Parameter counting for singular monopoles on ℝ3

Abstract

Abstract:We compute the dimension of the moduli space of gauge-inequivalent solutions to the Bogomolny equation on R 3 with prescribed singularities corresponding to the insertion of a finite number of 't Hooft defects. We do this by generalizing the methods of C. Callias and E. Weinberg to the case of R 3 with a finite set of points removed. For a special class of Cartan-valued backgrounds we go further and construct an explicit basis of L 2 -normalizable zero-modes. Finally we exhibit and study a two-parameter family of spherically symmetric singular monopoles, using the dimension formula to provide a physical interpretation of these configurations. This paper is the first in a series of three on singular monopoles, where we also explore the role they play in the contexts of intersecting D-brane systems and four-dimensional N = 2 super Yang-Mills theories.

Semiclassical framed BPS states

“…These boundary conditions allow for subleading terms that are still singular. As we argued in [30], this type of behavior is observed in explicit solutions and generally can occur for 't Hooft defects such that there exists a root α of the Lie algebra with | α, P n | = 1. For example, in the solutions of [50][51][52] the subleading behavior of the Higgs field is regular at the locus of a minimal SU (2) defect, while it is has 1/ √ r n behavior for the minimal SO(3) defect.…”

“…We normalize the E α so that {E ±α , iH α } forms a canonical sl(2) subalgebra. Further details on our Lie algebra conventions can be found in appendix A of [30].…”

“…We will see this explicitly when these solutions are reviewed in subsection 7.2. When θ 0 = 0, (2.13) reduce to the boundary conditions considered in [30], where the indicated subleading behavior of the fields was determined from imposing consistency of the defect boundary conditions with the variational principle for the action. These arguments can be generalized to the situation under consideration here; some of the details are presented in appendix B.…”

“…However we will obtain some new results even in the vanilla case. We focus on the case of pure 't Hooft defects, where the singular monopole moduli spaces studied in [5,30,[54][55][56][74][75][76] play an essential role.…”

“…Singular monopoles and their moduli spaces were first considered by Kronheimer [74], with further important results obtained in [5,[54][55][56]75]. See the introductory section of [30] for a more detailed account of the previous literature on singular monopoles and their moduli spaces.…”