Nonlinear local dual-Higgs magnetostatic flux cylinders

Use the local monopole fields, charges and superpotential with $a=A_D$, $c=u^{\mathrm{\prime }}(0)\mathrm{\ne }0$, $\mu \mathrm{\ne }0$, $v^2=\mathrm{\mid }\mu c\mathrm{\mid }\mathrm{/}\sqrt{2}$ and $\xi =2v^2$. Keep the holomorphic function $u$ nonlinear. Choose a closed disk $K_{\delta }=\{\mathrm{\mid }a\mathrm{\mid }\le \delta \}$ contained in its open valid patch with $u^{\mathrm{\prime }}(a)\mathrm{\ne }0$ throughout. Define

$m_{\ast }={\min }_{a\in K_{\delta }}\mathrm{\mid }\mu u^{\mathrm{\prime }}(a)\mathrm{\mid }>0,K={\sup }_{a\in K_{\delta }}\mathrm{\mid }\mu u^{\mathrm{\prime }\mathrm{\prime }}(a)\mathrm{\mid }<\mathrm{\infty }\mathrm{.}$

Continuity, compactness and the nonzero $c$ allow such a disk. No linear Taylor replacement of $u$ is made. In the already adopted constant-metric model put $b(a)=b_0=g_D^{-2}$. More generally, the same classical energy problem may be specified with a supplied smooth real function $b$ on a neighborhood of $K_{\delta }$ and constants $0<b_{-}\le b(a)\le b_{+}<\mathrm{\infty }$. This is additional explicit local Wilsonian metric data, not the vector-only logarithmically singular period metric inferred at a massless-monopole point. Let $b_0=b(0)$ and $g_0^2=b_0^{-1}$. Define

$H=\sqrt{2}M\tilde{M}+\mu u^{\mathrm{\prime }}(a),d=\mathrm{\mid }M{\mathrm{\mid }}^2-\mathrm{\mid }\tilde{M}{\mathrm{\mid }}^2,s=\mathrm{\mid }M{\mathrm{\mid }}^2+\mathrm{\mid }\tilde{M}{\mathrm{\mid }}^2,$

$V_b=b(a{)}^{-1}\mathrm{\mid }H{\mathrm{\mid }}^2+\frac{d^2}{2b(a)}+2\mathrm{\mid }a{\mathrm{\mid }}^{2}s\mathrm{.}$

Charged kinetic terms are canonical. The gauge kinetic coefficient is $b(a)$ and the neutral kinetic coefficient is $b(a)$. Work in the classical magnetostatic model with omitted or constant theta angle, no dynamical dual magnetic sources in the interior, static fields and $A_0=0$. No field-dependent-theta Gauss-law conclusion is included.

Fix $n\in \mathbb{Z}\setminus \{0\}$ and $e^{i\beta }=-\mu c\mathrm{/}(\sqrt{2}{v}^2)$. A planar admissible profile is a smooth real one-form $A=A_{1}d{x}^1+A_{2}d{x}^2$ and smooth complex fields $a,M,\tilde{M}$ on ${\mathbb{R}}^2$, with $a(x)\in K_{\delta }$, finite energy

$E_b={\int }_{{\mathbb{R}}^2}\{\frac{b(a)}{2}{B}^2+b(a)\mathrm{\mid }{\mathrm{\partial }}_{i}a{\mathrm{\mid }}^2+\mathrm{\mid }{D}_{i}M{\mathrm{\mid }}^2+\mathrm{\mid }{\tilde{D}}_i\tilde{M}{\mathrm{\mid }}^2+V_b\}{d}^{2}x,$

where $i=1,2$, $B={\mathrm{\partial }}_1{A}_2-{\mathrm{\partial }}_2{A}_1$, $D_i={\mathrm{\partial }}_i-i{A}_i$ and ${\tilde{D}}_i={\mathrm{\partial }}_i+i{A}_i$. Require uniform approach to the vacuum orbit at transverse infinity:

$a\to 0,\mathrm{\mid }M\mathrm{\mid },\mathrm{\mid }\tilde{M}\mathrm{\mid }\to v,M\tilde{M}\to e^{i\beta }{v}^2\mathrm{.}$

This is an independent boundary hypothesis, not inferred from finite energy. Outside a compact set $M$ is nonzero and its phase has winding $n$ on each sufficiently large circle; the phase of $\tilde{M}$ has winding $-n$. In the angular coordinate $\theta$, require the coefficient of $d\theta$ in $A-d\arg M$ to tend uniformly to zero, and the limiting flux to exist. Require also $D_{i}M,{\tilde{D}}_i\tilde{M}\to 0$ uniformly, as in the admitted flux-sector data. These conditions define the gauge-invariant class ${\mathcal{P}}_n$. Gauge transformations are arbitrary smooth maps $U:{\mathbb{R}}^2\to U(1)$ acting by $M\mapsto UM$, $\tilde{M}\mapsto U^{-1}\tilde{M}$ and $A\mapsto A-i{U}^{-1}dU$; there is no added asymptotic-constant gauge restriction. A canonical exterior representative has $M=m{e}^{in\theta }$ with $m>0$ real outside a compact set.

Write

${\tau }_n={\inf }_{P\in {\mathcal{P}}_n}{E}_b(P)\mathrm{.}$

This notation alone asserts neither nonemptiness nor attainment. The adopted electric/magnetic dictionary identifies the dual magnetic flux with the original electric flux only for the specified test-charge/flux boundary sector.

For fixed canonical exterior representatives $P_0,P_1\in {\mathcal{P}}_n$ and $L>0$, use coordinates $(x^1,x^2,x^3)$ on ${\mathbb{R}}^2\times [0,L]$ and write the cylinder gauge potential as $\mathcal{A}=A_{1}d{x}^1+A_{2}d{x}^2+A_{3}d{x}^3$. Define $F=d\mathcal{A}$, so $F_{ij}={\mathrm{\partial }}_i{A}_j-{\mathrm{\partial }}_j{A}_i$ for $1\le i,j\le 3$, and extend the charged covariant derivatives by $D_i={\mathrm{\partial }}_i-i{A}_i$ and ${\tilde{D}}_i={\mathrm{\partial }}_i+i{A}_i$ for $i=1,2,3$. The tangential end traces are $(A_{1}d{x}^1+A_{2}d{x}^2,a,M,\tilde{M})$ at $x^3=0,L$; no endpoint value is prescribed for the normal component $A_3$. Let ${\mathcal{C}}_n(L;P_0,P_1)$ be the smooth static fields on ${\mathbb{R}}^2\times [0,L]$ with these end traces, $a\in K_{\delta }$, and a planar admissible winding-$n$ profile on almost every slice. The flux $2\pi n$ is thus fixed throughout the cylinder, with no flux leakage to transverse infinity or interior magnetic source. Allow $A_3$ but choose $A_3=0$ in trial collars. Define

$E_{b,3}={\int }_0^L\text{\hspace{0.17em}⁣}{\int }_{{\mathbb{R}}^2}\{\frac{b(a)}{2}{\sum }_{i<j}{F}_{ij}^2+b(a){\sum }_{i=1}^3\mathrm{\mid }{\mathrm{\partial }}_{i}a{\mathrm{\mid }}^2+{\sum }_{i=1}^3(\mathrm{\mid }{D}_{i}M{\mathrm{\mid }}^2+\mathrm{\mid }{\tilde{D}}_i\tilde{M}{\mathrm{\mid }}^2)+V_b\}{d}^{2}xd{x}^3,$

and $I_n(L;P_0,P_1)={\inf }_{{\mathcal{C}}_n(L;P_0,P_1)}{E}_{b,3}$. Competitors have finite displayed energy and the stated flux/boundary data; no stationary solution or minimizer is assumed. Any further hard field cutoff is part of a different admissible class and must be checked to admit the profiles and interpolation paths used here. This functional is the specified local classical model; no error control to a microscopic action or quantum Wilson-loop interpretation is asserted.

Seiberg and Witten, §5.6.

Tong, §§3.1–3.2.

Local dual U(1) monopole-Higgs effective data

Finite-energy dual-Higgs flux-tube sector