Anyon fusion rule ($a\times b={\sum }_c{N}_{ab}^{c}c$)

An anyon fusion rule specifies which total anyon charges $c$ can result from combining charges $a,b$. The nonnegative integers $N_{ab}^c$ are anyon fusion multiplicities. The vacuum anyon is the identity for this product, and $N_{ab}^1={\delta }_{b,\bar{a}}$ identifies the antianyon.

A matrix encoding multiplication by charge $b$ in the anyon fusion rule. These matrices commute in a braided monoidal category. On a sphere, the dimension of the anyon fusion space with puncture charges $a_1,\dots ,a_n$ and total vacuum anyon charge is $(N_{a_1}\cdots N_{a_n}{)}_{11}$.

The dual charge to an anyon $a$, characterized by $N_{a\bar{a}}^1=1$. Reversing the orientation of an anyon strand replaces its label by its antianyon. Its quantum dimension satisfies $d_{\bar{a}}=d_a$.

The identity charge of an anyon fusion rule: $1\times a=a$. In an anyon fusion tree, a strand of this charge may be erased using the unit conventions.

The Hilbert space of ways to combine the specified anyon charges to the specified total charge. For two inputs its dimension is the anyon fusion multiplicity $N_{ab}^c$. Repeated anyon fusion is described by an anyon fusion tree.

The unitary operator implementing a chosen positive exchange of adjacent anyons $a,b$ in total charge $c$. In a multiplicity-free theory it is a scalar. An inverse crossing uses its inverse. Two positive exchanges give the anyon monodromy $R_c^{ab}{R}_c^{ba}$.

The full double braiding of two anyons. The anyon balancing identity gives $M_c^{ab}={\theta }_c/({\theta }_a{\theta }_b)$ in a permitted anyon fusion channel. Unlike a single anyon R-matrix entry, this scalar is invariant under an anyon vertex gauge transformation.

An anyon whose full anyon monodromy with every other charge is the identity in every allowed anyon fusion channel. The vacuum anyon is transparent. Transparent charges and their anyon fusion summands form a symmetric subcategory of the braided monoidal category. They can have quantum dimension greater than one.

Apply the anyon balancing identity to the vacuum anyon channel in $a\times \bar{a}$. Transparency gives $1=({\theta }_a{\theta }_{\bar{a}}{)}^{-1}={\theta }_a^{-2}$. Thus a transparent anyon has bosonic or fermionic topological spin. This does not require its quantum dimension to equal one.

For transparent anyons $a,b$, the hexagon identities for braiding for braiding express double exchange of $a\otimes b$ with any charge $x$ as the successive double exchanges of $a$ and $b$ with $x$. Both are identities. Naturality restricts the resulting identity to every simple anyon fusion summand $c$ of $a\otimes b$, proving that $c$ is transparent.

A basis diagram recording an order for successively combining anyon charges, with every edge labeled by its intermediate charge and every vertex by a permitted anyon fusion rule. Changes of parentheses are implemented by anyon F-matrices.

The convention in which the value of an unknotted, untwisted charge-$a$ loop is its quantum dimension $d_a$. For multiplicity-free anyon fusion, a split-fuse bubble with input charges $a,b$ and total charge $c$ gives $\sqrt{d_a{d}_b/d_c}$ times the identity strand; the corresponding completeness resolution has coefficient $\sqrt{d_c/(d_a{d}_b)}$.

The convention in which the anyon fusion trees form an orthonormal basis and a splitting vertex followed by its adjoint fusion vertex gives the identity on the total-charge strand. A closed normalized state has norm one. It differs from the isotopy normalization of anyon diagrams, in which an unknotted closed charge-$a$ strand evaluates to $d_a$.

A change of the chosen orthonormal basis of every anyon fusion space. In a multiplicity-free theory each basis change is an independent wave phase $u_c^{ab}$. Splitting vertices acquire $u_c^{ab}$ and adjoint fusion vertices acquire its inverse. A scalar diagram is invariant under these choices when its splitting and fusion vertex factors cancel. A closed diagram with independently specified unmatched vertex tensors need not be invariant.

The unitary matrix relating the anyon fusion trees $((ab{)}_{u}c{)}_t$ and $(a(bc{)}_v{)}_t$. Fix the convention $|((ab{)}_{u}c{)}_t⟩={\sum }_v[F_t^{abc}{]}_{uv}|(a(bc{)}_v{)}_t⟩$. An inverse change of parentheses uses the adjoint matrix, so its entries are complex conjugates with exchanged indices.

The dimension of the anyon fusion space for two anyons $a,b$ with total charge $c$. A multiplicity-free theory has $N_{ab}^c\in \{0,1\}$.