DefinitionDefinition: AI-adaptedProof: Not applicableprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Wilson loop on a finite lattice

Definition

Assume ${A C}_{ω}$ and take finite oriented lattice data with finite vertex set $V$ , positively oriented edge set $E^{+}$ , endpoint maps $s, t$ , compact group $G$ , link variables $U_{e}$ for $e \in E^{+}$ , and vertex action $U_{e} \mapsto h_{s (e)} U_{e} h_{t (e)}^{- 1}$ . Write $\overline{E^{+}}$ for the disjoint set of formal reverse edges, with endpoints interchanged, and set $U_{\overset{ˉ}{e}} = U_{e}^{- 1}$ . Fix a finite-dimensional complex vector space $W$ with positive-definite Hermitian form $h$ and a smooth group homomorphism $ρ : G \to U (W, h)$ , where $U (W, h)$ is the group of invertible complex-linear maps preserving $h$ ; put $d = \dim_{C} W \geq 1$ . A based closed lattice loop is a finite edge word $C = (e_{1}, \dots, e_{ℓ})$ with each $e_{i} \in E^{+} ⊔ \overline{E^{+}}$ , of integer length $ℓ \geq 0$ , and a chosen base vertex $v_{0}$ , such that consecutive endpoints match and, when $ℓ > 0$ , the first and last endpoints are $v_{0}$ ; when $ℓ = 0$ , the loop consists only of $v_{0}$ . Its ordered holonomy is $U_{C} : = U_{e_{1}} \dots U_{e_{ℓ}}$ , in traversal order with the first edge on the left; for $ℓ = 0$ , set $U_{C} = e$ . The reversed word is $C^{- 1} = ({\overset{ˉ}{e}}_{ℓ}, \dots, {\overset{ˉ}{e}}_{1})$ . The normalized Wilson loop is

W_{ρ} (C; U) : = \frac{1}{d} {Tr}_{W} ⁣ (ρ (U_{C})) .

The loop is dimensionless. Changing the base vertex by a cyclic rebase conjugates $U_{C}$ and leaves $W_{ρ}$ unchanged; reversing the loop gives $W_{ρ} (C^{- 1}; U) = \overline{W_{ρ} (C; U)}$ . On a fixed finite lattice with a specified finite product Haar probability and bounded action, $⟨ W_{ρ} (C) ⟩_{Λ}$ is the ordinary normalized finite-dimensional integral. No renormalized continuum Wilson loop or continuum limit is defined here.