Frobenius sewing and exact two-dimensional topological amplitudes

Fix an integer $r\ge 1$ and let $A={⨁}_{i=1}^r\mathbb{C}{e}_i$, with coordinatewise multiplication $e_i{e}_j={\delta }_{ij}{e}_i$ and unit $1_A={\sum }_{i=1}^r{e}_i$. Choose weights $w_i\in {\mathbb{C}}^{\times }$ and define $ϵ(e_i)=w_i$ and $\mathrm{\Delta }(e_i)=w_i^{-1}{e}_i\otimes e_i$. The Frobenius pairing $b(a,b)=ϵ(ab)$ is diagonal in the basis $(e_i)$ with entries $w_i$; $\mathrm{\Delta }$ is the inverse-pairing comultiplication. These maps define a finite-dimensional commutative Frobenius algebra over $\mathbb{C}$.

A closed oriented two-dimensional TQFT here is a strong symmetric monoidal functor from ${\mathsf{C}\mathsf{o}\mathsf{b}}_2^{\mathrm{c}\mathrm{l}}$ to ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathrm{f}\mathrm{d}}(\mathbb{C})$. The objects of ${\mathsf{C}\mathsf{o}\mathsf{b}}_2^{\mathrm{c}\mathrm{l}}$ 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 $(A,\mu ,\eta ,\mathrm{\Delta },ϵ)$ a closed oriented two-dimensional TQFT. Put $H=\mu \circ \mathrm{\Delta }$. For the connected closed surface ${\mathrm{\Sigma }}_g$ of genus $g\ge 0$, it assigns $Z_A({\mathrm{\Sigma }}_g)=ϵ(H^g(\eta (1)))$; 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.

Closed oriented two-dimensional TQFTs are commutative Frobenius algebras

The following typed composition displays the handle operator used by the classification: $\mathrm{\Delta }$ splits one state into two tensor factors and $\mu$ 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.

For the semisimple closed model defined by the stated idempotent data, let $H=\mu \circ \mathrm{\Delta }$ and let $g\ge 0$ be an integer. The connected closed genus-$g$ amplitude is $Z_A({\mathrm{\Sigma }}_g)={\sum }_{i=1}^r{w}_i^{1-g}$. The cylinder acts by $1_A$, and the torus amplitude is $r$.

Semisimple closed two-dimensional TQFT data

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.

For the semisimple closed model determined by the stated idempotent data, define $a^{\ast }={\sum }_i\overline{a_i}{e}_i$ when $a={\sum }_i{a}_i{e}_i$, and set $h(a,b)=ϵ(a^{\ast }b)$. The form $h$ is a Hermitian positive-definite inner product exactly when every weight $w_i$ is a positive real number. The algebraic closed oriented TQFT exists for every nonzero complex weight; positivity is an additional condition for this specified Hilbert-space interpretation.

Semisimple closed two-dimensional TQFT data

For the basis $(e_i)$, the associated sesquilinear form has diagonal matrix $\mathrm{diag}(w_1,\dots ,w_r)$. Positivity is a property of this explicit matrix and does not follow from the existence of the oriented closed TQFT.