  We provide a definition of Vafa-Witten invariants for projective surface
Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the
Vafa-Witten invariants for projective surfaces inspired by the S-duality
conjecture. We give calculations for a root stack over a general type quintic
surface, and quintic surfaces with ADE singularities. The relationship between
the Vafa-Witten invariants of quintic surfaces with ADE singularities and the
Vafa-Witten invariants of their crepant resolutions is also discussed.
