Parameter counting for singular monopoles on R^3

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.

61 pages, 2 figures; v2: references added, typo corrected

References in corpus (14)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• TASI Lectures on Solitons
• Magnetic Monopole Dynamics, Supersymmetry, and Duality
• Vortices and Impurities
• Instantons on Gravitons
• Moduli Spaces of Instantons on the Taub-NUT Space
• Brane bending and monopole moduli
• Semiclassical framed BPS states
• The 't Hooft-Polyakov Monopole in the Presence of a 't Hooft Operator
• Monopoles and Wilson Lines
• Geometric engineering of (framed) BPS states
• Singular Monopoles via the Nahm Transform
• Magnetic Charge Lattices, Moduli Spaces and Fusion Rules
• Dirac zero modes for Abelian BPS multimonopoles

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• A New SU(2) Anomaly
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• Brane bending and monopole moduli
• ADHM Revisited: Instantons and Wilson Lines
• Semiclassical framed BPS states
• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• Wall-crossing made smooth
• Monopole Bubbling via String Theory
• A Note On The Semiclassical Formulation Of BPS States In Four-Dimensional N=2 Theories
• 't Hooft Defects and Wall Crossing in SQM
• The M-theory origin of global properties of gauge theories
• Intersecting Branes, Domain Walls and Superpotentials in 3d Gauge Theories
• Little String Defects and Bala-Carter Theory
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Quivers, Line Defects and Framed BPS Invariants
• Wall Crossing from Dirac Zeromodes
• L^2-Kernels Of Dirac-Type Operators On Monopole Moduli Spaces
• Topologically Stratified Energy Minimizers in a Product Abelian Field Theory
• Holography for field theory solitons
• Yang-Mills instantons in Kaehler spaces with one holomorphic isometry
• Connecting Localization and Wall-Crossing via D-Branes
• Q-balls of Quasi-particles in a (2,0)-theory model of the Fractional Quantum Hall Effect
• An $L^2$-index formula of monopoles with Dirac-type singularities
• Simulating Magnetic Monopole-Defect Dynamics
• Instantons with continuous conformal symmetries: Hyperbolic and singular monopoles and more, oh my!