Write \(D\) for the positive diagonal operator on scalar functions with \(D_v=m^2+\sum _{e\ni v}w_e\), and write \(D_H\) for \(D\) acting separately on each fiber. The quadratic form in the connection background expands as \(\langle f,Q_Uf\rangle =\sum _v D_v\|f_v\|^2-\sum _e 2w_e\operatorname {Re}\langle f(h(e)),U_ef(s(e))\rangle \). The associated self-adjoint operator is \(Q_U=D_H-H_U\), where the only nonzero blocks of \(H_U\) are \(w_eU_e\) and \(w_eU_e^*\). Let \(K_U=D_H^{-1/2}H_UD_H^{-1/2}\). Then \(Q_U=D_H^{1/2}(I-K_U)D_H^{1/2}\), so \(\det Q_U=(\det D)^n\det (I-K_U)\). The identity connection has \(Q_I=Q_1\otimes I_F\), hence \((\det Q_1)^n=\det Q_I\). It is enough to compare \(I-K_U\) with \(I-K_I\).

The scalar adjacency \(A\) has the same off-diagonal weights. Its normalized operator \(K_1=D^{-1/2}AD^{-1/2}\) is self-adjoint, and \(K_I=K_1\otimes I_F\). The similar operator \(D^{-1}A\) has row sums \(\bigl (\sum _{e\ni v}w_e\bigr )/D_v<1\), because \(m>0\). The maximum absolute row sum is strictly less than \(1\), so the spectral radius of \(D^{-1}A\) is strictly less than \(1\). Since \(K_1=D^{1/2}(D^{-1}A)D^{-1/2}\), the operator \(K_1\) has the same spectral radius. Self-adjointness gives \(\|K_1\|=\rho (K_1)<1\).

The same comparison bounds the connection. For a section \(f\), set \(y_v=\|f_v\|\). Each fiber of \(H_U\) has operator norm \(w_e\), so \(\|(H_Uf)_v\|\le (Ay)_v\). Conjugating by \(D^{1/2}\) yields \(\|(K_Uf)_v\|\le (K_1y)_v\). Therefore \(\|K_Uf\|\le \|K_1y\|\le \|K_1\|\|f\|\), and \(\|K_U\|<1\). The eigenvalues of the self-adjoint operator \(K_U\) are real and lie in the open interval \((-1,1)\).

For every such eigenvalue, \(\log (1-\lambda )=-\sum _{k\ge 1}\lambda ^k/k\). Summing over eigenvalues gives \(\log \det (I-K_U)=-\sum _{k\ge 1}\frac 1k\operatorname {tr}(K_U^k)\). Expanding the trace in walks, a closed walk \(\gamma \) of length \(k\) contributes its positive weight \(w(\gamma )=\prod _{e\in \gamma }\frac {w_e}{\sqrt {D_{s(e)}D_{h(e)}}}\) times the trace of its holonomy. The holonomy is the ordered product of the fiber maps, with the first edge on the right, because parallel transport along \(e\) sends \(f(s(e))\) to \(U_ef(s(e))\). A unitary or orthogonal map has eigenvalues on the circle, so the real part of its trace is at most \(n\), with equality exactly when the map is the identity. The trace \(\operatorname {tr}(K_U^k)\) is real and equals the sum of those real parts against the walk weights. Hence \(\operatorname {tr}(K_U^k)\le \operatorname {tr}(K_I^k)=n\operatorname {tr}(K_1^k)\). Since each trace is at most the corresponding identity trace, \(\log \det (I-K_U)\ge \log \det (I-K_I)\). The logarithm is increasing, and \(\det (I-K_U)\ge \det (I-K_I)\). The powers of \(\det D\) cancel in the comparison. This is the asserted lower bound.

Equality of the determinants forces equality of the two series. For every \(k\ge 1\), \[\operatorname {tr}(K_U^k)=\operatorname {tr}(K_I^k).\] Their difference is the sum, over closed walks, of the positive weight times \(n-\operatorname {Re}\operatorname {tr}\operatorname {Hol}(\gamma )\). Each summand is nonnegative, so each vanishes. Every closed walk therefore acts as the identity on \(F\). Conversely, that condition makes every walk contribution agree with the identity connection, and the determinants agree.

These connections are exactly the identity orbit in the represented gauge group. Choose a root in each connected component and propagate \(\rho (U)\) along any walk from that root. The product is independent of the walk because every closed walk acts as the identity. The resulting vertex operators \(g_v\) satisfy \(U_e=g_{h(e)}g_{s(e)}^{-1}\). In the other direction, a gauge transform of the identity has determinant \(\det Q_I\) by the covariance already cited. A holonomy in the kernel of \(\rho \) is invisible to \(Q_U\): equality sees the action on \(F\), not a finer group element.