Define Operators Corresponding to Cobordisms Only Iff Each Connected Component of the Cobordism has Non-empty Outgoing Boundary. Drunken Risibility.

Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is OYZ = ExtH(X)(H(Y), H(Z)) – the cohomology, as usual, has complex coefficients, and H(Y) and H(Z) are regarded as H(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that OYZ is dual to OZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then OYZ is a symmetric algebra, and is not self-dual as a vector space.)

We can do better by defining a cochain complex O’YZ of morphisms by

O’YZ = BΩ(X)(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and BA(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution

Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · ·  —– (2)

in which the differential is given by

dƒ(a1 ⊗ · · · ⊗ ak ⊗ b) = 􏰝a1 ƒ(a2 ⊗ · · · ⊗ ak ⊗ b) + ∑(-1)i ƒ(a1 ⊗ · · · ⊗ aiai+1 ⊗ ak ⊗ b) + (-1)k ƒ(a1 ⊗ · · · ⊗ ak-1 ⊗ akb) —– (3)

whose cohomology is ExtA(B,C). This is different from OYZ = ExtH(X)(H(Y), H(Z)), but related to it by a spectral sequence whose E2-term is OYZ and which converges to H(O’YZ) = ExtΩ(X)(Ω(Y), Ω(Z)). But more important is that H(O’YZ) is the homology of the space PYZ of paths in X which begin in Y and end in Z. To be precise, Hp(O’YZ) ≅ Hp+dZ(PYZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A category B’. The corresponding composition of homology groups

Hi(PYZ) × Hj(PZW) → Hi+j−dZ(PYW) —— (4)

is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product PYZ × PZW with the concatenation map M → PYW.

Now let’s attempt to fit the closed string cochain algebra C to this A category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension dX of X, so that the cohomology Hi(C) is potentially non-zero for −dX ≤ i < ∞. There is a map Hi(X) → H−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L0X, based at any chosen point in X.

The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H(O’YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.

Philosophical Equivariance – Sewing Holonomies Towards Equal Trace Endomorphisms.

In d-dimensional topological field theory one begins with a category S whose objects are oriented (d − 1)-manifolds and whose morphisms are oriented cobordisms. Physicists say that a theory admits a group G as a global symmetry group if G acts on the vector space associated to each (d−1)-manifold, and the linear operator associated to each cobordism is a G-equivariant map. When we have such a “global” symmetry group G we can ask whether the symmetry can be “gauged”, i.e., whether elements of G can be applied “independently” – in some sense – at each point of space-time. Mathematically the process of “gauging” has a very elegant description: it amounts to extending the field theory functor from the category S to the category SG whose objects are (d − 1)-manifolds equipped with a principal G-bundle, and whose morphisms are cobordisms with a G-bundle. We regard S as a subcategory of SG by equipping each (d − 1)-manifold S with the trivial G-bundle S × G. In SG the group of automorphisms of the trivial bundle S × G contains G, and so in a gauged theory G acts on the state space H(S): this should be the original “global” action of G. But the gauged theory has a state space H(S,P) for each G-bundle P on S: if P is non-trivial one calls H(S,P) a “twisted sector” of the theory. In the case d = 2, when S = S1 we have the bundle Pg → S1 obtained by attaching the ends of [0,2π] × G via multiplication by g. Any bundle is isomorphic to one of these, and Pg is isomorphic to Pg iff g′ is conjugate to g. But note that the state space depends on the bundle and not just its isomorphism class, so we have a twisted sector state space Cg = H(S,Pg) labelled by a group element g rather than by a conjugacy class.

We shall call a theory defined on the category SG a G-equivariant Topological Field Theory (TFT). It is important to distinguish the equivariant theory from the corresponding “gauged theory”. In physics, the equivariant theory is obtained by coupling to nondynamical background gauge fields, while the gauged theory is obtained by “summing” over those gauge fields in the path integral.

An alternative and equivalent viewpoint which is especially useful in the two-dimensional case is that SG is the category whose objects are oriented (d − 1)-manifolds S equipped with a map p : S → BG, where BG is the classifying space of G. In this viewpoint we have a bundle over the space Map(S,BG) whose fibre at p is Hp. To say that Hp depends only on the G-bundle pEG on S pulled back from the universal G-bundle EG on BG by p is the same as to say that the bundle on Map(S,BG) is equipped with a flat connection allowing us to identify the fibres at points in the same connected component by parallel transport; for the set of bundle isomorphisms p0EG → p1EG is the same as the set of homotopy classes of paths from p0 to p1. When S = S1 the connected components of the space of maps correspond to the conjugacy classes in G: each bundle Pg corresponds to a specific point pg in the mapping space, and a group element h defines a specific path from pg to phgh−1 .

G-equivariant topological field theories are examples of “homotopy topological field theories”. Using Vladimir Turaev‘s two main results: first, an attractive generalization of the theorem that a two-dimensional TFT “is” a commutative Frobenius algebra, and, secondly, a classification of the ways of gauging a given global G-symmetry of a semisimple TFT.

Definition of the product in the G-equivariant closed theory. The heavy dot is the basepoint on S1. To specify the morphism unambiguously we must indicate consistent holonomies along a set of curves whose complement consists of simply connected pieces. These holonomies are always along paths between points where by definition the fibre is G. This means that the product is not commutative. We need to fix a convention for holonomies of a composition of curves, i.e., whether we are using left or right path-ordering. We will take h(γ1 ◦ γ2) = h(γ1) · h(γ2).

A G-equivariant TFT gives us for each element g ∈ G a vector space Cg, associated to the circle equipped with the bundle pg whose holonomy is g. The usual pair-of-pants cobordism, equipped with the evident G-bundle which restricts to pg1 and pg2 on the two incoming circles, and to pg1g2 on the outgoing circle, induces a product

Cg1 ⊗ Cg2 → Cg1g2 —– (1)

making C := ⊕g∈GCg into a G-graded algebra. Also there is a trace θ: C1  → C defined by the disk diagram with one ingoing circle. The holonomy around the boundary of the disk must be 1. Making the standard assumption that the cylinder corresponds to the unit operator we obtain a non-degenerate pairing

Cg ⊗ Cg−1 → C

A new element in the equivariant theory is that G acts as an automorphism group on C. That is, there is a homomorphism α : G → Aut(C) such that

αh : Cg → Chgh−1 —– (2)

Diagramatically, αh is defined by the surface in the immediately above figure. Now let us note some properties of α. First, if φ ∈ Ch then αh(φ) = φ. The reason for this is diagrammatically in the below figure.

If the holonomy along path P2 is h then the holonomy along path P1 is 1. However, a Dehn twist around the inner circle maps P1 into P2. Therefore, αh(φ) = α1(φ) = φ, if φ ∈ Ch.

Next, while C is not commutative, it is “twisted-commutative” in the following sense. If φ1 ∈ Cg1 and φ2 ∈ Cg2 then

αg212 = φ2φ1 —– (3)

The necessity of this condition is illustrated in the figure below.

The trace of the identity map of Cg is the partition function of the theory on a torus with the bundle with holonomy (g,1). Cutting the torus the other way, we see that this is the trace of αg on C1. Similarly, by considering the torus with a bundle with holonomy (g,h), where g and h are two commuting elements of G, we see that the trace of αg on Ch is the trace of αh on Cg−1. But we need a strengthening of this property. Even when g and h do not commute we can form a bundle with holonomy (g,h) on a torus with one hole, around which the holonomy will be c = hgh−1g−1. We can cut this torus along either of its generating circles to get a cobordism operator from Cc ⊗ Ch to Ch or from Cg−1 ⊗ Cc to Cg−1. If ψ ∈ Chgh−1g−1. Let us introduce two linear transformations Lψ, Rψ associated to left- and right-multiplication by ψ. On the one hand, Lψαg : φ􏰀 ↦ ψαg(φ) is a map Ch → Ch. On the other hand Rψαh : φ ↦ αh(φ)ψ is a map Cg−1 → Cg−1. The last sewing condition states that these two endomorphisms must have equal traces:

TrCh 􏰌Lψαg􏰍 = TrCg−1 􏰌Rψαh􏰍 —– (4)

(4) was taken by Turaev as one of his axioms. It can, however, be reexpressed in a way that we shall find more convenient. Let ∆g ∈ Cg ⊗ Cg−1 be the “duality” element corresponding to the identity cobordism of (S1,Pg) with both ends regarded as outgoing. We have ∆g = ∑ξi ⊗ ξi, where ξi and ξi ru􏰟n through dual bases of Cg and Cg−1. Let us also write

h = ∑ηi ⊗ ηi ∈ Ch ⊗ Ch−1. Then (4) is easily seen to be equivalent to

∑αhii = 􏰟 ∑ηiαgi) —– (5)

in which both sides are elements of Chgh−1g−1.

Closed String Algebra as a Graded-Commutative Algebra C: Cochain Complex Differentials: Part 2, Note Quote.

The most general target category we can consider is a symmetric tensor category: clearly we need a tensor product, and the axiom HY1⊔Y2 ≅ HY1 ⊗ HY2 only makes sense if there is an involutory canonical isomorphism HY1 ⊗ HY2 ≅ HY2 ⊗ HY1 .

A very common choice in physics is the category of super vector spaces, i.e., vector spaces V with a mod 2 grading V = V0 ⊕ V1, where the canonical isomorphism V ⊗ W ≅ W ⊗ V is v ⊗ w ↦ (−1)deg v deg ww ⊗ v. One can also consider the category of Z-graded vector spaces, with the same sign convention for the tensor product.

In either case the closed string algebra is a graded-commutative algebra C with a trace θ : C → C. In principle the trace should have degree zero, but in fact the commonly encountered theories have a grading anomaly which makes the trace have degree −n for some integer n.

We define topological-spinc theories, which model 2d theories with N = 2 supersymmetry, by replacing “manifolds” with “manifolds with spinc structure”.

A spinc structure on a surface with a conformal structure is a pair of holomorphic line bundles L1, L2 with an isomorphism L1 ⊗ L2 ≅ TΣ of holomorphic line bundles. A spin structure is the particular case when L1 = L2. On a 1-manifold S a spinc structure means a spinc structure on a ribbon neighbourhood of S in a surface with conformal structure. An N = 2 superconformal theory assigns a vector space HS;L1,L2 to each 1-manifold S with spinc structure, and an operator

US0;L1,L2: HS0;L1,L2 → HS1;L1,L2

to each spinc-cobordism from S0 to S1. To explain the rest of the structure we need to define the N = 2 Lie superalgebra associated to a spin1-manifold (S;L1,L2). Let G = Aut(L1) denote the group of bundle isomorphisms L1 → L1 which cover diffeomorphisms of S. (We can identify this group with Aut(L2).) It has a homomorphism onto the group Diff+(S) of orientation-preserving diffeomorphisms of S, and the kernel is the group of fibrewise automorphisms of L1, which can be identified with the group of smooth maps from S to C×. The Lie algebra Lie(G) is therefore an extension of the Lie algebra Vect(S) of Diff+(S) by the commutative Lie algebra Ω0(S) of smooth real-valued functions on S. Let Λ0S;L1,L2 denote the complex Lie algebra obtained from Lie(G) by complexifying Vect(S). This is the even part of a Lie super algebra whose odd part is Λ1S;L1,L2 = Γ(L1) ⊕ Γ(L2). The bracket Λ1 ⊗ Λ1 → Λ0 is completely determined by the property that elements of Γ(L1) and of Γ(L2) anticommute among themselves, while the composite

Γ(L1) ⊗ Γ(L2) → Λ0 → VectC(S)

takes (λ12) to λ1λ2 ∈ Γ(TS).

In an N = 2 theory we require the superalgebra Λ(S;L1,L2) to act on the vector space HS;L1,L2, compatibly with the action of the group G, and with a similar intertwining property with the cobordism operators to that of the N = 1 case. For an N = 2 theory the state space always has an action of the circle group coming from its embedding in G as the group of fibrewise multiplications on L1 and L2. Equivalently, the state space is always Z-graded.

An N = 2 theory always gives rise to two ordinary conformal field theories by equipping a surface Σ with the spinc structures (C,TΣ) and (TΣ,C). These are called the “A-model” and the “B-model” associated to the N = 2 theory. In each case the state spaces are cochain complexes in which the differential is the action of the constant section of the trivial component of the spinc-structure.

Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.

A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way.

If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0

….

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 = 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

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:

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

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).

Cobordism. Note Quote.

Objects and arrows in general categories can be very different from sets and functions. The category which quantum field theory concerns itself with is called nCob, the ‘n dimensional cobordism category’, and the rough definition is as follows. Objects are oriented closed (that is, compact and without boundary) (n − 1)-manifolds Σ, and arrows M : Σ → Σ′ are compact oriented n-manifolds M which are cobordisms from Σ to Σ′. Composition of cobordisms M : Σ → Σ′ and N : Σ′ → Σ′′ is defined by gluing M to N along Σ′.

Let M be an oriented n-manifold with boundary ∂M. Then one assigns an induced orientation to the connected components Σ of ∂M by the following procedure. For x ∈ Σ, let (v1,…,vn−1,vn) be a positive basis for TxM chosen in such a way that (v1,…,vn−1) ∈ TxΣ. It makes sense to ask whether vn points inward or outward from M. If it points inward, then an orientation for Σ is defined by specifying that (v1, . . . , vn−1) is a positive basis for TxΣ. If M is one dimensional, then x ∈ ∂M is defined to have positive orientation if a positive vector in TxM points into M, otherwise it is defined to have negative orientation.

Let Σ and Σ′ be closed oriented (n − 1)-manifolds. An cobordism from Σ to Σ′ is a compact oriented n-manifold M together with smooth maps

∑ →i M i’←∑’

where i is a orientation preserving diffeomorphism of Σ onto i(Σ) ⊂ ∂M, i′ is an orientation reversing diffeomorphism of Σ′ onto i′(Σ′) ⊂ ∂M, such that i(Σ) and i′(Σ′) (called the in- and out-boundaries respectively) are disjoint and exhaust ∂M. Observe that the empty set φ can be considered as an (n − 1)-manifold.

One can view M as interpolating from Σ to Σ′. An important property of cobordisms is that they can be glued together. Let M : Σ0 → Σ1 and M′ : Σ1 → Σ2 be cobordisms,

0 →i0 M i1 ← ∑1             ∑1i’1 M i2 ← ∑2

Then we can form a composite cobordism M′ ◦ M : Σ → Σ by gluing M to M′ using i′ ◦ i − 1 : ∂M → ∂M′:

Are Categories Similar to Sets? A Folly, if General Relativity and Quantum Mechanics Think So.

The fundamental importance of the path integral suggests that it might be enlightening to simplify things somewhat by stripping away the knot observable K and studying only the bare partition functions of the theory, considered over arbitrary spacetimes. That is, consider the path integral

Z(M) = ∫ DA e (i ∫M S(A) —– (1)

where M is an arbitrary closed 3d manifold, that is, compact and without boundary, and S[A] is the Chern-Simons action. Immediately one is struck by the fact that, since the action is topological, the number Z(M) associated to M should be a topological invariant of M. This is a remarkably efficient way to produce topological invariants.

Poincaré Conjecture: If M is a closed 3-manifold, whose fundamental group π1(M), and all of whose homology groups Hi(M) are equal to those of S3, then M is homeomorphic to S3.

One therefore appreciates the simplicity of the quantum field theory approach to topological invariants, which runs as follows.

1. Endow the space with extra geometric structure in the form of a connection (alternatively a field, a section of a line bundle, an embedding map into spacetime)
2. Compute a number from this manifold-with-connection (the action)
3. Sum over all connections.

This may be viewed as an extension of the general principle in mathematics that one should classify structures by the various kinds of extra structure that can live on them. Indeed, the Chern-Simons Lagrangian was originally introduced in mathematics in precisely this way. Chern-Weil theory provides access to the cohomology groups (that is, topological invariants) of a manifold M by introducing an arbitrary connection A on M, and then associating to A a closed form f(A) (for instance, via the Chern-Simons Lagrangian), whose cohomology class is, remarkably, independent of the original arbitrary choice of connection A. Quantum field theory takes this approach to the extreme by being far more ambitious; it associates to a connection A the actual numerical value of the action (usually obtained by integration over M) – this number certainly depends on the connection, but field theory atones for this by summing over all connections.

Quantum field theory is however, in its path integral manifestation, far more than a mere machine for computing numbers associated with manifolds. There is dynamics involved, for the natural purpose of path integrals is not to calculate bare partition functions such as equation (1), but rather to express the probability amplitude for a given field configuration to evolve into another. Thus one considers a 3d manifold M (spacetime) with boundary components Σ1 and Σ2 (space), and considers M as the evolution of space from its initial configuration Σ1 to its final configuration Σ2:

This is known mathematically as a cobordism from Σ1 to Σ2. To a 2d closed manifold Σ we associate the space of fields A(Σ) living on Σ. A physical state Ψ corresponds to a functional on this space of fields. This is the Schrödinger picture of quantum field theory: if A ∈ A(Σ), then Ψ(A) represents the probability that the state known as Ψ will be found in the field A. Such a state evolves with time due to the dynamics of the theory; Ψ(A) → Ψ(A, t). The space of states has a natural basis, which consists of the delta functionals Â – these are the states satisfying ⟨Â|Â′⟩ = δ(A − A′). Any arbitrary state Ψ may be expressed as a superposition of these basis states. The path integral instructs us how to compute the time evolution of states, by first expanding them in the Â basis, and then specifying that the amplitude for a system in the state Â1 on the space Σ1 to be found in the state Â2 on the space Σ2 is given by:

〈Â2|U|Â1〉= ∫A | ∑2 = A2 A | ∑1 = A1 DA e i S[A] —– (2)

This equation is the fundamental formula of quantum field theory: ‘Perform a weighted sum over all possible fields (connections) living on spacetime that restrict to A1 and A2 on Σ1 and Σ2 respectively’. This formula constructs the time evolution operator U associated to the cobordism M.

In this way we see that, at the very heart of quantum mechanics and quantum field theory, is a formula which associates to every space-like manifold Σ a Hilbert space of fields A(Σ), and to every cobordism M from Σ1 to Σ2 a time evolution operator U(M) : Σ1 – Σ2. To specify a quantum field theory is nothing more than to give rules for constructing the Hilbert spaces A(Σ) and the rules (correlation functions) for calculating the time evolution operators U(M). This is precisely the statement that a quantum field theory is a functor from the cobordism category nCob to the category of Hilbert spaces Hilb.

A category C consists of a collection of objects, a collection of arrows f:a → b from any object a to any object b, a rule for composing arrows f:a → b and g : b → c to obtain an arrow g f : a → c, and for each object A an identity arrow 1a : a → a. These must satisfy the associative law f(gh) = (fg)h and the left and right unit laws 1af = f and f1a = f whenever these composites are defined. In many cases, the objects of a category are best thought of as sets equipped with extra structure, while the morphisms are functions preserving the structure. However, this is neither true for the category of Hilbert spaces nor for the category of cobordisms.

The fundamental idea of category theory is to consider the ‘external’ structure of the arrows between objects instead of the ‘internal’ structure of the objects themselves – that is, the actual elements inside an object – if indeed, an object is a set at all : it need not be, since category theory waives its right to ask questions about what is inside an object, but reserves its right to ask how one object is related to another.

A functor F : C → D from a category C to another category D is a rule which associates to each object a of C an object b of D, and to each arrow f :a → b in C a corresponding arrow F(f): F(a) → F(b) in D. This association must preserve composition and the units, that is, F(fg) = F(f)F(g) and F(1a) = 1F(a).

1. Set is the category whose objects are sets, and whose arrows are the functions from one set to another.

2. nCob is the category whose objects are closed (n − 1)-dimensional manifolds Σ, and whose arrows M : Σ1 → Σ2 are cobordisms, that is, n-dimensional manifolds having an input boundary Σ1 and an output boundary Σ2.

3. Hilb is the category whose objects are Hilbert spaces and whose arrows are the bounded linear operators from one Hilbert space to another.

The ‘new philosophy’ amounts to the following observation: The last two categories, nCob and Hilb, resemble each other far more than they do the first category, Set! If we loosely regard general relativity or geometry to be represented by nCob, and quantum mechanics to be represented by Hilb, then perhaps many of the difficulties in a theory of quantum gravity, and indeed in quantum mechanics itself, arise due to our silly insistence of thinking of these categories as similar to Set, when in fact the one should be viewed in terms of the other. That is, the notion of points and sets, while mathematically acceptable, might be highly unnatural to the subject at hand!

Catastrophe

Since, natural phenomena are continuously battered by perturbations, any thematic of the fundament that classifies critical points of smooth functions becomes the defining parameter of catastrophe. If a natural system is defined by a function of state variables, then the perturbations are represented by control parameters on which the function depends. An unfolding of a function is such a family: it is a smooth function of the state variables with the parameters satisfying a specific condition. Catastrophe’s aim is then to detect properties of a function by studying its unfoldings.

Thom studied the continuous crossing from a variety (space) to another, the connections through common boundaries and points between spaces even endowed with different dimensions (a research on the so called “cobordism” (1) which yielded him the Field Medal in 1958), until he singled out few universal forms, that is mathematical objects representing catastrophes or abrupt, although continuous, transitions of forms: specific singularities appearing when an object is submitted to bonds, such as restrictions with regard to its ordinary dimensions, that it accepts except in particular points where it offers resistance by concentrating there, so to say, its structure. The theory is used to classify how stable equilibria change when parameters are varied, with points in parameter space at which qualitative changes affects behavior termed catastrophe points. Catastrophe theory should apply to any gradient system where the force can be written as the negative gradient of a potential, and the points where the gradient vanishes are what the theory prefers degenerate points. There are seven elementary types of catastrophes or generic singularities of an application and Thom decided to study their applications in caustics, surfaces lit according to different angle shots, reflections and refractions. Initially catastrophe theory was of use just to explain caustic formation and only afterwards many other phenomena, but without yielding quantitative solutions and exact predictions, rather qualitatively framing situations that were uncontrollable by only reductionistic quantitative methods summing up elementary units. The study of forms in irregular, accidental and even chaotic situations had truly in advance led scientists like Poincaré and Hadamard, to single out structurally invariable catastrophic evolutions in the most disparate phenomena, in terms of divergences due to sensitive dependence on little variations of the initial conditions. In such cases there were not exact laws, rather evolutionary asymptotic tendencies, which did not allow exact predictions, in case only statistic ones. While when exact predictions are possible, in terms of strict laws and explicit equations, the catastrophe ceases.

For Thom, catastrophe was a methodology. He says,

Mathematicians should see catastrophe theory as just a part of the theory of local singularities of smooth morphisms, or, if they are interested in the wider ambitions of this theory, as a dubious methodology concerning the stability or instability of natural systems….the whole of qualitative dynamics, all the ‘chaos’ theories talked about so much today, depend more or less on it.

Thom gets more philosophical when it comes to the question of morphogenesis. Stability for Thom is a natural condition to place upon mathematical models for processes in nature because the conditions under which such processes take place can never be duplicated and therefore must be invariant under small perturbations and hence stable. what makes morphogenesis interesting for Thom is the fact that locally, as the transition proceeds, the parameter varies, from a stable state of a vector field to an unstable state and back to a stable state by means of a process which locally models system’s morphogenesis. Furthermore, what is observed in a process undergoing morphogenesis is precisely the shock wave and resulting configuration of chreods (2) separated by strata of the shockwave, at each interval of time and over intervals of observation time. It then follows “that to classify an observed phenomenon or to support a hypothesis about the local underlying dynamic, we need in principle only observe the process, study the observed catastrophe or discontinuity set and try to relate it to one of the finitely many universal catastrophe sets, which would become then our main object of interest. Even if a process depends on a large number of physical parameters, as long as it is described by the gradient model, its description would involve one of seven elementary catastrophes; in particular one can give a relatively simple mathematical description of such apparently complicated processes even if one does not know what the relevant physical parameters are or what the physical mechanism of the process is. According to Thom, “if we consider an unfolding, we can obtain a qualitative intelligence about the behaviors of a system in the neighborhood of an unstable equilibrium point. this idea was not accepted widely and was criticized by applied mathematicians because for them only numerical exactness allowed prediction and therefore efficient action. After the work of Grothendieck, it is known that the theory of singularity unfolding is a particular case of a general category, the theory of flat deformations of an analytic set and for flat local deformations of an analytic set only the hyper surface case has a smooth unfolding of finite dimension. For Thom, this meant the if we wanted to continue the scientific domain of calculable exact laws, we would be justified in considering the instance where an analytic process leads to a singularity of codimension one in internal variables. Might we then not expect that the process be diffused and subsequently propagated in the unfolding according to a mode that is to be defined? Such an argument allows one to think that the Wignerian domain of exact laws can be extended into a region where physical processes are no longer calculable but where analytic continuation remains qualitatively valid.

Anyway, catastrophe theory studies forms as qualitative discontinuities though on a continuous substrate. In any case forms as mental facts are immersed in a matter which is still a thought object. The more you try to analyze it the more it appears as a fog, revealing a more and more complex and inexhaustible weaving the more it refines itself through the forms it assumes. In fact complexity is more and more ascertained until a true enigma is reached when you once for all want to define reality as a universe endowed with a high number of dimensions and then object of mental experiences to which even objective phenomena are at the end concretely reduced. Concrete reality is yet more evident than a scientific explanation and naïve ontology appears more concrete than the scientific one. It is steady and universal, while the latter is always problematic and revisable. Besides, according to Bachelard, while naïve explanation is immediately reflected into the ordinary language which is accessible to everybody, the claimed scientific explanation goes with its jargon beyond immediate experience, away from the life world which only we can know immediately.

As for example, the continuous character of reality, which Thom entrusts to a world intuition as a frame of the phenomenological discontinuities themselves, is instead contradicted by the present tendency to reduce all to discrete units of information (bits) of modern computing. Of course it has a practical value: an animal individuating a prey perceives it as an entity which is absolutely distinct from its environment, just as we discretize linguistic phonemata to learn speaking without confounding them. Yet a continuous background remains, notwithstanding the tendency of our brains to discretize. Such background is for example constituted by space and time. Continuum is said an illusion as exemplified by a film which appears continuous to us, while it is made of discrete frames. Really it is an illusion but with a true mental base, otherwise it would not arise at all, and such base is just the existence of continuum. Really we perceive continuum but need discreteness, finiteness in order to keep things under control. Anyway quantum mechanics seems to introduce discreteness in absolute terms, something we do not understand but which is operatively valid, as is shown by the possibility to localize or delocalize a wave packet by simply varying the value distributions of complementary variables as position and momentum or time and energy, according to Heisenberg’s principle of indetermination. Anyway, also the apparent quantum discontinuity hides a continuity which, always according to Heisenberg’s principle, may be only obscured and not cancelled in several phenomena. It is difficult to conceive but not monstrous. The hypothesis according to which we are finite and discrete in our internal structure is afterwards false for we are more than that. We have hundreds billions of neurons, which are in continuous movement, as they are constituted by molecules continuously vibrating in the space, so giving place to infinite possible variations in a considerable dimensions number, even though we are reduced to the smallest possible number of states and dimensions to deal with the system under study, according to a technical and algorithmic thought which is operatively effective, certainly practically motivated but unidentifiable with reality.

(1) Two manifolds M and N are said to be cobordant if their disjoint union is the boundary of some other manifold. Given the extreme difficulty of the classification of manifolds it would seem very unlikely that much progress could be made in classifying manifolds up to cobordism. However, René Thom, in his remarkable, if unreadable, 1954 paper (French), gave the full solution to this problem for unoriented manifolds, as well as many powerful insights into the methods for solving it in the cases of manifolds with additional structure. The key step was the reduction of the cobordism problem to a homotopy problem, although the homotopy problem is still far from trivial. This was later generalized by Lev Pontrjagin, and this result is now known as the Thom-Pontrjagin theorem.

(2) Every natural process decomposes into structurally stable islands, the chreods. The set of chreods and the multidimensional syntax controlling their positions constitute the semantic model. When the chreod is considered as a word of this multidimensional language, the meaning (signification) of this word is precisely that of the global topology of the associated attractor (or attractors) and of the catastrophes that it (or they) undergo. In particular, the signification of a given attractor is defined by the geometry of its domain of existence on the space of external variables and the topology of the regulation catastrophes bounding that domain. One result of this is that the signification of a form (chreod) manifests itself only by the catastrophes that create or destroy it. This gives the axiom dear to the formal linguists: that the meaning of a word is nothing more than the use of the word; this is also the axiom of the “bootstrap” physicists, according to whom a particle is completely defined by the set of interactions in which it participates.