Wilson loop is invariant under vertex gauge transformations

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $G$ be a group and let $(W,q)$ be a finite-dimensional complex Hermitian inner-product space. Define $U(W,q):=\{T\in {\mathrm{GL}}_{\mathbb{C}}(W):q(Tu,Tv)=q(u,v)\text{ for all }u,v\in W\}$, and let $\rho :G\to U(W,q)$ be a group homomorphism, with $d={\dim }_{\mathbb{C}}W\ge 1$. Let a finite oriented graph have vertices $\mathcal{V}$ and one chosen orientation ${\mathcal{E}}^{+}$ for each edge; for $e\in {\mathcal{E}}^{+}$ write $s(e),t(e)$ for its source and target. A link configuration assigns $U_e\in G$ to each $e\in {\mathcal{E}}^{+}$ and sets $U_{\bar{e}}=U_e^{-1}$. A based closed edge word $C=(e_1,\dots ,e_{\mathrm{\ell }})$ has length $\mathrm{\ell }\ge 0$, consecutive endpoints match, and when $\mathrm{\ell }>0$ it starts and ends at a chosen vertex $v_0$; for $\mathrm{\ell }=0$ it consists only of $v_0$. Put $U_C:=U_{e_1}\cdots U_{e_{\mathrm{\ell }}}$ in traversal order, with the empty product equal to the identity, and put $W_{\rho }(C;U):=d^{-1}{\mathrm{Tr}}_W(\rho (U_C))$. For every vertex tuple $k\in G^{\mathcal{V}}$ acting by $(k\cdot U{)}_e=k_{s(e)}{U}_e{k}_{t(e)}^{-1}$, one has $U_C(k\cdot U)=k_{v_0}{U}_C(U)k_{v_0}^{-1}$. Thus the normalized trace is gauge invariant and unchanged when the based word is cyclically rotated; for $\mathrm{\ell }>0$ reversal is $C^{-1}=({\bar{e}}_{\mathrm{\ell }},\dots ,{\bar{e}}_1)$, while the empty word reverses to itself, and $W_{\rho }(C^{-1};U)=\overline{W_{\rho }(C;U)}$.

Countable-choice assumption

Vertex gauge action on link variables

Ordered loop holonomy and normalized trace

Meyer, §1.2.3, printed pp. 11–12, on the endpoint transformation law of link transport and trace invariance

Lohmayer, §3.5.5, printed pp. 48–51, on traced closed lattice holonomy