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

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Semisimple closed two-dimensional TQFT data

Definition

Fix an integer $r \geq 1$ and let $A = ⨁_{i = 1}^{r} C e_{i}$ , with coordinatewise multiplication $e_{i} e_{j} = δ_{i j} e_{i}$ and unit $1_{A} = \sum_{i = 1}^{r} e_{i}$ . Choose weights $w_{i} \in C^{\times}$ and define $ϵ (e_{i}) = w_{i}$ and $Δ (e_{i}) = w_{i}^{- 1} e_{i} \otimes e_{i}$ . The Frobenius pairing $b (a, b) = ϵ (a b)$ is diagonal in the basis $(e_{i})$ with entries $w_{i}$ ; $Δ$ is the inverse-pairing comultiplication. These maps define a finite-dimensional commutative Frobenius algebra over $C$ .

A closed oriented two-dimensional TQFT here is a strong symmetric monoidal functor from ${C o b}_{2}^{c l}$ to ${V e c t}_{f d} (C)$ . The objects of ${C o b}_{2}^{c l}$ are finite ordered words of oriented circles; its morphisms are compact oriented smooth surfaces with parametrized incoming and outgoing circle boundaries and no free boundary, modulo orientation-preserving diffeomorphism relative to those boundary parameterizations. Composition is collar gluing and the monoidal operation is ordered disjoint union. The supplied closed classification associates to $(A, μ, η, Δ, ϵ)$ a closed oriented two-dimensional TQFT. Put $H = μ \circ Δ$ . For the connected closed surface $Σ_{g}$ of genus $g \geq 0$ , it assigns $Z_{A} (Σ_{g}) = ϵ (H^{g} (η (1)))$ ; the cylinder acts by $1_{A}$ .

This is a closed-sector construction. It asserts no open sector, Cardy datum, or fully extended point-level theory. Countable choice is inherited only through the root surface-normal-form input to the classification.

Definition references

Closed oriented two-dimensional TQFTs are commutative Frobenius algebras

Facts & Assumptions

Given: $r \geq 1$ , the basis $e_{1}, \dots, e_{r}$ , the nonzero weights $w_{1}, \dots, w_{r}$ , and the four displayed structure maps in the Definition.

[A1]

$A$ is the finite direct sum of $r$ copies of $C$ with coordinatewise multiplication, so its dimension is exactly $r$ and $e_{i} e_{j} = δ_{i j} e_{i}$ .

[F1]

Closed oriented two-dimensional TQFTs are commutative Frobenius algebras: over any field, finite-dimensional commutative Frobenius algebras correspond to strong symmetric monoidal functors on the closed oriented two-dimensional cobordism category.

[F2]

Closed oriented two-dimensional TQFTs are commutative Frobenius algebras: a connected closed surface of genus $g$ is evaluated by the counit after $g$ copies of $H = μ Δ$ after the unit; the arity-one cylinder conventions are identities.

Proof

1.1givenA1algebra

Coordinatewise multiplication is associative and commutative, and $1_{A} = \sum_{i} e_{i}$ is a unit because $e_{i} 1_{A} = e_{i}$ . The pairing matrix is $diag (w_{1}, \dots, w_{r})$ , whose determinant $\prod_{i} w_{i}$ is nonzero since every $w_{i} \neq 0$ ; hence $b (a, b) = ϵ (a b)$ is nondegenerate.

1.2givenA1algebra

The formulas for $Δ$ and $ϵ$ are linear maps. For each $i$ , both counit composites give $w_{i}^{- 1} ϵ (e_{i}) e_{i} = e_{i}$ , and both coassociative composites give $w_{i}^{- 2} e_{i} \otimes e_{i} \otimes e_{i}$ . On a basis tensor $e_{i} \otimes e_{j}$ , each of the two Frobenius composites and $Δ μ$ equals $0$ if $i \neq j$ and equals $w_{i}^{- 1} e_{i} \otimes e_{i}$ if $i = j$ . By bilinearity the counit, coassociativity, and Frobenius identities hold on all inputs, so these maps form a commutative Frobenius algebra.

2.1givenF1F2step 1.1step 1.2∎

Apply [F1] to the finite-dimensional commutative Frobenius algebra established in steps 1.1–1.2. The resulting functor has the genus- $g$ evaluation and cylinder action stated in the Definition by [F2]; no positivity or open-sector hypothesis is used.

Sewing diagram

The following typed composition displays the handle operator used by the classification: $Δ$ splits one state into two tensor factors and $μ$ joins those factors. The diagram records the map types and composition only; the algebraic identities and surface classification above establish the result independently of the drawing.

Depends on

Closed oriented two-dimensional TQFTs are commutative Frobenius algebras