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](https://arxiv.org/pdf/2111.01095.pdf), 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.



Practical consequences of the geometric cobordism hypothesis

> 
> 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.


