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 Incoming links: Anyon balancing identity

M_{c}^{ab} = θ_{c} / (θ_{a} θ_{b})

in a permitted anyon fusion channel. Unlike a single anyon R-matrix entry, this scalar is invariant under an anyon vertex gauge transformation.

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

a \otimes b

past

x

as successive braidings of its two factors past

x

. If

a, b

x

around

b

and then around

a

; both full moves are identities, hence

M_{a \otimes b, x} = 1.

This is the fusion closure of transparent anyons at the level of the composite object.

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

p i = 1_{c}

M_{c, x} = (p \otimes 1_{x}) M_{a \otimes b, x} (i \otimes 1_{x}) = 1_{c \otimes x} .

Thus

a, b transparent and N_{ab}^{c} > 0 ⟹ c transparent .

The 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 \overset{a}{ˉ}

1 = R_{1}^{a \overset{a}{ˉ}} R_{1}^{\overset{a}{ˉ} a} = \frac{θ _{1}}{θ _{a} θ _{\overset{a}{ˉ}}} = \frac{1}{θ _{a}^{2}} .

Here ribbon duality gives

θ_{\overset{a}{ˉ}} = θ_{a}

and

θ_{1} = 1

. Consequently

θ_{a} = + 1 or - 1.

Their 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.

a \times \overset{a}{ˉ}

. Transparency gives

1 = (θ_{a} θ_{\overset{a}{ˉ}})^{- 1} = θ_{a}^{- 2}

. Thus a transparent anyon has bosonic or fermionic topological spin. This does not require its quantum dimension to equal one.