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

In closed string theory the central object is the vector space C = C_S¹ 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 U_Σ,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 U_Σ,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 U_Σ,μ varies with μ is as follows:

If M_Σ 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

U_Σ : C^⊗p → Ω^∗(M_Σ, C^⊗q)

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

commutes, where the lower horizontal arrow is induced by the map M_Σ1 × M_Σ2 → M_{Σ2 ◦ 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 – O_ab, 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 O_ab × O_bc → O_ac we have a family of ways of composing, parametrized by the contractible space of conformal structures on the surface of the figure:

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 O_ab × O_bc × O_cd → O_ad, 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 M_Σ 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 M_Σ. 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 X^C 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 C_X of cochains, and conversely one can associate a space X_C 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 C_A 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(F_A, Z) → Hom(R_A, Z) → Ext¹(A, Z) → 0

derived from an exact sequence

0 → R_A → F_A → 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.