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As far as I understand, the cobordism hypothesis provides a construction of all (appropriately defined) fully-extended TQFTs. In particular, given a fully-dualizable object in a certain category, one can in principle construct the entire TQFT (e.g. the partition function on an arbitrary closed manifold of full dimensionality).

I recently came across the work of Grady and Pavlov, where they claim a proof of the geometric cobordism hypothesis, which should be an analogous statement for arbitrary quantum field theories (i.e. not necessarily topological ones).

My question is: does this lead to a more-or-less explicit construction of any non-trivial quantum field theories? If so, this would be extremely interesting since only a handful of interacting quantum field theories have been constructed in more than two dimensions, and none in more than three dimensions.

A related question is: Are there any tools in the work of Grady and Pavlov (or elsewhere in the literature) that allow one to compute any interesting quantities in non-trivial quantum field theories in more than two dimensions? An example of such a quantity would be the spectrum of the Hamiltonian on a given codimension-1 manifold.

I would be happy to hear any relevant comments, and dispelling of confusions I might have about what has been achieved in this line of developments.

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My question is: does this lead to a more-or-less explicit construction of any non-trivial quantum field theories? If so, this would be extremely interesting since only a handful of interacting quantum field theories have been constructed in more than two dimensions, and none in more than three dimensions.

One explicit example can be found on page 112 of the slides http://dmitripavlov.org/lecture-1.pdf. Given a Lie group $G$, a level for $G$, and an invariant polynomial of degree 2 on the Lie algebra of $G$, it constructs the prequantum Chern–Simons field theory as a fully extended 3-dimensional $G$-gauged functorial field theory. A paper with complete proofs is forthcoming.

Concerning quantization of functorial field theories, it was one of the motivations behind writing the two papers. The geometric cobordism hypothesis reduces the problem to the much more tractable computation of the right side, which amounts to computing certain mapping spaces of simplicial presheaves. In principle, we know what to do to compute the right side, and constructing a single point (as opposed to computing the entire space of quantum field theories) is even easier. Again, details are forthcoming. The only example that is written up so far is the case of quantum mechanics. See Section 6.1 in the paper, and also the introduction.

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    $\begingroup$ Thank you for the answer. However, I haven't quite understood from it what I was trying to get to: To what extent are the new results useful for analyzing non-topological theories of physical interest? Let me propose a specific hard open problem in quantum field theory: "Give an algorithm for computing the lowest nonzero energy-level of the critical 3D Ising model on the unit 2-sphere". Are the new results useful for addressing this question? $\endgroup$ Sep 25 at 23:28
  • $\begingroup$ Concerning the example of Chern-Simons theory, I was hoping for one where the theory is not topological. $\endgroup$ Sep 26 at 2:02
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    $\begingroup$ @ConfusedPhysicist: One can expect the quantization formalism to provide a somewhat explicit formula for the propagators (perhaps in terms of their integral kernels), and automatically prove that these propagators satisfy the Segal–Witten gluing axioms (which is the main point of the GCH). As for establishing spectral gaps, perhaps existing tools from operator theory could be applied, but as far as I am aware, the formalism itself does not provide anything on its own. $\endgroup$ Sep 28 at 22:32
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    $\begingroup$ @ConfusedPhysicist: The Chern–Simons theory is expected to become topological after quantization, but in order to prove this, quantization needs to be defined rigorously (in the mathematical sense), which is one of the motivations behind the project (probably not very interesting for physicsts, though). As another example, the prequantum functorial field theory for the Yang–Mills action can be constructed in a similar way. (Quantizing this theory and/or showing the existence of spectral gaps is, of course, a different story.) $\endgroup$ Sep 28 at 22:36

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