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Ising fusion data and chiral convention

Example

Use simple labels $1, ψ, σ$ with unit fusion and $σ \otimes σ = 1 \oplus ψ$ , $σ \otimes ψ = ψ \otimes σ = σ$ , and $ψ \otimes ψ = 1$ . In the three- $σ$ , total- $σ$ fusion space, the intermediate-channel basis is $(1, ψ)$ . In the pullback R convention of R-symbol as a channel exchange map, fix

$F_{σ}^{σ σ σ} = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}), R_{1}^{σ σ} = e^{- i π / 8}, R_{ψ}^{σ σ} = e^{3 i π / 8} .$

These are sector data only. This item makes no claim about an all-label Ising F/R table or global Ising ribbon coherence.

The phases above use the pullback convention in R-symbol as a channel exchange map. In a convention recording the eigenvalues of the object-level braiding itself, the eigenvalues are the inverses of the displayed phases. This convention note does not extend the sector data to any other Ising fusion space.

Facts & Assumptions

Given: the fusion rules and three- $σ$ , total- $σ$ matrices in the Example.

[F1]

Fusion multiplicity space: $N_{a b}^{c}$ is the dimension of $V_{a b}^{c} = Hom (a \otimes b, c)$ .

[F2]

F-symbol as a recoupling matrix: $F$ is the coordinate matrix between the two orthonormal tree bases.

[F3]

R-symbol as a channel exchange map: $R$ is the channel-exchange matrix in the stated pullback convention.

[F4]

Unitary fusion category: unitarity means preservation of the positive inner products on fusion spaces.

Verification

1.1F1givenalgebra

The three- $σ$ , total- $σ$ space has one left-tree basis vector with intermediate charge $1$ and one with intermediate charge $ψ$ : each channel in $σ \otimes σ = 1 \oplus ψ$ fuses with the remaining $σ$ to $σ$ . Thus this sector is two-dimensional, with the basis stated in the Example.

1.2F2F4givenalgebra

Let $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ . It is real and symmetric, and direct multiplication gives $H^{†} H = H^{2} = I_{2}$ . Hence the displayed F matrix is unitary in the stated basis.

1.3F3F4givenalgebra

Both displayed R entries have modulus one, and their ratio is $e^{3 i π / 8} / e^{- i π / 8} = e^{i π / 2} = i$ . Thus each channel exchange is unitary and the relative phase is exactly $i$ .

2.1step 1.1step 1.2step 1.3∎

Steps 1.1–1.3 verify the dimension, recoupling matrix, and exchange phases only in this three- $σ$ , total- $σ$ sector. No all-label associator, pentagon, hexagon, modularity, or ribbon claim is made here.

Depends on

Used by

Ising–Majorana exchange interface Definition
Three-Ising-anyon braid matrices are Clifford Example
Worked Fibonacci and Ising braid matrices Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)
David Penneys, Unitary Fusion Categories, Math 8800 lecture notes (standard reference, not scraped)