The Cobordism Hypothesis states, roughly, that the (∞,n)-category of cobordisms $Bord_n$ 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 monoidal (∞,n)-functor $Z : Bord_n \to C$, 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
which can be understood as the free infinite loop space on the point.
In (Lurie) a formalization and proof of the cobordism hypothesis is described.
For $\mathcal{C}$ a symmetric monoidal (∞,n)-category write $Core(\mathcal{C})$ for its core (the maximal ∞-groupoid in $\mathcal{C}$).
For $\mathcal{C}$, $\mathcal{D}$ two symmetric monoidal (∞,n)-categories, write $Fun^\otimes(\mathcal{D}, \mathcal{C} )$ for the (∞,n)-category of symmetric monoidal (∞,n)-functors between them.
Write $Bord_n^{fr}$ be the symmetric monoidal (∞,n)-category of cobordisms with n-framing.
Evaluation of any such functor $F$ on the point ${*}$
induces an (∞,n)-functor
such that
this factors through the core of $\mathcal{C}$;
the map
is an equivalence of (∞,n)-categories.
This is (Lurie, theorem 2.4.6).
The proof is based on
the Galatius-Madsen-Tillmann-Weiss theorem, which characterizes the geometric realization $|Bord_n^{or}|$ in terms of the suspension of the Thom spectrum;
Igusa’s connectivity result which he uses to show that putting “framed Morse functions” on cobordisms 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.
One of the striking consequences of theorem 1 is that it implies that
Every ∞-groupoid
of fully dualizable objects in a symmetric monoidal (∞,n)-category $\mathcal{C}$ carries a canonical ∞-action of (the ∞-group structure on the homotopy type of) the orthogonal group $O(n)$, induced by the action of $O(n)$ on the n-framing of the point in $Bord_n^{fr}$.
The action in corollary 1 is
for $n = 1$: the $O(1) = \mathbb{Z}/2\mathbb{Z}$ action action given by passing to dual objects;
for $n = 2$ the $O(2)$-action the Serre automorphism.
for $n = \infty$ the $O(n)$-action on n-fold loop spaces (see e.g. Gaudens-Menichi 07, section 5) (see also at orthogonal spectra).
(Lurie, examples 2.4.12, 2.4.14. 2.4.15)
For all $n \in \mathbb{N}$, the canonical $SO$-∞-action on
is trivial.
The action on a connective spectrum $\Omega^\infty X$ factors through the J-homomorphism
But on homotopy groups the image of J is pure torsion which means that for $\Omega^\infty X = B^n \mathbb{Z}$ the induced actions on homotopy groups are all trivial. From this and using the long exact sequence of homotopy groups it follows that the $\infty$-action itself is trivial.
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.
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 $m \leq n$. An $(X,\xi)$-structure on $N$ consists of the following data
A continuous function $f : N \to X$;
An isomorphism of vector bundles
between the fiberwise direct sum of the tangent bundle $T N$ with the trivial rank $(n-m)$ bundle and the pullback of $\xi$ along $f$.
This is (Lurie, notation 2.4.16).
The two extreme cases of def. 2 are the following
For $X = \ast$ the point and $\xi = \mathbb{R}^n$, then an $(X,\xi)$-structure is the same as an $n$-framing, hence
reproduces the $(\infty,n)$-category of framed cobordisms of def. 1.
For $X = B O(n)$ the classifying space of real vector bundles of rank $n$ (the delooping of the ∞-group $O(n)$ underlying the orthogonal group) and for $\xi = E O(n) \underset{O(n)}{\times} \mathbb{R}^n$ the vector bundle associated to the $O(n)$-universal bundle, then $(X,\xi)$-structure on $n$-dimensional manifolds is essentially no-structure (the maximal compact subgroup-inclusion $O(n)\to GL(n)$ is a weak homotopy equivalence). Cobordisms with this structure will also be called unoriented cobordisms
Accordingly, for $X = B SO(n)$ the delooping of the special orthogonal group, the corresponding $(X,\xi)$-structure makes oriented manifolds
Generally:
For $\chi \colon G \to O(n)$ a topological group mapping via a homomorphism to $O(n)$, then $X = B G$ and $\xi = \chi^\ast (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)$, the $(X,\xi)$-structure is G-structure.
This we get to below.
Let $X$ be a topological space and $\xi \to X$ an $n$-dimensional vector bundle. The (∞,n)-category $Bord_n^{(X, \xi)}$ is defined analogously to $Bord_n$ but with all manifolds equipped with $(X,\xi)$-structure, def. 2.
This is (Lurie, def. 2.4.17).
Let $\mathcal{C}$ 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
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). The following is some aspects of the idea of the proof in (Lurie, p. 57).
In the language of ∞-actions (as discussed there), the space $Top_{O(n)}(\tilde X, \tilde \mathcal{C})$ is that of horizontal maps fitting into
where the left map is the classifying map for $\xi$ and the right one is the canonical one out of the homotopy quotient.
Notice that for each point $x \colon \ast \to X$ there is an induced inclusion
of the framed cobordisms, def. 1, into those of $(X,\xi)$-structure, def. 3, including those cobordisms whose map to $X$ is constant on $X$, and observing that for these an $(X,\xi)$-structure is equivalently an $n$-framing. Moreover, by corollary 1 the induced point evaluation is $O(n)$-equivariant, hence yielding a morphism of ∞-groupoids
where $\tilde X$ denotes the $O(n)$-principal bundle to which $\xi$ is associated.
More generally, this is true for the pullback structure of $\xi$ along along any map $Y \to X$, yielding
By the previous comment, observe that $\alpha_Y$ is an equivalence for $Y = \ast$.
Now the codomain of this natural transformation sends (∞,1)-colimits in $Y$ over $X$ to (∞,1)-limits. (Lurie, theorem 3.1.8) shows that the same is true for the domain. Hence $\alpha_Y$ is an equivalence for all $Y$ that appear as (∞,1)-colimits of the point. But this is the case for all ∞-groupoids $Y$, by this proposition.
We consider now some special cases of the general definition of local structure-topological field theory
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 $\xi = \mathbb{R}^n \otimes X$.
In this case $\tilde X = O(n) \times X$. Write
Write $\Pi(X) \in$ ∞Grpd for the fundamental ∞-groupoid of $X$.
There is an equivalence in ∞Grpd
This is a special case of the above theorem.
Notice that one can read this as saying that $Cob_n(X)$ is roughly like the free symmetric monoidal (∞,n)-category on the fundamental ∞-groupoid of $X$ (relative to $\infty$-categories of fully dualizable objects at least).
We discuss the special case of the cobordism hypothesis for $(X,\xi)$-bundles (def. 3) for the special case of G-structure (example 4), hence for the 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 $\mathbf{B}O(n) \to$ Vect of $O(n)$ on $\mathbb{R}^n$, this induces accordingly a representation of $G$ on $\mathbb{R}^n$..
Let then
$X := B G$ be the classifying space for $G$;
$\xi_\chi := \mathbb{R}^n \times_G E G$ be the corresponding associated vector bundle to the universal principal bundle $E G \to B G$.
We say
is the $(\infty,n)$-category of cobordisms with $G$-structure.
See (Lurie, notation 2.4.21)
We have
For $G = 1$ the trivial group, a $G$-structure is just a framing and so
reproduces the $(\infty,n)$-category of framed cobordisms, def. 1.
For $G = SO(n)$ the special orthogonal group equipped with the canonical embedding $\chi : SO(n) \to O(n)$ a $G$-structure is an orientation
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,
See (Lurie, example 2.4.22).
Then we have the following version of the cobordism hypothesis for manifolds with $G$-structure.
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
between extended TQFTs on $n$-dimensional manifolds with G-structure and the ∞-groupoid homotopy invariants of the infinity-action of $G$ on $\tilde \mathcal{C}$ (which is induced by the evaluation on the point).
This is (Lurie, theorem 2.4.26).
Theorem 2 asserts that
Hence it remains to see that the right hand side are equivalently the homotopy invariants of the $G$-∞-action. This follows for instance with the discussion at ∞-action, by which
This yields
By the discussion at dependent product
which are the homotopy invariants.
The case that $\chi \colon G \longrightarrow O(n)$ is the identity is at the other extreme of the framed case, and turns out to be similarly fundamental.
For $\mathbf{H}$ an (∞,1)-topos, write $Corr_n(\mathbf{H})^\otimes$ for the (∞,n)-category of correspondences in $\mathbf{H}$. For $Phases \in DCat_n(\mathbf{H})$ an (∞,n)-category with duals internal to $\mathbf{H}$, write $Corr_n(\mathbf{H}_/{Phases})^{\otimes_{phases}}$ for the (∞,n)-category of correspondences over $Phases$ and equipped with the phased tensor product. There is the forgetful monoidal (∞,n)-functor
By the discussion at (∞,n)-category of correspondences these are (∞,n)-categories with duals and the canonical $O(n)$-∞-action on them, corollary 1, is trivial for $Corr_n(\mathbf{H})$. This means that an $O(n)$-homotopy fixed point in $Corr_n(\mathbf{H})$ is just an object of $\mathbf{H}$ equipped in turn with an $O(n)$-∞-action. Therefore
Local unoriented-topological field theory
are equivalent to objects $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action.
At least for $\mathbf{H} =$ ∞Grpd, then given such, the corresponding field theory $Z_{X/\!/O(n)}$ sends a cobordism $\Sigma$ to the space of maps
hence
In particular this means that the assignment to the point is again $X$ itself.
This is a slight rephrasing of the paragraph pp 58-59 in (Lurie).
At least for $\mathbf{H} =$ ∞Grpd, with $X \in \mathbf{H}$ an object equipped with an $O(n)$-∞-action, then horizontal lifts in
are equivalent to
This is (Lurie, prop. 3.2.8).
Via the interpretation of local field theories with coefficients in $Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$ as Local prequantum field theory, the statement of prop. 4 translates in quantum field theory jargon to the statement that “All background structures are fields.” This is essentially the slogan of general covariance.
Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an abelian ∞-group object, regarded as a (∞,n)-category with duals internal to $\mathbf{H}$.
At least if $\mathbf{H} =$ ∞Grpd, then local unoriented-topological field theories of the form
are equivalent to a choice
of $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action
a homomorphism of $O(n)$-∞-actions $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence (by the discussion at ∞-action) to a horizontal morphism in $\mathbf{H}$ fitting into the diagram
By prop. 3 the co-restriction
is equivalent to an ∞-action
Therefore by prop. 4 $Z$ is equivalent to
Finally, by theorem 2 and in view of remark 1, this is equivalent to maps of the form
By the discussion at Local prequantum field theory, these statements hold also for fields with moduli spaces in more general $(\infty,1)$-toposes $\mathbf{H}$ (one sufficient condition is that $\mathbf{H}$ has an (infinity,1)-site of definition all whose objects are etale contractible).
Some examples are discussed at prequantum field theory in the section Higher Chern-Simons field theory – Levels.
If in def. 3 one chooses $X = B SO(n) \times Y$ for any topological space $Y$, and $\xi$ the pullback of the canonical vector bundle bundle on $B SO$ to $B SO \times Y$, then an $(\infty,n)$-functor $Bord^{X}_n \to C$ is similar to what Turaev calls an HQFT over $Y$.
There is a vast generalization of the plain $(\infty,n)$-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 $k$ now corresponds to a new generator k-morphisms, and the (framed) $(\infty,n)$-category of cobordisms with singularities is now no longer the free symmetric monoidal $(\infty,n)$-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 $(n-1)$-boundary; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ with all dual are equivalent to choices of morphisms $1 \to A$, where $A \in \mathcal{C}$ 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.
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
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
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).
In particular this means that $Fun^\otimes(Bord_n^{fr} , C )$ is itself an $(\infinity,0)$-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 $Bord_n$. This is described at QFT with defects .
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 $SO(n)$-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 $\mathcal{B}G \simeq X$.
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.
duality between algebra and geometry in physics:
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 $n = 1$ is spelled out in detail in
Lecture notes and reviews 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 $(\infty,n)$-Categories and the Cobordism Hypothesis , in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, AMS 2011
Daniel Freed, The cobordism hypothesis (arXiv:1210.5100)
Chris Schommer-Pries, Dualizability in Low-Dimensional Higher Category Theory (arXiv:1308.3574)
Constantin Teleman, Five lectures on topological field theory, 2014 (pdf)
Discussion of the canonical $O(n)$-action on n-fold loop spaces (which may be thought of as a special case of the cobordism hypothesis) includes
Cobordisms with geometric structure are discussed in