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I'm really interested in Topological Quantum Field Theory (TQFT) and am currently planning to focus on it in my undergraduate thesis. My university, unfortunately, does not allow double majors in mathematics and physics or even a minor in physics, hence I fear that I do not have enough background in physics. I have taken the standard physics series though and have also read that a high level of understanding of quantum field theory is not a hard prerequisite to doing mathematical research in TQFTs. My questions then are primarily:

How then should I start learning and doing undergraduate-level research on TQFTs? Can I also ask for recommended readings on TQFTs for undergraduates and/or directions for undergraduates hoping to do research in the field?

We were told that we are not really expected to come up with a unique result or proof of our own given our lack of mathematical knowledge, and I'm really hopeful that this is doable. Although I will do my best to independently learn what I need to, I will understand if this is too advanced of a topic for just an undergraduate. Thank you!

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    $\begingroup$ Only a dabbler in TQFTs but my impression is you don't really need to know much (or any) physics to get into TQFTs. $\endgroup$
    – Asvin
    Feb 14 at 15:37
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    $\begingroup$ You might find the answers at mathoverflow.net/questions/178640/… helpful. $\endgroup$ Feb 14 at 16:30
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    $\begingroup$ I think the book link.springer.com/book/10.1007/978-3-319-49834-8 may help you! $\endgroup$
    – rfloc
    Feb 15 at 3:21
  • $\begingroup$ Even without majoring, why don't just audit a course about QFTs in the physics department? Must be a graduate course, sure the instructor would agree if you talk to them beforehand (or have your advisor drop them an email). $\endgroup$
    – Asaf
    Feb 15 at 14:55

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I am not sure if this is an answer or a request for clarification.

There are a lot of different topics in math (and in physics) that go by the name 'topological quantum field theory'. Beyond the initial hand-waving about '(some) observables not depending on the spacetime manifold's geometric structure', they aren't always that closely related. You can't reasonably cover all of them in a thesis.

So your first task would be narrow your list of topics down. Where do you want to focus on? Are you interested in higher category theory? Operads? Knots? Quantum groups? Vertex algebras? Group cohomology? Geometric representation theory? The existence of smooth structures on manifolds? The topology of moduli spaces? The algebraic geometry of moduli spaces? This isn't anything like an exhaustive list. (And it's probably 10 years out of date!)

I don't think you need to know physics, but it's helpful to at least try try reading physics lectures notes. I've found Greg Moore's various reviews have a mathematician-friendly perspective, FWIW.

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  • $\begingroup$ Would it be alright to then ask what focus you think might be the most accessible for an undergraduate student in math? thank you so much for this answer! $\endgroup$
    – Gab Romero
    Feb 20 at 15:40
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    $\begingroup$ @GabRomero That is a choice which is probably best made in consultation with an adviser at your university. You're going to need help at some point, and it'll matter more what your adviser is familiar with than which topic is on average easier to get into. $\endgroup$
    – user1504
    Feb 20 at 18:18
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I think the best way to go about any research project is to first identify the problem and then find a solution. The first step is actually the most important part. You want to study TQFT's. I wanted to study them in undergrad as well, though I have to admit my reasoning was because "it sounds cool and complicated." This isn't a great reason, so I hope you've come up with something more convincing for yourself. The first step for you will be to understand how QFT's are solved. What I mean is, if I show you a Lagrangian and you have no idea what you're supposed to do with it, then there's no point in complicating it with topology yet. So I would make sure you are familiar with the basics of that. Much of the intuition for why physicists care about TQFT's can be seen by looking at what Berry Curvature is, and how it leads to the Chern number. If you are not familiar with this, you can read Tong's lecture notes on it online:

http://www.damtp.cam.ac.uk/user/tong/qhe.html

Having done this, you may want to look into some of the topological terms that play an important role in physics. In the condensed matter setting, the "Theta-term" and the "Wess-Zumino-Witten term" are often considered. You may want to try to understand why these terms are used, and it may be especially helpful to compare them to Berry Curvature/Chern Number (see how they are similar and how they are different). Tong also discusses Chern-Simons theory which is a TQFT for the hall effect.

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    $\begingroup$ I have to disagree that "first step for you will be to understand how QFT's are solved". There are plenty of problems in mathematical TQFT that are entirely about algebra and topology, and OP might be interested in them, especially if they primarily have a math background. From their perspective, there might be no point in complicating the topology with physics! $\endgroup$ Feb 15 at 14:20
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    $\begingroup$ I think your answer is good advice for someone who is primarily interested in physical aspects of (T)QFTs, however. $\endgroup$ Feb 15 at 14:20

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