PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Handle operators give explicit genus amplitudes

Statement

For the semisimple closed model defined by the stated idempotent data, let $H = μ \circ Δ$ and let $g \geq 0$ be an integer. The connected closed genus- $g$ amplitude is $Z_{A} (Σ_{g}) = \sum_{i = 1}^{r} w_{i}^{1 - g}$ . The cylinder acts by $1_{A}$ , and the torus amplitude is $r$ .

Statement references

Semisimple closed two-dimensional TQFT data

Facts & Assumptions

Given: The algebra $A = ⨁_{i = 1}^{r} C e_{i}$ , $e_{i} e_{j} = δ_{i j} e_{i}$ , $η (1) = \sum_{i} e_{i}$ , $ϵ (e_{i}) = w_{i} \neq 0$ , $Δ (e_{i}) = w_{i}^{- 1} e_{i} \otimes e_{i}$ , and $H = μ \circ Δ$ from Semisimple closed two-dimensional TQFT data.

[F1]

Semisimple closed two-dimensional TQFT data: the idempotents form a basis, $H = μ Δ$ , and the classification assigns $Z_{A} (Σ_{g}) = ϵ (H^{g} (η (1)))$ to a connected closed genus- $g$ surface.

[F2]

Semisimple closed two-dimensional TQFT data: the cylinder convention is $1_{A}$ , and $H^{0} = 1_{A}$ .

Proof

1.1givenF1algebra

For every $i$ , $H (e_{i}) = μ (w_{i}^{- 1} e_{i} \otimes e_{i}) = w_{i}^{- 1} e_{i}$ because $e_{i}^{2} = e_{i}$ .

2.1givenF1F2step 1.1algebra

Since $η (1) = \sum_{i} e_{i}$ , repeated application of the diagonal action in step 1.1 gives $H^{g} (η (1)) = \sum_{i} w_{i}^{- g} e_{i}$ for each integer $g \geq 0$ ; at $g = 0$ this is the identity convention in [F2].

3.1givenF1F2step 2.1algebra∎

Apply $ϵ$ and use $ϵ (e_{i}) = w_{i}$ to obtain $Z_{A} (Σ_{g}) = ϵ (H^{g} (η (1))) = \sum_{i} w_{i}^{1 - g}$ . The cylinder is $1_{A}$ by [F2], and for $g = 1$ every summand is $1$ , so the torus amplitude is $r$ .

Sewing diagram

The arrows are the unit map, the $g$ -fold handle operator, and the counit. Their composite is the scalar assigned to the closed surface; the diagram is a type-checked summary of the formula, not a proof premise.

Depends on

Semisimple closed two-dimensional TQFT data