Knowlpedia

Definition

Bogomolny monopole equation

A first-order gauge equation equating the curvature of a connection on a three-manifold with the Hodge dual of the covariant derivative of a Higgs field.

Definition

Let P→YP\to Y be a with compact structure group over an oriented Riemannian three-manifold. A Bogomolny monopole is a pair (A,Φ)(A,\Phi) consisting of a AA and a Higgs field Φ∈Ω0(Y;ad⁡P)\Phi\in\Omega^0(Y;\operatorname{ad}P) satisfying the Bogomolny monopole equation

FA=∗dAΦ.F_A=*d_A\Phi.

Here FA∈Ω2(Y;ad⁡P)F_A\in\Omega^2(Y;\operatorname{ad}P) is the , dAΦd_A\Phi is the covariant derivative, and ∗:Ω1(Y)→Ω2(Y)*:\Omega^1(Y)\to\Omega^2(Y) is the . The equation is imposed modulo acting simultaneously on AA and Φ\Phi.

Energy decomposition

On a compact domain, or under decay conditions making every integral finite, the Yang–Mills–Higgs energy satisfies

12∫Y(∣FA∣2+∣dAΦ∣2)=12∫Y∣FA−∗dAΦ∣2+∫Y⟨FA∧dAΦ⟩.\frac12\int_Y\left(|F_A|^2+|d_A\Phi|^2\right) = \frac12\int_Y|F_A-*d_A\Phi|^2 +\int_Y\langle F_A\wedge d_A\Phi\rangle.

The final term becomes a boundary or topological charge after the is used. Solutions therefore saturate the associated Bogomolny bound. This completion-of-squares argument is the origin of the first-order equation Bogomolny, pp. 449–454.

Dimensional reduction

Let X=Y×RX=Y\times\mathbb R with the product metric and consider a translation-invariant connection

A~=A+Φ dt.\widetilde A=A+\Phi\,dt.

Its curvature is FA~=FA+dAΦ∧dtF_{\widetilde A}=F_A+d_A\Phi\wedge dt. With compatible product-orientation conventions, one of the four-dimensional self-duality equations reduces to FA=∗dAΦF_A=*d_A\Phi. This explains why monopoles inherit ellipticity modulo gauge and many structural features of instantons.

Examples and conventions

The abelian Dirac monopole satisfies the equation away from its singular point, with Φ\Phi proportional to the reciprocal radial coordinate. Smooth finite-energy SU(2)SU(2) monopoles provide the basic nonabelian examples.

References
  1. Evgeny B. Bogomolny, “The Stability of Classical Solutions,” Soviet Journal of Nuclear Physics 24 (1976), 449–454. CERN record. Relevant: the energy bound and first-order monopole equations.
  2. Michael Atiyah and Nigel Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton University Press, 1988. Publisher record. Relevant: Chapter 1, Bogomolny equations, gauge action, and dimensional reduction.