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
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"?