1.4. UNITARITY 17 of ∑ i di 2 that satisfies this identity, which is consistent with the S-matrix s00 = −√ 1 3 − φ Note that the Yang-Lee CFT is the minimal model M(2, 5). 1.4. Unitarity To apply the Jones algebroids to quantum physics, we need unitary theories. The definition of a unitary MTC can be found in Sec. 4.3 or [Tu]. In particular, all quantum dimensions must be positive real numbers. For Jones algebroids, when A = ±ie±2πi/4r, all quantum dimensions are positive, and the resulting MTCs are unitary. For specificity, we make the following choices: (1) When r is even, A = ie−2πi/4r, which is a primitive 4rth root of unity. (2) When r is odd and r = 1 mod 4, A = ie2πi/4r, which is a primitive 2rth root of unity. (3) When r is odd and r = 3 mod 4, A = ie−2πi/4r, which is also a primitive 2rth root of unity. When r is odd, the Jones algebroids are not modular. Theorem 1.28. (1) For any root of unity A, the Jones representation preserves the Markov pairing. (2) For the above choices of A, the Markov pairing is positive definite, hence the Jones representations are unitary. Proof. (1) Let σ ∈ Bn. It suﬃces to consider basis diagrams. We have ρA(σ)D1,ρA(σ)D2 = tr(D1ρA(σ)ρA(σ)D2) = tr(D1ρA(σ−1)ρA(σ)D2) = tr(D1D2) = D1,D2 . (2) Using Lem. 1.18, we can check that the Markov pairing on the basis in Lem. 1.19 is diagonal with positive norm. Although our theories are unitary, the F -matrices are not in general unitary. Unitary F -matrices are required for physical applications, hence we need to change bases to make the F -matrices unitary. Inspired by the Levin-Wen model [LW1], we choose the following normalizations: θu(a, b, c) = dadbdc. Since the norm of a trivalent vertex with colors a, b, c is Δ(a, b, c), our unitary normalization of a trivalent vertex is related to the default one by (1.29) a b c = √ dadbdc Δ(a, b, c) a b c

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