The Closed String Cochain Complex C is the String Theory Substitute for the de Rham Complex of Space-Time. Note Quote.

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In closed string theory the central object is the vector space C = CS1 of states of a single parameterized string. This has an integer grading by the “ghost number”, and an operator Q : C → C called the “BRST operator” which raises the ghost number by 1 and satisfies Q2 = 0. In other words, C is a cochain complex. If we think of the string as moving in a space-time M then C is roughly the space of differential forms defined along the orbits of the action of the reparametrization group Diff+(S1) on the free loop space LM (more precisely, square-integrable forms of semi-infinite degree). Similarly, the space C of a topologically-twisted N = 2 supersymmetric theory, is a cochain complex which models the space of semi-infinite differential forms on the loop space of a Kähler manifold – in this case, all square-integrable differential forms, not just those along the orbits of Diff+(S1). In both kinds of example, a cobordism Σ from p circles to q circles gives an operator UΣ,μ : C⊗p → C⊗q which depends on a conformal structure μ on Σ. This operator is a cochain map, but its crucial feature is that changing the conformal structure μ on Σ changes the operator UΣ,μ only by a cochain homotopy. The cohomology H(C) = ker(Q)/im(Q) – the “space of physical states” in conventional string theory – is therefore the state space of a topological field theory.

A good way to describe how the operator UΣ,μ varies with μ is as follows:

If MΣ is the moduli space of conformal structures on the cobordism Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, then we have a cochain map

UΣ : C⊗p → Ω(MΣ, C⊗q)

where the right-hand side is the de Rham complex of forms on MΣ with values in C⊗q. The operator UΣ,μ is obtained from UΣ by restricting from MΣ to {μ}. The composition property when two cobordisms Σ1 and Σ2 are concatenated is that the diagram

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commutes, where the lower horizontal arrow is induced by the map MΣ1 × MΣ2 → MΣ2 ◦ Σ1 which expresses concatenation of the conformal structures.

For each pair a, b of boundary conditions we shall still have a vector space – indeed a cochain complex – Oab, but it is no longer the space of morphisms from b to a in a category. Rather, what we have is an A-category. Briefly, this means that instead of a composition law Oab × Obc → Oac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

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In particular, any two choices of a composition law from the family are cochain homotopic. Composition is associative in the sense that we have a contractible family of triple compositions Oab × Obc × Ocd → Oad, which contains all the maps obtained by choosing a binary composition law from the given family and bracketing the triple in either of the two possible ways.

This is not the usual way of defining an A-structure. According to Stasheff’s original definition, an A-structure on a space X consists of a sequence of choices: first, a composition law m2 : X × X → X; then, a choice of a map

m3 : [0, 1] × X × X × X → X which is a homotopy between

(x, y, z) ↦ m2(m2(x, y), z) and (x, y, z) ↦ m2(x, m2(y, z)); then, a choice of a map

m4 : S4 × X4 → X,

where S4 is a convex plane polygon whose vertices are indexed by the five ways of bracketing a 4-fold product, and m4|((∂S4) × X4) is determined by m3; and so on. There is an analogous definition – applying to cochain complexes rather than spaces.

Apart from the composition law, the essential algebraic properties are the non-degenerate inner product, and the commutativity of the closed algebra C. Concerning the latter, when we pass to cochain theories the multiplication in C will of course be commutative up to cochain homotopy, but, the moduli space MΣ of closed string multiplications i.e., the moduli space of conformal structures on a pair of pants Σ, modulo diffeomorphisms of Σ which are the identity on the boundary circles, is not contractible: it has the homotopy type of the space of ways of embedding two copies of the standard disc D2 disjointly in the interior of D2 – this space of embeddings is of course a subspace of MΣ. In particular, it contains a natural circle of multiplications in which one of the embedded discs moves like a planet around the other, and there are two different natural homotopies between the multiplication and the reversed multiplication. This might be a clue to an important difference between stringy and classical space-times. The closed string cochain complex C is the string theory substitute for the de Rham complex of space-time, an algebra whose multiplication is associative and (graded)commutative on the nose. Over the rationals or the real or complex numbers, such cochain algebras model the category of topological spaces up to homotopy, in the sense that to each such algebra C, we can associate a space XC and a homomorphism of cochain algebras from C to the de Rham complex of XC which is a cochain homotopy equivalence. If we do not want to ignore torsion in the homology of spaces we can no longer encode the homotopy type in a strictly commutative cochain algebra. Instead, we must replace commutative algebras with so-called E-algebras, i.e., roughly, cochain complexes C over the integers equipped with a multiplication which is associative and commutative up to given arbitrarily high-order homotopies. An arbitrary space X has an E-algebra CX of cochains, and conversely one can associate a space XC to each E-algebra C. Thus we have a pair of adjoint functors, just as in rational homotopy theory. The cochain algebras of closed string theory have less higher commutativity than do E-algebras, and this may be an indication that we are dealing with non-commutative spaces that fits in well with the interpretation of the B-field of a string background as corresponding to a bundle of matrix algebras on space-time. At the same time, the non-degenerate inner product on C – corresponding to Poincaré duality – seems to show we are concerned with manifolds, rather than more singular spaces.

Let us consider the category K of cochain complexes of finitely generated free abelian groups and cochain homotopy classes of cochain maps. This is called the derived category of the category of finitely generated abelian groups. Passing to cohomology gives us a functor from K to the category of Z-graded finitely generated abelian groups. In fact the subcategory K0 of K consisting of complexes whose cohomology vanishes except in degree 0 is actually equivalent to the category of finitely generated abelian groups. But the category K inherits from the category of finitely generated free abelian groups a duality functor with properties as ideal as one could wish: each object is isomorphic to its double dual, and dualizing preserves exact sequences. (The dual C of a complex C is defined by (C)i = Hom(C−i, Z).) There is no such nice duality in the category of finitely generated abelian groups. Indeed, the subcategory K0 is not closed under duality, for the dual of the complex CA corresponding to a group A has in general two non-vanishing cohomology groups: Hom(A,Z) in degree 0, and in degree +1 the finite group Ext1(A,Z) Pontryagin-dual to the torsion subgroup of A. This follows from the exact sequence:

0 → Hom(A, Z) → Hom(FA, Z) → Hom(RA, Z) → Ext1(A, Z) → 0

derived from an exact sequence

0 → RA → FA → A → 0

The category K also has a tensor product with better properties than the tensor product of abelian groups, and, better still, there is a canonical cochain functor from (locally well-behaved) compact spaces to K which takes Cartesian products to tensor products.

Frobenius Algebras

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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, . . . , en) of A. Then by assumption the functionals (ei, ·) are a basis for A∗. Then there exist vectors e1,…en in A such that (ei, ej) = δji. Define the copairing i by setting

1 →  ∑i ei ⊗ ei

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

v = λiei → λiej ⊗ ej ⊗ ei → λiej(ej, ei) = λiei = v —– (1)

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

w = λiei → λiei ⊗ ej ⊗ ej → λi(ei, ej)ej = λiei = w —– (2)

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

By assumption this must be equal to v. This shows that (e1, . . . , en) 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, . . . , en), 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, . . . , en) for A, a corresponding dual basis (e1, . . . , en), which satisfies e(ei, ej) = δij,and which can be recovered from the decomposition  i(1) = ∑iei ⊗ 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).