Frobenius Algebras

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 S¹). 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(S¹) 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 Abrams work, which formulates the axioms for a Frobenius algebra in exactly the same way. The second way 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 Moore is to take advantage of the fact that it is relatively harmless to consider 2d cobordisms as embedded inside R³.

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:

1. (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.
2. (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 (e₁, . . . , e_n) of A. Then by assumption the functionals (e_i, ·) are a basis for A∗. Then there exist vectors e¹,…eⁿ in A such that (e_i, e^j) = δ^j_i. Define the copairing i by setting

1 →  ∑_i e_i ⊗ eⁱ

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

v = λⁱe_i → λiej ⊗ e^j ⊗ e_i → λⁱe_j(e^j, e_i) = λ_ie_i = v —– (1)

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

w = λ_ieⁱ → λ_ieⁱ ⊗ e_j ⊗ e^j → λ_i(eⁱ, e_j)e^j = λ_ieⁱ = w —– (2)

(b) ⇒ (a) . The  copairing  i  singles out a vector in A ⊗ A b y 1 → ∑ⁿ_i e_i ⊗ eⁱ for some vectors e_i, eⁱ ∈ 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 ⊗ eⁱ ⊗ v → e_i(eⁱ, v) —– (3)

By assumption this must be equal to v. This shows that (e₁, . . . , 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, eⁱ) = 0 ∀ i. But these scalars are exactly the coordinates in the ‘basis’ (e₁, . . . , 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 (e₁, . . . , e_n) for A, a corresponding dual basis (e¹, . . . , eⁿ), which satisfies e(e_i, e^j) = δ_i^j,and which can be recovered from the decomposition  i(1) = ∑_ie_i ⊗ eⁱ

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).

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