A compactness theorem for stable flat $SL(2,\mathbb{C})$ connections on $3$-folds

TL;DR

This paper establishes a compactness theorem for stable flat $SL(2,b{C})$ connections on 3-manifolds with non-degenerate flat $SU(2)$-connections, and shows the moduli space can be disconnected.

Contribution

It proves a Uhlenbeck-type compactness theorem for stable flat $SL(2,b{C})$ connections and demonstrates the moduli space's disconnectedness under certain conditions.

Findings

Proved a compactness theorem for stable flat $SL(2,b{C})$ connections.

Showed the moduli space of these connections can be disconnected.

Extended previous results to include $L^2$-bounded curvature connections.

Abstract

Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are $non$-$degenerate$. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a previous result in \cite{Huang}, we prove that the moduli space of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain technical assumptions.

Equations

$${\mathcal{F}}_{\mathcal{A}}=F_A-\frac{1}{2}[\varphi \wedge \varphi ]+\mathrm{i}{d}_A\varphi$$

$$F_A-\varphi \wedge \varphi =0,d_A\varphi =d_A^{\ast }\varphi =0.$$

$$\mathcal{M}(P,g):=\{(A,\varphi )\in {\mathcal{A}}_P\times {\Omega }^1({\mathfrak{g}}_P)\mid {\mathcal{F}}_{\mathcal{A}}=0,d_A^{\ast }\varphi =0\}/{\mathcal{G}}_P,$$

$$M(P,g):=\{\Gamma \in {\mathcal{A}}_P:F_{\Gamma }=0\}/{\mathcal{G}}_P$$

$${\int }_X|{\nabla }_A\varphi {|}^2+|F_A{|}^2\le c_0{\int }_X|\varphi {|}^2,$$

$${\Delta }_{\Gamma }:=d_{\Gamma }^{\ast }{d}_{\Gamma }+d_{\Gamma }{d}_{\Gamma }^{\ast }$$

$$\ker {\Delta }_{\Gamma }{|}_{{\Omega }^1(X,{\mathfrak{g}}_P)}=\{0\}.$$

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

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

$$dist(A,M(P,g)):=\underset{u\in {\mathcal{G}}_P,\Gamma \in M(P,g)}{\inf }\Vert u^{\ast }(A)-\Gamma {\Vert }_{L_1^2(Y)}\ge \tilde{C}$$

$$dist(A,M(P,g))\ge \tilde{C}$$

$${\Omega }^2(X)={\Omega }^{2,+}(X)\oplus {\Omega }^{2,-}(X).$$

$${\Omega }^{2,+}(X,{\mathfrak{g}}_P)={\Omega }^{2,+}(X,{\mathfrak{g}}_P)\oplus {\Omega }^{2,-}(X,{\mathfrak{g}}_P).$$

$$d_A^{+,\ast }B=0,F_A^{+}+\frac{1}{8}[B.B]=0,$$

$${\int }_X|{\nabla }_A\varphi {|}^2+{\int }_X⟨Ric\circ \varphi ,\varphi ⟩+2{\int }_X|F_A{|}^2=0.$$

$$p_1:Y\times S^1\to Y.$$

$$B=(1+{\ast }^X){\ast }^Y{p}_1^{\ast }(\varphi ),$$

$$\varphi ={\varphi }_1{e}^1+{\varphi }_2{e}^3+{\varphi }_3{e}^3.$$

$$\frac{1}{2}[\varphi \wedge \varphi ]=[{\varphi }_2,{\varphi }_3]e^{23}+[{\varphi }_3,{\varphi }_1]e^{31}+[{\varphi }_1,{\varphi }_2]e^{12}.$$

$$B={\varphi }_1(e^{01}+e^{23})+{\varphi }_2(e^{02}+e^{31})+{\varphi }_3(e^{03}+e^{12}),$$

$$-\frac{1}{4}[B.B]=[{\varphi }_2,{\varphi }_3](e^{01}+e^{23})+[{\varphi }_3,{\varphi }_1](e^{02}+e^{31})+[{\varphi }_1,{\varphi }_2](e^{0,3}+e^{12}).$$

$$\frac{1}{4}[B.B]=-\frac{1}{2}(1+{\ast }^X)[\varphi \wedge \varphi ].$$

$$2F_A^{+}=(1+{\ast }^X)F_A.$$

$$F_A^{+}+\frac{1}{8}[B.B]=(1+{\ast }^X)(F_A-\frac{1}{2}[\varphi \wedge \varphi ])=0.$$

$$d_A^{+,\ast }B={\ast }^Y{d}_A\varphi -e^0\wedge d_A^{\ast }\varphi =0.$$

$$\begin{array}{rl}\frac{1}{2}\Vert d_A^{+,\ast }B{\Vert }_{L^2(X)}^2& +\Vert F_A^{+}+\frac{1}{8}[B.B]{\Vert }_{L^2(X)}^2={\int }_X(\frac{1}{12}s|B{|}^2-\frac{1}{2}{W}^{+}⟨B\odot B⟩)\\ & +\frac{1}{2}\Vert F_A{\Vert }_{L^2(X)}^2+\frac{1}{4}\Vert {\nabla }_{A}B{\Vert }_{L^2(X)}^2+\frac{1}{16}\Vert [B.B]{\Vert }_{L^2(X)}^2+\frac{1}{2}{\int }_{X}tr(F_A\wedge F_A),\end{array}$$

$$S(\{A_i\}):=\bigcap _{r\in (0,\rho )}\{x\in X:\underset{i\to \infty }{\mathrm{liminf}}{\int }_{B_r(x)}|F_{A_i}{|}^2\ge {\epsilon }_0\},$$

$${\int }_{B_r(x)}|F_{A_i}{|}^2<{\kappa }^{-2}$$

$$\begin{array}{rl}{B}_r(y_0,{\theta }_0)& :=\{(y,\theta ):|y-y_0{|}_{g_Y}^2+|\theta -{\theta }_0{|}^2<r^2\}\\ & \subset (-r+{\theta }_0,r+{\theta }_0)\times B_r(y_0)\end{array}$$

$$\begin{array}{rl}\Vert F_{{\text{A}}_i}{\Vert }_{L^2(B_r(y_0,{\theta }_0))}^2& ={\int }_{B_r(y_0,{\theta }_0)}|F_{{\text{A}}_i}{|}^{2}dvo{l}_{g_Y}d\theta \\ & \le {\int }_{-r+{\theta }_0}^{r+{\theta }_0}d\theta {\int }_{B_r(y_0)}|F_{A_i}{|}^2\\ & \le 2r\underset{i}{\sup }\Vert F_{A_i}{\Vert }_{L^2(Y)}^2.\end{array}$$

Full text

††Teng Huang: School of Mathematical Sciences, University of Science and Technology of China; CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China; e-mail: htmath@ustc.edu.cn;htustc@gmail.com

Keywords. stable flat $SL(2,\mathbb{C})$ connections, Vafa-Witten equations, compactness theorem ††Mathematics Subject Classification (2010): 58E15;81T13

1 Introduction

Let $X$ be an oriented, closed, smooth $n$-dimensional manifold with smooth Riemannian metric $g$, and let $P$ be a principal $G$-bundle over $X$ with $G$ being a compact Lie group. We denote by $\mathcal{A}_{P}$ the set of all connections on $P$, and by $\Omega^{k}(X,\mathfrak{g}_{P})$ the set of $\mathfrak{g}_{P}$-valuled $k$-forms, where $\mathfrak{g}_{P}$ is the adjoint bundle of $P$. Suppose that $A$ is a connection on $P$ and its curvature is denote by $F_{A}\in\Omega^{2}(X,\mathfrak{g}_{P})$. For any connection $A$ on $P$ we have the covariant exterior derivatives $d_{A}:\Omega^{k}(X,\mathfrak{g}_{P})\rightarrow\Omega^{k+1}(X,\mathfrak{g}_{P})$. The curvature

[TABLE]

of the complex connection $\mathcal{A}:=A+\rm{i}\phi$ is a $2$-form with values in $P\times_{G}(\mathfrak{g}_{P}^{\mathbb{C}})$. We say that $\mathcal{A}=A+\rm{i}\phi$ is a complex flat connection with the moment map condition, if the pair $(A,\phi)$ satisfies,

[TABLE]

The system of the pairs $(A,\phi)$ is elliptic [4] For convenience, we call the solutions of the elliptic system as stable flat connections, see [1]. These equations are not only invariant under the actions of real gauge group $\mathcal{G}_{P}=C^{\infty}(P\times_{G}G)$, but also invariant under the actions of complex gauge group $\mathcal{G}^{\mathbb{C}}_{P}:=C^{\infty}(P\times_{G}G_{\mathbb{C}})$. The solution of stable flat connections on compact Riemannian surface $\Sigma$ is also a solution of Hitchin’s equation [6]. The moduli space of the solutions of Hitchin’s equations which satisfying $\int_{\Sigma}|\phi|^{2}\leq K$ is compact, see [4, Theorem 4.1]. Following [3, Proposition 2.2.3] or [10, Proposition 1.2.6], we know that the gauge-equivalence classes of flat connections over a connected manifold, $X$, are in one-to-one correspondence with the conjugacy classes of representations $\pi_{1}(X)\rightarrow G$.

The Uhlenbeck compactness theorem [17, 20] on the moduli space of the connections with $L^{p}$-bounds on curvature is one of the most fundamental theorems in the analytical aspect of the gauge theory. In [13], Taubes studied the Uhlenbeck style compactness problem for $SL(2,\mathbb{C})$ connections, including solutions to the above equations, on three-, four-manifold [13, 14, 15].

We denote by

[TABLE]

the moduli space of the stable flat $SL(2,\mathbb{C})$ connections. In particular, the moduli space of gauge-equivalence classes $[\Gamma]$ of flat connections $\Gamma$ on $P$,

[TABLE]

can be embedded into $\mathcal{M}(P,g)$ via the map $A\mapsto(A,0)$. The Uhlenbeck compactness theorem [17] shows that the moduli space $M(P,g)$ is compact.

One can see that the pair $(A,\phi)$ has the a priori estimate (see [4])

[TABLE]

where $c_{0}$ is a positive constant dependent on the metric $g$. Then the Uhlenbeck compactness theorem implies that the moduli space of solutions of stable flat $SL(2,\mathbb{C})$ connections satisfying (1.1) with $\int_{X}|\phi|^{2}\leq K$ is compact, for every given positive constant $K$. On the other hand, there are examples of sequences of solutions $(A_{i},\phi_{i})$ to (1.1) such that $\|\phi_{i}\|_{L^{2}(X)}$ diverges to infinity, therefore the moduli space of solutions to (1.1) is not always compact. An interesting question to ask if following:

let $K$ be a positive constant number, and consider the subset of $\mathcal{M}(P,g)$ consisting of $(A,\phi)$ such that $\|F_{A}\|_{L^{2}(X)}\leq K$ is this subset alawys compact?

In this article, we consider the case for the stable flat $SL(2,\mathbb{C})$ connections on a closed, smooth, oriented three-manifold $Y$. We will give a positive answer for this question if $Y,G,P$ satisfy certain conditions.

We denote by

[TABLE]

the self-dual operator with respect to a flat connection $\Gamma$. We recall the definition of non-degenerate flat connections as follows, see [2, Definition 2.4].

Definition 1.1**.**

Let $G$ be a compact Lie group, $P$ be a $G$-bundle over a closed, smooth manifold $X$ of dimension $n\geq 2$ and endowed with a smooth Riemannian metric $g$. The flat connection $\Gamma$ called $non$-$degenerate$ if

[TABLE]

The main observation of this article can be stated as follows.

Theorem 1.2**.**

(A compactness theorem for stable flat $SL(2,\mathbb{C})$ connections with bounded real curvatures). Let $Y$ be a closed, oriented, smooth Riemannian three-manifold, $P$ be a principal $SU(2)$ or $SO(3)$-bundle over $Y$. Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence of $C^{\infty}$-solutions of Equations (1.1). Suppose that all flat connections on the principal bundle $P$ are $non$-$degenerate$. If the $L^{2}$-norms of the curvatures $\|F_{A_{i}}\|_{L^{2}(Y)}$ are bounded, then there is a subsequence of $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{u_{i}\}_{i\in\Xi}$ such that $\{(u^{\ast}_{i}(A_{i}),u_{i}^{\ast}(\phi_{i}))\}_{i\in\Xi}$ converges to a pair $(A_{\infty},\phi_{\infty})$ obeying Equations (1.1) on $P$ in the $C^{\infty}$-topology. In particular, the moduli space of solutions of stable flat $SL(2,\mathbb{C})$ connections which obeys $\int_{Y}|F_{A}|^{2}\leq K$ is compact, for every positive constant $K$.

Remark 1.3**.**

Taubes [13] considered a sequence of complex connections $\mathcal{A}_{i}:=A_{i}+{\rm{i}}\phi_{i}$ such that the $L^{2}$-norms of $F_{\mathcal{A}_{i}}$ are bounded (in this article, the complex curvature is just zero). There are two possible cases: (1) if $\|\phi_{i}\|_{L^{2}(X)}$ has a bounded sequence, then Taubes proves $(u^{\ast}_{i}(A_{i}),u_{i}^{\ast}(\phi_{i}))\rightarrow(A_{\infty},\phi_{\infty})$ in the $C^{\infty}$-topology, (2) if $\|\phi_{i}\|_{L^{2}(X)}$ has no bounded sequence, then Taubes makes sense of the limit as a $\mathbb{Z}_{2}$ harmonic spinor. In particular, if the sequence $\|F_{A_{i}}\|_{L^{2}(Y)}$ is divergent to infinity, then following the inequality (1.2), one can see that the sequence $\|\phi_{i}\|_{L^{2}(Y)}$ is also divergent to infinity.

As a particular case of Theorem 1.2, we have an $L^{2}$-bound on the extra fields in the fibre direction at a connection $A_{0}$. Namely,

Corollary 1.4**.**

Let $Y$ be a closed, oriented, smooth Riemannian three-manifold, $P$ be a principal $SU(2)$ or $SO(3)$-bundle over $Y$. Suppose that all flat connections on the principal bundle $P$ are $non$-$degenerate$. Then for any sequence of solutions $\{(A_{0},\phi_{i})\}_{i\in\mathbb{N}}$ of Equations (1.1), there exists a subsequence $\Xi\subset\mathbb{N}$ and a positive constant $C>0$ such that $\int_{X}|\phi_{i}|^{2}\leq C$ for all $i\subset\Xi$.

The Corollary 1.4 is similar to the Vafa-Witten equations case, see[12, Corllary 1.4].

Following the notation of [3, Section 4.2.1], we denote by $([A,\phi])$ the equivalence class of a pair $(A,\phi)$, that is a point in $\mathcal{M}(P,g)$. We denote

[TABLE]

We can define a distance function on $\mathcal{M}(P,g)$ as follows:

[TABLE]

We can use the compactness theorem 1.2 to study the topological of the moduli space of stable flat $SL(2,\mathbb{C})$ connections.

Theorem 1.5**.**

Assume the hypotheses of Theorem 1.2. Suppose that all flat connections on the principal $G$-bundle $P$ are $non$-$degenerate$. If $(A,\phi)$ is a $C^{\infty}$-solution of Equations (1.1), then there is a positive constant $\tilde{C}=\tilde{C}(P,g)\in(0,1]$ such that

[TABLE]

unless $A$ is a flat connection.

Remark 1.6**.**

There are many combinations of conditions on $G,P,Y,g$ which imply that the flat connection is non-degenerate. For example, if $Y$ is a closed, oriented Riemannian three-manifold with the homology of $S^{3}$, $P$ is a principal $SU(2)$-bundle over $Y$, then every flat connection $\Gamma$ on $P$ is non-degenerate.

Corollary 1.7**.**

Let $Y$ be a closed, oriented, smooth Riemannian three-manifold with the homology of $S^{3}$, $P$ be a principal $SU(2)$-bundle. Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence of $C^{\infty}$-solutions of Equations (1.1). If the $L^{2}$-norms of the curvatures $\|F_{A_{i}}\|_{L^{2}(Y)}$ have a bound, then there is a subsequence of $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{u_{i}\}_{i\in\Xi}$ such that $\{(u^{\ast}_{i}(A_{i}),u^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi}$ converges to a pair $(A_{\infty},\phi_{\infty})$ obeying Equations (1.1) on $P$ in the $C^{\infty}$-topology. Furthermore, there is a positive constant $\tilde{C}=\tilde{C}(P,g)\in(0,1]$ such that

[TABLE]

unless $A$ is a flat connection.

The organization of this paper is as follows. In section 2, we first recall the compactness theorem of Vafa-Witten equations which is proved by Tanaka [12]. We also observe that the set of stable flat $SL(2,\mathbb{C})$ connections on a compact $3$-fold $Y$ is in one-to-one correspondence with solutions to a $S^{1}$-invariant Vafa-Witten equations on $Y\times S^{1}$. Then by Tanaka’s compactness theorem, we can prove a compactness theorem for stable flat $SL(2,\mathbb{C})$ connections. In section 3, we obtain a topological property of the moduli space of stable flat $SL(2,\mathbb{C})$ connections by our compactness theorem.

2 Compactness theorem for stable $SL(2,\mathbb{C})$ flat connections

2.1 Vafa-Witten equations and stable $SL(2,\mathbb{C})$ flat connections

In this section, we recall the compactness theorem of Vafa-Witten equations which is proved by Tanaka [12]. For an oriented $4$-dimensional Riemannian manifold $X$ with metric $g$, the Hodge star operator $\ast:\Omega^{2}(X)\rightarrow\Omega^{2}(X)$ induces the following splitting:

[TABLE]

Accordingly, the space of $\mathfrak{g}_{P}$-valued two-forms $\Omega^{2}(X,\mathfrak{g}_{P})$ splits as

[TABLE]

At first, we begin to define the Vafa-Witten equations [19]. One also can see Equations (2.4)–(2.5) in [12]. We call the pair $(A,B)\in\mathcal{A}_{P}\times\Omega^{2,+}(X,\mathfrak{g}_{P})$ a solution of Vafa-Witten equations, if $(A,B)$ satisfies

[TABLE]

where $[B.B]\in\Omega^{2,+}(X,\mathfrak{g}_{P})$ is defined in [11, Appendix A]. Vafa-Witten equations were introduced by Vafa and Witten to study S-duality in twist of $\mathcal{N}=4$ supersymmetric Yang-Mills theory [19]. By appropriately counting the number of points of the moduli space of Vafa-Witten equations, we hope to obtain a number $VW(P)$ called the Vafa-Witten invariant for the principal bundle $P\rightarrow X$ [11, Section 1.3]. These equations were also considered by Haydys [5] and Witten [21] from a different point of view.

Let $G$ be a Lie group and $P$ be a principal $G$-bundle over a smooth Riemannian manifold $X$. We recall the equivalent characterizations of flat bundles [10, Section 1.2], that is, bundles admitting a flat connection.

Proposition 2.1**.**

*([10, Proposition 1.2.6]) For a principal $G$-bundle $P$ over $X$, the following three conditions are equivalent:

(1) $P$ admits a flat structure,

(2) $P$ admits a flat connection,

(3) $P$ is defined by a representation $\rho:\pi_{1}(X)\rightarrow G$.*

Recall that, $\mathcal{F}_{\mathcal{A}}=F_{A}-\frac{1}{2}[\phi\wedge\phi]+{\rm{i}}d_{A}\phi$ is curvature of the complex connection $\mathcal{A}:=A+\rm{i}\phi$. The solutions of stable flat connections also satisfy the complex Yang-Mills equations [4]. We then have

Proposition 2.2**.**

If $(A,\phi)$ is a $C^{\infty}$-solution of (1.1) over a closed $n$-manifold $X$, then

[TABLE]

Now return to the setting of this article. Let $Y$ be an oriented, smooth, Riemannian three-manifold, $P$ be a $G$-principal bundle over $Y$ with $G$ being a compact Lie group. We denote by $X:=Y\times S^{1}$ the product manifold with the product metric. We pull back a connection $A$ on $P\rightarrow Y$ to $p_{1}^{\ast}(P)\rightarrow X$ via the canonical projection

[TABLE]

We define a section $B\in\Gamma(X,\Omega^{2,+}\otimes\mathfrak{g}_{p^{\ast}_{1}(P)})$ as follows

[TABLE]

where $\ast^{Y}$ (resp. $\ast^{X}$) is the Hodge star operator with respect to metric $g_{Y}$ (resp. $g_{X}$). We then have

Proposition 2.3**.**

The canonical projection gives a one-to-one correspondence between stable flat connections on $P$ and $S^{1}$-invariant Vafa-Witten equations on the pullback bundle $p^{\ast}_{1}(P)$.

Proof.

The proof is similar to [11, Lemma 8.2.2]. In a local coordinate $\{e^{1},e^{2},e^{3}\}$ of $T^{\ast}Y$, we can denote

[TABLE]

Note that

[TABLE]

Then by the definition of $B$, we get

[TABLE]

and

[TABLE]

Thus

[TABLE]

We also observe that

[TABLE]

Therefore, we have

[TABLE]

We also have an other equation

[TABLE]

∎

2.2 Compactness theorem for Vafa-Witten equations

Mares studied the analytic aspects of Vafa-Witten equations in [11]. We don’t have an $L^{2}$-bounded on the curvature $F_{A}$ of a connection $A$ which satisfies the Vafa-Witten equations as in the case of Hitchin-Simpson’s equations [6]. Mares observed that if $(A,B)$ is a solution of Vafa-Witten equations and the $L^{2}$-norm of $B$ has a uniform bound, then the curvature $F_{A}$ also has a uniform bound in $L^{2}$-norm by the following identity

[TABLE]

where $\odot$ denotes some bilinear on $\Omega^{2,+}(X,\mathfrak{g}_{P})\otimes\Omega^{2,+}(X,\mathfrak{g}_{P})$, $s$ is the scalar curvature of the metric, and $W^{+}$ is the self-dual part of the Weyl curvature of the metric (see [12, Page 1204] or [11, Section B.4] for more details). Following Uhlenbeck compactness theorem, he obtained a compactness theorem of Vafa-Witten equations under the extra fields $B$ have a bound in $L^{2}$-norm [11].

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 which is defined as in [12]. This set $S(\{A_{i}\})$ describes the singular set of a sequence of connections $\{A_{i}\}$. In [12], Tanaka observed that under the particular circumstance where the connections are non-concentrating and the limiting connection is non-locally reducible, one obtains an $L^{2}$-bound on the extra fields. Here, we say that a connection $A$ on a principal $SU(2)$ or $SO(3)$ bundle $P$ is locally reducible if the vector bundle $\mathfrak{g}_{P}$ has a one-dimensional subbundle that is $A$-covariantly constant, See [12, Definition 2.1]. Note that a connection on a principal $SU(2)$ or $SO(3)$ bundle $P$ being locally reducible is the same as being honestly reducible if $X$ is simply connected. The following is an analogue of the second part of [16, Theorem 1.1], but under the assumption that $S(\{A_{i}\})$ is empty.

Theorem 2.4**.**

([16, Theorem 1.2] and [12, Proposition 4.4] ) Let $\{(A_{i},B_{i})\}$ be a sequence solutions of Vafa-Witten equations, set $r_{i}:=\|\phi_{i}\|_{L^{2}(X)}$. Let $\delta$ denote the injectivity radius of $X$. Suppose that there exist $r\in(0,\delta)$ and a sequence $\Xi\subset\mathbb{N}$ such that

[TABLE]

*for every $i\in\Xi$ and $x\in X$. Assume that the sequence $\{r_{i}\}_{i\in\mathbb{N}}$ has no bounded subsequence. Then there exist a closed, nowhere dense set $Z\subset X$, a real line bundle $\mathcal{I}\rightarrow X-Z$, a section $\nu\in\Gamma(X-Z,\mathcal{I}\otimes\Omega^{2,+})$, a connection $A_{\Delta}$ on $P\mid_{X-Z}$, and an isometric bundle homomorphism $\sigma_{\Delta}:\mathcal{I}\rightarrow\mathfrak{g}_{P}$. Their properties are listed below:

(a) $Z$ is the zero locus of $|\nu|$,

(b) The function $|\nu|$ is Hölder continuous $C^{0,1/\kappa}$ on $X$,

(c) The section $v$ is harmonic in the sense of $d\nu=0$,

(d) $|\nabla\nu|$ is an $L^{2}$-function on $X-Z$ that extends as an $L^{2}$-function on $X$,

(e) The curvature tensor of $A_{\Delta}$ is anti-self-dual,

(f) The homomorphism $\sigma_{\Delta}$ is $A_{\Delta}$-covariantly constant.

In addition, there exist a subsequence $\Lambda\subset\Xi$ and a sequence $u_{i}$ of automorphisms from $P$ such that

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

(ii) The sequence $\{r^{-1}_{i}u^{\ast}_{i}(B_{i})\}$ converges to $v\otimes\sigma_{\Delta}$ in the $L^{2}_{1}$-topology on compact subset in $X-Z$ and the $C^{0}$-topology on $X$. Meanwhile, $\{r_{i}^{-1}|B_{i}|\}$ converges to $|\nu|$ in the weakly $L^{2}_{1}$-topology and the $C^{0}$-topology on the whole of $X$.*

2.3 Proof of our results

In this section, we give the proof of our main result. At first, we observe that

Proposition 2.5**.**

Let $Y$ be a closed, oriented, smooth Riemannian three-manifold, $P$ be a principal $G$-bundle with $G$ being a compact Lie group. Let $\{A_{i}\}_{i\in\mathbb{N}}$ be a sequence $C^{\infty}$-connections on $P$ with the $L^{2}$-norms of the curvatures $\|F_{A_{i}}\|_{L^{2}(X)}$ have a uniform bound. We denote by $\{\textbf{A}_{i}\}$ the pullback $S^{1}$-invariant connections. Then the set $S(\{\textbf{A}_{i}\})$ is empty, where $S(\cdot)$ is defined in Equation (2.1).

Proof.

For a point $(y_{0},\theta_{0})\in Y\times S^{1}$, we denote by

[TABLE]

the geodesic ball on $Y\times S^{1}$. Hence, we have

[TABLE]

We can choose $r$ sufficiently small such that

[TABLE]

where $\kappa$ is the constant on Theorem 2.4. Therefore, we complete this proof. ∎

Following the idea in the proof of [15, Theorem 1.2], we can obtain a compactness theorem for the stable flat $SL(2,\mathbb{C})$ connections on three-manifold.

Theorem 2.6**.**

*Let $Y$ be a closed, oriented, smooth Riemannian three-manifold, $P$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence of $C^{\infty}$-solutions of Equations (1.1), set $r_{i}:=\|\phi_{i}\|_{L^{2}(X)}$. Suppose that the $L^{2}$-norms of the curvatures $\|F_{A_{i}}\|_{L^{2}(Y)}$ have a uniform bound and the sequence $\{r_{i}\}_{i\in\mathbb{N}}$ has no bounded subsequence. Then there exist a closed, nowhere dense set $Z_{Y}\subset Y$, a real line bundle $\mathcal{I}_{Y}\rightarrow Y-Z_{Y}$, a section $\nu\in\Gamma(Y-Z_{Y},\mathcal{I}_{Y}\otimes\Omega^{1})$, a connection $A_{\Delta}$ on $P\mid_{Y-Z_{Y}}$, and an isometric bundle homomorphism $\sigma_{\Delta}:\mathcal{I}_{Y}\rightarrow\mathfrak{g}_{P}$. Their properties are listed below:

(a) $Z_{Y}$ is the zero locus of $|\nu|$,

(b) The section $v$ is harmonic in the sense of $d\nu=d^{\ast}\nu=0$,

(c) The curvature tensor of $A_{\Delta}$ is flat,

(d) The homomorphism $\sigma_{\Delta}$ is $A_{\Delta}$-covariantly constant.

In addition, there exist a subsequence $\Lambda\subset\Xi$ and a sequence $u_{i}$ of automorphisms from $P$ such that

(i) $\{u_{i}^{\ast}(A_{i})\}$ converges to $A_{\Delta}$ in the $L^{2}_{1}$ topology on compact subset in $Y-Z_{Y}$ and

(ii) The sequence $\{r^{-1}_{i}u^{\ast}_{i}(\phi_{i})\}$ converges to $\nu\otimes\sigma_{\Delta}$ in the $L^{2}_{1}$ topology on compact subset in $Y-Z_{Y}$ and the $C^{0}$-topology on $Y$. Meanwhile, $\{r_{i}^{-1}|\phi_{i}|\}$ converges to $|\nu|$ in the weakly $L^{2}_{1}$-topology and the $C^{0}$-topology on the whole of $Y$.*

Proof.

As explained momentarily, this theorem constitutes a special case to Theorem 2.4. To obtain Theorem 2.6 from Theorem 2.4, we take $X$ in Theorem 2.6 to be the product $Y\times S^{1}$ with the metric being the product metric. The pull-back of the principal $G$-bundle $P$ on $Y$ to $X$ via the projection map to $Y$ defines a principal $G$-bundle over $X$, the latter is denoted also by $P$. Let $\{(A_{i},\phi_{i})\}$ be a sequence solutions of stable flat $SL(2,\mathbb{C})$ connections over $Y$. For simplicity we keep the same notations for objects on $Y$ and their pullbacks to $X$. We denote $B_{i}=(1+\ast^{X})\ast^{Y}\phi_{i}$. If suppose that the sequence $\{r_{i}:=\|\phi_{i}\|_{L^{2}(Y)}\}_{i\in\mathbb{N}}$ has no bounded subsequence, then $\|B_{i}\|_{L^{2}(X)}$ also has no bounded subsequence. A similar sort of argument can be used to prove that Theorem 2.4’s set $Z$ is the product of $S^{1}$ and a closed set $Z_{Y}\subset Y$ and that Theorem 2.4’s real line bundle $\mathcal{I}$ is isomorphic to the pull-back via the projection map of a real line bundle defined on the complement in $Y$ of $Z_{Y}$, this denoted for now by $\mathcal{I}_{Y}$. Moreover, such an isomorphism identifies Theorem 2.4’s version of $\nu$ with the pull-back of a harmonic, $\mathcal{I}_{Y}$ valued 1-form on $Y-Z_{Y}$ with $Z_{Y}$ denoting the locus where its norm is zero. ∎

Remark 2.7**.**

Taubes considered a sequence of complex connections $\{\mathcal{A}_{i}:=A_{i}+{\rm{i}}\phi_{i}\}$ such that the $L^{2}$-norms of $\{F_{\mathcal{A}_{i}}\}$ are bounded. If $\{\|\phi_{i}\|_{L^{2}(X)}\}$ has no bounded sequence, then Taubes makes sense of the limit as a $\mathbb{Z}_{2}$ harmonic spinor [15]. In our result, we add the conditions that all connections $\{\mathcal{A}_{i}\}$ are stable flat $SL(2,\mathbb{C})$ connection and the real curvatures $\{F_{A_{i}}\}$ have $L^{2}$-bounded, then we prove the limit as a decoupled stable flat $SL(2,\mathbb{C})$ connection.

The next theorem is a special case of Theorem 1.1b in [15]. It implies among other things that $Z_{Y}$ has measure zero. To set the notation for this upcoming theorem. A point $p\in Z_{Y}$ is a point of discontinuity for $\mathcal{I}_{Y}$, if $\mathcal{I}_{Y}$ is not isomorphic to the product bundle on the complement of $Z_{Y}$ in any neighborhood of $p$ [15].

Theorem 2.8**.**

([15, Theorem 1.1b]) Let $Z_{Y}$ and $\mathcal{I}_{Y}$ be as described in Theorem 2.6. The set $Z_{Y}$ has Hausdorff dimension at most 1, and moreover, the set of the points of discontinuity for $\mathcal{I}_{Y}$ (defined in the preceding paragraph) are the points in the closure of an open subset of $Z_{Y}$ that is an embedded $C^{1}$ curve in $Y$ denoted by $\Sigma$.

Uhlenbeck’s [17] theorem applies to the connections on $P$ and in particular makes the following assertion:

Uhlenbeck’s Theorem: Let $\{A_{i}\}_{i\in\mathbb{N}}$ be a sequence of connections on $P$ over a closed, oriented, $3$-manifold. If $L^{2}$-norms of the curvatures $F_{A_{i}}$ of the connections $\{A_{i}\}$ have a uniform bound, then there is a subsequence $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{u_{i}\}_{i\in\Xi}$ such that $\{u^{\ast}_{i}(A_{i})\}$ converges weakly in the $L^{2}_{1}$-topology to a connection $A_{\infty}$ on $P$.

By the priori estimate (1.2), we then have

Theorem 2.9**.**

([15, Theoreom 1.1a]) Let $Y$ be a closed, oriented, smooth Riemannian three-manifold, $P$ be a principal $SU(2)$ or $SO(3)$-bundle over $Y$. Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence of $C^{\infty}$-solutions of Equations (1.1). Suppose that the sequence $\{\|\phi_{i}\|_{L^{2}(Y)}\}$ has a bounded subsequence. Then there is a subsequence of $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{u_{i}\}_{i\in\Xi}$ such that $\{(u^{\ast}_{i}(A_{i}),u_{i}^{\ast}(\phi_{i}))\}_{i\in\Xi}$ converges to a pair $(A_{\infty},\phi_{\infty})$ obeying Equations (1.1) on $P$ in $C^{\infty}$-topology.

We are finally ready to use the above results in the following proposition.

Proposition 2.10**.**

*Let $Z_{Y}$ and $\mathcal{I}_{Y}$ be as described in Theorem 2.6, so that $\sigma_{\Delta}$ and $A_{\Delta}$ are defined over $Y-Z_{Y}$. Then

(1) There exists a smooth flat connection $A_{\infty}$ defined over all of $Y$, and a Sobolev class $L^{2}_{2}$ gauge transformation $u_{\infty}$ defined over $Y-Z_{Y}$ such that $u_{\infty}^{\ast}(A_{\infty})$ is restriction to $Y-Z_{Y}$ of $A_{\infty}$. Defining $\sigma_{\infty}:=u_{\infty}^{\ast}(\sigma_{\Delta})$ over $Y-Z_{Y}$, then $\nabla_{A_{\infty}}\sigma_{\infty}=0$.

(2) The bundle $\mathcal{I}_{Y}$ over $Y-Z_{Y}$ extends to a bundle defined over all of $Y$, which we again denote by $\mathcal{I}_{Y}$,

(3) There exist extensions of both $v\in\Gamma(\mathcal{I}_{Y}\otimes\Omega^{1})$ and $\sigma_{\infty}:\mathcal{I}_{Y}\rightarrow\mathfrak{g}_{P}$ to all of $Y$. We again denote these by $v$ and $\sigma_{\infty}$. The extensions satisfy $dv=0$ and $\nabla_{A_{\infty}}\sigma_{\infty}=0$.*

Proof.

The idea of our proof is similar to [12, Proposition 4.6].

We first prove item 1. Following weak Uhlenbeck compactness theorem (see [20, Theorem A]), for any sequence $\{A_{i}\}_{i\in N}$ with bounded $L^{2}$-curvature $\{F_{A_{i}}\}_{i\in\mathbb{N}}$ on a principal $G$-bundle over a closed three-manifold, there exists a subsequence (again denote $\{A_{i}\}_{i\in\mathbb{N}}$ ) and a sequence of gauge transformations $\{u_{i}\}_{i\in\mathbb{N}}$ such that $u^{\ast}_{i}(A_{i})$ converges weakly to a limit connection $A_{\infty}$ over all of $Y$ in $L^{2}_{1}$. Recall from Theorem 2.6 that $A_{\Delta}$ is the $L^{2}_{1}$ limit over compact subset $Y-Z_{Y}$ of gauge equivalent connections. Since weakly $L^{2}_{1}$ limits preserve $L^{2}_{2}$ gauge equivalence, it follows there exists a Sobolev-class $L^{2}_{2}$ gauge transformation $u_{\infty}$ such that $u_{\infty}^{\ast}(A_{\Delta})=A_{\infty}$.

Note that $A_{\Delta}$ is flat and gauge-equivalent over the complement of $Z_{Y}$ to $A_{\infty}$. Thus, $A_{\infty}$ is a $L^{2}_{1}$ connection whose curvature is $L^{2}$, and vanishes on the complete of $Z_{Y}$, which by Theorem 2.8 is a set of measure zero. Hence the curvature of $A_{\infty}$ is flat and so a standard elliptic regularity argument can be used to prove that there is an $L^{2}_{2}$ and $C^{0}$ automorphism of $P$ that transforms $A_{\infty}$ into a smooth flat connection. After possibly composing $u_{\infty}$ with such an automorphism, we may assume without loss of generality that $A_{\infty}$ is smooth and that $u_{\infty}$ is continuous. That $\nabla_{A_{\infty}}\sigma_{\infty}=0$ follows from Theorem 2.6 since $\sigma_{\Delta}$ is $A_{\Delta}$-covariantly constant. This establishes the item 1.

We next prove the item 2 that $\mathcal{I}_{Y}$ extends over $Z_{Y}$. Let $\Sigma\subset Z_{Y}$ denote the $C^{1}$ submanifold that is described by Theorem 2.8. It is enough to prove that $\Sigma$ is empty. For this purpose, assume to the contrary that $\Sigma\neq\emptyset$ and let $S\subset\Sigma$ be a component. This is a $C^{1}$ embedded curve. Fix a point $p\in S$. Since $S$ is $C^{1}$, there is an embedded disk $D\subset Y$ closure intersects $S$ transversally at a single point which is $p$. This is also its only intersection point with $Z_{Y}$ since $S$ is an open subset of $Z_{Y}$. Since $p$ is a point of discontinuity for the bundle $\mathcal{I}_{Y}$, the restriction of $\mathcal{I}_{Y}$ to $D-\{p\}$ is not isomorphic to the product line bundle. In particular, parallel transport by $A_{\Delta}$ of $\sigma_{\Delta}$ along any circle in $D-\{p\}$ which wraps once around $p$ gives $-\sigma_{\Delta}$. However, $A_{\Delta}$ is gauge-equivalent to a connection which is smooth over all of $D$. This parallel transport around sufficiently small bounded interval will be arbitrarily close to $+\sigma_{\Delta}$, which is a contradiction.

Finally, we prove item 3 by showing that both $v$ and $\sigma_{\infty}$ extend to all of $Y$ as $L^{2}_{1}$ sections. Granted this extension, we may argue as in item 1 that both $dv$ and $\nabla_{A_{\infty}}\sigma_{\infty}$ are $L^{2}$ sections which vanish almost everywhere, and hence by elliptic regularity, $v$ and $\sigma_{\infty}$ are smooth and satisfy $dv=0$ and $\nabla_{A_{\infty}}\sigma_{\infty}=0$ over all of $Y$. ∎

Following above results, we can prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature.

Proof of Theorem 1.2.

We set $r_{i}:=\|\phi_{i}\|_{L^{2}(Y)}$. At first, we can prove that there exists a subsequence $\Xi\subset\mathbb{N}$ such that $\{r_{i}\}_{i\in\Xi}$ have a uniform bound. If not, then the sequence $\{r_{i}\}_{i\in\mathbb{N}}$ has no bounded subsequence. We denote $A_{\infty}$, $\nu$ and $\sigma_{\infty}$ as described in Proposition 2.10. Hence, following Proposition 2.10, we have

[TABLE]

and

[TABLE]

Since $\nu\otimes\sigma_{\infty}\in\Omega^{1}(Y,\mathfrak{g}_{P})$, the hypothesis of the flat connection $A_{\infty}$ implies that

[TABLE]

Following the item 1 in Proposition 2.10, there exist a continuous Sobolev-class $L^{2}_{2}$ gauge transformation $u_{\infty}$ defined over $Y-Z_{Y}$ such that

[TABLE]

and

[TABLE]

Hence

[TABLE]

on $Y-Z_{Y}$. The zero locus of the extension of $|\nu|$ is the set $Z_{Y}$. And we can set $\sigma_{\Delta}$ is a unit length, $A_{\Delta}$-covariantly constant homomorphism over $Y-Z_{Y}$. Hence, we can say $|\nu|=0$ on $Y$.

On the other hand, following the last item in Theorem 2.6, there exist a subsequence $\Xi\subset\mathbb{N}$ and a sequence $\{u_{i}\}_{i\in\Xi}$ of automorphisms from $P$ such that $\{r^{-1}_{i}u^{\ast}_{i}(\phi_{i})\}$ converges to $\nu\otimes\sigma_{\Delta}$ in the $L^{2}_{1}$-topology on compact subset in $Y-Z_{Y}$ and the $C^{0}$-topology on $Y$. Meanwhile, $\{r_{i}^{-1}|\phi_{i}|\}$ converges to $|\nu|$ in the weakly $L^{2}_{1}$-topology and the $C^{0}$-topology on the whole of $Y$. Hence

[TABLE]

It’s contradicting the fact $r^{-1}_{i}\|\phi_{i}\|_{L^{2}(Y)}=1$, $\forall i\in\mathbb{N}$. In particular, the preceding argument shows that there exists a subsequence $\Xi\subset\mathbb{N}$ such that $\{r_{i}\}_{i\in\Xi}$ have a uniform bound. Thus following Theorem 2.9, there is a subsequence (again denote by $\{(A_{i},\phi_{i})\}_{i\in\Xi}$) and a sequence of gauge transformation $\{u_{i}\}_{i\in\Xi}$ such that $\{(u^{\ast}_{i}(A_{i}),u^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi}$ converges to a pair $(A_{\infty},\phi_{\infty})$ on $P$ in the $C^{\infty}$-topology. ∎

Corollary 2.11**.**

Let $Y$ be a closed, oriented Riemannian three-manifold with the homology of $S^{3}$, $P$ be a principal $SU(2)$-bundle. Let $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ be a sequence of $C^{\infty}$ solutions of Equations (1.1). If the $L^{2}$-norms of the curvatures $\|F_{A_{i}}\|_{L^{2}(Y)}$ have a uniform bound, then there is a subsequence of $\Xi\subset\mathbb{N}$ and a sequence of gauge transformations $\{u_{i}\}_{i\in\mathbb{N}}$ such that $\{(u^{\ast}_{i}(A_{i}),u^{\ast}_{i}(\phi_{i}))\}_{i\in\Xi}$ converges to a pair $(A_{\infty},\phi_{\infty})$ obeying Equations (1.1) on $P$ in the $C^{\infty}$-topology.

3 Disconnectedness of the moduli space $\mathcal{M}(P,g)$

3.1 A lower positive bound of extra fields

We call a stable flat connection $\mathcal{A}:=A+\rm{i}\phi$ decoupled, if the real connection $A$ is flat and the extra field $\phi$ is a harmonic $\mathfrak{g}_{P}$-$1$-form with respect to $d_{A}+d^{\ast}_{A}$, i.e,

[TABLE]

Using a result of Uhlenbeck [18], the author observed that if the stable flat connection $\mathcal{A}$ over a closed, smooth, Riemannian three-manifold $Y$, then the $L^{2}$-norm of extra fields has a uniform positive lower bound unless the real connection is flat.

Theorem 3.1**.**

([7]) Let $Y$ be a closed, oriented, Riemannian three-manifold and endowed with a smooth Riemannian metric $g$, $P$ be a principal $G$-bundle with $G$ being a compact Lie group. If $(A,\phi)$ is a $C^{\infty}$-solution of equations (1.1), then there is a positive constant $C=C(X,g,G)$ such that

[TABLE]

unless $A$ is a flat connection.

Suppose that all flat connections on $P$ are non-degenerate. Then the extra fields vanish if the stable flat connection $A+\rm{i}\phi$ is decoupled over a closed Riemannian manifold. Following Theorem 3.1, we then have

Corollary 3.2**.**

Assume the hypothesis of Theorem 3.1. Suppose that all flat connections on the principal bundle $P$ are $non$-$degenerate$. If $(A,\phi)$ is a $C^{\infty}$-solution of equations (1.1), then either there exists a positive constant $C=C(X,g,G)$ such that

[TABLE]

or $\phi$ vanishes.

3.2 A lower positive bound of curvatures

One can see that $M(P,g)$ is the space of real flat connections and $\mathcal{M}^{\prime}(P,g):=\mathcal{M}(P,g)\backslash M(P,g)$ is the space of real connections are non-flat. Hence we can denote by

[TABLE]

the distance between $M(P,g)$ and $\mathcal{M}^{\prime}(P,g)$. Following Theorem 3.1. We can obtain a topological property of the moduli space $\mathcal{M}(P,g)$.

Proposition 3.3**.**

( Disconnectedness of the moduli space $\mathcal{M}(P,g)$). Let $Y$ be a closed, oriented, smooth, Riemannian three-manifold, $P$ be a principal $G$-bundle with $G$ being a compact Lie group. Suppose that all flat connections on the principal bundle $P$ are $non$-$degenerate$. If the moduli spaces $M(P,g)$ and $\mathcal{M}^{\prime}(P,g):=\mathcal{M}(P,g)\backslash M(P,g)$ are all non-empty, then the moduli space $\mathcal{M}(P,g)$ is disconnected.

Proof.

Under the hypothesis of the flat connection, following the Corollary 3.2, the $L^{2}$-norm of the extra field $\|\phi\|_{L^{2}(X)}$ has a lower bound unless $\phi$ vanishes. If the moduli spaces $M(P,g)$ and $\mathcal{M}(P,g)\backslash M(P,g)$ are all non-empty, then

[TABLE]

where $C=C(Y,g)$ is the positive constant in Corollary 3.2, i.e., the moduli space $\mathcal{M}(P,g)$ is disconnected. ∎

We extend the idea in [9] to stable flat $SL(2,\mathbb{C})$ connection case, we prove a gap result of the real curvature following the compactness theorem 1.2.

Proposition 3.4**.**

Let $Y$ be a closed, oriented, smooth, Riemannian three-manifold, $P$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. Suppose that all flat connections on the principal bundle $P$ are $non$-$degenerate$ . If the pair $(A,\phi)$ is a $C^{\infty}$-solution of equations (1.1), then there is a positive constant $C=C(Y,g,P)$ such that

[TABLE]

unless the real connection $A$ is flat.

Proof.

Suppose that the constant $C$ does not exist. We may then choose a sequence $\{(A_{i},\phi_{i})\}_{i\in\mathbb{N}}$ such that $\|F_{A_{i}}\|_{L^{2}(Y)}\rightarrow 0\ as\ i\rightarrow\infty$, and $\{A_{i}\}_{i\in\mathbb{N}}$ are all non-flat. Thus the compactness Theorem 1.2 implies that there exists a pair $(A_{\infty},\phi_{\infty})$ obeys the equations (1.1) and there is a sequence of gauge transformations $\{u_{i}\}_{i\in\Xi}$ such that $(u_{i}^{\ast}(A_{i}),u_{i}^{\ast}(\phi_{i}))\rightarrow(A_{\infty},\phi_{\infty})$ in $C^{\infty}$ over $Y$. Following Theorem 3.1, the $L^{2}$-norm of extra field $\|\phi_{\infty}\|_{L^{2}X}$ has a positive lower bound. Therefore, we have

[TABLE]

where ${C}={C}(Y,g)$ is a positive constant.

On the other hand, since $\|F_{A_{i}}\|_{L^{2}(Y)}\rightarrow 0$, the weak Uhlenbeck compactness theorem implies that the connection $A_{\infty}$ on $P$ is flat. Hence $A_{\infty}$ is non-degenerate by the hypothesis on this proposition. It implies that the extra field $\phi_{\infty}\equiv 0$. It’s contradiction with $\|\phi_{\infty}\|_{L^{2}(Y)}$ has a uniform positive lower bound. ∎

Remark 3.5**.**

The solutions of stable flat connections also satisfy the complex Yang-Mills equations [4]. The author proved that if the pair $(A,\phi)$ is a smooth solution of stable flat connection over a closed, smooth, Riemannian $n$-manifold $X$, the curvature $F_{A}$ of non-flat connection $A$ has a uniform positive lower $L^{p}$-bound under the condition that all flat connections are all non-degenerate, See [8, Theorem 1.2].

Proof of Theorem 1.5.

At first, we give the priori estimate for the curvature of connection. Since

[TABLE]

we have

[TABLE]

The last inequality, we use the Sobolev embedding $L^{2}_{1}\hookrightarrow L^{p}$ for $2\leq p\leq 6$ with embedding constant $C_{S}$. Here

[TABLE]

due to the fact that

[TABLE]

for all $U,V\in T_{y}(Y)$. Combining the preceding inequalities yields

[TABLE]

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

If $\|a\|_{L^{2}_{1}(Y)}\leq 1$, then

[TABLE]

Therefore, we have

[TABLE]

where $C$ is the positive constant in Proposition 3.4. We set $\tilde{C}:=\min\{1,\frac{C}{2c}\}$, thus

[TABLE]

We complete this proof. ∎

Acknowledgements

We would like to thank the anonymous referees for careful reading of my manuscript and helpful comments. I would like to thank Y. Tanaka for kind comments regarding this and it companion article [12]. This work was supported in part by NSF of China (11801539) and the Fundamental Research Funds of the Central Universities (WK3470000019), the USTC Research Funds of the Double First-Class Initiative (YD3470002002).