Static SU(2) adjoint-Higgs Prasad–Sommerfield monopole sector

Assume the Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$. Use natural units $\mathrm{\hslash }=c=1$ on Minkowski spacetime ${\mathbb{R}}^{1,3}$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$, orientation $dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and spatial orientation ${ϵ}_{123}=1$. Fix the global gauge group $SU(2)$, not $SO(3)$, and the trivial right principal $SU(2)$-bundle over each spatial slice. Let

$T_a=-\frac{i}{2}{\sigma }_a(a=1,2,3),\kappa (X,Y)=-2{\mathrm{Tr}}_2(XY),$

where ${\sigma }_a$ are the Pauli matrices and $\mathfrak{s}\mathfrak{u}(2)$ has basis $T_1,T_2,T_3$. Thus $[T_a,T_b]={ϵ}_{abc}{T}_c$ and $\kappa (T_a,T_b)={\delta }_{ab}$. The positive coupling $g$ is dimensionless. The scalar field is one real adjoint Higgs field: a smooth $\mathfrak{s}\mathfrak{u}(2)$-valued map $\mathrm{\Phi }$ on ${\mathbb{R}}^3$, viewed as the real form in the complexified adjoint representation. A gauge connection is a smooth $\mathfrak{s}\mathfrak{u}(2)$-valued one-form $A=A_{i}d{x}^i$. In a local section change $s_j=s_i{h}_{ij}$, use the right-action convention $A_j={\mathrm{Ad}}_{h_{ij}^{-1}}{A}_i+h_{ij}^{-1}d{h}_{ij}$ and ${\mathrm{\Phi }}_j={\mathrm{Ad}}_{h_{ij}^{-1}}{\mathrm{\Phi }}_i$. Define

$F_{ij}={\mathrm{\partial }}_i{A}_j-{\mathrm{\partial }}_j{A}_i+[A_i,A_j],D_i\mathrm{\Phi }={\mathrm{\partial }}_i\mathrm{\Phi }+[A_i,\mathrm{\Phi }],B_i=\frac{1}{2}{ϵ}_{ijk}{F}_{jk}\mathrm{.}$

The static spacetime fields are the smooth maps $\tilde{\mathrm{\Phi }},{\tilde{A}}_{\mu }:\mathbb{R}\times {\mathbb{R}}^3\to \mathfrak{s}\mathfrak{u}(2)$, $\mu =0,1,2,3$, defined by $\tilde{\mathrm{\Phi }}(t,x)=\mathrm{\Phi }(x)$, ${\tilde{A}}_i(t,x)=A_i(x)$, and ${\tilde{A}}_0(t,x)=0$; thus ${\mathrm{\partial }}_t{\tilde{A}}_i={\mathrm{\partial }}_t\tilde{\mathrm{\Phi }}=0$. The spacetime connection is $\tilde{A}={\tilde{A}}_{\mu }d{x}^{\mu }$, with $x^0=t$. This Prasad–Sommerfield sector has no other dynamical fields and identically zero Higgs potential. Its adopted energy functional is

$E[A,\mathrm{\Phi }]=\frac{1}{2g^2}{\int }_{{\mathbb{R}}^3}{\sum }_{i=1}^3(\kappa (B_i,B_i)+\kappa (D_i\mathrm{\Phi },D_i\mathrm{\Phi }))d^{3}x\mathrm{.}$

The field dimensions are $[A_i]=[\mathrm{\Phi }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$ and $[F_{ij}]=[B_i]=[D_i\mathrm{\Phi }]={\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-2}$, so $E$ has dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$. A configuration is finite-energy when this integral is finite.

The boundary data fix $v>0$ of dimension ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-1}$ and require $\mathrm{\Phi }(R\cdot )\mathrm{/}v\to n_{\mathrm{\infty }}$ in $C^{\mathrm{\infty }}(S^2,\mathfrak{s}\mathfrak{u}(2))$, where $S^2=\{\omega \in {\mathbb{R}}^3:\mathrm{\mid }\omega \mathrm{\mid }=1\}$ has its outward orientation and $\mathrm{\mid }{n}_{\mathrm{\infty }}{\mathrm{\mid }}_{\kappa }=1$; here $\mathrm{\mid }X{\mathrm{\mid }}_{\kappa }^2:=\kappa (X,X)$. In particular $\mathrm{\mid }\mathrm{\Phi }(R\omega ){\mathrm{\mid }}_{\kappa }\to v$ uniformly. The stabilizer subgroup of $T_3$ is $U(1)=\{\exp (\theta T_3):\theta \in \mathbb{R}\mathrm{/}(4\pi \mathbb{Z})\}$. Let $D_N,D_S$ be the closed northern and southern hemispheres. Specify smooth lifts $u_{\alpha }:U_{\alpha }\to SU(2)$ on open neighborhoods $U_{\alpha }\supset D_{\alpha }$ with ${\mathrm{Ad}}_{u_{\alpha }}{T}_3=n_{\mathrm{\infty }}$. On their common equator $C$, oriented as $\mathrm{\partial }{D}_N$, the transition is $u_S=u_{N}h$, where $h(\phi )=\exp (\theta (\phi )T_3)$, $\phi \in [0,2\pi ]$ runs positively around $C$, and $\theta :[0,2\pi ]\to \mathbb{R}$ is a specified smooth lift.

For $R>0$, put ${\iota }_R(\omega )=R\omega$ and $F=\frac{1}{2}{F}_{ij}d{x}^i\wedge d{x}^j$. Define the adapted angular forms

${\mathcal{A}}_{\alpha ,R}={\mathrm{Ad}}_{u_{\alpha }^{-1}}({\iota }_R^{\ast }A)+u_{\alpha }^{-1}d{u}_{\alpha },{\mathcal{F}}_{\alpha ,R}={\mathrm{Ad}}_{u_{\alpha }^{-1}}({\iota }_R^{\ast }F)\mathrm{.}$

Require pointwise convergence ${\mathcal{A}}_{\alpha ,R}\to {\mathcal{A}}_{\alpha ,\mathrm{\infty }}$ on $D_{\alpha }$, with ${\mathcal{A}}_{\alpha ,\mathrm{\infty }}$ extending as a $C^1$ one-form to $U_{\alpha }$. Require its curvature ${\mathcal{F}}_{\alpha ,\mathrm{\infty }}:=d{\mathcal{A}}_{\alpha ,\mathrm{\infty }}+\frac{1}{2}[{\mathcal{A}}_{\alpha ,\mathrm{\infty }}\wedge {\mathcal{A}}_{\alpha ,\mathrm{\infty }}]$ to extend as a $C^1$ two-form to $U_{\alpha }$, and ${\mathcal{F}}_{\alpha ,R}\to {\mathcal{F}}_{\alpha ,\mathrm{\infty }}$ uniformly on $D_{\alpha }$. Norms and convergence use the round metric of the unit sphere and $\kappa$. Define the limiting tangential covariant derivative by its local expression

$D_{\mathrm{t}\mathrm{a}\mathrm{n}}{n}_{\mathrm{\infty }}{\mathrm{\mid }}_{D_{\alpha }}:={\mathrm{Ad}}_{u_{\alpha }}(d{T}_3+[{\mathcal{A}}_{\alpha ,\mathrm{\infty }},T_3])={\mathrm{Ad}}_{u_{\alpha }}[{\mathcal{A}}_{\alpha ,\mathrm{\infty }},T_3],$

where $d{T}_3=0$ and the bracket is evaluated on tangent vectors. The transition law makes these expressions agree on the equator. Require $D_{\mathrm{t}\mathrm{a}\mathrm{n}}{n}_{\mathrm{\infty }}=0$. Let $d\mathrm{\Omega }$ be the outward round area form on $S^2$ and define $B_{\mathrm{\infty }}:S^2\to \mathfrak{s}\mathfrak{u}(2)$ by

${\mathrm{Ad}}_{u_{\alpha }}{\mathcal{F}}_{\alpha ,\mathrm{\infty }}=B_{\mathrm{\infty }}d\mathrm{\Omega }\text{on }{D}_{\alpha }\mathrm{.}$

Curvature covariance makes this definition agree on $C$. Equivalently, $B_{\mathrm{\infty }}(\omega )={\lim }_{R\to \mathrm{\infty }}{R}^2{\sum }_i{\omega }_i{B}_i(R\omega )$ uniformly on each hemisphere, since ${\iota }_R^{\ast }F=R^2{\sum }_i{\omega }_i{B}_i(R\omega )d\mathrm{\Omega }$. It is the scaled radial magnetic field, not the unscaled spatial field. Require $[B_{\mathrm{\infty }}(\omega ),n_{\mathrm{\infty }}(\omega )]=0$, meaning that its value belongs to the pointwise stabilizer Lie algebra $\{X\in \mathfrak{s}\mathfrak{u}(2):[X,n_{\mathrm{\infty }}(\omega )]=0\}$. These convergence and reduction conditions are adopted boundary hypotheses, not consequences asserted for every finite-energy field. The surface limit below is also required to exist.

The admissible external Wilson probes include the fundamental doublet, on which $T_3$ has weights $\pm \frac{1}{2}$; this specifies the electric probe lattice without adding a propagating fundamental matter field. Define the classical magnetic surface charge and its energy normalization by

$m:={\lim }_{R\to \mathrm{\infty }}{\int }_{S_R^2}\kappa (\mathrm{\Phi },B_i){\widehat{x}}_{i}dS,T:=g^{-2}m,$

where $\widehat{x}=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ is the outward unit normal and $dS$ is Euclidean area. The transition increment $\theta (2\pi )-\theta (0)$ is not assumed to be quantized in this definition. These are adopted classical model data; they assert no empirical law, quantum continuum construction, supersymmetry representation, or quantum BPS state.

David Tong, §2.1, model and monopole conventions

Jeffrey A. Harvey, §§1.4 and 2.1–2.2, SU(2) Higgs model and monopole conventions

Non-Abelian gauge group, bundle, and representation data

Principal connection and right-action convention

Lie-algebra-valued form bracket convention

Associated covariant derivative

Smooth adjoint representation

Normalized invariant form on SU(2)