nLab
cobordism hypothesis

Context

Manifolds and cobordisms

Quantum field theory

Contents

Idea

The Cobordism Hypothesis roughly states that the (∞,n)-category of cobordisms Bord nBord_n is the free symmetric monoidal (∞,n)-category with duals on a single object.

Since a fully extended topological quantum field theory may be identified with an (∞,n)-functor Z:Bord nCZ : Bord_n \to C, this implies that all these TQFTs are entirely determined by their value on the point: “the n-vector space of states” of the theory.

As motivation, notice that by Galatius-Tillmann-Madsen- 09Weiss we have that the loop space of the geometric realization of the framed cobordism category is equivalent to the sphere spectrum

ΩCob d frlim nMaps *(S n,S n)Ω S \Omega \Vert Cob_d^{fr} \Vert \simeq \lim_{\to_{n \to \infty}} Maps_*(S^n, S^n) \simeq \Omega^\infty S^\infty

which can be understood as the free infinite loop space on the point.

Formalization

In (Lurie) a formalization and proof of the cobordism hypothesis is described.

For framed cobordisms

Statement

Definition

Let CC by a symmetric monoidal (∞,n)-category with duals and Core(C)Core(C) its core (the maximal ∞-groupoid in CC).

Let Bord n frBord_n^{fr} be the symmetric monoidal (∞,n)-category of cobordisms with n-framing.

Finally let Fun (Bord n fr,C)Fun^\otimes(Bord_n^{fr} , C ) be the (∞,n)-category of symmetric monoidal (∞,n)-functors from bordisms to CC.

Theorem (cobordism hypothesis, framed version)

Evaluation of any such functor FF on the point *{*}

FF(*) F \mapsto F({*})

induces an (∞,n)-functor

pt *:Fun (Bord n fr,C)C. pt^* : Fun^\otimes(Bord_n^{fr} , C ) \to C .

such that

  • this factors through the core of CC;

  • the map

    pt *:Fun (Bord n fr,C)Core(C) pt^* : Fun^\otimes(Bord_n^{fr} , C ) \to Core(C)

    is an equivalence of (∞,n)-categories.

This is (Lurie, theorem 2.4.6).

Proof

The proof is based on

  1. the Galatius-Madsen-Tillmann-Weiss theorem, which characterizes the geometric realization |Bord n or||Bord_n^{or}| in terms of the suspension of the Thom spectrum;

  2. Igusas connectivity result which he uses to show that putting “framed Morse functions” on cobrdisms doesn’t change their homotopy type (theorem 3.4.7, page 73)

In fact, the Galatius-Madsen-Weiss theorem is now supposed to be a corollary of Lurie’s result.

Implications – The canonical O(n)O(n)-∞-action

One of the striking consequences of theorem 1 is that it implies that

Corollary

Every ∞-groupoid

𝒞 fd𝒞 \mathcal{C}^{fd} \hookrightarrow \mathcal{C}

of fully dualizable objects in a symmetric monoidal (∞,n)-category 𝒞\mathcal{C} carries a canonical ∞-action of (the ∞-group structure on the homotopy type of) the orthogonal group O(n)O(n), induced by the action of O(n)O(n) on the n-framing of the point in Bord n frBord_n^{fr}.

(Lurie, corollary 2.4.10)

Example

The action in corollary 1 is

(Lurie, examples 2.4.12, 2.4.14. 2.4.15)

For cobordisms with extra topological structure

We discuss the cobordism hypothesis for cobordisms that are equipped with the extra structure of maps into some topological space equipped with a vector bundle. This is the case for which an extended TQFT is (the local refinement of) what has also been called an HQFT.

Definition

Let XX be a topological space and ξX\xi \to X a real vector bundle on XX of rank nn. Let NN be a smooth manifold of dimension mnm \leq n. An (X,ξ)(X,\xi)-structure on NN consists of the following data

This is (Lurie, notation 2.4.16).

Definition

Let XX be a topological space and ξX\xi \to X an nn-dimensional vector bundle. The (∞,n)-category Bord n(X,ξ)Bord_n(X, \xi) is defined analogously to Bord nBord_n but with all manifolds equipped with (X,ξ)(X,\xi)-structure.

This is (Lurie, def. 2.4.17).

Theorem

Let CC be a symmetric monoidal (∞,n)-category with duals, let XX be a CW-complex, let ξX\xi \to X be an nn-dimensional vector bundle over XX equipped with an inner product, and let X˜X\tilde X \to X be the associated O(n)-principal bundle of orthonormal frames in ξ\xi.

There is an equivalence in ∞Grpd

Fun (Bord n (X,ξ),C)Top O(n)(X˜,C˜), Fun^\otimes(Bord_n^{(X,\xi)}, C) \simeq Top_{O(n)}(\tilde X, \tilde C) \,,

where on the right we regard C˜\tilde C as a topological space carrying the canonical O(n)O(n)-action discussed above.

This is (Lurie, theorem. 2.4.18).

We consider some special cases of this general definition

For framed cobordisms in a topological space

We discuss the special case of the cobordism hypothesis for (X,ξ)(X,\xi)-cobordisms (def. 3) for the case that the vector bundle ξ\xi is the trivial vector bundle ξ= nX\xi = \mathbb{R}^n \otimes X.

In this case X˜=O(n)×X\tilde X = O(n) \times X. Write

Bord n fr(X):=Bord n (X,X× n). Bord_n^{fr}(X) := Bord_n^{(X,X \times \mathbb{R}^n)} \,.

Write Π(X)\Pi(X) \in ∞Grpd for the fundamental ∞-groupoid of XX.

Corollary

There is an equivalence in ∞Grpd

Fun (Bord n fr(X),C)(,n)Cat(Π(X),C˜)Grpd(Π(X),Core(C˜)), Fun^\otimes(Bord^{fr}_n(X), C) \simeq (\infty,n)Cat(\Pi(X), \tilde C) \simeq \infty Grpd(\Pi(X), Core(\tilde C)) \,,

This is a special case of the above theorem.

Notice that one can read this as saying that Cob n(X)Cob_n(X) is roughly like the free symmetric monoidal (∞,n)-category on the fundamental ∞-groupoid of XX (relative to \infty-categories of fully dualizable objects at least).

For cobordisms with GG-structure

We discuss the special case of the cobordism hypothesis for (X,ξ)(X,\xi)-bundles (def. 3) for the special case that XX is the classifying space of a topological group.

Let GG be a topological group equipped with a homomorphism χ:GO(n)\chi : G \to O(n) to the orthogonal group. Notice that via the canonical linear representation BO(n)\mathbf{B}O(n) \to Vect of O(n)O(n) on n\mathbb{R}^n, this induces accordingly a representation of GG on n\mathbb{R}^n..

Let then

Definition

We say

Bord n G:=Bord n (BG,ξ χ). Bord^G_n := Bord_n^{(B G, \xi_\chi)} \,.

is the (,n)(\infty,n)-category of cobordisms with GG-structure.

See (Lurie, notation 2.4.21)

Definition

We have

  • For G=1G = 1 the trivial group, a GG-structure is just a framing and so

    Bord n (1,ξ)Bord n fr Bord_n^{(1,\xi)} \simeq Bord_n^{fr}

    reproduces the (,n)(\infty,n)-category of framed cobordisms, def. 1.

  • For G=SO(n)G = SO(n) the special orthogonal group equipped with the canonical embedding χ:SO(n)O(n)\chi : SO(n) \to O(n) a GG-structure is an orientation

    Bord n (SO(n))Bord n or. Bord_n^{(SO(n))} \simeq Bord_n^{or} \,.
  • For G=O(n)G = O(n) the orthogonal group itself equipped with the identity map χ:O(n)O(n)\chi : O(n) \to O(n) a GG-structure is no structure at all,

    Bord n O(n)Bord n. Bord_n^{O(n)} \simeq Bord_n \,.

See (Lurie, example 2.4.22).

Then we have the following version of the cobordism hypothesis for manifolds with GG-structure.

Theorem

For GG an ∞-group equipped with a homomorphism GO(n)G \to O(n) to the orthogonal group (regarded as an ∞-group in ∞Grpd), then evaluation on the point induces an equivalence

Fun (Bord n G,𝒞)(𝒞˜) G Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq (\tilde {\mathcal{C}})^{G}

between extended TQFTs on nn-dimensional manifolds with G-structure and the ∞-groupoid homotopy invariants of the infinity-action of GG on 𝒞˜\tilde \mathcal{C} (which is induced by the evaluation on the point).

This is (Lurie, theorem 2.4.26).

For HQFTs

If in def. 3 one chooses X=BSO(n)×YX = B SO(n) \times Y for any topological space YY, and ξ\xi the pullback of the canonical vector bundle bundle on BSOB SO to BSO×YB SO \times Y, then an (,n)(\infty,n)-functor Bord n XCBord^{X}_n \to C is similar to what Turaev calls an HQFT over YY.

For cobordisms with singularities (boundaries/branes and defects/domain walls)

There is a vast generalization of the plain (,n)(\infty,n)-category of cobordisms (with topological structure) considered above given by allowing the cobordisms to be equipped with various types of singularities (Lurie 09, Definition Sketch 4.3.2).

Each type of singularity in dimension kk now corresponds to a new generator k-morphisms, and the (framed) (,n)(\infty,n)-category of cobordisms with singularities is now no longer the free symmetric monoidal (,n)(\infty,n)-category freely generated from just a point (a 0-morphisms), but freely generated from these chosen generators. This general version is (Lurie 09, Theorem 4.3.11).

For instance if the generator on top of the point *\ast is a 1-morphism of the form *\emptyset \to \ast, then this defines a type of codimension (n1)(n-1)-boundary; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal (,n)(\infty,n)-category 𝒞\mathcal{C} with all dual are equivalent to choices of morphisms 1A1 \to A, where A𝒞A \in \mathcal{C} is the fully dualizable object assigned to the point, as before, and now equipped with a morphism from the tensor unit into it. Indeed, this is the usual datum that describes branes in QFT (see for instance at FRS formalism).

For more on this see at QFT with defects.

For noncompact cobordisms

One important variant of the category of cobordisms is obtained by discarding all those morphisms which have non-empty incoming (say, dually one could use outgoing) bounrary component. Then a representation of this category imposes on its values “cups but no caps”, hence only half of the data of a dualizable object in the given degree.

Accordingly, in this case the cobordism hypothesis says that such a functor is given not quite by a fully dualizable object, but by a weaker structure called a Calabi-Yau object (see there for more).

2-dimensional TQFT of this form is known as TCFT, see there for more

For cobordisms with geometric structure

A non-topological quantum field theory is a representation of a cobordism category for cobordisms equipped with extra stuff, structure, property that is “not just topological”, meaning roughly not of the form of def. 3.

The theory for this more general case is not as far developed yet.

Remarks

Morphisms of TQFTs

In particular this means that Fun (Bord n fr,C)Fun^\otimes(Bord_n^{fr} , C ) is itself an (,0)(\infinity,0)-category, i.e. an ∞-groupoid.

When interpreting symmetric monoidal functors from bordisms to CC as TQFTs this means that TQFTs with given codomain CC form a space, an ∞-groupoid. In particular, any two of them are either equivalent or have no morphism between them.

According to Chris Schommer-Pries interesting morphisms of TQFTs arise when looking at transformations only on sub-categories on all of Bord nBord_n. This is described at QFT with defects .

Invariants determined from the point

The theorem does say that the invariant attached by an extended TQFT to the point determines all the higher invariants – however it is important to notice that there are strong constraints on what is assigned to the point. For an nn-dimensional framed theory one needs to assign a fully dualizable object, and the meaning of the term “fully dualizable” depends on nn, and gets increasingly hard to satisfy as n grows..

For an nn-dimensional unoriented theory, the object assigned to the point has to be a fixed point for the O(n)O(n)- action on fully dualizable objects that is obtained from the framed case of the theorem.

In the 1d case, this O(1)O(1) action on dualizable objects takes every object to its dual, and an O(1)O(1) fixed point is indeed a vector space with a nondegenerate symmetric inner product.

For an oriented theory nn-dimensional theory need an SO(n)SO(n)-fixed point, which for n=1n=1 is nothing but for n=2n=2 ends up meaning a Calabi-Yau category (in the case the target 2-category is that of categories).

In fact something more general is true: if one wants a theory that takes values on manifolds equipped with a GG-structure, for GG any group mapping to O(n)O(n) (such as for instance orientation already discussed or its higher versions Spin structure or String structure or Fivebrane structure or …) one needs to assign to the point a GG-fixed point in dualizable objects in your category (with GG acting through O(n)O(n)).

This beautifully includes all the above plus for example manifolds with maps (up to homotopy) to some auxiliary (connected) space XX – here we take GG to be the loop space ΩX\Omega X of XX (mapping trivially to O(n)O(n)), so that a reduction of the structure group of the manifold to GG involves a map to the delooping GX\mathcal{B}G \simeq X.

Such theories are classified by XX-families of fully dualizable objects.

Notice that there is an important subtlety of Lurie’s theorem in the case of manifolds with GG-structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the GG-fixed points for the GG-action on fully dualizable objects, but rather they are the homotopy fixed points. This is very important because a homotopy fixed point is not just a property. It is additional structure. Depending on GG, this additional structure is often encoded in the higher dimensional portion of the field theory.

One can see this in the 1 dimensional case: there is no property of vector spaces which automatically endows them with an inner product, but it is extra structure.

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

References

The original hypothesis is formulated in

The formalization and proof is described in

This is almost complete, except for one step that is not discussed in detail. But a new (unpublished) result by Søren Galatius bridges that step in particular and drastically simplifies the whole proof in general.

The comparatively simple case of n=1n = 1 is spelled out in detail in

Lecture notes on the topic of the cobordisms hypothesis include

Another review is in

Cobordisms with geometric structure are discussed in

Revised on September 2, 2014 15:55:24 by David Corfield (91.125.67.210)