nLab
cobordism hypothesis

Idea

The Cobordism Hypothesis roughly states that the (∞,n)-category of cobordisms ${\mathrm{Bord}}_n$ 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 $Z:{\mathrm{Bord}}_n\to C$, this implies that all these TQFTs are entirely determined by their value on the point: “the n-vector space of states” of the theory.

Simple version on geometric realization

By GalatiusTillmannMadsenWeiss (see cobordism category) we have that the loop space of the geometric realization of the framed cobordism category is equivalent to the sphere spectrum

$$\Omega \parallel {\mathrm{Cob}}_d^{\mathrm{fr}}\parallel \simeq \underset{{\to }_{n\to \mathrm{\infty }}}{\lim }{\mathrm{Maps}}_{*}(S^n,S^n)\simeq {\Omega }^{\mathrm{\infty }}{S}^{\mathrm{\infty }}$$\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.

Therefore…

Formalization

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

For framed cobordisms

Definition

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

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

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

Theorem (cobordism hypothesis, framed version)

Evaluation of any such functor $F$ on the point $*$

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

induces an (∞,n)-functor

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

such that

• this factors through the core of $C$;

• the map

  $${\mathrm{pt}}^{*}:{\mathrm{Fun}}^{\otimes }({\mathrm{Bord}}_n^{\mathrm{fr}},C)\to \mathrm{Core}(C)$$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 $\mid {\mathrm{Bord}}_n^{\mathrm{or}}\mid$ 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.

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.

Definition

Let $X$ be a topological space and $\xi \to X$ a real vector bundle on $X$ of rank $n$. Let $N$ be a smooth manifold of dimension $m\le n$. An $(X,\xi )$-structure on $N$ consists of the following data

• A continuous function $f:N\to X$;

• An isomorphism of vector bundles

  $$TN\oplus {ℝ}^{n-m}\simeq f^{*}\xi$$T N \oplus \mathbb{R}^{n-m} \simeq f^* \xi

  between the fiberwise direct sum of the tangent bundle $TN$ with the trivial rank $(n-m)$ bundle and the pullback of $\xi$ along $f$.

This is (Lurie, notation 2.4.16).

Definition

Let $X$ be a topological space and $\xi \to X$ an $n$-dimensional vector bundle. The (∞,n)-category ${\mathrm{Bord}}_n(X,\xi )$ is defined analogously to ${\mathrm{Bord}}_n$ but with all manifolds equipped with $(X,\xi )$-structure.

This is (Lurie, def. 2.4.17).

Theorem

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

There is an equivalence in ∞Grpd

$${\mathrm{Fun}}^{\otimes }({\mathrm{Bord}}_n^{(X,\xi )},C)\simeq {\mathrm{Top}}_{O(n)}(\tilde{X},\tilde{C}),$$Fun^\otimes(Bord_n^{(X,\xi)}, C) \simeq Top_{O(n)}(\tilde X, \tilde C) \,,

where on the right we regard $\tilde{C}$ as a topological space carrying the canonical $O(n)$-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 $(X,\xi )$-cobordisms (def. 3) for the case that the vector bundle $\xi$ is the trivial vector bundle $\xi ={ℝ}^n\otimes X$.

In this case $\tilde{X}=O(n)\times X$. Write

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

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

Corollary

There is an equivalence in ∞Grpd

$${\mathrm{Fun}}^{\otimes }({\mathrm{Bord}}_n^{\mathrm{fr}}(X),C)\simeq (\mathrm{\infty },n)\mathrm{Cat}(\Pi (X),\tilde{C})\simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),\mathrm{Core}(\tilde{C})),$$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 ${\mathrm{Cob}}_n(X)$ is roughly like the free symmetric monoidal (∞,n)-category on the fundamental ∞-groupoid of $X$ (relative to $\mathrm{\infty }$-categories of fully dualizable objects at least).

For cobordisms with $G$-structure

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

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

Let then

• $X:=BG$ be the classifying space for $G$;

• ${\xi }_{\chi }:=({ℝ}^n{\times }_{G}EG$ be the corresponding associated vector bundle to the universal principal bundle $EG\to BG$.

Definition

We say

$${\mathrm{Bord}}_n^G:={\mathrm{Bord}}_n^{(BG,{\xi }_{\chi })}.$$Bord^G_n := Bord_n^{(B G, \xi_\chi)} \,.

is the $(\mathrm{\infty },n)$-category of cobordisms with $G$-structure.

See (Lurie, notation 2.4.21)

Definition

We have

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

  $${\mathrm{Bord}}_n^{(1,\xi )}\simeq {\mathrm{Bord}}_n^{\mathrm{fr}}$$Bord_n^{(1,\xi)} \simeq Bord_n^{fr}

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

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

  $${\mathrm{Bord}}_n^{(\mathrm{SO}(n))}\simeq {\mathrm{Bord}}_n^{\mathrm{or}}.$$Bord_n^{(SO(n))} \simeq Bord_n^{or} \,.

• For $G=O(n)$ the orthogonal group itself equipped with the identity map $\chi :O(n)\to O(n)$ a $G$-structure is no structure at all,

  $${\mathrm{Bord}}_n^{O(n)}\simeq {\mathrm{Bord}}_n.$$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 $G$-structure.

Theorem

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

$${\mathrm{Fun}}^{\otimes }({\mathrm{Bord}}_n^G,𝒞)\simeq (\widetilde{𝒞}{)}^G$$Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq (\tilde {\mathcal{C}})^{G}

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

This is (Lurie, theorem 2.4.26).

For HQFTs

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

(…)

For noncompact cobordisms

See Calabi-Yau object.

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.

• steps towards classification of quantum field theories with super-Euclidean structure are discussed at

  • (1,1)-dimensional Euclidean field theories and K-theory

  • (2,1)-dimensional Euclidean field theories and tmf

• a general definition of a cobordism category of cobordisms equipped with geometric structure given by a morphism into, roughly, a smooth infinity-groupoid of structure is discussed in (Ayala).

Remarks

Morphisms of TQFTs

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

When interpreting symmetric monoidal functors from bordisms to $C$ as TQFTs this means that TQFTs with given codomain $C$ 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 ${\mathrm{Bord}}_n$. 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 $n$-dimensional framed theory one needs to assign a fully dualizable object, and the meaning of the term “fully dualizable” depends on $n$, and gets increasingly hard to satisfy as n grows..

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

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

For an oriented theory $n$-dimensional theory need an $\mathrm{SO}(n)$-fixed point, which for $n=1$ is nothing but for $n=2$ 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 $G$-structure, for $G$ any group mapping to $O(n)$ (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 $G$-fixed point in dualizable objects in your category (with $G$ acting through $O(n)$).

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

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

Notice that there is an important subtlety of Lurie’s theorem in the case of manifolds with $G$-structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the $G$-fixed points for the $G$-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 $G$, 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.

Related concepts

• generalized tangle hypothesis

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

References

The original hypothesis is formulated in

• John Baez, James Dolan, Higher dimensional algebra and Topological Quantum Field Theory J.Math.Phys. 36 (1995) 6073-6105 (arXiv)

The formalization and proof is described in

• Jacob Lurie, On the Classification of Topological Field Theories

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 $n=1$ is spelled out in detail in

• Yonatan Harpaz, The Cobordism Hypothesis in Dimension 1 (arXiv:1210.0229)

Lecture notes on the topic of the cobordisms hypothesis include

• Jacob Lurie, TQFT and the Cobordism Hypothesis (video, notes)

• Julie Bergner, UC Riverside Seminar on Cobordism and Topological Field Theories (2009).

• Julie Bergner, Models for $(\mathrm{\infty },n)$-Categories and the Cobordism Hypothesis , in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory

Another review is in

• Dan Freed, The cobordism hypothesis (arXiv:1210.5100)

Cobordisms with geometric structure are discussed in

• David Ayala, Geometric cobordism categories (pdf)