Vafa-Witten invariants of projective surfaces

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In 1994 the physicists Vafa and Witten introduced new "invariants" of four dimensional spaces. These invariants "count" solutions of a certain equation (the N=4 supersymmetric Yang-Mills equations) over the four dimensional space, and should tell us something about the space. There is one for every integer charge of the Yang-Mills field.Motivated by a generalisation 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.Until now mathematicians have been unable to make sense of how this "counting" should be done without getting infinity. This project gives a definition for any space which is "projective", and for any charge (including ones for which troublesome "semistable" or "reducible" solutions appear). We will then compute the invariants for many such spaces with negative curvature. We will also produce "refined" Vafa-Witten invariants containing more information. These should be the invariants sought by physicists aiming to describe "topologically twisted maximally supersymmetric 5d super Yang-Mills theory".