Abstract: This is report on joint work with Martijn Kool. Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces. Sheaves can be viewed as generalizations of vector bundles. Moduli spaces are algebraic varieties that parametrize the objects in which we are interested.
While the Vafa-Witten formula cannot be literally true as a formula for the Euler numbers in general, we give an interpretation of it in terms of virtual topological invariants, and confirm it in many cases. We extend the results to finer topological invariants and also to rank 3.
Such virtual invariants occur everywhere in modern enumerative geometry, inspired by physics, like Gromov-Witten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. Basically the idea is that the moduli spaces have very bad properties, but they "want to be" smooth, and the virtual invariants are those that they would have if they were smooth.
We will try to give elementary explanations of all these concepts and their background, as much as is possible. Most of the time will be spent to just trying to explain the problem and the result. If there is time, some hints will be given about the methods.
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