Gauge stabilizers and properness of the Sobolev action

A gauge section $u$ stabilizes $A$ exactly when the local action formula gives $du=uA-Au$, that is, $d_{A}u=0$. This equation says that $u$ is parallel for the induced connection on $\mathrm{Ad}P$. Since $M$ is connected, parallel transport determines $u$ everywhere from its value at $p_m$, so evaluation is injective on the stabilizer. A parallel section's value commutes with transport around every based loop, hence lies in the holonomy centralizer.

Let $B_j=A_j^{u_j}$ and work in the fixed finite atlas. The local action formula gives $d{u}_j=u_j{B}_j-A_j{u}_j$. Since $u_j$ is unitary, its pointwise matrix norm is bounded uniformly. The convergent sequences $A_j,B_j$ are bounded in $H^k$, so the equation gives a uniform $H^1$ bound on $u_j$. If a uniform $H^r$ bound holds for some $0\le r\le k$, [F6] bounds both products on the right in $H^r$; hence $d{u}_j$ is bounded in $H^r$ and $u_j$ is bounded in $H^{r+1}$. Induction through $r=k$ gives a uniform $H^{k+1}$ bound.

Conversely, let $g\in SU(N)$ commute with every element of ${\mathrm{Hol}}_{p_m}(A)$. Transport $g$ from $p_m$ along any path to a point $x$ and conjugate it by the path's parallel-transport matrix. Two paths give the same value because their concatenation has holonomy commuting with $g$. This defines a parallel section $u$ with $u(p_m)=g$. To verify its endpoint regularity, fix $x_0$ and a coordinate ball $V$ with a smooth bundle trivialization. Choose a fixed piecewise smooth path from $x_m$ to $x_0$, then join $x_0$ to each $x\in V$ by radial paths that depend smoothly on $(t,x)$. On each varying segment the transport matrix solves $U(t,x)=U_0-{\int }_0^{t}C(s,x)U(s,x)ds$, where $C(s,x)=A_{{\gamma }_x(s)}({\dot{\gamma }}_x(s))$ is $C^1$ in $(s,x)$ because the local coefficients of $A$ are $C^1$. On a compact subball, for a coordinate direction $e_j$, the difference quotients $Q_h(t,x)=(U(t,x+h{e}_j)-U(t,x))\mathrm{/}h$ satisfy $Q_h(t,x)=-{\int }_0^{t}C(s,x+h{e}_j)Q_h(s,x)ds-{\int }_0^t\frac{C(s,x+h{e}_j)-C(s,x)}{h}U(s,x)ds\mathrm{.}$ The coefficients in this linear integral equation converge uniformly as $h\to 0$ to $C$ and ${\mathrm{\partial }}_{x_j}C$. The equation $V_j(t,x)=-{\int }_0^t(C(s,x)V_j(s,x)+{\mathrm{\partial }}_{x_j}C(s,x)U(s,x))ds$ has a unique continuous solution by successive approximation for the corresponding linear ODE. Because $\mathrm{\parallel }(C(s,x+h{e}_j)-C(s,x))\mathrm{/}h\mathrm{\parallel }$ is uniformly bounded by $\mathrm{\parallel }{\mathrm{\partial }}_{x_j}C{\mathrm{\parallel }}_{\mathrm{\infty }}$, the equation for $Q_h$ first gives $q_h(t)\le b+L{\int }_0^t{q}_h(s)ds$ with one $b$ independent of $h$. Iterating $n$ times gives $q_h(t)\le b{\sum }_{q=0}^{n-1}(Lt{)}^q\mathrm{/}q!+{\sup }_s{q}_h(s)(Lt{)}^n\mathrm{/}n!$; for each $h$, $q_h$ is bounded on the compact interval, so letting $n\to \mathrm{\infty }$ yields $q_h(t)\le b{e}^{LT}$ uniformly in $h$. Subtracting the equation for $V_j$ from that for $Q_h$ gives $f_h(t)\le a_h+L{\int }_0^t{f}_h(s)ds$, where $f_h(t):=\mathrm{\parallel }{Q}_h(t,x)-V_j(t,x)\mathrm{\parallel }$ and $a_h:=T(\mathrm{\parallel }C(\cdot ,x+h{e}_j)-C(\cdot ,x){\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{V}_j{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }(C(\cdot ,x+h{e}_j)-C(\cdot ,x))\mathrm{/}h-{\mathrm{\partial }}_{x_j}C{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }U{\mathrm{\parallel }}_{\mathrm{\infty }})\to 0$ uniformly on the compact cylinder. Iterating this inequality $n$ times gives $f_h(t)\le a_h{\sum }_{q=0}^{n-1}(Lt{)}^q\mathrm{/}q!+{\sup }_s{f}_h(s)(Lt{)}^n\mathrm{/}n!$. The already established uniform bound on $Q_h$ bounds the remainder; letting $n\to \mathrm{\infty }$ yields $f_h(t)\le a_h{e}^{Lt}\to 0$. Thus $Q_h\to V_j$ uniformly. The same difference estimate applied at nearby $x$, together with continuity of $C$, ${\mathrm{\partial }}_{x_j}C$, and $U$, shows $V_j$ is continuous in $(t,x)$. Hence $U$ is $C^1$ in the endpoint. Concatenating finitely many fixed and varying segments and changing smooth bundle frames preserve this regularity, so the path-independent section has $C^1$ local representatives. Since $M$ is compact, these representatives lie in $H^1$ in the finite-atlas norms. If $u\in H^r$ for some $1\le r\le k$, then the right side of $du=uA-Au$ is in $H^r$ by [F6], so $du\in H^r$ and $u\in H^{r+1}$. Induction gives $u\in H^{k+1}$, so $u\in {\mathcal{G}}_{k+1}$ and stabilizes $A$. Thus evaluation has exactly the stated image.

By [F4], a subsequence converges strongly in $H^k$ to a section $u$. The $H^k\hookrightarrow C^0$ embedding preserves the pointwise $SU(N)$ equations and the bundle transition law, so $u$ is a group-bundle section. In the gauge equation, $A_j\to A$, $B_j\to B$, and $u_j\to u$ in $H^k$; because $H^k$ is an algebra, $u_j{B}_j-A_j{u}_j\to uB-Au$ in $H^k$. Thus $d{u}_j$ converges in $H^k$ and the local coefficients converge in $H^{k+1}$. The limit satisfies $du=uB-Au$, equivalently $B=A^u$, and $u\in {\mathcal{G}}_{k+1}$.

On the standard ${\mathbb{C}}^N$, the holonomy matrices are unitary. If the representation is irreducible, [F5] makes every matrix commuting with the representation a scalar $\lambda I_N$. Membership in $SU(N)$ gives $\mathrm{\mid }\lambda \mathrm{\mid }=1$ and ${\lambda }^N=1$, hence its centralizer is $Z_N$. The stabilizer equality from steps 1.1 and 2.1 therefore gives full stabilizer $Z_N$ and trivial reduced stabilizer.

For each $1\le r<N$, represent an $r$-plane in ${\mathbb{C}}^N$ by its Hermitian orthogonal projection $Q$, so the Grassmannian is $\{Q:Q^{\ast }=Q,Q^2=Q,\mathrm{tr}Q=r\}$ in the finite-dimensional matrix space. Such projections have norm one. Regard any sequence of them as constant curves on $[0,1]$; [F3] gives a uniformly convergent subsequence, and the three defining equations pass to its limit. Thus each Grassmannian is compact. Suppose $A_j\to A$ and each $A_j$ is reducible. A subsequence has a fixed invariant dimension $r$, and by compactness another subsequence of invariant projections converges to $Q$. For a fixed based piecewise smooth loop $\gamma$, subdivide it into finitely many segments lying in bundle charts. On one segment, let $C_j(t)$ and $C(t)$ be the local matrix coefficients in the parallel-transport equations for $A_j$ and $A$. Since $H^k\hookrightarrow C^1$ by [F1], $A_j\to A$ in $H^k$ implies $C_j\to C$ uniformly. There is a common bound $\mathrm{\parallel }{C}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\le L$, and the transport matrices $U_j,U$ have norm one. Their integral equations, with the same initial matrix on the first segment, give for $W_j=U_j-U$ the inequality $\mathrm{\parallel }{W}_j(t)\mathrm{\parallel }\le T{\delta }_j+L{\int }_0^t\mathrm{\parallel }{W}_j(s)\mathrm{\parallel }ds$, where $T$ is the segment length and ${\delta }_j=\mathrm{\parallel }{C}_j-C{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$. Iterating as in step 2.1 yields ${\sup }_t\mathrm{\parallel }{W}_j(t)\mathrm{\parallel }\le T{e}^{LT}{\delta }_j\to 0$. At each later segment the initial matrices already converge, and the same inequality adds their vanishing initial difference to the right side. Induction over the finite subdivision proves ${\mathrm{Hol}}_{p_m}(A_j,\gamma )\to {\mathrm{Hol}}_{p_m}(A,\gamma )$. Since the holonomy matrices are unitary and preserve $\mathrm{ran}{Q}_j$, they commute with $Q_j$; passing to the limit shows that ${\mathrm{Hol}}_{p_m}(A,\gamma )$ commutes with $Q$ for every loop. Therefore $\mathrm{ran}Q$ is a nonzero proper invariant subspace and $A$ is reducible. The reducible locus is closed, so ${\mathcal{A}}_k^{\ast }$ is open.

The action map is continuous by the preceding Sobolev-calculus lemma. For a compact subset of ${\mathcal{A}}_k^{\ast }\times {\mathcal{A}}_k^{\ast }$, any sequence in its inverse image has a subsequence whose connection and transformed-connection coordinates converge in that compact set. Steps 1.2 and 2.2 give a further subsequence of gauge classes converging in ${\bar{\mathcal{G}}}_{k+1}$, while step 3.1 gives freeness. The inverse image is closed because the action map is continuous, so sequential compactness in these metrizable spaces is compactness. This proves properness. By step 3.1 the reduced action is free.