Anyon fusion tree

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.