We initiate explicit computations of Vafa–Witten invariants of $3$-manifolds, analogous to Floer groups in the context of Donaldson theory. In particular, we explicitly compute the Vafa–Witten invariants of $3$-manifolds in a family of concrete examples relevant to various surgery operations (the Gluck twist, knot surgeries, log-transforms). We also describe the structural properties that are expected to hold for general $3$-manifolds, including the modular group action, relation to Floer homology, infinite-dimensionality for an arbitrary $3$-manifold, and the absence of instantons.

中文翻译：

---

$3$-流形和 Vafa-Witten 理论

我们开始对 $3$ 流形的 Vafa-Witten 不变量进行显式计算，类似于 Donaldson 理论背景下的 Floer 群。特别是，我们在与各种手术操作（格鲁克扭转、结手术、对数变换）相关的一系列具体示例中明确计算了 $3$ 流形的 Vafa-Witten 不变量。我们还描述了一般 3$ 流形的结构特性，包括模群作用、与 Floer 同源性的关系、任意 3$ 流形的无限维性以及瞬子的缺失。