Let $C$ be a symmetric monoidal $n$-category. An extended framed $C$-valued TQFT is a symmetric monoidal functor from the framed bordism category $\mathrm{Cob}^{fr}_n(n)$ to $C$.

The cobordism hypothesis, first formally written down by Baez--Dolan, states that an extended framed $C$-valued TQFT is determined up to isomorphism by its value at the point. It has been proved by Lurie.

I understand that this is an interesting statement from the topological and $n$-categorical points of view. My question is: are there any corollaries of this hypothesis that are of physical interest? Or is there at least some sort of physical motivation/plausibility argument for cobordism hypothesis?


1 Answer 1


Yes. The physical motivation is that topological field theories, as examples of quantum field theories, should be fully local, meaning that one should be able to calculate any information about a (fully extended) TQFT $Z$ on a manifold $M$ by cutting $M$ into pieces, formulating $Z$ on these pieces, and gluing. The takeaway in physics is that in any class of systems thought to be described by topological field theories, one should be able to determine the TQFT for a particular system from how the system behaves on a neighborhood of a point.

Even though we don't know how to define quantum field theories precisely, physicists know they must satisfy a few important axioms, including locality. This means that all of the data in a quantum field theory formulated on a manifold $M$ satisfies a sheaflike axiom: one must be able to glue the fields together from information on an arbitrary small open cover of $M$. So a principal bundle with connection on $M$ is OK, as is a differential form, but a CW structure is not. Similarly, things such as the Lagrangian or Hamiltonian must be expressible in terms of data satisfying this condition. This implies that information one can calculate about the QFT, such as state spaces or partition functions, also satisfies a gluing axiom: they are determined somehow from the information entering the theory, which we can formulate on small balls around any point, satisfying some kind of gluing condition.

In functorial TQFT, we're not in general provided with fields or Hamiltonians, but only things like partition functions and state spaces, and their lower-dimensional analogues. But TQFTs are QFTs, so they must satisfy locality. This motivated Atiyah's original definition: the partition function $Z(M)$ must be local, so we can cut $M$ into a sequence of cobordisms and compose the maps between state spaces to compute partition functions. This is only expressing locality in one direction out of $n$, so we should be able to repeat this process in the other $n-1$ directions to compute the partition function in terms of fully local data, i.e. describing $M$ as glued together from many copies of $\mathbb R^n$, then using whatever $Z(\mathbb R^n)$ means to compute the partition function of $M$.

From this perspective, we see $\mathbb R^n$ instead of points, which may seem strange. The reason is that in a cobordism category of Riemannian or Lorentz manifolds, which you'd use for functorial non-topological QFT, one has to add a collar to manifolds in codimension one, so when fully extending, one has a two-dimensional collar in codimension 2, and so on. In this way full locality is seen by $\mathbb R^n$, an $n$-dimensional collar around a point. But for cobordisms between smooth manifolds, a collar is redundant data, so the axioms of fully extended TQFT drop the collars and just use the higher-codimension manifolds, whence $Z(\mathrm{pt})$: the cobordism hypothesis says a TQFT should be determined by fully local data, which would mean cutting any manifold into copies of $\mathbb R^n$, and the axioms of TQFT replace $Z(\mathbb R^n)$ by $Z(\mathrm{pt})$.

That's part of the cobordism hypothesis; it also calculates what kinds of objects $Z(\mathrm{pt})$ can be. Unfortunately I'm not sure of the physical motivation for that part.

Moving on to consequences in physics. I don't know of a general statement, but I'll discuss an interesting example in the theory of topological phases of matter. These are condensed-matter systems which display surprising topological properties, and condensed-matter theorists are interested in classifying them (in a fixed dimension $n$, say). It's not yet known what the mathematical definition of a topological phase of matter is, but physicists often formulate them in terms of "lattice field theories": discretized QFTs on manifolds equipped with something like a triangulation, and such that the fields, Hamiltonian, etc. are also suitably discretized in terms of purely combinatorial information. (The locality axiom is different in this setting, expressed in terms of graph distance.) It is believed that the low-energy physics of such systems is described by fully extended TQFTs, but even formulating a precise conjecture, never mind a proof, is currently open at both physical and mathematical levels of rigor, as far as I know. Moreover, the low-energy TQFT should classify the lattice system in some strong sense. This provides a (conjectural) approach to classifying topological phases of matter: given an $n$-dimensional TQFT $Z$, build a topological phase with $Z$ as its low-energy TQFT, and then classify $n$-dimensional TQFTs.

Of course, explicating the cobordism hypothesis in dimensions greater than 2 is tricky, but this approach appears to be working. For example, the $\mathrm{SO}_3$-homotopy fixed points in the 3-category of $\mathbb C$-linear monoidal categories are conjectured to be spherical fusion categories (one direction is known, due to Douglas-Schommer-Pries-Snyder), and the TQFT associated to a spherical fusion category $\mathsf C$ is the Turaev-Viro-Barratt-Westbury (TVBW) TQFT for $\mathsf C$. In physics, topological phases of matter in (spacetime) dimension 3 on oriented manifolds are believed to be classified by the Levin-Wen model associated to a spherical fusion category, and Kirrilov Jr proves a result which provides strong evidence that the low-energy theory of the Levin-Wen model for $\mathsf C$ is the TVBW TQFT for $\mathsf C$.

Another example is the classification of symmetry-protected topological (SPT) phases. These are (conjecturally) the topological phases of matter whose low-energy theories are invertible TQFTs. The cobordism hypothesis implies that these are classified by homotopy classes of maps out of Madsen-Tillmann spectra (though there's now a proof of this fact independent of the cobordism hypothesis, due to Schommer-Pries), and one can compare this classification with other classifications of SPTs. This was taken up by Freed-Hopkins, who found agreement between this and other kinds of classifications for a variety of dimensions and symmertry groups.

For some more detail on the motivation for the cobordism hypothesis, as well as some other applications in physics, I recommend Dan Freed's expository article.


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