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This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy.

Arithmetic topology is based on an analogy between number fields and 3-manifolds where primes are something like knots, the Legendre symbol is something like a linking number, etc. In quantum topology, on the other hand, one way to study 3-manifolds is to study 3d TQFTs, e.g. functors $Z : 3\text{Cob} \to \text{Vect}$. These functors assign to every 3-manifold, interpreted as a cobordism from the empty 2-manifold to itself, a morphism $k \to k$ where $k$ is the base field, and therefore give $k$-valued invariants of 3-manifolds.

If the analogy between number fields and 3-manifolds is strong enough, there might conceivably exist an "arithmetic cobordism category" whose morphisms are number fields and whose objects are... whatever boundaries of number fields are in arithmetic topology. (One might need to adapt this construction depending on whether number fields are considered to have "boundaries" at all.) It might conceivably be possible to adapt constructions of 3d TQFTs to the arithmetic case and therefore to find "quantum invariants" of number fields.

So is any construction like this possible, or am I just talking nonsense?

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  • $\begingroup$ Something TQFT-like in number theory cropped up here - londonnumbertheory.wordpress.com/2010/05/10/… $\endgroup$
    – David Corfield
    Commented May 26, 2010 at 9:03
  • $\begingroup$ This goes into that direction: math.uiuc.edu/K-theory/0547 $\endgroup$
    – Thomas Riepe
    Commented May 26, 2010 at 9:28
  • $\begingroup$ In the analogy 3manifolds = numberfields, 1-manifolds = finitefields, you're asking 2manifolds = ?. You could also ask 0manifolds = ?. $\endgroup$
    – André Henriques
    Commented May 26, 2010 at 12:45
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    $\begingroup$ Number fields are not (compact) 3-manifolds. Rings of integers and S-integers are. Local fields are 2-manifolds, the boundary around a knot. Rings of integers in them are the tubular neighborhood of the knot. That doesn't give a lot of 2-manifolds to work with. Not enough for Heegaard splittings. But you could try to approach the Casson invariant some other way, without mentioning Heegaard splittings or the TQFT more generally. $\endgroup$
    – Ben Wieland
    Commented Jul 10, 2010 at 3:31

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Unfortunately, I think we're still quite far away from knowing the answer to your question. Fortunately, a lot of progress has been made without answering your question.

The key advance here was Minhyong Kim's arithmetic Chern-Simons theory, which answers and doesn't answer parts of your question – it's key insight is that an analogue of a path integral from Chern-Simons theory gives interesting invariants of number fields.

Other references you might find interesting: Kirthi Joshi has written about surgery on number fields using anabelian ideas; a multi-author paper that I'm having trouble finding now "extends" arithmetic CS theory one dimension down, in the sense of TQFT; and the "Shifted" project of Clark Barwick and Peter Haine is trying to use condensed / pyknotic theory to push some of this further.

But I don't think we have a truly natural answer for what a cobordism is – the artificial answer, I think, is a chain of number fields (or more broadly arithmetic schemes) with "arithmetic surgeries" between them, i.e. isomorphism between some of their local Galois groups.

And I don't think we have answers to other questions that I'd like to know, for example: is there a notion of non-topological arithmetic QFT? In other words, is there a notion of "metric" on the ring of integers of a number field, viewed as a 3-manifold?

I last followed this a number of months ago, so I stand to be corrected if any of this post is incorrect!

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