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(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.)

Recently, I've been reading a lot about topics in quantum field theory (QFT) and topological quantum field theory (TQFT). I've consulted many previous threads asking for book and reference recommendations (e.g: here from 2010), and I've thoroughly enjoyed learning about this area of math. It's beautiful stuff, but I have a couple of questions after learning more and more about this subject (by "subject", I mean TQFT).

Question(s): What exactly "comes after" TQFT mathematically? What has it been applied to in recent (say, 5-10 but this is not strict) years? Is TQFT still actively researched, or is it a "dead" subject and if so, what new things have been built off of it?

I’ve tried scouring the internet in search of an answer to this question, but I haven't seemed to find a satisfactory one yet. Most of the resources (and papers that I've found thus far) on TQFT are at least over a decade old and all seem to contain the same material, so part of me has been wondering if I’ve been working towards a dead end? This area of math is so interesting that I sincerely hope that this isn’t the case, but I wanted to ask on MathOverflow in the hopes that someone could provide "next steps" after learning about TQFT (by "next steps", things like papers/books to read, new genres to take a look at, lectures to watch, or even some historical remarks are what I'm referring to here). If (hypothetically) there are simply no such resources available, is there a reason for this sudden drop in usage of the subject? What would be better (or more productive) routes to take in that case?

Thanks for the help! A thoughtful answer to this question would be greatly appreciated.

Nicholas James
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    I'm not certain what you mean by "comes after". Are you referring to a specific person's temporal narrative? What generally comes "after" is whatever you take an interest in, i.e. the linear narrative should be your own. – Ryan Budney Jun 26 '23 at 03:17
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    @RyanBudney: Thank you for the reply! What I meant by "comes after" was this: what has it been applied to in recent years (and are there resources to learn about these new applications?), is it actively researched today and if so in what areas, what new genres or topics have been built off of it in recent years (and are there any resources to learn about these new advances?), etc. Hopefully that clears things up, I hope my question is less ambiguous and easier to answer now. – Nicholas James Jun 26 '23 at 03:24
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    Presumably you're familiar with the relationship between string theory and TQFT? – Alec Rhea Jun 26 '23 at 04:01
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    The holy grail of TQFT would be a prescription for the duality that interchanges the operator quantum numbers and topological quantum numbers of an arbitrary QFT. More prosaically, there is the sub-problem of finding dualities between classes of perturbative intractable QFTs and (inherently nonperturbative) TQFTs. – Buzz Jun 26 '23 at 04:09
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    Along similar lines to the above comment, find your favourite TQFT paper, type its title into Google Scholar, and look at "cited by". Repeat. – JP McCarthy Jun 26 '23 at 08:29
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    I think a good resource to learn what is currently going on in the area of TQFT is https://scgcs.berkeley.edu/ and in particular the "events" tab. For example you can find a recent summer school listed there http://categorified.net/TQFT2023/ with lecture notes etc. There are more events coming up, so you might want to ask the organisers if you could attend. – Abdelmalek Abdesselam Jun 26 '23 at 09:59
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    A big topic is a restricted notion of TQFT: and extended topological field theory. The main activity here is to try to give a complete proof of Laurie’s cobordism hypothesis. This is mostly under the auspice of homotopy theory and differential topology. https://en.m.wikipedia.org/wiki/Cobordism_hypothesis – Ian Agol Jun 26 '23 at 15:29
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    The question asks about “new genres to take a look at”, one example is nontopological functorial field theory, see https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/survey_submitted1.pdf, and, more recently, https://arxiv.org/abs/2011.01208 and https://arxiv.org/abs/2111.01095. – Dmitri Pavlov Jun 26 '23 at 20:08
  • @IanAgol when you say "Laurie" do you mean "Lurie"? Also, the Cobordism Hypothesis is due to Baez and Dolan, but Lurie sketched a proof of it. – David White Jul 03 '23 at 00:49

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