Monopole moduli spaces for compact 3-manifolds
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.
scientific article; zbMATH DE number 1179884