The Expansions of the Nahm Pole Solutions to the Kapustin-Witten Equations

TL;DR

This paper investigates the detailed asymptotic behavior of Nahm pole solutions to the Kapustin-Witten equations on 3-manifolds, revealing a link between boundary regularity and Einstein geometry.

Contribution

It establishes a precise condition under which the sub-leading terms of Nahm pole solutions are smooth at the boundary, connecting geometric properties of the manifold to solution regularity.

Findings

Sub-leading terms are smooth at the boundary iff the 3-manifold is Einstein.

Provides a characterization of boundary regularity for Nahm pole solutions.

Links geometric structure of the manifold to analytical properties of solutions.

Abstract

For a 3-manifold $X$ and compact simple Lie group $G$, we study the expansions of polyhomogeneous Nahm pole solutions to the Kapustin-Witten equations over $X\times (0,+\infty)$. Let $y$ be the coordinate of $(0,+\infty)$, we prove that the sub-leading terms of a polyhomogeneous Nahm pole solution is smooth to the boundary when $y\to 0$ if and only if $X$ is an Einstein 3-manifold.