nLab cobordism hypothesis

Idea

Plain cobordism hypothesis

What is known as the Cobordism Hypothesis is, in its most basic form, the conjecture that, roughly, the (∞,n)-category of cobordisms $Bord_n^{fr}$ with framing is equivalently the free symmetric monoidal (∞,n)-category with duals on a single object.

The idea is due to Baez & Dolan (1995), who stated it in the further generality of $k$-tuply monoidal $n$-categories (i.e. for $k \leq \infty$) under the name Tangle Hypothesis (p. 23), before definitions of these higher category theoretical notions were available. The term “cobordism hypothesis” is used by Baez and Dolan only in passing and without definition (p. 25) but was popularized by Lurie (2009) who gave a detailed though still incomplete sketch of a proof by appeal to the notion of $n$-fold complete Segal spaces as a geometric definition of higher categories, thereby breaking the impasse caused by previous attempts (e.g. tac:18-10) which instead tried algebraic definitions of higher categories.

Since a (framed) fully extended topological quantum field theory (TQFT) may be identified with a symmetric monoidal $(\infty,n)$-functor $Z \colon Bord^{fr}_n \to \mathcal{C}^\otimes$, the cobordism hypothesis implies (and should be implied by, via a higher Yoneda lemma) that fully extended TQFTs are entirely determined by their value on the point (the “n-vector space of states” of the theory), and conversely that fully dualizable objects in $\mathcal{C}^{\otimes}$ equivalently encode such fully extended TQFTs. As such the hypothesis is actually called the “Extended TQFT Hypothesis” by Baez & Dolan (1995) (p. 28).

From the cobordism hypothesis for framing structure it follows fairly readily that the $\infty$-groupoid of fully dualizable objects in a symmetric monoidal $(\infty,n)$-category carries an action of the orthogonal group $O(n)$ and that cobordisms with tangential G-structure for $G \hookrightarrow O(n)$ are similarly classified by the homopy fixed loci of the corresponding $G$-action [Lurie (2009, Thm. 2.4.26)]. This yields the statement of the hypothesis for common notions of cobordisms, such as with orientation (SO(n)-structure), spin structure (Spin(n)-structure), etc.

The idea of the cobordism hypothesis is that:

1. cylinders count as trivial (identity) cobordisms between their boundary components, corresponding to how local spacetime-evolution in a topological quantum field theory is trivial (corresponding to the physics jargon that “their Hamiltonian vanishes”),

2. so that all the non-trivial information in cobordism and in TQFT-propagation is in the structure of handle body-attachments (in physics known as: “topology change”) which obey the zig-zag rules known from adjunctions, 2-adjunctions and their further higher dimensional analogs,

3. which means that the point – regarded as a 0-dimensional element of an $n$-dimensional cobordism – is equipped with the structure of, first of all, a dualizable object under the relevant tensor product of cobordisms (which is disjoint union), and next that of a $k$-dualizable object for all $k \leq n$.

Geometric cobordism hypothesis

More generally, the geometric cobordism hypothesis [Grady & Pavlov (2021)] asserts that general non-topological geometric extended functorial field theories (such as conformal field theory as well as the usual relativistic QFT on curved spacetimes) are similarly encoded by (1.) local data depending only on the notion of geometry and (2.) global topological data encoded only in higher dualizability structure.

For example, the FRS-theorem on rational 2d CFT is, at least in spirit, a realization of this idea: It states that rational 2d conformal field theories, when defined on all possible worldsheet surfaces, are fully encoded (1.) by local geometric data encoded in vertex operator algebras or conformal nets or similar and (2.) global purely topological data (a suitable Frobenius algebra object internal to the modular tensor category of representations of that vertex operator algebra/conformal net).

Conversely, the plain (topological) cobordism hypothesis is the special case of the geometric cobordism hypothesis of no geometric structure and in this sense very much captures the idea which in physics is known as “TQFTs are those QFTS without local degrees of freedom”.

In general, where the notion of extended (geometric) functorial field theory is one candidate for a definition of (non-topological) non-perturbative quantum field theory, the (geometric) cobordism hypothesis may be regarded as the first step in establishing the construction of such theories, reducing the problem entirely to the local behaviour of the theory.

Notice that the local behaviour of quantum field theory is still immensely rich: It keeps busy quantum field theorists most of whom never encounter a topological effect in their life. For example the original statement of the Wightman axioms/Haag-Kastler axioms are entirely concerned only with the local behaviour of Lorentzian QFTs (namely on Minkowski spacetime) and yet the construction of any non-trivial examples in dimensions $d \geq 4$ remains wide open.

In this vein, much in the spirit of Freyd’s dictum on the role of categorical algebra in general, one may read the (geometric) cobordism hypothesis as trivializing the trivial part of non-perturbative quantum field theory.

A proof of the geometric cobordism hypothesis is thus the important first step towards the currently wide-open and yet crucial problem of mathematically constructing non-perturbative quantum field theories.

Formalization

In [Lurie 2009] a sketch of an approach to the formalization and proof of the cobordism hypothesis is described.

For framed cobordisms

Statement

For $\mathcal{C}$ a symmetric monoidal (∞,n)-category with duals 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.

Definition

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

Theorem (cobordism hypothesis, framed version)

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

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

induces an (∞,n)-functor

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

such that

• this factors through the core of $\mathcal{C}$;

• the map

  $$pt^* \;\colon\; Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to Core(\mathcal{C})$$

  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 $|Bord_n^{or}|$ in terms of the suspension of the Thom spectrum;

2. 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.

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

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

Corollary

Every ∞-groupoid

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

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 $Bord_n^{fr}$.

(Lurie, corollary 2.4.10)

Example

The action in corollary is

• for $n = 1$: the $O(1) = \mathbb{Z}/2\mathbb{Z}$ action given by passing to dual objects;

• for $n = 2$: the $O(2)$-action of 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)

Proposition

For all $n \in \mathbb{N}$, the canonical $SO$-∞-action on

$$B^n \mathbb{Z} \in Ab_\infty(\infty Grpd) \hookrightarrow (\infty,n)CatWithDuals$$

is trivial.

Proof

The action on a connective spectrum $\Omega^\infty X$ factors through the J-homomorphism

$$SO \times \Omega^\infty X \stackrel{(J,id)}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X \,.$$

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.

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.

For cobordisms with any structure (“$(X,\xi)$-structure”)

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 $m \leq n$. An $(X,\xi)$-structure on $N$ consists of the following data

• A continuous function $f : N \to X$;

• An isomorphism of vector bundles

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

  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. are the following

Example

For $X = \ast$ the point and $\xi = \mathbb{R}^n$, then an $(X,\xi)$-structure is the same as an $n$-framing, hence

$$(Bord_n^{(\ast, \mathbb{R}^n)}) \simeq Bord_n^{fr}$$

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

Example

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

$$Bord_n^{un} \coloneqq Bord_n^{(B O(n), E O(n)\underset{O(n)}{\times} \mathbb{R}^n)} \,.$$

Accordingly, for $X = B SO(n)$ the delooping of the special orthogonal group, the corresponding $(X,\xi)$-structure makes oriented manifolds

$$Bord_n^{or} \coloneqq Bord_n^{(B SO(n), E SO(n)\underset{SO(n)}{\times} \mathbb{R}^n)} \,.$$

Generally:

Example

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.

Definition

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. .

This is (Lurie, def. 2.4.17).

Theorem

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

$$Fun^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \simeq Top_{O(n)}(\tilde X, \tilde \mathcal{C}) \,,$$

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). The following is some aspects of the idea of the proof in (Lurie, p. 57).

Remark

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

$$\array{ X && \longrightarrow && \tilde X//O(n) \\ & \searrow && \swarrow \\ && B SO(n) }$$

where the left map is the classifying map for $\xi$ and the right one is the canonical one out of the homotopy quotient.

Idea of Proof of theorem .

Notice that for each point $x \colon \ast \to X$ there is an induced inclusion

$$Bord_n^{fr} \stackrel{x}{\longrightarrow} Bord_n^{(X,\xi)}$$

of the framed cobordisms, def. , into those of $(X,\xi)$-structure, def. , 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 the induced point evaluation is $O(n)$-equivariant, hence yielding a morphism of ∞-groupoids

$$\alpha \;\colon\; Func^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X, \tilde \mathcal{C}) \,,$$

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

$$\alpha_Y \;\colon\; Func^\otimes(Bord_n^{(Y,\xi|Y)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X\underset{X}{\times} Y, \tilde \mathcal{C}) \,.$$

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

For framed cobordisms in a topological space

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-cobordisms (def. ) 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

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

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

Corollary

There is an equivalence in ∞Grpd

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

This is a special case of the above theorem.

Notice that one can read this as saying that $Cob_n(X)$ 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).

For cobordisms with $G$-structure

We discuss the special case of the cobordism hypothesis for $(X,\xi)$-bundles (def. ) for the special case of G-structure (example ), 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 $E G \to B G$.

Definition

We say

$$Bord^G_n := Bord_n^{(B G, \xi_\chi)} \,.$$

is the $(\infty,n)$-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

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

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

• 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

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

• 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,

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

See (Lurie, example 2.4.22).

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

Corollary

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

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

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

This is (Lurie, theorem 2.4.26).

Proof

Theorem asserts that

$$Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq Maps_{G}(E G , \tilde C) \,.$$

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

$$Maps_G(V,W)\simeq \infty Grpd_{/B G}(V/\!/G, W/\!/G) \,.$$

This yields

$$Maps_{G}(E G , \tilde C) \simeq \infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \,.$$

By the discussion at dependent product

$$\infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \simeq \underset{B G}{\prod} (\tilde C /\!/B G)$$

which are the homotopy invariants.

For (un-)oriented cobordisms

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

$$Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes$$

By the discussion at (∞,n)-category of correspondences these are (∞,n)-categories with duals and the canonical $O(n)$-∞-action on them, corollary , 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

Proposition

Local unoriented-topological field theory

$$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H})^\otimes$$

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

$$\array{ \Pi(\Sigma) && \longrightarrow && X//O(n) \\ & {}_{\mathllap{T \Sigma \oplus \mathbb{R}^{n-dim(\Sigma)}}}\searrow && \swarrow \\ && B O(n) }$$

hence

$$Z_{X//O(n)} \colon \Sigma \mapsto [\Pi(\Sigma),X]^{O(n)} \,.$$

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).

Proposition

At least for $\mathbf{H} =$ ∞Grpd, with $X \in \mathbf{H}$ an object equipped with an $O(n)$-∞-action, then horizontal lifts in

$$\array{ && Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \\ & \nearrow & \downarrow \\ Bord_n^\sqcup &\underset{X//O(n)}{\longrightarrow}& Corr_n(\mathbf{H})^\otimes }$$

are equivalent to

$$(Bord_n^{(X//O(n), X \underset{O(n)}{\times}\mathbb{R}^n)})^{\sqcup} \longrightarrow Phases^\otimes \,.$$

This is (Lurie, prop. 3.2.8).

Remark

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. translates in quantum field theory jargon to the statement that “All background structures are fields.” This is essentially the slogan of general covariance.

Corollary

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

$$Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$$

are equivalent to a choice

1. of $X \in \mathbf{H}$ equipped with an $O(n)$-∞-action

2. 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

$$\array{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow &\swArrow_\simeq& \swarrow \\ && B O(n) } \,.$$

Proof

By prop. the co-restriction

$$Bord_n^\sqcup \stackrel{Z}{\longrightarrow} Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes$$

is equivalent to an ∞-action

$$\array{ X &\longrightarrow& X//O(n) \\ && \downarrow \\ && B O(n) }$$

Therefore by prop. $Z$ is equivalent to

$$(Bord_n^{(X//O(n), X \underset{O(n)}{\times} \mathbb{R}^n) })^\sqcup \longrightarrow Phases^\otimes \,.$$

Finally, by theorem and in view of remark , this is equivalent to maps of the form

$$\array{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow && \swarrow \\ && B O(n) } \,.$$

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.

For HQFTs

If in def. 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 $Y$.

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

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.

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) boundary 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).

(Lurie, section 4.2)

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. .

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).

A full-blown geometric cobordism statement is due to Grady & Pavlov 2021.

Remarks

Morphisms of TQFTs

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 .

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 $SO(n)$-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.

Related concepts

• generalized tangle hypothesis

• conformal cobordism category

• opetopic type theory

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

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

References

Topological cobordisms

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:q-alg/9503002]

A sketch of an approach to the formalization and proof of the cobordism hypothesis is described in:

• Jacob Lurie, On the Classification of Topological Field Theories, Current Developments in Mathematics 2008 (2009) 129-280 [arXiv:0905.0465]

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

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

and aspects of the case $n = 2$ (see also at 2d TQFT) are further discussed in

• Chris Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories (arXiv:1112.1000)

Lecture notes and reviews on the topic of the cobordisms hypothesis:

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

• Jacob Lurie, The Classification of Extended Topological Field Theories (video)

• 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, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (arXiv:1210.5100, doi:10.1090/S0273-0979-2012-01393-9)

• Chris Schommer-Pries, Dualizability in Low-Dimensional Higher Category Theory (arXiv:1308.3574)

• Constantin Teleman, Five lectures on topological field theory, 2014 (pdf)

An approach to the proof of the cobordism hypothesis via factorization homology is in

• David Ayala, John Francis, The cobordism hypothesis, (arXiv:1705.02240)

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

• Gerald Gaudens, Luc Menichi, section 5 of Batalin-Vilkovisky algebras and the $J$-homomorphism, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365–374 (arXiv:0707.3103)

Geometric cobordisms

Higher cobordisms with geometric structure (i.e. for non-topological extended quantum field theory) are discussed in

• David Ayala, Geometric cobordism categories (arXiv:0811.2280)

• Daniel Grady, Dmitri Pavlov, Extended field theories are local and have classifying spaces [arXiv:2011.01208]

and full proof of the cobordism hypothesis in this geometric generality (hence for non-topological extended FQFT such as conformal field theory) is claimed in:

• Daniel Grady, Dmitri Pavlov, The geometric cobordism hypothesis [arXiv:2111.01095]

based on Grady & Pavlov 2020.

Review:

• Dmitri Pavlov, The geometric cobordism hypothesis, talk at QFT and Cobordism, CQTS (Mar 2023) [pdf, pdf, web, video: YT]