Topological Field Theories

Topological Field Theories RTG Graduate Summer School Geometry of Quantum Fields and Strings University of Pennsylvania, Philadelphia June 15-19, 2009 Lectures by John Francis Notes by Alberto Garcia-Raboso 1 Cobordism Historically, it all began with cobordism, before topological field theories were even defined. Pontrjagin invented cobordism theory in the 1930 s in order to compute homotopy groups of spheres. Back then they knew that π i S n = 0 for i < n (proof by cellular approximation) and that π n S n = Z, classified by degree (for n = 1 it is the winding number; the (n 1)th suspension takes π 1 S 1 = πn S n ). For spheres of different dimensions, there are no obvious invariants: a map S n S m for n m is zero at the level of homology. Let f : S n+k S n be a smooth map (can always homotope to one) and x S n a regular value of f (surjective on tangent spaces; the set of regular values is a dense subset of S n by Sard s theorem) so that f 1 (x) is a smooth submanifold of S n+k. Take a homotopy H between two regular values x and y of f and pull back to S n+k : f 1 H(0) f 1 H([0, 1]) f 1 H(1) {0} [0, 1] {1} M x W M y x H y S n+k f S n+k Then W is a manifold with boundary W = M x M y : this is a cobordism between M x and M y. There is extra structure: take D ε a disc around x S n and pull it back to f 1 D ε, a tubular neighborhood of M x diffeomorphic to the normal bundle N i of i : M x S n+k. Moreover, we can produce a global trivialization N i = Mx R n from the map f, called a framing. We have constructed a map π n+k S n Ω fr k, where Ωfr k consists of k-manifolds embedded in R n+k with a framing of the normal bundle, modulo cobordism. Pontrjagin proved this is an isomorphism. He also erroneously calculated some homotopy groups to be zero; later he corrected his mistake. Problem: most manifolds are not framed. How do we modify the right hand side to get rid of the framing, or incorporate more information (e.g., orientation, spin structure,... )? Accordingly, we have to modify the left hand side. This was solved by René Thom in his thesis (1950 s). Definition 1.1. Extra structure is a sequence of maps B n diagram commutes: B n B n+1 f n BO(n) (= Grn R ) such that the following f n BO(n) BO(n + 1) 1

Definition 1.2. Let V M be a vector bundle of rank k, with classifying map ν : M BO(k). A B k -structure on V is the homotopy class of a lift of ν to B k : B k M ν f k BO(k) A (stable) B-structure on V is the equivalence class of a B k -structure on V, where g, g : M B k are equivalent if they coincide as B k+m -structures on V R m for some m 0. Examples 1.3. of V. a) B n = BSO(n) (= Gr n R, oriented n-planes in R ): a B k -structure on V is an orientation b) B n = BSpin(n): given an orientation on V, a B k -structure is a spin structure. c) B 2n = B 2n+1 = BU(n) (= Gr n C ) codifies complex structures on real vector bundles (remember the inclusion U(n) O(2n), yielding BU(n) BO(2n)). d) B n (make B n BO(n) into a fibration in the homotopy category): if you can lift M ν BO(n), then V has to be trivial, so a B k -structure is a global trivialization of V. Definition 1.4. Let B be extra structure. We define Ω B n to be the cobordism classes of n-manifolds with B- structure, i.e., closed compact n-manifolds M ν R n+k with a B k -structure on N ν, modulo cobordism and stable equivalence: we say M Ω B n is equivalent to zero if there exists W n+1 such that W n+1 η R n+k+1 W = M ν R n+k and N η has a B-structure that restricts to the B-structure on N ν. Theorem 1.5 (Thom). π n MB = Ω B n, where MB is the Thom spectrum of {B n } Let us now define the elements appearing in the statement of this theorem. Definition 1.6. A spectrum is a sequence of pointed spaces {X(n)} n Z with maps ΣX(n) X(n + 1) The importance of spectra stems from the fact that they are in one-to-one correspondence with generalized homology and cohomology theories. Examples 1.7. a) X(n) = K(Z, n), the Eilenberg-Maclane spectrum, denoted HZ. We know the homotopy equivalences ΩK(Z, n + 1) K(Z, n); taking the adjoint gives ΣK(Z, n) K(Z, n + 1) (warning: these are not homotopy equivalences; look at examples K(Z, 0) Z, K(Z, 1) S 1, K(Z, 2) CP,... ). The associated generalized cohomology theory is good ole singular cohomology H n (Y ) = [Y, K(Z, n)]. b) X(n) = S n, the sphere spectrum S, with the obvious maps ΣS n = S n+1. The associated homology theory is stable homotopy theory: π s ny := lim k π n+k Σ k Y. 2

c) X(n) = M B(n), the Thom spectrum of manifolds with B-structure, which we will define shortly. The associated homology theory is Ω B n (Y ), consisting of n-manifolds M ν R n+k with B-structure and a map M f Y, modulo cobordism and stable equivalence: define M 0 if there is W n+1, a cobordism with B-structure compatible with that of M = W and a map W f Y such that f W f. What we defined before, Ω B n, is evidently Ω B n (). Definition 1.8. Let V X be a vector bundle. The Thom space of V, denoted Th(V ), is the disc bundle of V with its boundary identified to a point. If X is compact, Th(V ) is equivalent to the one-point compactification of V. We have Th(X R n ) Σ n X + and Th(V R n ) Σ n Th(V ) (exercise: check these). If γ n Gr n R is the universal n-plane bundle, the spaces Th(γ n ) assemble into a spectrum MO (which will be the Thom spectrum for B n = BO(n)): the calssifying map Gr n R Gr n+1 R of the bundle γ n R Gr n R induces a lengthpreserving bundle map γ n R γ n+1 Gr n R Gr n+1 R which gives a map ΣTh(γ n ) Th(γ n R) Th(γ n+1 ). More generally, there is a natural map f nγ n R f n+1 γn+1 B n B n+1 inducing ΣTh(f nγ n ) Th(f n+1 γn+1 ) (remember f n : B n BO(n)). This defines the Thom spectrum MB of {B n }. Definition 1.9. Let E be a spectrum. We define π n E := lim k π n+k E(k). Example 1.10. π n S = π s ns 0 = π n+k S k for sufficiently large k (Freudhental suspension theorem). Definition 1.11. Let E be a spectrum. The generalized homology theory associated to E assigns to a topological space X the abelian groups E X := lim π ( ) +k E(k) X+ k Remark 1.12. π E = E () Theorem 1.13 (Thom). Ω B n (X) = (MB) n X = lim k π n+k ( MB(k) X+ ). We sketch the proof of the previous version of the theorem, which corresponds to the case X in the latter formulation. Proof. First we construct a map Ω B n lim k π n+k MB(k). We begin with an n-manifold M ν R n+k with a stable B-structure on the normal bundle N ν, i.e., a lift to B k of the classifying map M BO(k). Identifying 3

N ν with a tubular neighborhood of M, there is a natural map j defined by the following diagram: N ν ν T R n+k j = R n+k R n+k + R n+k M R n+k For ε small enough, j is an embedding; we thus get a map S n+k Th(N ν ) by taking p S n+k to j 1 (p) if p im j or to the distinguished point of Th(N ν ) otherwise. On the other hand, there is a pullback diagram N ν f k γk M B k which induces a map Th(N ν ) Th(fk γk ) = MB(k). Composing these maps yields an element of π n+k MB(k) (exercise: check that this construction is stable, and that it only depends on the B-cobordism class of M). To construct an inverse to this map, choose a representative S n+k α MB(k) of lim k π n+k MB(k) and compose it with the map MB(k) = Th(fk γk ) Th(γ k ) = MO(k) coming from the bundle map f k γk γ k B k f k BO(k) Deform this composition to make it transverse to the zero section of the map MO(k) Gr k R. Then the inverse image of this zero section is a manifold of dimension n embedded in S n+k (exercise: check this); a lifting argument gives the desires B-structure. Now we give a general recipe for defining cobordism invariants. A spectrum E defines both a generalized homology theory E and a generalized cohomology theory E. Given a map of spectra MB α E (i.e., maps M B(n) E(n) commuting with suspensions up to homotopy; this gives natural transformations of the associated homology and cohomology theories) we can produce cobordism invariants of B-manifolds in E () (e.g., for the Eilenberg-Maclane spectrum, E () = H 0 () = Z). We need two ingredients: The fundamental class: for M n a B-manifold there is a distinguished point in Ω B n (M), namely M itself with the identity map, which we denote by M. Using the map of spectra α we can push its class forward: Ω B (M) α E (M). Let B = lim k B k. For any class p E (B), consider the pullback ν p E (M) under the map ν : M B k B. Theorem 1.14. p(m) := ν p, α [M], where E (M) E (M) B-cobordism invariant., E () is the natural pairing, is a 4

Proof. Because of the additivity of the definition it suffices to show p(m) = 0 if M W (respecting the B-structure). Assume such a W exists and consider the relative fundamental class [W, W ] Ω B n+1 (W, W ). Remember the long exact sequence in homology and its functoriality: Ω B n+1 (W, W ) δ Ω B n (B) Ω B n (W ) α E n+1 (W, W ) δ α E n (B) α E n (W ) where M W W, and notice that the compatibility of B-structures implies the commutativity of the following diagram: ν M Moreover, δ[w, W ] = [M]. Then, p(m) = ν p, α [M] = ν p, α δ[w, W ] W ν = ν p, α δ[w, W ] (by naturality of the pairing) = ν p, α δ[w, W ] (by functoriality) = 0 (because δ = 0) B k Example 1.15. Take E = HZ to be the Eilenberg-Maclane spectrum. The pairing is then the usual pairing of cocycles and cycles. For B = BSO, a version of the Hurewicz theorem yields a map MB = MSO α HZ. Recall that H (BSO)/torsion = Z[p 1, p 2,...] (Pontrjagin classes). Choose p = p n i i ; α [M] is the usual fundamental class, and ν p is a Pontrjagin class of the stable normal bundle. Then p(m) are the so-called Pontrjagin numbers of M. We can do the same with Z/2 coefficients and B = BO, yielding the Stiefel-Whitney numbers of M. Remark 1.16. In the last example, it is usually the tangent bundle that is used to define the Pontrjagin numbers of M instead of the stable normal bundle. In the stable category, these bundles are inverses to each other, so they determine each other s invariants. 2 TFTs: first attem Definition 2.1. A tensor (a.k.a, symmetric monoidal) structure on a category C is a functor C C C satisfying appropiate symmetry and associativity conditions, and with a unit element. A tensor functor between tensor categories is a functor respecting. Examples 2.2. a) Set can be given two different tensor structures: disjoint union and cartesian product. b) Vect k with the usual tensor product k. c) Let Cob B n be the category whose objects are closed compact (n 1)-manifolds with B-structure, and whose morphisms Hom Cob B n (M n 1, N n 1 ) are diffeomorphism classes of cobordisms W n (i.e., W = M N) with B-structure compatible with the B-structures on M and N. It is a tensor category with respect to disjoint union of (n 1)-manifolds. 5

Definition 2.3 (Atiyah). An n-dimensional topological field theory is a tensor functor Cob B F n C, where C is a tensor category. Remark 2.4. Atiyah s original definition had C = Vect k. Our first attem is to associate to M n 1 the cochain complex of singular cochains on M. This is well-defined on objects. What about morphisms? We would need W n to give a map C (M) C (N). M i W C (W ) j N C (M) i j C (N) The map i goes in the wrong direction; we would need something like i. The idea is to do some kind of pushpull over cobordisms but, in some sense, complexes are too rigid. By increasing the categorical level we can use the existence of adjoint functors. 3 TFTs: second attem Notice that the singular cochains on a topological space form a commutative dg-algebra. Definition 3.1. Let A cdgalg k. An A-module is a cochain complex M C (k) with an action A k M M satisfying appropiate associativity conditions. The category of A-modules is denoted A Mod. Given a map of dg-algebras A f B, we can define the usual restriction and extension of scalars functors A Mod Ind f Res f B Mod Here we do have a push-pull scheme. Hence, take C = Cat k to be the (2-)category of all k-linear categories (i.e., categories enriched over C (k)), where there is a notion of, and define F : Cob n Cat k taking M C (M) Mod. Then, M i W j N C (M) Mod Res i C (W ) Mod Ind j Res i Ind j C (N) Mod The problem is that this construction does distinguish between cobordisms that are diffeomorphic., but they are identified in Cob n. Definition 3.2. Cobn is the category whose objects are the same as those of Cob n, and whose morphisms Hom Cobn (M, N) are classifying spaces for bordisms between M and N. Remark 3.3. Cob n is a topological category, i.e., its Hom-sets are actually topological spaces. Notationally, this is conveyed by the underlining of Hom s To describe the Hom spaces more concretely, notice that π 0 Hom Cob (M, N) = Hom Cobn (M, N). Each connected component is a classifying space for the diffeomorphism group of any given representative of the diffeo- n morphism class, i.e., Hom Cob (M, N) = BDiff(W ) n [W ] Hom Cobn (M,N) 6

Let X be a connected topological space. A map f : X Hom Cob n (M, N) has image contained in one connected component, say [W ], so that it can be considered as a map f : X BDiff(W ). This classifies a bundle f EDiff(W ) X with fiber Diff(W ). Considering the associated bundle E := EDiff(W ) Diff(W ) W and its pullback f E, we can say that maps f : X BDiff(W ) classify bundles of bordisms diffeomorphic to W. Remark 3.4. From this construction, we see diffeomorphism groups acting on C ( ) Mod. However, these actions are through homotopy automorphisms. Hence we cannot expect this construction to shed any light on things like the Poincaré conjecture. We now want to define our TFT as a tensor functor F : Cob n Cat k. For consistency, we shall want to turn Cat k into a topological category as well: given D, D Cat k, discard all noninvertible morphisms in the category Hom Catk (D, D ) = Fun(D, D ), obtaining a groupoid, and consider its classifying space (i.e., the geometric realization of its nerve). We are now left with two problems: Our definition of Hom Cob n (M, N) does not make composition associative on the nose: it is necessary to modify the smooth structures of the cobordisms in a collar neighborhood of their boundary to glue them to a smooth cobordism. There are two solutions: one is strictification (highly unnatural and contrary to our philosophy here); the other is to take the -categorical route. More importantly, M C (M) Mod does not respect the tensor structure! We will now concentrate on solving this last issue by using the topological category structures just defined on Cob n and Cat k. 4 Tensor structures Let X is a topological category, and X X. Then Hom X (X, ) is a functor from X to Top, and we define X : Top X as its left adjoint, should it exist. Concretely, given a topological space M we have an object M X of X, together with natural isomorphisms Hom X (M X, Y ) = Map(M, Hom X (X, Y )) for every Y X. If such a left adjoint exists for every X X we say that X is a tensored topological category. Examples 4.1. a) For X = Top, we can take M X = M X. b) X = Set can be thought of as a topological category, by considering the Hom sets as discrete topological spaces. Then, M X = π 0 M X satisfies the requirements. c) X = C (Z) is also a topological category: the Hom sets are chain complexes, and we have the following sequence of functors: C (Z) Γ N sab forgetful free sset Sing( ) The first pair of adjoint functors is the Dold-Kan correspondence; the second one consists of the forgetful functor and its left adjoint, the free simplicial abelian group functor; in the third pair we have the geometric realization and the singular set functors. The composition of these functors yields the geometric realization functor, on one hand, and the functor associating to a topological space M the complex of singular chains on it, C (M), on the other. The tensor structure is then given by M X = C (M) k X. Top 7

d) Let X = cdgalg k. There is a forgetful functor to the category of chain complexes over k, with left adjoint the free commutative dg-algebra functor, Sym( ) : C (k) cdgalg k. Since the Hom sets in cdgalg k are commutative dg-algebras themselves, a version of the Dold-Kan correspondence (see the last example) gives a topological category structure. As for the tensor structure, we give one example: for a free commutative dg-algebra Sym V we have i.e., M Sym V = Sym(C (M) k V ). Hom cdgalgk (M Sym V, A) = Map(M, Hom cdgalgk (Sym V, A)) = Map(M, Hom C (k)(v, A)) = Hom C (k)(c (M) k V, A) = Hom cdgalgk (Sym(C (M) k V ), A) Remark 4.2. From this point on, either char k = 0 or we need to substitute commutativity by an E -structure. Guided by out previous attem at a TFT, we now study the case of the commutative dg-algebra C (X). Proposition 4.3. M C (X) = C (Map(M, X)) We need a few elements before outlining the proof. Remarks 4.4. a) If M is a discrete space and A cdgalg k, we have M A = M A, where the last expression is the coproduct of copies of A indexed by M. b) A : Top cdgalg k preserves (homotopy) colimits. c) C ( ) takes (homotopy) colimits to (homotopy) limits. Example 4.5. The sphere S 1 can be realized as a homotopy pushout: S S 1 0 = hocolim What this means is that we take the usual colimit after replacing the maps S 0 by cofibrations S 0 D 1 : S S 1 0 = hocolim = colim S0 D 1 Then, S 1 A S 0 A = hocolim A D 1 A A k A = hocolim A A = A L A k A which is the complex calculating the Hochschild homology of A. Abusing notation, we also denote this dg-algebra by HH (A). Remark 4.6. More generally, we have ΣX = hocolim X and ΩX = holim X A 8

Proof (of the proposition). Notice that the statement is true for M a discrete space. In this case, by 4.4a, M C (X) = M C (X); on the other hand, a Künneth-type formula gives C (Map(M, X)) = C (X M ) = M C (X). Furthermore, both side preserve homotopy colimits (remarks 4.4b and 4.4c). Hence it is true in general, for any topological space can be generated (up to homotopy) by homotopy colimits of finite sets (the case of spheres follows from the remarks above, and we can then use CW approximation). Example 4.7. H (Map(S 1, X)) = HH (C (X)). Proposition 4.8. For A cdgalg k, the functor ( ) Mod : cdgalg k Cat k preserves tensor products. Proof. First of all, notice that for C Cat k we have Fun (A Mod, C) = Hom cdgalgk (A, End C (1 C )). Indeed, a tensor functor preserves the unit, so that A 1 C and A = Hom cdgalgk (A, A) Hom C (1 C, 1 C ). Moreover, A Mod is generated under colimits by A, and tensor functors also preserve colimits. Then, Hom Catk ( M (A Mod), C ) = Map ( M, HomCatk (A Mod, C) ) = Map ( M, Hom cdgalgk (A, End C (1 C )) ) = Hom cdgalgk ( M A, EndC (1 C ) ) = Hom Catk ( (M A) Mod, C ) Corollary 4.9. M (C (X) Mod) C ( Map(M, X) ) Mod. 5 TFTs: third attem Given a k-linear category we can now define F : Cob n Cat k by taking M M C. We have M i W j N M C i C W C j C N C The functor j C preserves colimits, so by general nonsense it has a right adjoint, giving the desired pull-push construction. The upshot is that now F preserves the tensor structure: F (M M ) = (M M ) C = (M C) (M C) Example 5.1. Let C = C (X) Mod. Writing X M for Map(M, X) to lighten the notation, we have (M M ) C (X) Mod C (X M M ) Mod C (X M X M ) Mod ( C (X M ) C (X M ) ) Mod C (X M ) Mod C (X M ) Mod (M C (X) Mod) (M C (X) Mod) In retrospect, we see what was wrong with M C (M) Mod: in a sense, we were trying to define a σ-model without a target! 9

We now study in detail the following geometric example: let X be algebraic (a scheme or stack), obtained by gluing affine schemes U α = Spec A α, and consider the ( -version of the derived) category of quasicoherent sheaves on X: QC X lim α QC Uα lim α A α Mod We have M QC Uα M ( A α Mod ) (M A α ) Mod, where M A α is a commutative dg-algebra; defining X M by QC X M = M QC X, we see that QC X M lim α (M A α ) Mod, i.e., X M is a derived algebraic object (in general, a stack). A point M gives an evaluation map X M ev X X, which in turn yields a pair of adjoint functors: QC X M ev ev QC X In the affine case X = Spec A, these functors are nothing but restriction and extension of scalars: M A Mod Res (M A) L A A Mod This identifies QC X M with the category Mod ev O X M (QC X) of elements of QC X with an action of ev O X M. Example 5.2. If M = S 1 and X = Spec A is affine, example 4.5 gives Γ ( ) O X S 1 = S 1 A = HH (A), so that QC X S 1 HH (A) Mod. More generally, from the homotopy pushout square S 0 S 1 we obtain the homotopy pullback X S1 ev X X and ev O X S 1 ev X X X S0 X X = ev ev O X = O X = HH (O X ), the Hochschild homology sheaf, and QC X S 1 Mod HH (O X )(QC X ) Consider the n-disc D n as a cobordism from S n 1 to the emy (n 1)-manifold. Notice that X Dn = X and X =, and denote X Dn s X Sn 1. Then, We have the following result. QC X S n 1 s QC X Proposition 5.3. Let S, T be algebraic and sufficiently nice. Then Γ X k Mod QC S T = Fun(QC S, QC T ) In other words, any functor QC S to QC T is of the form F q (p F K) for some K QC S T, where p : S T S and q : S T T are the canonical projections. These functors are called integral transforms with kernel K. 10

Using the projection formula and the adjunction between s and s, we can calculate the kernel corresponding to the functor Γ X s : Γ X (s F) = Hom OX (O X, s F) = Hom OX (s O X S n 1, s F) = Hom OX S n 1 (O X Sn 1, s s F) = Γ X S n 1 (F s O X ) On the other hand, D n can also be considered as a cobordism in the opposite direction. It can easily be shown that it corresponds to the functor s O X : k Mod QC X Sn 1. Moreover, these calculations should completely determine the whole TFT, since any manifold can be assembled from discs. For example, cutting S n into two discs and composing the previous calculations yields a functor k Mod k Mod given by tensoring with Γ X (s s O X ) = Γ X S n 1 (s O X ). Alternatively, we can directly calculate the functor associated to a closed n-manifold to be Γ X M (O X M ) 6 Extending TFTs [Missing] 11