d^{+}_{A_{\infty}}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}}=0$. Following the result in [4, Section 3], one know that the connections on

[TABLE]

also obey $\ker{d^{+}_{A_{\infty}}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}}=0$ for suitable constant $\varepsilon$.

Proposition 3.11**.**

Let $G$ be a compact Lie group, $P\rightarrow X$ be a principal $G$-bundle over a closed, connected, four-dimensional manifold $X$ with a smooth Riemannian metric $g$. Suppose that the connections on $\bar{M}_{ASD}$ are all $regular$, i.e., $\ker{d^{+}_{A_{\infty}}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}}=0$ for any connection $[A_{\infty}]\in\bar{M}_{ASD}$, then there are positive constants $\varepsilon=\varepsilon(P,g)$ and $\mu=\mu(P,g)$ such that

[TABLE]

where $\mu(A)$ is as in Definition 2.2.

We then recall a result of extra fields $\phi$ which proved by author [13].

Theorem 3.12**.**

Assume the hypotheses of Proposition 3.11. Suppose that the connections on $\bar{M}_{ASD}$ are all $regular$, i.e., $\ker{d^{+}_{A_{\infty}}|_{\Omega^{2,+}(X,\mathfrak{g}_{P})}}=0$ for any connection $[A_{\infty}]\in\bar{M}_{ASD}$, then there is a positive constant $c=c(P,g)$ with following significance. If $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations, then the extra filed $\phi$ obeys

[TABLE]

unless $A$ is anti-self-dual with respect to $g$.

Now, we begin to consider a sequence smooth solutions $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ of Kapustin$\textendash$Witten equations. If we suppose that $\|\phi_{i}\|_{L^{2}(X)}$ has no bounded subsequence, following the compactness theorem 3.8 proved by Taubes, we only know that the connection $A_{\Delta}$ has harmonic curvature. Moreover if we suppose that $F_{A_{i}}^{+}$ converge to zero in $L^{2}$-topology, then we can claim that $A_{\Delta}$ is an anti-self-dual connection on the complement of $Z\cup\Theta\cup\Sigma$.

Corollary 3.13**.**

Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence smooth solutions of Kapustin$\textendash$Witten equations, set $r_{i}:=\|\phi_{i}\|_{L^{2}(X)}$. Suppose that $\{F^{+}_{A_{i}}\}_{i\in\mathbb{N}}$ converge to zero in $L^{2}$-topology and the sequence $\{r_{i}\}_{i=1,2,\ldots}$ has no bounded subsequence. Let $Z$, $\Theta$ and $\mathcal{I}$ be as described in Theorem 3.8, so that $\sigma_{\Delta}$ and $A_{\Delta}$ are defined over $X-(Z\cup\Theta\cup\Sigma)$. Then the connection $A_{\Delta}$ is anti-self-dual connection on $P_{\Delta}$.

Before the proof of Corollary 3.13, we should recall a key a compactness theorem due to Sedlacek, see [17, Theorem 4.3] or [3, Theorem 35.17] .

Theorem 3.14**.**

*Let $G$ be a compact Lie group and P a principal $G$-bundle over a closed, smooth, oriented, four-dimensional Riemannian manifold $X$ with a Riemannian metric $g$. If $\{A_{i}\}_{i\in\mathbb{N}}$ is a sequence $C^{\infty}$ connection on $P$ and the sequence $\{F^{+}_{A_{i}}\}_{i\in\mathbb{N}}$ converges to zero in $L^{2}$-topology, then there exists

(1) An integer $L$ and a finite set of points, $\Sigma=\{x_{1},\ldots,x_{L}\}\subset X$, $\Sigma$ maybe empty,

(2) A smooth anti-self-dual $A_{\infty}$ on a principal $G$-bundle $P_{\infty}$ over $X$ with $\eta(P_{\infty})=\eta(P)$,

(3) A subsequence $\Xi\subset\mathbb{N}$, we also denote by $\{A_{i}\}_{i\in\Xi}$, a sequence gauge transformation $\{g_{i}\}_{i\in\Xi}$ such that, $g^{\ast}_{i}(A_{i})$ weakly converges to $A_{\infty}$ in $L^{2}_{1}$ on $X-\Sigma$, and $g_{i}^{\ast}(F_{A_{i}})$ weakly converges to $F_{A_{\infty}}$ in $L^{2}$ on $X-\Sigma$.*

Proof of Corollary 3.13..

We can apply Theorem 3.14 to the sequence $\{A_{i}\}_{i\in\mathbb{N}}$. This yields a subsequence $\Xi\subset\mathbb{N}$, a sequence of gauge transformations $\{g_{i}\}_{i\in\Xi}$ and a anti-self-dual connection $A_{\infty}$ on a principal $G$-bundle $P_{\infty}$ which is the weakly $L^{2}_{1}$ limit of $\{g_{i}^{\ast}(A_{i})\}_{i\in\Xi}$ over $X-\Sigma$, $\Sigma$ is the set of some points on $X$. If the sequence $r_{i}$ don’t has a bounded subsequence. Apply Theorem 3.8 to the subsequence $\{(A_{i},\phi_{i})\}_{i\in\Xi}$. Recall from Theorem 3.8 that $A_{\Delta}$ is the limit over compact subset of $X-(Z\cup\Theta)$ of gauge transformations of $\{A_{i}\}_{i\in\Xi}$. In particular, both $A_{\infty}$ and $A_{\Delta}$ are weakly $L^{2}_{1}$ limits over $X-(Z\cup\Theta\cup\Sigma)$ of gauge-equivalent connections. Since weakly $L^{2}_{1}$ limits preserve $L^{2}_{2}$ gauge equivalence, it follows that there exists a Sobolev-class $L^{2}_{2}$ gauge transformation $g_{\Delta}$ such that $g_{\Delta}^{\ast}(A_{\Delta})=A_{\infty}$. Thus $A_{\Delta}$ is anti-self-dual on the complement of $Z\cup\Theta\cup\Sigma$. ∎

The result due to Tanaka [19, Theorem 1.3] for the sequence $\{A_{i}\}_{i\in\mathbb{N}}$ obeys $\|F_{A_{i}}^{+}\|_{L^{2}(X)}$ converges to zero in $L^{2}$-topology will invalid, since $S({A_{i}})$ may be not empty. But we could prove a compactness theorem for a sequence solutions of Kapustin$\textendash$Witten equations by using the Lemma 3.3.

Proposition 3.15**.**

*Let $X$ be a closed, oriented, simply-connected, four-dimensional manifold with a smooth Riemannain metric $g$, $P\rightarrow X$ be a principal $SU(2)$ or $SO(3)$-bundle with $p_{1}(P)$ negative. Suppose that the connections on $\bar{M}_{ASD}(P,g)$ are irreducible. If $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ is a sequence smooth solutions of Kapustin$\textendash$Witten equations and the curvatures $F^{+}_{A_{i}}$ converge to zero in $L^{2}$-topology, then

(1) An integer $L$ and a finite set of points, $\Sigma=\{x_{1},\ldots,x_{L}\}\subset X$,;

(2) A subsequence $\Xi\subset\mathbb{N}$, we also denote by $\{A_{i}\}$, a sequence gauge transformatin $\{g_{i}\}_{i\in\Xi}$ such that, $(g^{\ast}_{i}(A_{i}),g_{i}^{\ast}(\phi_{i}))$ converges to $(A_{\infty},0)$ in $C^{\infty}$ on $X-\Sigma$, where $A_{\infty}$ is an anti-self-dual connection on a principal $P_{\infty}$.*

Proof.

At first, we claim that the sequence $r_{i}:=\|\phi_{i}\|_{L^{2}(X)}$ has a bounded subsequence. If not, the sequence $r_{i}$ don’t has a bounded subsequence. Following Theorem 3.8, we have $Z,\Theta$ and $\sigma_{\Delta}$, $v$ which described in Theorem 3.8. We define $\sigma_{\infty}:=g_{\Delta}^{\ast}(\sigma_{\Delta})$ over $X-(Z\cup\Theta\cup\Sigma)$, where $g_{\Delta}$, $\Sigma$ are as described in the proof of Corollary 3.13. Following Theorem 3.8, we then have

[TABLE]

Thus we have a section $s:=v\otimes\sigma_{\infty}$ on $P_{\infty}|_{X-(Z\cup\Theta\cup\Sigma)}$ and one can see $v\otimes\sigma_{\infty}$ is non-zero all over $X-(Z\cup\Theta\cup\Sigma)$. We can rewrite $s$ to $s=\tilde{\sigma}\otimes\tilde{v}$, where $\tilde{\sigma}\in\Gamma(X-(Z\cup\Theta\cup\Sigma),\mathfrak{g}_{P_{\infty}})$ and $\tilde{v}\in\Omega^{1}(X-(Z\cup\Theta\cup\Sigma))$. We also setting $\langle\tilde{\sigma},\tilde{\sigma}\rangle=1$, thus $\langle d_{A_{\infty}}\tilde{\sigma},\tilde{\sigma}\rangle=0$ along $X-(Z\cup\Theta\cup\Sigma)$. In a direct calculate, we have

[TABLE]

thus $d\tilde{v}=0$. It follows that $d_{A_{\infty}}\tilde{\sigma}\wedge v=0$. Since $\tilde{v}$ is nowhere zero along $X-(Z\cup\Theta\cup\Sigma)$, we must have $d_{A_{\infty}}\tilde{\sigma}=0$. According to Lemma 3.3, $A$ is irreducible along a open set of $X-(Z\cup\Theta\cup\Sigma)$, then $\tilde{\sigma}=0$. This contradict $s$ is non-zero on $X-(Z\cup\Theta\cup\Sigma)$. Hence we prove that the sequence $\{r_{i}\}_{i=1,2,\ldots}$ must has a bounded subsequence. Then following the compactness theorem 3.7, there exist a pair $(A_{\Delta},\phi_{\Delta})\in\mathcal{A}_{P_{\Delta}}\times\Omega^{1}(X,\mathfrak{g}_{P_{\Delta}})$ that obeys the Kapustin$\textendash$Witten equations and there has a subsequence $\Xi^{\prime}\subset\Xi$ and a sequence $\{g_{i}\}_{i\in\Xi^{\prime}}$ of automorphisms of $P_{\Delta}$ such that $\{(g_{i}^{\ast}(A_{i}),g^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi^{\prime}}$ converges to $(A_{\Delta},\phi_{\Delta})$ in the $C^{\infty}$ topology on $X-\{x_{1},\cdots,x_{L}\}$. Under the assumption of $\{A_{i}\}_{i\in\mathbb{N}}$, the connection $A_{\Delta}$ is an irreducible anti-self-dual connection, then the vanish theorem ensures $\phi_{\Delta}=0$. ∎

Proof of Theorem 1.3..

Now we begin to proof Theorem 1.3. Suppose that the constant $\delta$ does not exist. We may choose a sequence of solutions $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ of Kapustin$\textendash$Witten equations such that $\|F^{+}_{A_{i}}\|_{L^{2}(X)}\rightarrow 0$. Following Proposition 3.15, there exist a subsequence $\Xi\in\mathbb{N}$ and a sequence transformation $\{g_{i}\}_{i\in\Xi}$ such that $(g^{\ast}_{i}(A_{i}),g^{\ast}_{i}(\phi_{i}))\rightarrow(A_{\infty},0)$ in $C^{\infty}$ over $X-\Sigma$, $\Sigma:=\{x_{1},\cdots,x_{L}\}$. There also exist a positive constant $C$, such that $\|\phi_{i}\|_{L^{2}(X)}\leq C$. We then have

[TABLE]

where $c=c(g)$ is a positive constant. Therefore

[TABLE]

This contradict $\|\phi_{i}\|_{L^{2}(X)}$ has a uniform positive lower bound, see Theorem 3.12. The preceding argument shows that the desired constant $\delta$ exists.

If we denote by $A_{\infty}$ an ASD connection on $P$, then the curvature $F_{A}$ of a connection $A:=A_{\infty}+a$ has an estimate

[TABLE]

where $C=C(g)$ is a positive constant. If $\|a\|_{L^{2}_{1}(X)}\leq 1$, then

[TABLE]

We also suppose that $A$ is not anti-self-dual with respect to $g$, then

[TABLE]

where $\delta$ is the positive constant as above. So we can set $\tilde{\delta}:=\min\{1,\frac{\delta}{2C}\}$, hence

[TABLE]

unless $A$ is ASD with respect to metric $g$. ∎

3.5 Some examples

In this section, we give some conditions on the topology of manifold, the metric of manifold and the topology of principle bundle to ensure that the connections on $\bar{M}_{ASD}$ are all $generic$. For a compact four-manifold $X$, the compacitification $\bar{M}_{ASD}(P,g)$ of $M_{ASD}(P,g)$ contained in the disjoint union

[TABLE]

We denote by $\eta(P)$ the element in $H^{2}(X,\mathbb{R})$ which defined as [17, Definition 2.1]. Following [17, Theorem 5.5] , every principal $G$-bundle, $M(P_{l},g)$ over $X$ appearing in (3.3) has the property that $\eta(P_{l})=\eta(P)$.

Proposition 3.16**.**

Let $X$ be a closed, oriented, simply-connected, four-dimensional manifold with a $generic$ Riemannain metric $g$, $P\rightarrow X$ be a principal $SU(2)$ or $SO(3)$-bundle with $p_{1}(P)$ negative. If $b^{+}(X)>0$, then the connection $[A]\in M_{ASD}$ is irreducible.

Proof.

If $G=SU(2)$ or $SO(3)$, $X$ is a simply-connected four-manifold and $b^{+}(X)>0$, then following [2, Corollary 4.3.15] , the only reducible anti-self-dual connection on a principal $G$-bundle over $X$, is the product connection on the product bundle $P=X\times G$ if only if the anti-self-dual connection is flat connection, i.e., $p_{1}(P)=0$. Hence under our assumption on $X,P,G$, then the anti-self-dual connection must be irreducible. ∎

In Proposition 3.16 we mean by $generic$ $metric$ the metrics in a second category subset of the space of $C^{k}$ metrics for some fixed $k>2$ ([2, Section 4] and [4, Corollary 2]). It may reassure the reader to know that for all practical purposes one can work with an open dense subset of the smooth metrics, or even real analytic metrics.

Proposition 3.17**.**

Let $X$ be a closed, oriented, simply-connected, four-dimensional manifold with a $generic$ $Riemannain$ $metric$ $g$, $P\rightarrow X$ be a principal $SO(3)$-bundle with $p_{1}(P)$ negative. If $b^{+}(X)>0$ and the second Stiefel-Whitney class $w_{2}(P)\neq 0$, then the connections $[A]\in\bar{M}_{ASD}$ are all $generic$.

Proof.

Under the hypothesis on this proposition, we know that the connections $[A]\in\bar{M}_{ASD}$ are all $regular$, i.e., $\mu(A)>0$ where $\mu(A)$ is as in Definition 2.2, see [4, Corollary 3.9] . If $G=SO(3)$, following [17, Theorem 2.4] , we have $\eta(P)=w_{2}(P)$. We then obtain that if $w_{2}(P)$ is non-trivial, then every principal $G$-bundle, $M(P_{l},g)$, over $X$ appearing in (3.3) has the property that $w_{2}(P_{l})$ is non-trivial. We would claim that the ASD connections on $M(P_{l},g)$ are irreducible. Since an reducible ASD connection on $P_{l}$ ensures that the bundle $P_{l}$ is trivial bundle. This contradict $w_{2}(P_{l})\neq 0$. By the similar method in Proposition 4.5, for any $[A]\in\bar{M}_{ASD}(P,g)$, we have $\lambda(A)>0$, where $\lambda(A)$ is as in Definition (4.2), i.e., $[A]$ is irreducible. We complete the proof of this proposition. ∎

Following Theorem 1.3, we then have

Corollary 3.18**.**

*Assume the hypothesis on Proposition 3.17. Suppose that $b^{+}(X)>0$ and the second Stiefel-Whitney class $w_{2}(P)\neq 0$. There is a positive constant $\delta=\delta(P,g)$ with following significance. If the pair $(A,\phi)$ is a smooth solution of Kapustin$\textendash$Witten equations, then one of following must hold:

(1) $F^{+}_{A}=0$ and $\phi=0$;

(2) the pair $(A,\phi)$ satisfies*

[TABLE]

In particular, there is a positive constant $\tilde{\delta}=\tilde{\delta}(g,P)$ such that

[TABLE]

unless $A$ is anti-self-dual with respect to $g$.

4 Kähler surfaces

In this section, we now denote by $X$ a compact Kähler surface with a Kähler form $\omega$ and $E$ a principal $G$-bundle over $P$ with structure group $G$. We also set $d_{A}=\partial_{A}+\bar{\partial}_{A}$, $d^{\ast}_{A}=\partial^{\ast}_{A}+\bar{\partial}^{\ast}_{A}$ and $\phi=\sqrt{-1}(\theta-\theta^{\ast})$, where $\theta\in\Omega^{1,0}(X,adE)$. Thus, Tanaka observed that Kapustin$\textendash$Witten equations on a closed Kähler surface are the same as Hitchin$\textendash$Simpson’s equations, see [21, Proposition 3.1] .

Proposition 4.1**.**

Let $X$ be a closed Kähler surface, the Kapustin$\textendash$Witten equations have the following form that asks $(A,\theta)\in\mathcal{A}_{E}\times\Omega^{1,0}(X,adE)$ to satisfies

[TABLE]

Hence the bundle $E$ on $X$ is holomorphic and $\theta$ is a holomorphic section of $End(E)\otimes\Omega^{1,0}(X)$, i.e., the bundle $(E,\theta)$ is a Higgs bundle. Following the Kobayashi$\textendash$Hitchin corresponding for Higgs bundle, see [12, 18], we know that the Higgs bundle $(E,\theta)$ is stable since the pair $(A,\phi)$ satisfies the Hitchin$\textendash$Simpson equations.

4.1 Irreducible connections

In this section, we first recall a definition of irreducible connection on a principal $G$-bundle. Given a connection $A$ on a principal $G$-bundle $E$ over $X$. We can define the stabilizer $\Gamma_{A}$ of $A$ in the gauge group $\mathcal{G}_{E}$ by

[TABLE]

one also can see [2, 6]. A connection $A$ called reducible if the connection $A$ whose stabilizer $\Gamma_{A}$ is larger than the centre $C(G)$ of $G$. Otherwise, the connections are irreducible, they satisfy $\Gamma_{A}\cong C(G)$. For the cases $G=SU(2)$ or $SO(3)$, it’s easy to see that a connection $A$ is irreducible when it admits no nontrivial covariantly constant Lie algebra-value [math]-form, i.e.,

[TABLE]

We can defined the least eigenvalue $\lambda(A)$ of $d^{\ast}_{A}d_{A}$ as follow.

Definition 4.2**.**

For $A\in\mathcal{A}_{E}$, define

[TABLE]

is the lowest eigenvalue of $d^{\ast}_{A}d_{A}$.

By the similar method of the proof of [4, Proposition A.3] or [3, Proposition 35.14] , we would also show that the least eigenvalue $\lambda(A)$ of $d^{\ast}_{A}d_{A}$ with respect to connection $A$ is $L^{p}_{loc}$-continuity $2\leq p<4$.

Proposition 4.3**.**

Let $X$ be a closed, connected, oriented, smooth four-manifold with Riemannian metric, $g$. Let $\Sigma=\{x_{1},x_{2},\ldots,x_{L}\}\subset X$ ($L\in\mathbb{N}^{+}$) and $\rho=\min_{i\neq j}dist_{g}(x_{i},x_{j})$, let $U\subset X$ be the open subset give by

[TABLE]

Let $G$ be a compact Lie group, $A_{0}$, $A$ are $C^{\infty}$ connections on the principal $G$-bundles $E_{0}$ and $E$ over $X$ and $p\in[2,4)$. There is an isomorphism of principal $G$-bundles, $u:E\upharpoonright X\backslash\Sigma\cong E_{0}\upharpoonright X\backslash\Sigma$, and identify $E\upharpoonright X\backslash\Sigma$ with $E_{0}\upharpoonright X\backslash\Sigma$ using this isomorphism. Then $\lambda(A)$ satisfies upper bound

[TABLE]

and the lower bound,

[TABLE]

where $c$ is a positive constant depends on $g,p$.

We now have the useful

Corollary 4.4**.**

Let $G$ be a compact Lie group and P a principal $G$-bundle over a closed, smooth, oriented, four-dimensional Riemannian manifold $X$ with a Riemannian metric $g$. If $\{A_{i}\}_{i\in\mathbb{N}}$ is a sequence $C^{\infty}$ connection on $P$ and the sequence $\{F^{+}_{A_{i}}\}_{i\in\mathbb{N}}$ converges to zero in $L^{2}$-topology, then

[TABLE]

where $\lambda(A)$ is as in Definition 4.2.

Proof.

The proof is similar to the least eigenvalue of operator $d_{A}^{+}d_{A}^{+,\ast}|_{\Omega^{2,+}(X,adE)}$ with respect to connection $A$ (see [3, Corollary 35.16] ). ∎

For a compact Kähler surface $X$ we have a moduli space of ASD connections $M(E,g)$. The compacitification $\bar{M}(E,g)$ of $M(E,g)$ contained in the disjoint union

[TABLE]

Following [2, Theorem 4.4.3] , the space $\bar{M}(E,g)$ is compact.

Proposition 4.5**.**

Let $X$ be a compact, Kähler surface with smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle with $c_{2}(E)=c$ positive. Suppose that $g$ is a $c$-$general$ Kähler metric, then there are positive constants $\varepsilon=\varepsilon(E,g)$ and $\lambda_{0}=\lambda_{0}(E,g)$ with following significance. If $A$ is a connection on $E$ such that

[TABLE]

and $\lambda(A)$ is as in (4.1), then

[TABLE]

Proof.

If $g$ is $c$-$generic$ metric, $\lambda(A)>0$ for $[A]\in M(E_{l},g)$ and $E_{l}$ is a principal $SU(2)$-bundle over $X$ appearing in the Uhlenbeck compactification. Since the function $\lambda(A)$, $A\in\bar{M}(E,g)$ is continuous by Corollary 4.4 and $\bar{M}(E,g)$ is compact, there is a uniform positive constant $\lambda$ such that $\lambda(A)\geq\lambda$ for $[A]\in\bar{M}(E,g)$. Suppose that the constant $\varepsilon$ does not exist. We may then choose a minimizing sequence $\{A_{i}\}_{i\in\mathbb{N}}$ of connections on $E$ such that $\|F_{A_{i}}^{+}\|_{L^{2}(X)}\rightarrow 0$ and $\lambda(A_{i})\rightarrow 0$ as $i\rightarrow\infty$. According to Corollary 4.4, $A_{i}$ converges to an ASD connection, $A_{\infty}$, on $L^{2}_{1,loc}(X)$. Then $\lim_{i\rightarrow\infty}\lambda(A_{i})=\lambda(A_{\infty})>0$, contradicting our initial assumption regarding the sequence $\{A_{i}\}_{i\in\ N}$. ∎

Fix an algebraic surface $S$ and an ample line bundle $L$ on $S$. For every integer $c$ we have defined the moduli space $\mathcal{M}_{c}(S,L)$ of $L$-stable rank two holomorphic vector bundles $V$ on $S$ and such that $c_{1}(V)=0$, $c_{2}(V)=c$. Taubes [23] has shown that, if $X$ is an arbitrary closed $4$-manifold and $g$ is any Riemannian metric on $X$, then the moduli space of all irreducible $g$-ASD connections on $P$ with $c_{2}(P)=c$ moduli gauge equivalence which denote by $\mathcal{M}(P_{c},g)$ is nonempty if $c$ is sufficiently large. In the case of an algebraic surface $S$ and a Hodge metric corresponding to the ample line bundle $L$, similar existence results are due to Maruyama and Gieseker [10]. Friedman-Morgan ([7, Chapter IV, Theorem 4.7]) given a very general result along these lines is the following:

Theorem 4.6**.**

With $S$ and $L$ as above, for all $c\geq 2p_{g}(S)+2$, the modulis space $\mathcal{M}_{c}(S,L)$ is nonempty.

Next let us determine when a Hodge metric with Kähler form $\omega$ admits reducible ASD connections. Corresponding to such a connection is an associated ASD harmonic $1$-form $\alpha$, well-defined up to $\pm 1$, representing an integral cohomology class, which by the description of $\Omega^{2,-}(X)$ is of type $(1,1)$ and orthogonal to $\omega$. Thus Friedman-Morgan proved for an integer $c>0$, there exist Hodge metric $g$ over $S$ is $c$-$generic$ in the sense of Definition 1.4 (see [7, Chapter IV, Proposition 4.8]). For the convenience of readers, we give a detailed proof.

Proposition 4.7**.**

Fix $c>0$. Then there is an open dense subset $\mathcal{D}$ of the cone of ample divisors on $S$ such that if $g$ is a Hodge metric whose Kähler form lies in $\mathcal{D}$, $g$ is a $c$-$generic$ metric in the sense of Definition 1.4.

Proof.

By standard argument, the set of $\omega$ in the ample cone which are orthogonal to an integral class $\alpha$ with $0<-\alpha^{2}\leq c$ is the intersection of the ample cone with a collection of hyperplanes in $H^{2}(S;\mathbb{R})$ which is locally finite on the ample cone. This result is immediate from this. ∎

4.2 Approximate ASD connections

In this section, we will give a general criteria under which an approximate ASD connection $A\in\mathcal{A}_{E}^{1,1}$ can be deformation into an other approximate ASD connection $A_{\infty}$ which obeys the equation

[TABLE]

Let $A$ be a connection on a principal $G$-bundle over $X$. The equation (4.3) for a second connection $A_{\infty}:=A+a$, where $a\in\Omega^{1}(X,adE)$ is a bundle valued $1$-form, can be written as:

[TABLE]

We seek a solution of the equation (4.4) in the form

[TABLE]

where $s\in\Omega^{0}(X,adE)$ is a section on $ad(E)$. Then equation (4.4) becomes to a second order equation:

[TABLE]

For convenience, we define a map

[TABLE]

It’s easy to check, we have the pointwise bound:

[TABLE]

where $C$ is a uniform positive constant. We want to prove that if $\Lambda_{\omega}F_{A}$ is small in an appropriate sense, then there is a small solution $s$ to equation (4.5).

Theorem 4.8**.**

Let $X$ be a compact Kähler surface with a Kähler metric $g$, $E$ a principal $SU(2)$-bundle over $X$. There are positive constants $\varepsilon=\varepsilon(E,g)\in(0,1)$ and $\lambda=\lambda(E,g)\in(0,\infty)$ with following significance. If $A\in\mathcal{A}_{E}^{1,1}$ obeys

[TABLE]

*then there is a section $s\in\Omega^{0}(X,adE)$ such that the connection $A_{\infty}:=A+\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$ satisfies

(1) $\Lambda_{\omega}F_{A_{\infty}}=0$

(2) $\|s\|_{L^{2}_{2}(X)}\leq C\|\Lambda_{\omega}F_{A}\|_{L^{2}(X)}$,

(3) $\|F^{0,2}_{A_{\infty}}\|_{L^{2}(X)}\leq C\|\Lambda_{\omega}F_{A}\|^{2}_{L^{2}(X)}$.

where $C=C(\lambda,g)\in[1,\infty)$ is a positive constant.*

Now, we begin to prove Theorem 4.8. The method of proof Theorem 4.8 is base on Taubes’ ideas [22]. At first, suppose that $s$ and $f$ are sections of $adE$ with

[TABLE]

The first observation is

Lemma 4.9**.**

If $\lambda(A)\geq\lambda>0$, then there exists a unique $C^{\infty}$ solution to equation (4.6). Furthermore, we have

[TABLE]

where $c=c(\lambda,g)$ is a positive constant.

Proof.

Since $\nabla_{A}^{\ast}\nabla_{A}$ is an elliptic operator of order $2$, then for each $k\geq 0$, there is a positive constant $C_{k}$ so that for all section $v$ of $adE$, see [2] (A8),

[TABLE]

We take $v=s$ and $k=0$, then

[TABLE]

where $c=c(\lambda,g)$ is a positive constant. By the Sobolev inequality in four dimension,

[TABLE]

where $C$ is a positive constant. Hence we complete the proof of this lemma. ∎

Lemma 4.10**.**

If $d_{A}^{\ast}d_{A}s_{1}=f_{1}$, $d_{A}^{\ast}d_{A}s_{2}=f_{2}$, then

[TABLE]

We can prove the existence of a solution of (4.5) by the contraction mapping principle. We write $s=(d_{A}^{\ast}d_{A})^{-1}f$ and (4.5) becomes an equation for $f$ of the from

[TABLE]

where $S(f,g):=B((d_{A}^{\ast}d_{A})^{-1}f,(d_{A}^{\ast}d_{A})^{-1}g)$. By Lemma 4.10,

[TABLE]

We denote $g_{k}=f_{k}-f_{k-1}$ and $g_{1}=f_{1}$, then

[TABLE]

and

[TABLE]

It is easy to show that, under the assumption of $\Lambda_{\omega}F_{A}$, the sequence $f_{k}$ defined by

[TABLE]

starting with $f_{1}=\Lambda_{\omega}F_{A}$, is Cauchy with respect to $L^{2}$, and so converges to a limit $f$ in the completion of $\Gamma(adE)$ under $L^{2}$.

Proposition 4.11**.**

There are positive constant $\varepsilon\in(0,1)$ and $C\in(1,\infty)$ with following significance. If the connection $A$ satisfies

[TABLE]

then each $g_{k}$ exists and is $C^{\infty}$. Further for each $k\geq 1$, we have

[TABLE]

Proof.

The proof is by induction on the integer $k$. The induction begins with $k=1$, one can see $g_{1}=\Lambda_{\omega}F_{A}$. The induction proof if completed by demonstrating that if (4.8) is satisfied for $j<k$, then it also satisfied for $j=k$. Indeed, since

[TABLE]

Now, we provide the constants $\varepsilon$ sufficiently small and $C$ sufficiently large to ensures that

[TABLE]

i.e., $\frac{2c}{1-C\|\Lambda_{w}F_{A}\|_{L^{2}(X)}}\leq C$, hence we complete the proof of this proposition. ∎

Proof of Theorem 4.8..

The sequence $g_{k}$ is Cauchy in $L^{2}$, the limit

[TABLE]

is a solution to (4.7). Using Lemma 4.9 and Proposition 4.11, we have

[TABLE]

we provide $\varepsilon$ and $C$ to ensure $C\varepsilon\leq\frac{1}{2}$, hence

[TABLE]

We denote $A_{\infty}:=A+\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$, then

[TABLE]

where $c$ is a positive constant. We complete the proof of Theorem 4.8. ∎

4.3 A topological property for the moduli space of stable Higgs bundles

In this section, we use the inequality in Theorem 4.8 to prove that the Higgs filed has a positive lower bounded if the Kähler metric $g$ is $c$-$generic$. Following Theorem 3.4, we have a vanishing result of Higgs fields as follows.

Corollary 4.12**.**

Let $X$ be a closed Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle with $c_{2}(E)=c$ positive. Suppose that $g$ is a $c-generic$ metric in the sense of Definition 1.4. If the Higgs pair $(A,\theta)$ is a smooth solution of the equations: $\Lambda_{\omega}F_{A}=0,\ [\theta\wedge\theta^{\ast}]=0$, then the Higgs field $\theta$ vanishes.

Proof Theorem 1.5..

Let $(A,\theta)\in\mathcal{A}^{1,1}_{E}\times\Omega^{1,0}(X,\mathfrak{g}_{E})$ be a smooth solution of Hitchin$\textendash$Simpson equations. We denote $\phi=\sqrt{-1}(\theta-\theta^{\ast})\in\Omega^{1}(X,adE)$. Suppose that the constant $C$ does not exist. We may provide a positive constant $\varepsilon_{0}$ such that

[TABLE]

where $c=c(g)$ is a positive constant and $\varepsilon$ is a constant as Proposition 4.5. Thus $\lambda(A)>0$. Following Theorem 4.8, there exist a connection $A_{\infty}$ which satisfies $\Lambda_{w}F_{A_{\infty}}=0$ and

[TABLE]

for some positive constant $c=c(g,P)$. We have two integrable inequalities:

[TABLE]

[TABLE]

To obtain the second identity, we use the Weitzenböck formula [6] Equation (6.25):

[TABLE]

We also observe that $\langle F^{0,2}_{A_{\infty}}+F^{2,0}_{A_{\infty}},[\phi,\phi]\rangle_{L^{2}(X)}=0$ since $[\phi,\phi]\in\Omega^{1,1}(X,adE)$.

Combining the preceding inequalities gives

[TABLE]

for a positive constant $c=c(g)$. If we provide $C=1/(2c)$ such that $\|\phi\|^{2}_{L^{2}(X)}\leq C$, then $\Lambda_{\omega}F_{A}\equiv 0$. Its contradiction to our initial assumption regarding the $\Lambda_{\omega}F_{A}$. The preceding argument shows that the desired constant $C$ exists.

We also suppose $X$ is simply-connected, if the constant $\tilde{C}$ does not exist. We may choose a sequence $\{(A_{i},\theta_{i})\}_{i\in\mathbb{N}}$ of Hitchin$\textendash$Simpson equations such that $\|\Lambda_{\omega}F_{A_{i}}\|_{L^{2}(X)}\rightarrow 0$. We set $\phi_{i}:=\sqrt{-1}(\theta_{i}-\theta^{\ast}_{i})$.Following the idea of Theorem 1.3, the Proposition 3.15 implies that there exist a subsequence $\Xi\in\mathbb{N}$ and a sequence transformation $\{g_{i}\}_{i\in\Xi}$ such that $(g^{\ast}(A_{i}),g^{\ast}(\phi_{i}))\rightarrow(A_{\infty},0)$ in $C^{\infty}$ over $X-\Sigma$, $\Sigma:=\{x_{1},\cdots,x_{L}\}$. We also have $\lim_{i\rightarrow\infty}\int_{X}|\phi_{i}|^{2}=0$. This contradict $\|\phi_{i}\|_{L^{2}(X)}$ has a uniform positive lower bounded. The preceding argument shows that the desired constant $\tilde{C}$ exists. ∎

Corollary 4.13**.**

Let $X$ be a compact, simply-connected, Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle over $X$. Suppose that $(E,\theta)$ is a stable Higgs bundle with $c_{2}(E)=c$, then there is a positive integer $c$ with following significance. If $g$ is a $c$-$generic$ metric in the sense of Definition (1.4), then the moduli space of stable Higgs bundle is non-connected.

Proof.

Taubes [23] has shown that, if $X$ is an arbitrary closed $4$-manifold and $g$ is any Riemannian metric on $X$, $P$ be a $SU(2)$-bundle with $c_{2}(E)=c$ sufficiently large, then there exist an irreducible ASD connection on $P$. The moduli space $M_{ASD}(E,g)$ is non-empty under our conditions. Since the map $(A,\theta)\mapsto\|\theta\|_{L^{2}}$ is continuous, then the moduli space $M_{HS}$ is not connected. ∎

Following Proposition 4.7 and Corollary 4.13, we have

Corollary 4.14**.**

Let $S$ be a compact, simply-connected, algebraic surface with a Hodge metric $g$, $(E,\theta)$ be a Higgs bundle over $X$. There is an open dense subset $\mathcal{D}$ of the cone of ample divisor on $S$ and a positive integer $c$ with following significance. If the Kähler form of $g$ lies in $\mathcal{D}$ and $(E,\theta)$ is stable with $c_{2}(E)=c$, then the moduli space of stable Higgs bundle is non-connected.

4.4 Vafa-Witten equaions

We next consider analogue of the Hitchin$\textendash$Simpson equations, called Vafa$\textendash$Witten equations [27]. Let $X$ be a closed, oriented, smooth, four-manifold, $E$ be a principal $G$-bundle with $G$ being a compact Lie group over $X$. We denote by $A$ a connection on $E$, $B$ a section of the associated bundle $\Omega^{2,+}\otimes adE$ and $C$ a section of $adE$. A triple $(A,B,C)$ is a solution of Vafa$\textendash$Witten equations if $(A,B,C)$ satisfies

[TABLE]

where $[B.B]\in\Gamma(X,\Omega^{2,+}\otimes adE)$ is defined through the Lie brackets of $adE$ and $\Omega^{2,+}$, see [16, Appendix A] . We take $X$ to be a compact Kähler surface with Kähler form $\omega$. In a local orthonomal coordinates, we can write $B=\beta-\beta^{\ast}+B_{0}\omega$, where $\beta\in\Omega^{2,0}(X,adE)$, $B_{0}\in\Omega^{0}(X,adE)$ and $\gamma=C-\sqrt{-1}B_{0}$, one can see [16, Chapter 7] for details. We have

Theorem 4.15**.**

([16, Theorem 7.1.2] ) On a compact Kähler surface, the Vafa$\textendash$Witten equations (4.9) have the following form that asks $(A,\beta,\gamma)\in\mathcal{A}_{E}^{1,1}\times\Omega^{2,0}(X,adE)\times\Omega^{0}(X,adE)$ to satisfies

[TABLE]

Mares also discussed a relation between the existence of a solution to the equations and a stability of vector bundles, see [16, 19].

Definition 4.16**.**

For any $\beta\in\Omega^{2,0}(X,EndE)\cong\Omega^{0}(Hom(E,E\otimes K))$, we say that a subbundle $E^{\prime}\subset E$ is $\beta$-invariant, if

[TABLE]

A holomorpic bundle $E$ is $\beta$-stable (semi-stable) if all $\beta$-invariant holomorphic subbundle $E^{\prime}\subset E$ satisfies

[TABLE]

We also recall a bound on $\|\beta\|_{L^{\infty}}$ in terms of $\|\beta\|_{L^{2}}$. One can see [16] for detail.

Theorem 4.17**.**

Let $X$ be a compact, Kähler surface with a smooth Kähler metric $g$, $E$ a principal $G$-bundle over $X$ with $G$ being a compact Lie group. If the pair $(A,\beta,\gamma)$ is a smooth solution of Equations (4.10), there is a positive constant $C=C(g)$ such that

[TABLE]

We extent the idea of Hitchin$\textendash$Simpson equations to the case of the Vafa$\textendash$Witten equations on the compact Kähler surface with a $c$-$generic$ Kähler metric.

Theorem 4.18**.**

*Let $X$ be a compact, Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle with $c_{2}(E)=c$ positive. Suppose that $g$ is a $c$-$generic$ metric in the sense of Definition (1.4), then there is a positive constant $C=C(g,P)$ with following significance. If the triple $(A,\beta,\gamma)\in\mathcal{A}_{E}^{1,1}\times\Omega^{2,0}(X,adE)\times\Omega^{0}(X,adE)$ is satisfies equations (4.10), then one of following must hold:

(1) $\Lambda_{\omega}F_{A}=0$, or

(2) the extra field $\beta$ satisfies*

[TABLE]

Proof.

For $(A,\beta,\gamma)$ is a solution of equations (4.10), we denote $\beta_{0}:=\beta-\beta^{\ast}\in\Omega^{2,+}(X,adE)$, hence $d_{A}^{+,\ast}\beta_{0}=\ast(\bar{\partial}_{A}\beta-\partial_{A}\bar{\beta})=0$. Suppose that the constant $C$ does not exist. We may provide a positive constant $\varepsilon_{0}$ such that

[TABLE]

where $c=c(g)$ is a positive constant and $\varepsilon$ is a constant as Proposition 4.5. Thus $\lambda(A)>0$. Following Theorem 4.8, there exist a connection $A_{\infty}$ such that

[TABLE]

for a positive constant $c=c(g,P)$. By the Weizenböck formula (6.26) in [6], we have two integrable identities:

[TABLE]

[TABLE]

We also observe that

[TABLE]

for a positive constant $c=c(g)$. Combing the preceding inequalities gives

[TABLE]

for a positive constant $c=c(g)$. We provide $\|\beta_{0}\|^{2}_{L^{2}(X)}\leq 1/(2c)$, thus $\Lambda_{\omega}F_{A}=0$. This contradicting our initial assumption regarding the $\Lambda_{\omega}F_{A}$. The preceding argument shows that the desired constant $C$ exists. ∎

We denote by

[TABLE]

the moduli space of solutions of Vafa$\textendash$Witten equations. We apply a vanishing theorem of extra fields of Vafa$\textendash$Witten equations due to Mare [16, Theorem 4.2.] 1 to prove that

Theorem 4.19**.**

Let $X$ be a simply-connected, Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$ or $SO(3)$-bundle. Let the triple $(A,\beta,\gamma)$ be a solution of the $decoupled$ Vafa$\textendash$Witten equations:

[TABLE]

If $A$ is an irreducible connection, then $\beta=\gamma=0$.

Following the idea of Hitchin$\textendash$Simpson equations, we also have

Corollary 4.20**.**

Let $X$ be a compact, simply-connected, Kähler surface with a smooth Kähler metric $g$, $E$ be a principal $SU(2)$-bundle. Suppose that $(E,\beta)$ is $\beta$-stable bundle with $c_{2}(E)=c$, then there is a positive integer $c$ with following significance. If $g$ is a $c$-$generic$ metric in the sense of Definition (1.4), then the moduli space of $\beta$-stable bundle is non-connected.

Acknowledgment

I would like to thank the anonymous referee for careful reading of my manuscript and helpful comments. I would like to thank Professor Clifford Taubes for helpful comments regarding his article [24]. This work is supported by Nature Science Foundation of China No. 11801539 and Postdoctoral Science Foundation of China No. 2017M621998, No. 2018T110616.