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As Richard Thomas has written (we paraphrase just slightly), mathematical physicists Vafa and Witten introduced new "invariants" of four-dimensional spaces in a paper:

A Strong Coupling Test of S-Duality (1994),

Cumrun Vafa, Edward Witten

https://arxiv.org/abs/hep-th/9408074 Nucl.Phys.B431:3-77,1994

These invariants "count" solutions of a certain equation (the N=4 supersymmetric Yang-Mills equations) over the four dimensional space, and should tell us (both physicists and mathematicians) something about the space. There is one for every integer charge of the Yang-Mills field.

Motivated by a generalization of electromagnetic duality in string theory, Vafa and Witten predicted that on a fixed space, one could put all these invariants together in a generating series (a Taylor series or Fourier series, with coefficients the Vafa-Witten invariants) and get a very special function called a "modular form". In particular, the invariants should have hidden symmetries that mean that only a finite number of them determine all the rest.

If I was told correctly, until now mathematicians have been unable to make sense of how this Vafa-Witten "counting" should be done without getting infinity.

question 1: So what are Vafa-Witten invariants meant for mathematicians in your research fields or subfields? (questionable since the mathematical "counting" so far involves getting infinity.)

question 2: Are there similar invariants describing "topologically twisted maximally supersymmetric 5d Yang-Mills theory"?

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    $\begingroup$ Are you aware of arxiv.org/abs/1702.08487, arxiv.org/abs/1702.08488 ? $\endgroup$
    – ssx
    Commented Dec 18, 2019 at 23:05
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    $\begingroup$ Why does the text of that grant application so closely match the text of this question? $\endgroup$
    – Sam Hopkins
    Commented Dec 19, 2019 at 0:03
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    $\begingroup$ (FWIW, the comment I made was in reference to a now-deleted comment by the question-asker linking to gow.epsrc.ukri.org/NGBOViewGrant.aspx?GrantRef=EP/R013349/1) $\endgroup$
    – Sam Hopkins
    Commented Dec 19, 2019 at 1:26
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    $\begingroup$ I was just being lazy to make similar statements -- thus so trust more about the statements made by the experts like R. Thomas. Let us focus on the scientific content of the questions. My statement is inspired by reading that text over there. (I am not writing another grant proposal, just to clarify.) Again, let us focus on the scientific content of the questions. $\endgroup$
    – wonderich
    Commented Dec 19, 2019 at 2:11
  • $\begingroup$ @Simpleton, thanks for the Refs. $\endgroup$
    – wonderich
    Commented Dec 19, 2019 at 17:05

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I'm not trying to give an answer, I just want to make a few remarks.

question 2: Are there similar invariants describing "topologically twisted maximally supersymmetric 5d Yang-Mills theory"? No. A crucial aspect of the Vafa-Witten computation is supersymmetry. For supersymmetry to exist, a globally covariant section of the spinor bundle (the quare root of the canonical bundle) is needed; in other words, the manifold should be a local model of some even-dimensional Calabi-Yau manifold.

Five dimensional supersymmetric theories exist. But always embedded on higher dimensional spaces.

question 1: So what are Vafa-Witten invariants meant for mathematicians in your research fields or subfields? (questionable since the mathematical "counting" so far involves getting infinity.)

You can think on the Vafa-Witten theory as a lower dimensional analog of the Donaldson-Thomas theory. Indeed, the cases explicitly analized in his famous paper were choosen because on those cases, the Vafa-Witten partition function localizes to the Euler characteristic of the moduli space of instantons for the relevant twisted version of $N=4$ super Yang-Mill theory. In mathematical terms that means that aforementioned invariants provide an enummerative theory of ideal sheaves on a given 4-manifold with spin structure.

For an explicit computation of the Vafa-Witten invariants as "lower dimensional analogue" of the DT ones see: Crystals and intersecting branes.

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