Finite holonomy witnesses for irreducibility

By [F1], ${\mathcal{A}}_{\mathrm{\infty }}^{\ast }$ is exactly the set of connections whose holonomy action has no common proper nonzero invariant subspace.

For each $V\in {\mathcal{G}}_N$, irreducibility supplies a based piecewise-smooth loop ${\mathrm{\ell }}_V$ whose holonomy $g_V={\mathrm{Hol}}_A({\mathrm{\ell }}_V)$ does not preserve $V$. If $P_V$ is the orthogonal projection onto $V$, then $(I-P_V)g_V{P}_V\mathrm{\ne }0$. For each fixed loop $\mathrm{\ell }$, the set of $V$ with $(I-P_V){\mathrm{Hol}}_A(\mathrm{\ell })P_V\mathrm{\ne }0$ is open, since the projector topology and matrix multiplication are continuous. These open sets cover ${\mathcal{G}}_N$.

Compactness in [F2] gives a finite subcover, hence finitely many loops ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_r$ for which no proper nonzero subspace is invariant under every $g_j={\mathrm{Hol}}_A({\mathrm{\ell }}_j)$. The case $r=0$ cannot cover the nonempty space ${\mathcal{G}}_N$, so $r\ge 1$.

Define the continuous function $h_A(P)={\sum }_{j=1}^r\mathrm{\parallel }(I-P)g_{j}P{\mathrm{\parallel }}_F$ on ${\mathcal{G}}_N$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_F$ is the Frobenius norm. It is positive at every $P$ by step 2.1. By [F3] it has a minimum $\delta >0$, and $h_A(P)\ge \delta$ for every $P\in {\mathcal{G}}_N$.

On each smooth segment of a chosen loop, in a smooth trivialization along that segment, parallel transport satisfies $U^{\mathrm{\prime }}=-B(t)U$, $U(a)=I_N$, where $B:[a,b]\to \mathfrak{s}\mathfrak{u}(N)$ is continuous. Identify $M_N(\mathbb{C})$ with ${\mathbb{R}}^{2N^2}$ by taking real and imaginary parts of all entries. Left multiplication $X\mapsto -B(t)X$ is then a real-linear endomorphism represented by a continuous matrix $C(t)\in M_{2N^2}(\mathbb{R})$. Regard the realified transport vector as the first column of a $2N^2\times 2N^2$ real matrix with its other columns zero; it satisfies the first-column equation of the square matrix IVP $Y^{\mathrm{\prime }}=C(t)Y$. Thus [F4] gives its unique solution on the entire compact segment, and the same realification applies to a nearby coefficient $B+b$. Since $b(t)\in \mathfrak{s}\mathfrak{u}(N)$, both $B$ and $B+b$ are in $\mathfrak{s}\mathfrak{u}(N)$. For either complex coefficient $B_0\in \{B,B+b\}$ and its solution $Y^{\mathrm{\prime }}=-B_{0}Y$, $Y(a)=I_N$, skew-Hermiticity gives $(Y^{\ast }Y{)}^{\mathrm{\prime }}=Y^{\ast }{B}_{0}Y-Y^{\ast }{B}_{0}Y=0$, and Jacobi's formula gives $(\det Y{)}^{\mathrm{\prime }}=-\mathrm{tr}(B_0)\det Y=0$. Thus $Y(t)\in SU(N)$. If $\tilde{U}$ solves the equation with coefficient $B+b$, put $W=U^{-1}\tilde{U}$. Then $W^{\mathrm{\prime }}=-U^{-1}bUW$, $W(a)=I_N$, and $W(t)\in SU(N)$. The integral equation and unitary invariance of the Frobenius norm give $\mathrm{\parallel }W(b)-I_N{\mathrm{\parallel }}_F\le {\int }_a^b\mathrm{\parallel }b(t){\mathrm{\parallel }}_{F}dt$. A $C^1$ perturbation of the connection makes each of the finitely many segment integrals arbitrarily small; finite composition therefore makes all $g_j$ vary continuously in the stated $C^1$ topology. Choose a neighborhood with $\mathrm{\parallel }{g}_j(A^{\mathrm{\prime }})-g_j(A){\mathrm{\parallel }}_F<\delta \mathrm{/}(2r)$ for every $j$.

For any orthogonal projection $P$, both $\mathrm{\parallel }P{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ and $\mathrm{\parallel }I-P{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ are at most $1$. Thus $\mid \mathrm{\parallel }(I-P){\mathrm{Hol}}_{A^{\mathrm{\prime }}}({\mathrm{\ell }}_j)P{\mathrm{\parallel }}_F-\mathrm{\parallel }(I-P)g_{j}P{\mathrm{\parallel }}_F\mid \le \mathrm{\parallel }{\mathrm{Hol}}_{A^{\mathrm{\prime }}}({\mathrm{\ell }}_j)-g_j{\mathrm{\parallel }}_F\mathrm{.}$ Summing over $j$ and using step 4.1 together with the uniform positive margin from step 3.1 gives $h_{A^{\mathrm{\prime }}}(P)>\delta \mathrm{/}2$ uniformly in $P$. Hence the same finite tuple has no common invariant proper nonzero subspace for every such $A^{\mathrm{\prime }}$.

A $C^1$ neighborhood on the compact base is specified by finitely many continuous section seminorms; each is continuous in the $C^{\mathrm{\infty }}$ section topology. Therefore the neighborhood in step 4.1 is also a $C^{\mathrm{\infty }}$ neighborhood, and ${\mathcal{A}}_{\mathrm{\infty }}^{\ast }$ is open in the $C^{\mathrm{\infty }}$ topology. Step 2.1 gives the finite holonomy witness at $A$.