Vafa-Witten invariants of projective surfaces

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".

We expect our proposal to lead to a blossoming of links between geometry and theoretical physics. The interaction between these subjects is well-documented in many areas, and has led to breakneck progress in various fields. But in Vafa-Witten theory it has been held up by the lack of a mathematical definition of the invariants. With this in place we expect rapid progress. From experience of this interaction in the past, we can expect exciting and dramatic predictions from physics that drive the mathematics forward faster.

This research project should forge new links between enumerative algebraic geometry, geometric representation theory and string theory. In the long run physicists expect a refined version of Vafa-Witten theory to have an impact on 4-manifold topology. We will work towards this goal by producing this refined theory on projective surfaces; our understanding of this case should eventually lead to a theory on all 4-manifolds.