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Unitary Artin braid representation on fixed-charge fusion spaces

Statement

For a unitary braided fusion category $C$ , simple representatives $x, c \in Irr (C)$ as in Multiplicity-aware fusion-tree basis, $n \geq 2$ , and a nonzero fixed-total fusion space, the operators in Braid action on a fixed-charge fusion space define a unitary group homomorphism $ρ_{c} : B_{n} ⟶ U (V (x^{n} \to c)) .$ In particular, the adjacent Artin relations and distant commutation relations hold in each fixed-charge block.

Facts & Assumptions

Given: A unitary fusion category with a specified unitary braiding, the simple representatives $x, c$ , the nonzero fixed-charge fusion space $V (x^{n} \to c)$ , and the generator matrices $T_{i}^{- 1} R_{i} T_{i}$ .

[F1]

In a braided monoidal category, the local braidings on $X^{\otimes n}$ give a canonical homomorphism from the Artin group; the associator transports the action to a fixed parenthesization. An object of a braided category carries canonical braid actions

[F2]

The current QC25 bodies define finite semisimple orthogonal fusion-tree spaces, unitary associators and braidings, and unitary F/R coordinates; their pentagon and hexagon coordinate identities match the categorical coherence maps. The fusion-tree dimension lemma supplies complete bases. Unitary fusion category Multiplicity-aware fusion-tree basis Fusion-tree basis and dimension recursion Braiding F-symbol unitarity R-symbol unitarity Pentagon in fusion-tree coordinates Hexagon in F/R coordinates

Proof

1.1givenF2algebra

By [F2], each generator matrix is a product of unitary operators: $T_{i}$ and $T_{i}^{- 1}$ are unitary changes of orthonormal fusion-tree basis, and $R_{i}$ is unitary on every orthogonal pair-charge summand. Therefore $ρ_{c} (σ_{i})^{*} ρ_{c} (σ_{i}) = I$ and every generator, as well as its inverse, is unitary.

2.1F1F2step 1.1givenalgebra∎

By [F1], let $b_{i}$ be the local categorical braidings on the fixed parenthesized tensor power; they satisfy the Artin relations. On the multiplicity space $V = Hom (x^{\otimes n}, c)$ define $B_{b} (f) = f \circ b^{- 1}$ . For object automorphisms $g, h$ , $B_{h} (B_{g} (f)) = f \circ g^{- 1} \circ h^{- 1} = f \circ (h \circ g)^{- 1} = B_{h \circ g} (f)$ . Thus inverse precomposition transports the object-level relations to the $B_{b_{i}}$ . In the fusion-tree coordinates of [F2], reassociating to the pair-first tree, applying the pullback R-symbol, and reassociating back gives exactly $T_{i}^{- 1} R_{i} T_{i}$ from Braid action on a fixed-charge fusion space. Hence this generator assignment satisfies the Artin presentation and defines a group homomorphism, unitary by step 1.1.

Depends on

Used by

Projective braid gate and operator metric Definition
A triple braid acts identically on fusion spectators Lemma
Arbitrary-size triple-encoded braid universality Theorem
Arbitrary-size triple-encoded projective braid approximation Theorem

Cited to discharge well-definedness by Braid action on a fixed-charge fusion space.

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Sources

Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
John Preskill, Lecture Notes for Physics 219: Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)