ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Holonomy and discrete geometric interfaces — triangular graph Wilson trace

Statement

Take the oriented triangle $v_{0} \overset{e_{1}}{\to} v_{1} \overset{e_{2}}{\to} v_{2} \overset{e_{3}}{\to} v_{0}$ , assign $g_{e_{1}} = g_{e_{2}} = g_{e_{3}} = I_{2}$ , and use the defining representation $ρ : S U (2) \to U (C^{2})$ given by $ρ (g) v : = g v$ for column vectors $v \in C^{2}$ . Let $k = diag (i, - i) \in S U (2)$ and take the vertex gauge assignment $h_{v_{0}} = k$ , $h_{v_{1}} = h_{v_{2}} = I_{2}$ . The first and third edge matrices change to $k^{- 1}$ and $k$ , respectively, while the loop product remains $I_{2}$ and its Wilson trace remains $2$ .

Statement references

Finite SU(2) graph holonomy model

Wilson traces are invariant under vertex gauge transformations

SU(2) is a compact Hausdorff matrix group

Facts & Assumptions

Given: The directed triangle, identity edge assignment, defining representation, and vertex gauge rule of the finite graph model.

[F1]

Finite SU(2) graph holonomy model: $g_{e}^{'} = h_{t (e)} g_{e} h_{s (e)}^{- 1}$ and the loop product is later-left ordered.

[F2]

Wilson traces are invariant under vertex gauge transformations: closed-loop Wilson traces are invariant under every vertex gauge assignment.

[F3]

SU(2) is a compact Hausdorff matrix group: the matrices in $S U (2)$ form a group under matrix multiplication.

Proof

technique · direct verification

1.1givenF1F3algebra

For every $g \in S U (2)$ , $g^{†} g = I_{2}$ , so the defining action on $C^{2}$ preserves its standard inner product; matrix multiplication gives $ρ (g_{1} g_{2}) = ρ (g_{1}) ρ (g_{2})$ , and the matrix entries vary continuously in the Euclidean matrix topology. Thus each orbit map $g \mapsto ρ (g) v$ is continuous and $ρ$ is a strongly continuous finite-dimensional unitary representation as required by the model. The matrix $k$ is unitary and has determinant $i (- i) = 1$ , so $k \in S U (2)$ ; also $k \neq I_{2}$ and $k^{- 1} \neq I_{2}$ . By [F1], $g_{e_{1}}^{'} = I_{2} I_{2} k^{- 1} = k^{- 1}$ , $g_{e_{2}}^{'} = I_{2}$ , and $g_{e_{3}}^{'} = k I_{2} I_{2} = k$ . Thus individual edge matrices change.

2.1step 1.1F1F2F3algebra∎

Their later-left loop product is $g_{e_{3}}^{'} g_{e_{2}}^{'} g_{e_{1}}^{'} = k I_{2} k^{- 1} = I_{2}$ , whose defining-representation trace is $2$ . The original product is also $I_{2}$ and has trace $2$ ; this concrete calculation agrees with [F2].

Remarks

The directed graph and gauge action are depicted below. The arrows set the multiplication order; the displayed verification is the proof of the trace identity.

Holonomy and discrete geometric interfaces — triangular graph Wilson trace

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources