We define Vafa-Witten invariants via formal virtual localization. First we review localization and virtual localization with respect to $C^*$-actions. We describe the fixpoint locus of the $C^*$ action on the Vafa-Witten moduli space, and its decomposition into components corresponding to the partitions of the rank r.
We introduce the Vafa-Witten generating function, and its decomposition corresponding to the components of the fix point locus. In particular we have the horizontal component, corresponding to the partition (r), which is the moduli space of stable sheaves, and the vertical component, corresponding to the partition (1,..,1). We show that the contribution to the horizontal component to the VW invariants is the virtual Euler number of the moduli space of sheaves.
We state the Vafa-Witten predictions for the modularity of the Vafa-Witten generating function.
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