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In a ribbon anyon theory, twisting a composite charge $c\in a\times b$ is equivalent to twisting each input and applying their full anyon monodromy. The topological spin obeys ${\theta }_1=1$ and ${\theta }_{\bar{a}}={\theta }_a$; the latter is the ribbon duality convention, with no complex conjugation of the scalar.

The framing of a link diagram obtained by pushing a component slightly in the plane of the diagram. Diagram curls change this framing; arbitrary removal of curls changes an anyon diagram by its topological spin.

The net chiral central charge of an edge theory. For a bosonic unitary modular tensor category, its residue modulo eight is determined by the topological spins and quantum dimensions through $e^{2\pi ic/8}={\mathcal{D}}^{-1}{\sum }_a{d}_a^2{\theta }_a$. This phase corrects the signature dependence in the Witten-Reshetikhin-Turaev invariant.

For a bosonic Abelian anyon theory, a Lagrangian subgroup of Abelian anyons is a subgroup of types of anyons closed under anyon fusion, with trivial topological spin for every member and trivial mutual braiding among members. Every anyon outside the subgroup must braid nontrivially with at least one member; pairwise compatibility alone is insufficient.

These conditions describe a maximal condensable collection for an appropriate fully gapped boundary. In the toric code, the vacuum together with the electric anyon gives one choice, and the vacuum together with the magnetic anyon gives another. Electric and magnetic anyons cannot both belong to the same collection because their mutual braiding is nontrivial. Additional physical context, including a nonchiral bulk and any imposed symmetry, matters when using this data to construct a boundary.^[ref]

Write $c_{u,v}$ for braiding and $M_{u,v}=c_{v,u}{c}_{u,v}$ for anyon monodromy. Transparency says $M_{a,x}=1$ for every $x$. The hexagon identities for braiding express braiding a composite $a\otimes b$ past $x$ as successive braidings of its two factors past $x$. If $a,b$ are both transparent anyons, move $x$ around $b$ and then around $a$; both full moves are identities, henceThis is the fusion closure of transparent anyons at the level of the composite object.

For completeness, let $i:c\to a\otimes b$ and $p:a\otimes b\to c$ be an inclusion and projection for an allowed anyon fusion summand, with $pi=1_c$. Naturality of braiding givesThusThe vacuum anyon belongs to this collection, and taking duals preserves transparency, so it is also closed under antianyons.

There is a further restriction on the spin of a transparent anyon. Apply the anyon balancing identity to the vacuum anyon channel of $a\times \bar{a}$:Here ribbon duality gives ${\theta }_{\bar{a}}={\theta }_a$ and ${\theta }_1=1$. ConsequentlyTheir topological spins are those of bosons or fermions. This does not imply $d_a=1$: a transparent anyon can have several anyon fusion channels, with nontrivial exchange action within them, even though every full anyon monodromy is trivial. If the theory is additionally a modular tensor category, the only transparent anyon is the vacuum anyon; Question 1 need not impose that extra hypothesis.

The two crossings of $j,h$ make their full anyon monodromy. For a transparent anyon $h$, this is the identity, so that pair of crossings can be removed together. No single-crossing exchange phase, and no topological spin from a twist, is introduced.

After this removal the upper and lower anyon fusion trees differ only in their intermediate labels. In the physics normalization of anyon diagrams, successive split-fuse bubbles give the Kronecker deltas ${\delta }_{bf}$ and ${\delta }_{cg}$. The remaining closed normalized state has norm one. Hence, for the admissible labelings of a,In particular, every admissible diagram satisfying the label matching in a evaluates to $1$. There is no factor $d_a$ here: the source explicitly permits the physics normalization of anyon diagrams, in which a closed normalized state has value $1$.

The composite is $f=e\times m$. Exchanging two composites exchanges the two $e$ constituents and the two $m$ constituents and produces one mutual winding between unlike constituents. The identical-particle phases are both $+1$, while the mutual contribution is $-1$, henceThe topological spin obeys $e^{2\pi i{s}_f}=e^{i{\theta }_{ff}}$, soThis is precisely the two-dimensional spin-statistics relation for a fermion, so the result is consistent.

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.

The topological spin ${\theta }_a$ is the phase produced by a $2\pi$ rotation of anyon $a$, equivalently its self-exchange phase in an Abelian anyon theory. The two-dimensional spin-statistics relation is ${\theta }_a=e^{2\pi i{s}_a}$.