A Gluing Theorem for the Kapustin-Witten Equations with a Nahm Pole

TL;DR

This paper develops a gluing construction for Nahm pole solutions to the Kapustin-Witten equations on manifolds with boundaries, identifying obstructions and local models, and demonstrates existence of solutions on various boundary types.

Contribution

It introduces a new gluing method for Nahm pole solutions, including an obstruction class and local Kuranishi model, expanding the understanding of solutions on manifolds with boundaries.

Findings

Existence of Nahm pole solutions on compact four-manifolds with specific boundaries.

Identification of an obstruction class for gluing solutions.

Construction of a local Kuranishi model for the gluing process.

Abstract

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 four-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 four-manifold with hyperbolic boundary under some topological assumptions.