Vafa-Witten invariants and S-duality
For a real four manifold M, the S-duality conjecture of Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons on M to the Langlands dual gauge group SU(r)/Z_r-invariants counting SU(r)/Z_r-instantons on M; and both of the invariants satisfy modularity properties. This is a generalization of electro-magnetic duality in physics. On mathematics side the SU(r)-Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth complex projective surface underlying M. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Z_r-Vafa-Witten invariants, and provide the proposal to prove the S-duality conjecture of Vafa-Witten for algebraic surfaces. A particular case of K3 surface is proved under this proposal.