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. CONTENTS 2.1. Surface DM stacks, examples 2.2. Moduli space of semistable sheaves on surface DM stacks 2.3. The moduli space of Higgs pairs 3. Deformation theory and the Vafa-Witten invariants 3.1. Deformation theory 3.2. Families and the moduli space 3.3. The U(rk) Vafa-Witten invaraints 3.4. SU(rk) Vafa-Witten invariants 3.5. C ˚-fixed loci 4. Calculations 4.1. Root stack on quintic surfaces 4.2. Quintic surfaces with ADE singularities 4.3. Discuss on the crepant resolutions Appendix A. The perfect obstruction theory, following Tanaka-Thomas A.1. The perfect obstruction theory for U(rk)-invariants A.2. Deformation of Higgs fields 1 2

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In this article, we study the existence of tautological families on a Zariski open set of the coarse moduli space parametrizing certain Galois covers over projective spaces. More specifically, let (1) H n.r.d (resp. M n,r,d ) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic covers of degree r over the projective space P n branched along a divisor of degree r d ≥ 4, and (2) d 2 ) be the stack (resp. coarse moduli) of smooth cyclic triple covers over P 1 with 2d 1 − d 2 ≥ 4 and 2d 2 − d 1 ≥ 4. In the former case, we show that such a family exists if and only if gcd(r d,n +1) | d while in the latter case, we show that it always exists. We further show that even when such a family exists, often it cannot be extended to the open locus of objects without extra automorphisms. The existence of tautological families on a Zariski open set of its coarse moduli can be interpreted in terms of rationality of the stack if the coarse moduli space is rational. Combining our results with known results on the rationality of the coarse moduli of points on P 1 and the coarse moduli of plane curves (see [BG10], [BGK09]), we determine the rationality of H 1,r,d (resp. H 2,r,d ) for r d ≥ 4 (resp. r d ≥ 49). On the other hand H 1,3,d 1 ,d 2 is unirational, and we show that its coarse moduli M 1,3,d 1 ,d 2 is unirational and fibred over a rational base by homogeneous varieties which are rational if char(k) = 0. Our study is motivated by [GV09] and [GV08].