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Wilson traces are invariant under vertex gauge transformations

Statement

For every finite SU(2) graph configuration, continuous finite-dimensional unitary representation $ρ$ , closed graph path $ℓ$ , and vertex gauge assignment $h$ , the Wilson trace of $ℓ$ is unchanged by $g_{e} \mapsto h_{t (e)} g_{e} h_{s (e)}^{- 1}$ .

Statement references

Finite SU(2) graph holonomy model

Facts & Assumptions

Given: The edge orientation, path ordering, representation, and vertex action of Definition Finite SU(2) graph holonomy model.

[F1]

Finite SU(2) graph holonomy model: each edge variable transforms as $g_{e}^{'} = h_{t (e)} g_{e} h_{s (e)}^{- 1}$ , and a path product is ordered with later edges on the left.

Proof

technique · direct cancellation

1.1givenF1algebra

The vertex action preserves the reverse-edge condition because $g_{\overset{ˉ}{e}}^{'} = h_{s (e)} g_{e}^{- 1} h_{t (e)}^{- 1} = (h_{t (e)} g_{e} h_{s (e)}^{- 1})^{- 1} = (g_{e}^{'})^{- 1}$ . Thus the transformed variables are again a graph configuration.

1.2givenF1algebra

If the path is empty, its product is $I_{2}$ before and after the gauge action, so its trace is unchanged. For the rest of the proof assume the closed path has $n \geq 1$ edges and write it as $e_{1} : v_{0} \to v_{1}, \dots, e_{n} : v_{n - 1} \to v_{n} = v_{0}$ .

2.1givenF1algebrastep 1.2

The transformed product is $g_{e_{n}}^{'} \dots g_{e_{1}}^{'} = h_{v_{0}} g_{e_{n}} h_{v_{n - 1}}^{- 1} h_{v_{n - 1}} g_{e_{n - 1}} \dots h_{v_{1}}^{- 1} h_{v_{1}} g_{e_{1}} h_{v_{0}}^{- 1} = h_{v_{0}} (g_{e_{n}} \dots g_{e_{1}}) h_{v_{0}}^{- 1}$ ; all adjacent intermediate factors cancel because $h_{v}^{- 1} h_{v} = I$ .

3.1givenF1step 2.1algebra∎

Applying the homomorphism $ρ$ carries this conjugation to conjugation by $ρ (h_{v_{0}})$ . For finite matrices, $Tr (A B) = \sum_{i, j} A_{i j} B_{j i} = Tr (B A)$ ; take $A = ρ (h_{v_{0}})$ and $B = ρ (g_{e_{n}} \dots g_{e_{1}}) ρ (h_{v_{0}})^{- 1}$ to obtain the same trace as before. Thus the Wilson trace is invariant.

Depends on

Finite SU(2) graph holonomy model