Monopole moduli spaces for compact 3-manifolds
type: Item
instance of: scholarly article
title: Monopole moduli spaces for compact 3-manifolds
zbMATH Open document ID: 0920.58011
author: Marc Pauly
published in: Mathematische Annalen
publication date: 1998-10-18T00:00:00Z
review text: The author investigates the moduli space \(\mathcal M\) of solutions \((\nabla,\Phi)\) to the \(\text{SU}(2)\)-Bogomolny monopole equation \(F_\nabla=*d_\nabla\Phi\) on a three-dimensional compact manifold \(X\), with singularities of prescribed charges \((k_1, \dots, k_n)\) at marked points \((p_1, \dots, p_n)\). Here, ``charge \(k\) at \(p\)'' means that a solution on a small ball \(B^3\) around \(p\) lifts to an anti-selfdual connection on an \(\text{SU}(2)\)-bundle over \(B^4\) via the Hopf fibration \(S^1\to B^4\to B^3\), such that \(S^1\) acts on the singular fiber over \(q\) with weights \(\pm k\). The main result of the paper is the following Theorem 1. The virtual dimension of the moduli space \(\mathcal M\) of singular monopoles on \(X\) with singularities at \((p_1, \dots, p_n)\) and charges \((k_1, \dots, k_n)\) equals \(4\sum k_i\). As an example, some solutions with \(n=1\), \(k_1=1\) on \(S^3\) are computed. It is shown that the corresponding moduli space is regular and four-dimensional as expected.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 1179884
zbMATH Keywords: Bogomolnyi monopole equation
zbMATH Keywords: singular monopoles
MaRDI profile type: Publication
full work available at URL: s002080050180
OpenAlex ID: W2034658174
DOI: 10.1007/S002080050180
description: scientific article; zbMATH DE number 1179884