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.

中文翻译：

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通过表面Deligne-Mumford堆的Tanaka–Thomas的Vafa–Witten不变量

我们提供了射影表面Deligne-Mumford堆栈的Vafa-Witten不变量的定义，归纳了受$ S $-对偶性猜想启发，在射影表面的Vafa-Witten不变量上构造Tanaka-Thomas的情况。我们给出了一般类型五次曲面和具有ADE奇点的五次曲面上的根堆栈的计算。还讨论了具有ADE奇异性的五项曲面的Vafa-Witten不变量与其新分辨率的Vafa-Witten不变量之间的关系。