1.5. ISING AND FIBONACCI THEORY 21 This theorem is from [Jo3]. We have already seen two different normalizations of the Jones representation at r = 4 or level k = 2: the Jones normalization and the Kauffman bracket nor- malization. There is also a third normalization related to the γ-matrices: ργ A (σi) = e π 4 γi+1γi = 1 2 (1 + γi+1γi). The orders of ρJ A (σi), ργ A (σi), and ρA(σi) are 4, 8, and 16 respectively. Although all three normalizations have the same projective image, their linear images are in general different. For example, for the γ-matrix representation of PBn, instead of E1 n−1 , the image is E−1 n−1 [FRW]. In ν = 5 /2 FQH liquid, since the quasiparticle has charge e /4, the Nayak-Wilczek representation is the Jones normalization. In terms of γ-matrices, ρJ A (σi) = eπi/4ργ A (σi) = eπi/4e π 4 γi+1γi [NW]. In physics, the Jones braid group representation is understood as automor- phisms of Majorana fermions [I]: ρA(σi)(γj) J = ρA(σi)γjρA(σi)−1.JJ Then γi γi+1, γi+1 −γi, γj γj if j = i, i + 1. 1.5.2. Fibonacci theory. If A = ie2πi/20, k = 3, then the Jones algebroid, which is not modular, has label set L = {0, 1, 2, 3}. The subcategory consisting of only even labels {0, 2} is called the Fibonacci theory. The established notation for the two labels {0, 2} is {1,τ}. Let φ = 1+ 5 2 be the golden ratio. Label set: L = {1,τ} Fusion rules: τ 2 = 1 τ Quantum dimensions: {1,φ} Twist: θ1 = 1, θτ = e4πi/5 S-matrix: S = 1 2 + φ Å 1 φ φ −1 ã Braidings: R1 ττ = e−4πi/5, ττ = e3πi/5 F -matrices: τττ = Å φ−1 φ−1/2 φ−1/2 −φ−1 ã The Fibonacci theory is related to the Yang-Lee theory by a Galois conjugate. They have the same fusion rules, hence all degrees of the braid group representations are the same Fibonacci numbers. As a consequence of the density of the braid group representations of the Fibonacci theory, it is possible to design a universal TQC using τ’s. Fibonacci theory can also be realized directly using the quantum groups G2 and F4. The representation of B4 on Hom(1,τ ⊗4 ) has a basis 1 τ τ τ τ τ τ 1 1
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