This is joint work with Richard Thomas. We consider a set of gauge-theoretic equations on closed four-manifolds, introduced by Vafa and Witten back to around '94 on the study of S-duality conjecture in N=4 super Yang-Mills theory in four dimensions. In this talk, we define Vafa-Witten invariants for projective surfaces from the moduli space of solutions to the equations by localisation techniques with respect to its natural C^* action on the moduli space. We then calculate the partition functions of them in examples; and see their matches with predictions made by Vafa and Witten more than twenty years ago. If time allows, we also mention how to categorify these in the style of team Joyce on the theory of the Donaldson-Thomas invariants.
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