Finite-energy dual-Higgs flux-tube sector

Use the dual $U(1{)}_D$ fields and potential of the preceding local monopole-Higgs model. Let $x^1,x^2$ be coordinates on the transverse plane ${\mathbb{R}}^2$ and let $x^3$ be the coordinate along a straight static tube. Write ${\tilde{A}}_i$ for the spatial dual gauge potential, ${\tilde{F}}_{12}={\mathrm{\partial }}_1{\tilde{A}}_2-{\mathrm{\partial }}_2{\tilde{A}}_1$, $D_{i}M=({\mathrm{\partial }}_i-i{\tilde{A}}_i)M$, and $D_i\tilde{M}=({\mathrm{\partial }}_i+i{\tilde{A}}_i)\tilde{M}$.

A finite-energy dual vortex profile of winding $n$ is a smooth static configuration $(A_D,M,\tilde{M},{\tilde{A}}_1,{\tilde{A}}_2)$ on ${\mathbb{R}}^2$ with $\mathrm{\mid }{A}_D(x)\mathrm{\mid }<r$ at every point, where $\mathrm{\mid }{A}_D\mathrm{\mid }<r$ is the preceding model's open holomorphic patch, for which the transverse energy

$\sigma ={\int }_{{\mathbb{R}}^2}(\frac{{\tilde{F}}_{12}^2}{2g_D^2}+g_D^{-2}\mathrm{\mid }{\mathrm{\partial }}_i{A}_D{\mathrm{\mid }}^2+\mathrm{\mid }{D}_{i}M{\mathrm{\mid }}^2+\mathrm{\mid }{D}_i\tilde{M}{\mathrm{\mid }}^2+V_{\mu }-V_{\mathrm{v}\mathrm{a}\mathrm{c}})d^{2}x$

is finite. The fields approach a supersymmetric vacuum at infinity satisfying $A_D=0$, $\mathrm{\mid }M\mathrm{\mid }=\mathrm{\mid }\tilde{M}\mathrm{\mid }=v>0$, $\sqrt{2}M\tilde{M}+\mu u^{\mathrm{\prime }}(0)=0$, and $V_{\mathrm{v}\mathrm{a}\mathrm{c}}=0$. The phase map $M\mathrm{/}\mathrm{\mid }M\mathrm{\mid }$ restricted to a sufficiently large circle has degree $n\in \mathbb{Z}\setminus \{0\}$ as a map to the unit circle; the phase map $\tilde{M}\mathrm{/}\mathrm{\mid }\tilde{M}\mathrm{\mid }$ has degree $-n$. Require also that $D_{i}M$ and $D_i\tilde{M}$ tend to zero on that circle, that the coefficient of $d\theta$ in the pulled-back one-form $\tilde{A}-d\arg M$ tends uniformly in $\theta$ to zero as the circle radius tends to infinity, and that the limiting flux exists. The profile is a translation-invariant tube along $x^3$; its tension is the transverse integral $\sigma$. This Definition includes no assertion that such a configuration exists or is a stationary solution of the field equations.

The patch restriction makes $u(A_D)$ and $V_{\mu }$ defined throughout the profile. For physical use, retain the preceding model's supplied ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}>0$, $0<\epsilon <1$, background domain ${\mathcal{U}}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, and application excitation bound $E_{\mathrm{a}\mathrm{p}\mathrm{p}}\le \epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, with $v<\epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$. Background values must lie in that supplied domain. Finite transverse energy alone establishes none of these applicability qualifications or a microscopic error bound. These inputs impose no hard Fourier-support requirement on this classical profile class. A separately imposed hard cutoff on the variational class must specify its constraints and check the admissibility of any profiles or interpolation paths used in a consuming result.

In the adopted electric/magnetic frame, the original electric flux of this sector means the dual magnetic flux ${\mathrm{\Phi }}_D={\lim }_{R\to \mathrm{\infty }}{\int }_{\mathrm{\mid }x\mathrm{\mid }\le R}{\tilde{F}}_{12}{d}^{2}x$. The flux dictionary is part of the selected duality model; this Definition does not state that every electric line charge is represented by a vortex. In natural units, $[{\tilde{A}}_i]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $[{\tilde{F}}_{12}]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, and $[\sigma ]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$.

Seiberg and Witten, §5.6.

Tong, §4.6.2, “Monopoles”.

Dual U(1) monopole-Higgs effective data: Local dual U(1) monopole-Higgs effective data.

Pure $N=2$ SU(2) electric/magnetic frame: Pure $N=2$ SU(2) Coulomb-branch and period data.