nLab
cobordism hypothesis

some material that should go here is, for some reason, still to some extent at generalized tangle hypothesis. You should go there to learn more. Better yet, you go there and then come back here to create a better entry on the cobordisms hypothesis.

Idea

The Cobordism Hypothesis roughly states that

The n-category ${\mathrm{Cob}}_n$ of extended cobordisms (when made precise) is the free stable $n$-category with duals on one object (the point).

In extended topological quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect:

Extended TQFT Hypothesis

An $n$-dimensional unitary extended TQFT is a weak $n$-functor, preserving all levels of duality, from the $n$-category $n\mathrm{Cob}$ of cobordisms to $n\mathrm{Hilb}$, the $n$-category of $n$-Hilbert spaces?.

Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain:

The primacy of the point

An $n$-dimensional unitary extended TQFT is completely described by the $n$-Hilbert space it assigns to a point.

Lurie’s formulation and proof of the cobordism hypothesis

In

• Jacob Lurie On the Classification of Topological Field Theories

a formalization and proof of the cobordism hypothesis is indicated.

This uses the languge of (∞,n)-categories modeled as n-fold complete Segal spaces and a concrete realization of the (∞,n)-category of cobordisms in this context.

What is not yet described in detail it what it means for an $(\mathrm{\infty },n)$-category to be a symmetric monoidal category but by comparison with the defintion of symmetric monoidal (∞,1)-category one can see what that would be.

The precise formulation of the cobordism hypothesis then takes the form of the following

(theorem 2.4.6 in On the Classification of Topological Field Theories)

Theorem (cobordism hypothesis, framed version)

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.

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

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

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

induces an $(\mathrm{\infty },n)$-functor

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

The statement is 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 (infinity,n)-categories.

Proof

The proof is based on

1. the Galatius-Madsen-Tillmann-Weiss theorem (thm 2.7.4, page 50) 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.

Implication: morphisms of TQFTs

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

Remarks

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.

References

The original hypothesis is in

• John Baez, James Dolan, Higher dimensional algebra and Topological Quantum Field Theory (arXiv)

The sketch of the proof is in

• Jacob Lurie, On the Classification of Topological Field Theories

Parts of the above text have been taken from blog discussion here.

Recorded lectures

A videa of Lurie lecturing on his theorem is here:

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

• The UC Riverside Seminar on Cobordism and Topological Field Theories, organized by Julie Bergner in the Fall of 2009.