FQFT and cohomology
Since a fully extended topological quantum field theory may be identified with an monoidal (∞,n)-functor , this implies that all these TQFTs are entirely determined by their value on the point: “the n-vector space of states” of the theory.
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.
Evaluation of any such functor on the point
induces an (∞,n)-functor
This is (Lurie, theorem 2.4.6).
The proof is based on
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
of fully dualizable objects in a symmetric monoidal (∞,n)-category carries a canonical ∞-action of (the ∞-group structure on the homotopy type of) the orthogonal group , induced by the action of on the n-framing of the point in .
The action in corollary 1 is
For all , the canonical -∞-action on
But on homotopy groups the image of J is pure torsion which means that for the induced actions on homotopy groups are all trivial. From this and using the long exact sequence of homotopy groups it follows that the -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.
This is (Lurie, notation 2.4.16).
The two extreme cases of def. 2 are the following
For the point and , then an -structure is the same as an -framing, hence
reproduces the -category of framed cobordisms of def. 1.
For the classifying space of real vector bundles of rank (the delooping of the ∞-group underlying the orthogonal group) and for the vector bundle associated to the -universal bundle, then -structure on -dimensional manifolds is essentially no-structure (the maximal compact subgroup-inclusion is a weak homotopy equivalence). Cobordisms with this structure will also be called unoriented cobordisms
This we get to below.
This is (Lurie, def. 2.4.17).
Let be a symmetric monoidal (∞,n)-category with duals, let be a CW-complex, let be an -dimensional vector bundle over equipped with an inner product, and let be the associated O(n)-principal bundle of orthonormal frames in .
In the language of ∞-actions (as discussed there), the space is that of horizontal maps fitting into
where the left map is the classifying map for and the right one is the canonical one out of the homotopy quotient.
Notice that for each point there is an induced inclusion
of the framed cobordisms, def. 1, into those of -structure, def. 3, including those cobordisms whose map to is constant on , and observing that for these an -structure is equivalently an -framing. Moreover, by corollary 1 the induced point evaluation is -equivariant, hence yielding a morphism of ∞-groupoids
More generally, this is true for the pullback structure of along along any map , yielding
By the previous comment, observe that is an equivalence for .
Now the codomain of this natural transformation sends (∞,1)-colimits in over to (∞,1)-limits. (Lurie, theorem 3.1.8) shows that the same is true for the domain. Hence is an equivalence for all that appear as (∞,1)-colimits of the point. But this is the case for all ∞-groupoids , by this proposition.
We consider some special cases of this general definition
We discuss the special case of the cobordism hypothesis for -cobordisms (def. 3) for the case that the vector bundle is the trivial vector bundle .
In this case . Write
This is a special case of the above theorem.
Notice that one can read this as saying that is roughly like the free symmetric monoidal (∞,n)-category on the fundamental ∞-groupoid of (relative to -categories of fully dualizable objects at least).
be the classifying space for ;
is the -category of cobordisms with -structure.
See (Lurie, notation 2.4.21)
For the trivial group, a -structure is just a framing and so
For the orthogonal group itself equipped with the identity map a -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 -structure.
This is (Lurie, theorem 2.4.26).
Theorem 2 asserts that
By the discussion at dependent product
which are the homotopy invariants.
The case that is the identity is at the other extreme of the framed case, and turns out to be similarly fundamental.
For an (∞,1)-topos, write for the (∞,n)-category of correspondences in . For an (∞,n)-category with duals internal to , write for the (∞,n)-category of correspondences over 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 -∞-action on them, corollary 1, is trivial for . This means that an -homotopy fixed point in is just an object of equipped in turn with an -∞-action. Therefore
Local unoriented-topological field theory
are equivalent to objects equipped with an -∞-action.
At least for ∞Grpd, then given such, the corresponding field theory sends a cobordism to the space of maps
In particular this means that the assignment to the point is again itself.
This is a slight rephrasing of the paragraph pp 58-59 in (Lurie).
are equivalent to
This is (Lurie, prop. 3.2.8).
At least if ∞Grpd, then local unoriented-topological field theories of the form
are equivalent to a choice
By prop. 3 the co-restriction
is equivalent to an ∞-action
Therefore by prop. 4 is equivalent to
There is a vast generalization of the plain -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 now corresponds to a new generator k-morphisms, and the (framed) -category of cobordisms with singularities is now no longer the free symmetric monoidal -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 is a 1-morphism of the form , then this defines a type of codimension -boundary; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal -category with all dual are equivalent to choices of morphisms , where 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.
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
In particular this means that is itself an -category, i.e. an ∞-groupoid.
When interpreting symmetric monoidal functors from bordisms to as TQFTs this means that TQFTs with given codomain form a space, an ∞-groupoid. In particular, any two of them are either equivalent or have no morphism between them.
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 -dimensional framed theory one needs to assign a fully dualizable object, and the meaning of the term “fully dualizable” depends on , and gets increasingly hard to satisfy as n grows..
For an -dimensional unoriented theory, the object assigned to the point has to be a fixed point for the - action on fully dualizable objects that is obtained from the framed case of the theorem.
In the 1d case, this action on dualizable objects takes every object to its dual, and an fixed point is indeed a vector space with a nondegenerate symmetric inner product.
For an oriented theory -dimensional theory need an -fixed point, which for is nothing but for 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 -structure, for any group mapping to (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 -fixed point in dualizable objects in your category (with acting through ).
This beautifully includes all the above plus for example manifolds with maps (up to homotopy) to some auxiliary (connected) space – here we take to be the loop space of (mapping trivially to ), so that a reduction of the structure group of the manifold to involves a map to the delooping .
Such theories are classified by -families of fully dualizable objects.
Notice that there is an important subtlety of Lurie’s theorem in the case of manifolds with -structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the -fixed points for the -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 , 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.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|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|
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 is spelled out in detail in
Lecture notes and reviews on the topic of the cobordisms hypothesis include
Julie Bergner, Models for -Categories and the Cobordism Hypothesis , in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory, AMS 2011
Discussion of the canonical -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