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A genus-two RT dimension from factorization

Statement

Assume full AC, inherited from the weighted RT state-space construction. For the established $ℓ = 4$ ribbon category and RT factorization, the unmarked genus-two state space has dimension $10$ .

Facts & Assumptions

Given: Full AC, $ℓ = 4$ and the modular data and factorization of RT torus operators and the state-space dimension formula.

[F1]

For $ℓ = 4$ , the labels are $0, 1, 2$ and $S_{0 x} = (1 / 2, 1 / \sqrt{2}, 1 / 2)$ (Truncated sl2 fusion data give the sine Hopf-link matrix).

[F2]

The unmarked genus- $g$ formula is $\dim V (Σ_{g}) = \sum_{x} S_{0 x}^{2 - 2 g}$ (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 full weighted RT functor and geometric factorization are established (The weighted Reshetikhin–Turaev TQFT and modular-functor factorization).

Proof

1.1F1F2A1givenalgebra

At genus $g = 2$ and with no marked labels, [F2] gives $\dim V (Σ_{2}) = \sum_{x = 0}^{2} S_{0 x}^{- 2} .$

2.1F1step 1.1algebra∎

The three terms are $(1 / 2)^{- 2} = 4$ , $(1 / \sqrt{2})^{- 2} = 2$ , and $(1 / 2)^{- 2} = 4$ . Their sum is $10$ .

Depends on

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Sources

Bakalov and Kirillov, Lectures on Tensor Categories and Modular Functor, §§3.3 and 5.8 (standard reference, not scraped)
Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity, §3.3 (standard reference, not scraped)