  Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have
developed a theory of Vafa-Witten invariants for projective surfaces using the
moduli space of Higgs sheaves. Their definition and calculation prove the
S-duality prediction of Vafa-Witten in many cases in the side of gauge group
$SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture
in physics by Vafa-Witten and the definition of Vafa-Witten invariants for
smooth projective surfaces and surface Deligne-Mumford stacks. We make a
prediction that the Vafa-Witten invariants for Deligne-Mumford surfaces may
give the generating series for the Langlands dual group
$^{L}SU(r)=SU(r)/\zz_r$. We survey a check for the projective plane $\pp^2$.
