Conditional monopole Higgs vacuum after a small $N=1$ perturbation

Use natural units in which the speed of light and reduced Planck constant are one, with metric $\eta =\mathrm{diag}(-1,1,1,1)$, and the local monopole patch of the selected pure-SU(2) $N=2$ source-proposed low-energy model. Let $u(A_D)$, $c=u^{\mathrm{\prime }}(0)\mathrm{\ne }0$, $M$, $\tilde{M}$, $g_D$, and $\mu$ be the data in the local dual-Higgs definition, with $\mu \mathrm{\ne }0$ and $(\mathrm{\mid }\mu c\mathrm{\mid }\mathrm{/}\sqrt{2}{)}^{1\mathrm{/}2}<\epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ for the supplied cutoff and margin of that definition. Physical applicability also retains its supplied background domain and excitation bound, without an inferred microscopic error estimate. Restrict to a neighborhood of $A_D=0$ on which $u^{\mathrm{\prime }}(A_D)\mathrm{\ne }0$. The supersymmetric vacuum equations in this patch have exactly one $U(1{)}_D$ gauge orbit, represented by

$A_D=0,M\tilde{M}=-\frac{\mu c}{\sqrt{2}},\mathrm{\mid }M\mathrm{\mid }=\mathrm{\mid }\tilde{M}\mathrm{\mid }=v:={\left(\frac{\mathrm{\mid }\mu c\mathrm{\mid }}{\sqrt{2}}\right)}^{1\mathrm{/}2}>0.$

Thus the local monopole fields have nonzero gauge-fixed amplitudes and $U(1{)}_D$ is fully Higgsed. With canonical charged-hypermultiplet kinetic terms in the connection-normalized local model, the dual vector mass is

$m_D^2=2g_D^2(\mathrm{\mid }M{\mathrm{\mid }}^2+\mathrm{\mid }\tilde{M}{\mathrm{\mid }}^2)=4g_D^2{v}^2>0.$

For an actual generic nonlinear confinement conclusion, use the precise magnetostatic flux-boundary class of the nonlinear cylinder definition, with the full $u$, a closed neutral disk on which $u^{\mathrm{\prime }}\mathrm{\ne }0$, positive metric bounds, constant or absent theta, uniformly vacuum-approaching profiles, and fixed canonical end traces. For this energy branch assume full AC, as required by its integration suppliers. The planar energy infimum ${\tau }_n$ is positive and finite for every $n\mathrm{\ne }0$, and the full static-energy infimum obeys

${\lim }_{L\to \mathrm{\infty }}\frac{I_n(L)}{L}={\tau }_n>0.$

This proves asymptotically linear confinement of the specified original electric flux in that classical local effective model without assuming a noncritical stationary vortex or any attained minimum. It makes no generic $O(1)$ remainder claim.

For a separate conditional straight-tube interpretation, fix a nonzero original electric test-charge sector and, for each separation $L\ge L_0>0$, a nonempty supplied class ${\mathcal{B}}_L$ of admissible configurations with the same prescribed external-charge and endpoint data. Supply its finite classical energy map ${\mathcal{E}}_L:{\mathcal{B}}_L\to \mathbb{R}$ and put $V(L)={\inf }_{Q\in {\mathcal{B}}_L}{\mathcal{E}}_L(Q)$. Assume that for every $L\ge L_0$ this infimum is attained by a straight tube with an admitted transverse vortex profile $P_L$ of nonzero winding, with

$V(L)=L\sigma (P_L)+e_L,\mathrm{\mid }{e}_L\mathrm{\mid }\le C_0,$

where $\sigma (P)$ is the transverse energy of the flux-tube definition and $C_0$ is independent of $L$. This is an explicit tube/minimization and endcap assumption, not an existence conclusion. Each such profile has quantized original electric flux and $\sigma (P_L)>0$. The profile may vary with $L$: these assumptions give $V(L)={\sigma }_{L}L+O(1)$ with ${\sigma }_L=\sigma (P_L)>0$, and alone give neither a constant limiting slope nor a positive lower bound uniform in $L$.

A positive constant slope with an $O(1)$ remainder follows under the following additional, length-reusable comparison hypothesis. Let $\mathcal{S}$ be a nonempty family of admitted transverse profiles in one fixed nonzero-winding sector, containing the transverse profile of such a minimizing tube for every $L\ge L_0$. Define

${\tau }_{\mathcal{S}}={\inf }_{P\in \mathcal{S}}\sigma (P),0<{\tau }_{\mathcal{S}}<\mathrm{\infty }\mathrm{.}$

Assume there is a sequence $P_k\in \mathcal{S}$ with $\sigma (P_k)\to {\tau }_{\mathcal{S}}$ and a constant $C_1<\mathrm{\infty }$ such that for every $k$ and every $L\ge L_0$ an actual competitor $Q_{k,L}\in {\mathcal{B}}_L$ satisfies

${\mathcal{E}}_L(Q_{k,L})\le L\sigma (P_k)+C_1\mathrm{.}$

The endpoint data defining ${\mathcal{B}}_L$ are the same for these competitors and the assumed minimizers; only the separation changes. The constants $C_0,C_1,L_0$ are independent of $k$ and $L$. These are explicit uniform endcap/comparison assumptions, not claims that arbitrary external charges admit such competitors. Then

${\tau }_{\mathcal{S}}L-C_0\le V(L)\le {\tau }_{\mathcal{S}}L+C_1,$

so $V(L)={\tau }_{\mathcal{S}}L+O(1)$, without requiring that any profile attain ${\tau }_{\mathcal{S}}$ or that the minimizing profiles be independent of $L$. If $\mathcal{S}$ lies in the preceding precise nonlinear planar class, its positive lower bound, under full AC, already gives ${\tau }_{\mathcal{S}}>0$; the uniform competitor bound remains an additional assumption. A single fixed minimizing transverse profile with uniformly reusable endcaps is a sufficient special case. None of these conditional assumptions asserts that a generic nonlinear model has an attained vortex minimum.

In the separately specified constant-metric linear truncation $u=u_m+c{A}_D$, assume full AC and take the actual canonical fixed-flux cylinder class defined above. The critical vortex supplier now proves existence and an absolute transverse energy minimum for every nonzero winding and every effective divisor of degree $\mathrm{\mid }n\mathrm{\mid }$, with arbitrary centres and multiplicities. Actual fixed-width full-model collars exist for any fixed smooth canonical end traces in the stated closed valid neutral disk. Hence its full static-energy infimum is $4\pi v^2\mathrm{\mid }n\mathrm{\mid }L+O(1)$, with an exact translation-invariant minimum for matching minimizing traces. This exact critical conclusion no longer assumes a supplied solution or endcap bound; it does not transfer BPS saturation or an $O(1)$ remainder to generic nonlinear u.

Seiberg and Witten, §5.6, equations (5.17)–(5.21).

Bilal, §5.3.

Tong, §4.6.2.

The generic nonlinear classical fixed-flux duty is now proved by energy comparison and asymptotic infima; it does not need a noncritical planar or three-dimensional minimizer. Uniform vacuum approach, a closed valid neutral disk with nonzero $u^{\mathrm{\prime }}$, explicit positive metric bounds and the magnetostatic constant-theta boundary problem are part of the stated model. The exact critical/BPS existence and equality duty is stronger and is now supplied by the actual reviewed arbitrary-divisor mathematical theorem; it remains a linear-superpotential result. A generic formula $I_n(L)={\tau }_{n}L+O(1)$, unrestricted behavior at a singular or unbounded metric, microscopic matching/error control, arbitrary line-charge dictionaries, and quantum confinement are not conclusions of this theorem. They are not silently substituted for the original local classical confinement contract.