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The sl2 modular data at ell=4

Statement

Assume full AC, inherited from the weighted RT torus-space construction. For the established standard $ℓ = 4$ root-of-unity ribbon quotient, the labels are $0, 1, 2$ , the dimensions are $(1, \sqrt{2}, 1)$ , and

$S = (\begin{array}{ccc} \frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \\ \frac{1}{\sqrt{2}} & 0 & - \frac{1}{\sqrt{2}} \\ \frac{1}{2} & - \frac{1}{\sqrt{2}} & \frac{1}{2} \end{array}), (θ_{0}, θ_{1}, θ_{2}) = (1, e^{3 π i / 8}, - 1) .$

The fusion products are $1 \otimes 1 = 0 \oplus 2$ , $1 \otimes 2 = 1$ , and $2 \otimes 2 = 0$ ; the torus state dimension is $3$ .

Facts & Assumptions

Given: Full AC, $ℓ = 4$ , $q = e^{π i / 4}$ , and the positive-trace convention.

[F1]

The actual quotient has $d_{a} = \sin ((a + 1) π / 4) / \sin (π / 4)$ and $θ_{a} = q^{a (a + 2) / 2}$ by The standard root-of-unity sl2 tilting ribbon quotient is modular. Applying the conditional sine calculation to these established data gives $S_{a b} = \sqrt{1 / 2} \sin ((a + 1) (b + 1) π / 4)$ (Truncated sl2 fusion data give the sine Hopf-link matrix).

[F2]

The actual quotient has the truncated fusion rule of The rank-one tilting quotient is the truncated fusion category; at $ℓ = 4$ its retained range gives the three products in the Statement.

[F3]

The torus state dimension is $∣ I ∣$ under the proved TQFT factorization theorem (RT torus operators and the state-space dimension formula).

[A1]

Full AC (The Axiom of Choice) is assumed and supplies the inherited countable choice. The actual tilting quotient and full weighted RT functor are established by their exact suppliers (The standard root-of-unity sl2 tilting ribbon quotient is modular, The weighted Reshetikhin–Turaev TQFT and modular-functor factorization).

Proof

1.1F1givenalgebra

Evaluate the sine dimension formula at $a = 0, 1, 2$ : it gives $1, \sqrt{2}, 1$ . The twist exponents $a (a + 2) / 2$ are $0, 3 / 2, 4$ , so $θ = (1, e^{3 π i / 8}, e^{π i}) = (1, e^{3 π i / 8}, - 1)$ . Substitution into the sine matrix formula gives the displayed entries.

2.1F2F3A1step 1.1algebra∎

The truncated fusion rule gives $1 \otimes 1$ outputs $0, 2$ , $1 \otimes 2$ output $1$ , and $2 \otimes 2$ output $0$ , each with multiplicity one. Since there are three simple labels, [F3] gives torus dimension $3$ .

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Sources

Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §3.3 (standard reference, not scraped)
Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §3.3 (standard reference, not scraped)