Definition. The finite Wilson action and counting law [fdip-006O]

✍️sourceAGENTDRAFTED

For the finite connection data and a fixed real \(\beta \ge 0\), define the Wilson action \[S_W(U)=\beta \sum _{p\in \mathcal P} \left (1-\frac {\operatorname {Re}\chi _\rho (U_p)}d\right ).\] Since \(\rho (U_p)\) is unitary, its eigenvalues have modulus one and \(-d\le \operatorname {Re}\chi _\rho (U_p)\le d\). Thus \(S_W(U)\ge 0\). The action is gauge invariant and does not depend on starting vertices or boundary orientations.

Use unnormalized counting measure on each edge group, and therefore on \(G^E\). The partition function, probability and expectation are \[\begin {aligned} Z_W&=\sum _{U\in G^E}e^{-S_W(U)},\\ \mathbb P_W(U)&=Z_W^{-1}e^{-S_W(U)},\\ \langle F\rangle _W&=Z_W^{-1}\sum _{U\in G^E}F(U)e^{-S_W(U)}. \end {aligned}\] These finite sums have \(0<Z_W<\infty \). Matrix-valued observables are averaged entry by entry. The finite group and chosen representation are inputs to this law, alongside the square complex and \(\beta \).

The constant term in the action matters for the partition function. With \(P=|\mathcal P|\), put \[\begin {aligned} \mathcal Z(\beta ) &=\sum _{U\in G^E}\exp \left ( \frac \beta d\sum _p\operatorname {Re}\chi _\rho (U_p)\right ),\\ Z_W(\beta )&=e^{-\beta P}\mathcal Z(\beta ). \end {aligned}\] The two weights define the same normalized expectation. Replacing each edge's counting measure by uniform probability measure instead gives \(|G|^{-|E|}Z_W\); that is a different partition-function normalization, again with the same expectation. In particular \(Z_W(0)=|G|^{|E|}\). The action and expectation agree with the conventions in [wingate2025lattice, equations (6.20)–(6.21), p. 8]; that source uses normalized Haar integration for its compact-group setting, whereas the finite counting convention here is explicit.