In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin–Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing the two solutions together along the cylindrical end. In addition, we establish a local Kuranishi model for this gluing picture. As an application, we show that over any compact 4‐manifold with $S^3$ or $T^3$ boundary, there exists a Nahm pole solution to the obstruction perturbed Kapustin–Witten equations. This is also the case for a 4‐manifold with hyperbolic boundary under some topological assumptions.

中文翻译：

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具有Nahm极的Kapustin–Witten方程的胶合定理

在本文中，我们为带有边界和圆柱端的流形上的Kapustin-Witten方程的Nahm极点解建立了粘合构造。给定两个Nahm极点解在圆柱端有一些收敛性假设，我们证明存在一个障碍类，可以沿着圆柱端将两个解粘合在一起。此外，我们为该粘贴图片建立了本地的仓石模型。作为应用，我们证明了在任何紧凑的4流形上${\mathrm{小号}}^3$ 要么 ${Ť}^3$在边界处，存在一个扰动了Kapustin-Witten方程的Nahm极点解。在某些拓扑假设下具有双曲边界的4流形也是这种情况。