Gauge covariance of fusion-space braid actions

For three consecutive charges $A,B,C$, naturality of the braiding in its first variable, applied to $c_{A,B}:A\otimes B\to B\otimes A$, gives $c_{B\otimes A,C}\circ (c_{A,B}\otimes {\mathrm{id}}_C)=({\mathrm{id}}_C\otimes c_{A,B})\circ c_{A\otimes B,C}\mathrm{.}$ This is an equality from $((A\otimes B)\otimes C)$ to $C\otimes (B\otimes A)$. Reassociating the target by ${\alpha }_{C,B,A}^{-1}$ and expanding each $c_{X\otimes Y,Z}$ by the second hexagon in [F2] gives the two adjacent Artin composites $\begin{array}{rl}{\displaystyle L}& {\displaystyle =(c_{B,C}\otimes {\mathrm{id}}_A)\circ {\alpha }_{B,C,A}^{-1}\circ ({\mathrm{id}}_B\otimes c_{A,C})\circ {\alpha }_{B,A,C}\circ (c_{A,B}\otimes {\mathrm{id}}_C),}\\ {\displaystyle R}& {\displaystyle ={\alpha }_{C,B,A}^{-1}\circ ({\mathrm{id}}_C\otimes c_{A,B})\circ {\alpha }_{C,A,B}\circ (c_{A,C}\otimes {\mathrm{id}}_B)\circ {\alpha }_{A,C,B}^{-1}\circ ({\mathrm{id}}_A\otimes c_{B,C})\circ {\alpha }_{A,B,C}\mathrm{.}}\end{array}$ The cancelled factor is ${\alpha }_{C,B,A}^{-1}\circ {\alpha }_{C,B,A}=\mathrm{id}$; the remaining factors are exactly the displayed $L$ and $R$. Hence the adjacent braid relation holds as a typed equality of maps. Coherence [F7] identifies the canonical rebracketings of these three factors when they sit inside a longer tree; [F3] records the resulting equality in every fusion-tree block.

If two crossings involve disjoint adjacent pairs, reparenthesize the ordered factors as $W_L\otimes (A\otimes B)\otimes M\otimes (C\otimes D)\otimes W_R$, where $W_L,W_R$ collect the outer factors and $M$ is the (possibly empty) intervening tensor product. Let $g:A\otimes B\to B\otimes A$ and $h:C\otimes D\to D\otimes C$ be the two crossings. Suppressing only the canonical parentheses, bifunctoriality gives the typed identity $({\mathrm{id}}_{B\otimes A}\otimes {\mathrm{id}}_M\otimes h)\circ (g\otimes {\mathrm{id}}_M\otimes {\mathrm{id}}_{C\otimes D})=g\otimes {\mathrm{id}}_M\otimes h=(g\otimes {\mathrm{id}}_M\otimes {\mathrm{id}}_{D\otimes C})\circ ({\mathrm{id}}_{A\otimes B}\otimes {\mathrm{id}}_M\otimes h)\mathrm{.}$ Tensoring with the outer identities gives the same equality on the full word. Coherence [F7] transports it back to the chosen tree, so distant braid generators commute, including when one or more intervening factors are present.

For $n$ identical charges $a$, reassociate each object-level crossing $b_i$ to act on factors $i,i+1$, and let its fusion-space operator be $B_i(f)=f\circ b_i^{-1}$. The identity $B_{h\circ g}=B_h\circ B_g$ follows by composition: $B_h(B_g(f))=(f\circ g^{-1})\circ h^{-1}=f\circ (h\circ g{)}^{-1}$. Consequently the object-level relators proved in steps 1.1–1.2 hold for the $B_i$ as well. By [F5], the generator assignment extends uniquely to $\rho :B_n\to U(V)$. Reassociations are unitary by [F6], and the local exchange matrices are products of unitary F/R blocks from [F6]; hence each $B_i$ and every braid word is unitary.

In any parenthesized tree, let $G_T$ be the direct sum over admissible internal labels of the tensor products of the vertex gauge matrices. It is unitary because each factor is unitary and the tree basis is orthonormal by [F4]. The formulas in [F4] show that every F/R factor in a crossing matrix changes by inserting the inverse gauge at its target and the gauge at its source. Along a composite word, adjacent intermediate factors cancel, leaving $U_{\beta }^{\mathrm{\prime }}=G_{L^{\mathrm{\prime }}}^{-1}{U}_{\beta }{G}_L$. This also holds for colored words whose source and target lists differ.

For identical charges, $L^{\mathrm{\prime }}=L$ and the formula of step 2.2 is conjugation by the single unitary $G_L$, so ${\rho }^{\mathrm{\prime }}(\beta )=G_L^{-1}\rho (\beta )G_L$ for every $\beta \in B_n$. If ${\psi }^{\mathrm{\prime }}=G_L^{-1}\psi$ and $M^{\mathrm{\prime }}=G_L^{-1}M{G}_L$, then $⟨{\psi }^{\mathrm{\prime }},M^{\mathrm{\prime }}{\psi }^{\mathrm{\prime }}⟩=⟨G_L^{-1}\psi ,G_L^{-1}M\psi ⟩=⟨\psi ,M\psi ⟩$ because $G_L$ is unitary. Hence all expectation values, including measurement probabilities represented this way, are unchanged.