16 1. TEMPERLEY-LIEB-JONES THEORIES (5) F -matrix: i j k m l = n F ijk l nm i j k n l {F ijk l nm } can be determined by various tracings of the above identity. Definition 1.26. A RFC is a modular tensor category (MTC ) if det S = 0. Theorem 1.27. Let r ≥ 3. (1) If A is a primitive 4rth root of unity, then S is nondegenerate, hence VA is modular. (2) If r is odd, and A is a primitive 2rth root of unity, then ˜ = ˜ even ⊗( 1 1 1 1 ), where ˜ even is the submatrix of ˜ indexed by even labels. Furthermore, det ˜ even = 0. Part (1) is well-known, and part (2) can be found in [FNWW]. For each odd level k, all the even labels Leven = {0, 2,...,k − 1} form a closed tensor subcategory, which is modular. We denote this even subcategory by VA even . 1.3. Yang-Lee theory When A = eπi/5 for level k = 3, the even subtheory of VA has label set Leven = {0, 2}. We will rename them as 0 = 1, 2 = τ to conform to established notation. This will be our first nontrivial MTC. It corresponds to a famous non-unitary CFT in statistical mechanics, called the Yang-Lee singularity, hence its name. The data for this theory is summarized as below. Obvious data such as 1 ⊗ τ = τ and Fτ 1ττ = 1 are omitted φ = 1+ √ 5 2 is the golden ratio. Label set: L = {1,τ} Fusion rules: τ 2 = 1 ⊕ τ Quantum dimensions: {1, 1 − φ} Twist: θ1 = 1, θτ = e−2πi/5 S-matrix: S = −√ 1 3 − φ Å 1 1 − φ 1 − φ −1 ã Braidings: R1 ττ = e2πi/5, Rτ ττ = eπi/5 F -matrices: Fτ τττ = Å −φ 2 − φ −1 − 2φ φ ã We remark on a subtle point about central charge. The Yang-Lee CFT has central charge c = −22/5. It is known that the topological central charge ctop of the corresponding MTC satisfies the identity ctop = c mod 8 through p+ D = e πi 4 c where p±1 = ∑ i∈L θ±1d2 i i and D2 = ∑ i∈L d2 i [FG]. It is common to choose D as the positive root of ∑ i∈L di 2 . But for the Yang-Lee theory, it is the negative root

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