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A student who specialises in algebraic geometry has contacted me to ask if they could collaborate with me on some problem which relates mathematical physics and algebraic geometry. I think his idea is that he knows more about algebraic geometry and I know more about physics and QFT, so he is asking if there is some problem which combines both where we would both contribute decisively.

I would be keen to stay away from string theory for now if possible, so I was wondering if there is some interesting research problem at the moment which combined both QFT and algebraic geometry (I can be more specific about the type of algebraic geometry which he does if necessary if the problem needs to be narrowed down further).

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    $\begingroup$ The problem of categorification can be seen as in the intersection of TQFT and algebraic geometry, see here. $\endgroup$
    – Sebastien Palcoux
    Commented Jun 7, 2020 at 17:44
  • $\begingroup$ This older paper of a classmate of mine was in this intersection of areas: arxiv.org/pdf/1112.6208.pdf and may be of interest. $\endgroup$
    – B. Bischof
    Commented Jun 8, 2020 at 1:04
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    $\begingroup$ There has been a lot of work on the algebraic geometry of Feynman integrals (just search for "periods, motives, Feynman integrals"). You might also consider the direction started by Arkani-Hamed et al. in arxiv.org/abs/1212.5605 $\endgroup$
    – Vivek Shende
    Commented Jun 8, 2020 at 4:13
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    $\begingroup$ If you count matrix models or more general integrable systems as QFT, these have a lot to do with the algebraic geometry of the spectral curve. See the review article arxiv.org/abs/1510.04430 . $\endgroup$
    – Sylvain Ribault
    Commented Jun 8, 2020 at 18:46

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