Theorem:

To give an open string theory is equivalent to giving a Frobenius algebra A inside Vect. To give a closed string theory is equivalent to giving a commutative Frobenius algebra B inside Vect.

The algebra A (B) is defined on the vector space which is the image under Z of the interval I (circle S^{1}). To prove that a open/closed string theory defines a Frobenius algebra on these vector spaces is easy, especially after one reformulates the definition of a Frobenius algebra in a categorical or ‘topological’ way. To prove the converse, that every Frobenius algebra arises as Z(I) or Z(S^{1}) for some open/closed Topological Quantum Field Theory (TQFT) Z is the more interesting result. There are three different ways of proving this fact.

The first and perhaps most modern way (elegantly set forth in * Kock’s work*) is to express 2Cob and OCob using generators and relations, and to use a result of

*, which formulates the axioms for a Frobenius algebra in exactly the same way. The*

**Abrams work***is to use the Atiyah-style definition of a TQFT, where the burden of proof is to show that, given a Frobenius algebra A, one can define the vectors Z(M) ∈ Z(∂M) in a consistent way, i.e. the definition is independent of the cutting of M into smaller pieces (this is called consistency of the sewing in conformal field theory). The third way has been implicitly suggested by*

**second way***is to take advantage of the fact that it is relatively harmless to consider 2d cobordisms as embedded inside R*

**Moore**^{3}.

Definition:

Frobenius algebras are classical algebras that were once, shamefully, called ‘Frobeniusean algebras’ in honour of the Prussian mathematician Georg Frobenius. They have many equivalent definitions; but before we list them it is worthwhile to record the following fact.

Lemma:

Suppose A is an arbitrary vector space equipped with a bilinear pairing ( , ) : A ⊗ A → C. Then the following are equivalent:

- (a) A is finite dimensional and the pairing is nondegenerate; i.e. A is finite dimensional and the map A → A∗ which sends v → (v, ·) is an isomorphism.
- (b) A is self dual in the rigid monoidal sense; i.e. there exists a copairing i : C → A ⊗ A which is dual to the pairing e : A ⊗ A → C given by e(a, b) = ε(ab).

Proof:

(a) ⇒ (b). Choose a basis (e_{1}, . . . , e_{n}) of A. Then by assumption the functionals (e_{i}, ·) are a basis for A∗. Then there exist vectors e^{1},…e^{n} in A such that (e_{i}, e^{j}) = δ^{j}_{i}. Define the copairing i by setting

1 → ∑_{i} e_{i} ⊗ e^{i}

Then a general vector v = λ^{i}e_{i} goes through the composite V →^{i⊗id} V ⊗ V ⊗ V →^{e⊗id} V – as:

v = λ^{i}e_{i} → λiej ⊗ e^{j} ⊗ e_{i} → λ^{i}e_{j}(e^{j}, e_{i}) = λ_{i}e_{i} = v —– (1)

Similarly, w = λ_{i}e^{i} goes through the composite V →^{id⊗i} V ⊗ V ⊗ V →^{e⊗id} V as:

w = λ_{i}e^{i} → λ_{i}e^{i} ⊗ e_{j} ⊗ e^{j} → λ_{i}(e^{i}, e_{j})e^{j} = λ_{i}e^{i} = w —– (2)

(b) ⇒ (a) . The copairing i singles out a vector in A ⊗ A b y 1 → ∑^{n}_{i} e_{i} ⊗ e^{i} for some vectors e_{i}, e^{i} ∈ A and some number n (note that we have not used finite dimensionality here). Now take an arbitrary v ∈ A and send it through the composite V →^{i⊗id} V ⊗ V ⊗ V →^{e⊗id} V:

v → e_{i} ⊗ e^{i} ⊗ v → e_{i}(e^{i}, v) —– (3)

By assumption this must be equal to v. This shows that (e_{1}, . . . , e_{n}) spans A, so A is finite dimensional. Now we show that v → (v, ·) is injective, and hence an isomorphism. Suppose (v, ·) is the zero functional. Then in particular (v, e^{i}) = 0 ∀ i. But these scalars are exactly the coordinates in the ‘basis’ (e_{1}, . . . , e_{n}), so that v = 0.

This lemma translates the algebraic notion of nondegeneracy into category language, and from now on we shall use the two meanings interchangeably. It also makes explicit that a nondegenerate pairing allows one to construct, from a basis (e_{1}, . . . , e_{n}) for A, a corresponding dual basis (e^{1}, . . . , e^{n}), which satisfies e(e_{i}, e^{j}) = δ_{i}^{j},and which can be recovered from the decomposition i(1) = ∑_{i}e_{i} ⊗ e^{i}

A Frobenius algebra is

(a) A finite dimensional algebra A equipped with a nondegenerate form (also called trace) ε : A → C.

(b) A finite dimensional algebra (A, β) equipped with a pairing β : A ⊗ A → C which is nondegenerate and associative.

(c) A finite dimensional algebra (A, γ) equipped with a left algebra isomorphism to its dual γ : A → A∗.

Observe that if A is an algebra, then there is a one-to-one correspondence between forms ε : A → C and associative bilinear pairings (·, ·) : A ⊗ A → C. Given a form, define the pairing by (a, b) = ε(ab), this is obviously associative. Given the pairing, define a form by ε(a) = (1,a) = (a,1); these are equal since the pairing is associative. This establishes the equivalence of (a) and (b).