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Abstract: We describe the components corresponding of the fixpoint locus on the Vafa-Witten moduli space.
We introduce the Vafa-Witten generating function, and its decomposition corresponding to the components of the fixpoint 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.
We state the structure theorem of Laaracker for the vertical Vafa-Witten invariants, and give conjectural formulas for the vertical Vafa-Witten invariants.
Finally we conjecture an analogue of Laaracker's structure theorem also for the horizontal invariants.