Bogomolny-Prasad-Sommerfield monopole

A Bogomolny-Prasad-Sommerfield monopole is a finite-energy Yang-Mills-Higgs configuration satisfying $B_i=\pm D_i\Phi$. Its magnetic charge is the degree of the normalized Higgs field at spatial infinity.

For the trace inner product and adjoint covariant derivative, compatibility gives $\Delta |\Phi {|}^2=2|D\Phi {|}^2+2⟨\Phi ,D_i{D}_i\Phi ⟩$. The gauge-theory Bianchi identity and $B_i=\pm D_i\Phi$ imply $D_i{D}_i\Phi =0$. Therefore $\Delta |\Phi {|}^2=2|D\Phi {|}^2$, equal to the energy density $|B{|}^2+|D\Phi {|}^2$ on a Bogomolny-Prasad-Sommerfield monopole. If the energy density is instead defined with an overall factor $1/2$, this Laplacian is twice that density.

For zero scalar potential, the static field equations are $D_i{D}_i\Phi =0$ and $D_i{F}_{ij}=[\Phi ,D_j\Phi ]$. If $B_i=-D_i\Phi$, the gauge-theory Bianchi identity gives the first equation. For the second,using $F_{ik}={ϵ}_{ik\ell }{B}_{\ell }$ and ${ϵ}_{ijk}{ϵ}_{ik\ell }=-2{\delta }_{j\ell }$. Thus the first-order equations imply the full second-order equations locally, independently of the radial ansatz.

A spherically symmetric SU(2) monopole can be written with the internal Higgs direction aligned with the spatial radial direction and the gauge field built from the invariant tensor ${ϵ}_{aij}$.

For a Bogomolny-Prasad-Sommerfield monopole, the dimensionless radial Bogomolny equations are $dK/d\rho =-KH$ and $dH/d\rho =(1-K^2)/{\rho }^2$. The displayed profiles solve both equations: differentiating $\rho /\sinh \rho$ gives $K(1/\rho -\coth \rho )=-KH$, while differentiating $H$ gives $1/{\rho }^2-{\mathrm{csch}}^2\rho =(1-K^2)/{\rho }^2$. At the origin $K=1-{\rho }^2/6+O({\rho }^4)$ and $H=\rho /3+O({\rho }^3)$, giving a regular core. At infinity $K$ decays exponentially and $H=1-1/\rho +O(e^{-2\rho })$, giving the vacuum Higgs magnitude and a long-range Abelian magnetic field.

For ${\Phi }^a=f(r)x^a/r$ and $A_i^a={ϵ}_{iaj}{x}^j\alpha (r)$ with $[e_a,e_b]={ϵ}_{abc}{e}_c$, set $K=1+r^2\alpha$. The equation $B=-D\Phi$ equates radial coefficients to give $f^{\prime }=-2\alpha -r^2{\alpha }^2$ and transverse coefficients to give $r{\alpha }^{\prime }+2\alpha =-f/r-r\alpha f$. These are exactly the displayed two radial equations. The sign depends on the chosen orientation and on the order of indices in the gauge ansatz.