Definition. A connection background and its positive precision [fdip-006D]
AGENTDRAFTED
Fix a finite or compact Hausdorff group \(G\) and a finite-dimensional real or
complex representation \(\rho :G\to \operatorname {GL}(F)\), continuous
in the compact case. Supply a positive inner product for which \(\rho \)
is orthogonal or unitary. If necessary, start from any positive inner
product and average it:
\[\langle x,y\rangle _G
=\int _G\langle \rho (g)x,\rho (g)y\rangle _0\,dg.\]
Here \(dg\) is normalized right Haar measure; for a finite group it is the
uniform average. Haar existence and inversion from left to right are
given in [gleason2010haar, Theorem 4.3, pp. 6–9; Proposition 4.1, pp. 4–5].
Compactness makes its nonzero total mass finite, allowing normalization.
Right invariance gives invariance of the displayed
pairing. The integrand is continuous and strictly positive for
\(x=y\ne 0\), so the average is positive. The invariant metric is part
of the data used below.
On a finite incidence presentation, a
connection assigns \(U_e\in G\) to every chosen oriented
edge. Reversing that edge assigns \(U_{\bar e}=U_e^{-1}\).
Set \(H=F^V\) and \(K=F^E\), with the sums of the supplied fiber inner
products. Write \(U_e\) also for \(\rho (U_e)\) in operator formulas.
For scalar weights \(w_e>0\) and mass \(m>0\), define
\[\begin {aligned}
(d_Uf)(e)&=f(h(e))-U_ef(s(e)),\\
W&=\operatorname {diag}(w_eI_F),\\
L_U&=d_U^*Wd_U,\qquad Q_U=L_U+m^2I_H.
\end {aligned}\]
The adjoint uses these declared inner products. Directly,
\[\langle f,Q_Uf\rangle
=\sum _e w_e\|f(h(e))-U_ef(s(e))\|^2+m^2\|f\|^2.\]
Thus \(Q_U\ge m^2I_H>0\). This is the unitary connection version of
the difference energy; the adjoint qualification is also
explicit in [kenyon2011spanning, Section 3.3, p. 6].
At \(m=0\), the same sum shows
\[\ker L_U=\ker d_U
=\{f:f(h(e))=U_ef(s(e))\text { for every }e\}.\]
Its elements are called parallel sections. The unpinned
massless Gaussian is normalizable exactly when this space is zero:
then \(L_U\) is positive definite; otherwise the density is constant
along a nonzero kernel direction and its integral diverges. This is
separate from the pinned construction. Nontrivial group
holonomy need not eliminate all parallel vectors in a higher-dimensional
or nonfaithful representation.
Mathlib's LinearMap.ker_adjoint_comp_self gives
\(\ker (A^*A)=\ker A\) for finite-dimensional inner-product spaces.
Apply it to \(A=W^{1/2}d_U\); positive weights make \(W^{1/2}\)
invertible. This supports the kernel identity, not the connection,
holonomy or normalizability assertions.