1.5. ISING AND FIBONACCI THEORY 19 {Ui} by ei = 1 d Ui. In the unitary cases, ei 2 = e† i = ei eiej = ejei, |i − j| ≥ 2 eiei±1ei = d−2ei Then ρA(σi) J = −1 + (1 + q)ei is the original Jones representation, where q = A−4. Since ei has eigenvalues among {0, 1}, ρJ A (σi) has eigenvalues among {−1,q}. In the Kauffman bracket, ρA(σi) = A · id + A−1Ui = −A ( −1 + (1 + q)ei ) . Hence ρA(σi) = −AρA(σi). J The representation of Bn given by σi → −A is abelian. Hence Jones representations ρA, ρA J in two different normalizations are projectively the same. But as linear representations, the orders of the braid generators are different. 1.5. Ising and Fibonacci theory Throughout the book, we will focus on two theories: Ising and Fibonacci. Besides their mathematical simplicity and beauty, they are conjectured physically to model non-abelian states in FQH liquids at filling fraction ν = 5 /2 and ν = 12 /5. 1.5.1. Ising theory. Ising theory is the A = ie−2πi/16, level k = 2 Jones algebroid. We will also call the resulting unitary MTC the Ising MTC, and the unitary TQFT the Ising TQFT. The label set for the Ising theory is L = {0, 1, 2}. It is related to the Witten-SU(2)-Chern-Simons theory at level k = 2, but not the same [RSW]. In physics, the three labels 0, 1, 2 are named 1,σ,ψ, and we will use this notation. The explicit data of the Ising theory: Label set: L = {1,σ,ψ} Fusion rules: σ2 = 1 + ψ, ψ2 = 1, ψσ = σψ = σ Quantum dimensions: d1 = 1, dσ = √ 2, dψ = 1 Twist: θ1 = 1, θσ = e2πi/16, θψ = −1 S-matrix: S = 1 2 Ñ 1 √ 2 1 2 0 − 2 1 − 2 1 é Braidings: Rψψ 1 = −1, Rψσ σ = Rσψ σ = −i R1 σσ = e−πi/8, Rψ σσ = e3πi/8 F -matrices: F σσσ σ = 1 √ 2 Å 1 1 1 −1 ã , F ψσψ σ = F σψσ ψ = −1 Currently non-abelian Ising anyons are closest to experimental realization: the non-abelian anyon σ is believed to be realized by half quantum vortex in p + ip superfluids [RG], and by the charge e /4 quasiparticle in ν = 5 /2 FQH liquids [MR, GWW]. The simple object ψ is a Majorana fermion, which has not been detected in physics. Ising theory is related to the Ising model in statistical mechanics, chiral superconductors, and FQH liquids at ν = 5 /2. For application to TQC, we need to know the closed images of the resulting braid group representations. In this regard,

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