Fixed-boundary anyonic fusion-path space

Let $\mathcal{C}$ be a fusion category over $\mathbb{C}$, so it is finite semisimple, rigid, monoidal, and has simple unit Fusion and multifusion categories. Let $a,b,c$ be simple objects and let $n\ge 0$. Fix one representative of each simple isomorphism class for the internal path labels; the external objects $a,b,c$ need not equal their representatives. Set $T_0=b$ and recursively fix the left-associated parenthesization $T_{j+1}:=T_j\otimes a$. The fixed-boundary anyonic path space is

$H_{a,n;b,c}:={\mathrm{Hom}}_{\mathcal{C}}(c,T_n)\mathrm{.}$

For simple labels $x,y$, define the split multiplicity space $W_{x,a}^y:={\mathrm{Hom}}_{\mathcal{C}}(y,x\otimes a)$. For $n\ge 1$, a fusion path is a sequence of simple labels $x_0=b,x_1,\dots ,x_n=c$, with $x_1,\dots ,x_{n-1}$ drawn from the fixed representative set, together with, for every $1\le j\le n$, one basis index ${\mu }_j$ in a chosen basis $B_{x_{j-1},a}^{x_j}$ of $W_{x_{j-1},a}^{x_j}$. The basis index is retained even when distinct indices have the same adjacent labels. Fusion rules are the nonnegative integers $N_{xa}^y$ in $[x][a]={\sum }_y{N}_{xa}^y[y]$ Fusion rules.

For $n=0$, there is no path if $b\cong ̸c$. If $b\cong c$, fix an isomorphism $u_{c,b}:c\to b$ and take it as the single empty-path vector (use $u_{b,b}={\mathrm{id}}_b$ when the objects coincide). In the unitary setting choose this isomorphism unitary.

For $n\ge 1$, if $i_j\in B_{x_{j-1},a}^{x_j}$ is the selected inclusion $x_j\to x_{j-1}\otimes a$, define the path composite by ${\gamma }_0:={\mathrm{id}}_b$ and ${\gamma }_j:=({\gamma }_{j-1}\otimes {\mathrm{id}}_a)\circ i_j:x_j\to T_j$. For $x_n=c$, ${\gamma }_n$ is a vector in $H_{a,n;b,c}$. The next lemma proves these composites form a basis and computes the dimension. No Hilbert-space or physical Hamiltonian structure is part of this definition.

For three sites the diagram records the split direction $c\to b\otimes a\otimes a\otimes a$ and the recursively parenthesized tree. Each vertex also carries its basis index ${\mu }_j$; the picture omits no mathematical datum used in the definition.

Each internal label $x_j$ is simple, and each ${\mu }_j$ ranges over a basis of the displayed split Hom space. A periodic closed chain is obtained by summing the fixed-boundary spaces over representative boundary labels $b=c$; its path labels satisfy the cyclic boundary condition.