Local dual U(1) monopole-Higgs effective data

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 duality frame near one monopole singularity of the selected pure $SU(2)$, four-dimensional rigid $\mathcal{N}=2$ model. Let $A_D$ be the complex scalar coordinate in the local dual $U(1{)}_D$ vector multiplet, and let $M,\tilde{M}$ be complex scalar fields of dual electric charges $+1$ and $-1$ in two $\mathcal{N}=1$ chiral multiplets. These are emergent infrared monopole fields in a dual frame of the independent pure ultraviolet model, not microscopic matter hypermultiplets. The dual gauge group acts by

$A_D\mapsto A_D,M\mapsto e^{i\alpha }M,\tilde{M}\mapsto e^{-i\alpha }\tilde{M},\alpha \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})\mathrm{.}$

Let $u(A_D)$ be the supplied holomorphic coordinate function on an open patch $\mathrm{\mid }{A}_D\mathrm{\mid }<r$ about $A_D=0$, with $u(0)=u_m$ and $c:=u^{\mathrm{\prime }}(0)\mathrm{\ne }0$. Let $\mu \mathrm{\ne }0$ be the coefficient of the small adjoint mass perturbation, let $g_D>0$ be the dimensionless dual gauge coupling, and use connection normalization for $A_D$, canonical kinetic terms for $M,\tilde{M}$, and zero microscopic D-type Fayet--Iliopoulos parameter. Freeze the positive local vector-multiplet metric at $b=g_D^{-2}$; this is a specified constant-coupling two-derivative approximation within the patch, not the full field-dependent Wilsonian metric. Define

$W_{\mu }(A_D,M,\tilde{M})=\sqrt{2}{A}_{D}M\tilde{M}+\mu u(A_D),D=g_D(\mathrm{\mid }M{\mathrm{\mid }}^2-\mathrm{\mid }\tilde{M}{\mathrm{\mid }}^2)\mathrm{.}$

$V_{\mu }=g_D^2\mathrm{\mid }{\mathrm{\partial }}_{A_D}{W}_{\mu }{\mathrm{\mid }}^2+\mathrm{\mid }{\mathrm{\partial }}_M{W}_{\mu }{\mathrm{\mid }}^2+\mathrm{\mid }{\mathrm{\partial }}_{\tilde{M}}{W}_{\mu }{\mathrm{\mid }}^2+\frac{1}{2}{D}^2\mathrm{.}$

Let ${\tilde{F}}_{\mu \nu }={\mathrm{\partial }}_{\mu }{\tilde{A}}_{\nu }-{\mathrm{\partial }}_{\nu }{\tilde{A}}_{\mu }$ and ${\tilde{D}}_{\mu }{A}_D={\mathrm{\partial }}_{\mu }{A}_D$, ${\tilde{D}}_{\mu }M=({\mathrm{\partial }}_{\mu }-i{\tilde{A}}_{\mu })M$, and ${\tilde{D}}_{\mu }\tilde{M}=({\mathrm{\partial }}_{\mu }+i{\tilde{A}}_{\mu })\tilde{M}$. In metric $\eta =\mathrm{diag}(-1,1,1,1)$, the connection-normalized bosonic kinetic Lagrangian is

${\mathcal{L}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=-{\eta }^{\mu \nu }(g_D^{-2}{\mathrm{\partial }}_{\mu }\overline{A_D}{\mathrm{\partial }}_{\nu }{A}_D+\overline{{\stackrel{~}{D}}_{\mu }M}{\tilde{D}}_{\nu }M+\overline{{\stackrel{~}{D}}_{\mu }\stackrel{~}{M}}{\tilde{D}}_{\nu }\tilde{M})-\frac{1}{4g_D^2}{\tilde{F}}_{\mu \nu }{\tilde{F}}^{\mu \nu }\mathrm{.}$

The mass dimensions are $[A_D]=[M]=[\tilde{M}]=[\mu ]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $[u]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, $[W_{\mu }]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^3$, $[F_{A_D}]=[F_M]=[F_{\tilde{M}}]=[D]=[{\tilde{F}}_{\mu \nu }]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, $[{\tilde{A}}_{\mu }]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, $[c]=\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}$, and $[g_D]=1$.

Supply a positive effective cutoff ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ of mass dimension one and a dimensionless control margin $0<\epsilon <1$. This is independently chosen applicability data, not an identification with the dynamical scale $\mathrm{\Lambda }$ of the Coulomb-branch model. Define the small-perturbation regime by

$v:={\left(\frac{\mathrm{\mid }\mu c\mathrm{\mid }}{\sqrt{2}}\right)}^{1\mathrm{/}2}<\epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}},0<\mathrm{\mid }\mu \mathrm{\mid }<\frac{\sqrt{2}{\epsilon }^2{\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}^2}{\mathrm{\mid }c\mathrm{\mid }}\mathrm{.}$

The two inequalities are equivalent because $c\mathrm{\ne }0$; $\mu \mathrm{\ne }0$ remains required. Write $a=A_D$ and supply a gauge-invariant open background-validity domain ${\mathcal{U}}_{\mathrm{E}\mathrm{F}\mathrm{T}}\subset \{(a,M,\tilde{M})\in {\mathbb{C}}^3:\mathrm{\mid }a\mathrm{\mid }<r\}$ containing the vacuum orbit $a=0$, $\mathrm{\mid }M\mathrm{\mid }=\mathrm{\mid }\tilde{M}\mathrm{\mid }=v$, $M\tilde{M}=-\mu c\mathrm{/}\sqrt{2}$. The supplied regime may impose additional mass, coupling or background-field restrictions; no universal choice of this domain is inferred from the cutoff inequality.

To specify an application energy range, use the standard coordinate basis of Minkowski space ${\mathbb{R}}^{1,3}$ with the displayed metric. Supply a nonempty set ${\mathcal{K}}_{\mathrm{a}\mathrm{p}\mathrm{p}}\subset {\mathbb{R}}^4$ of excitation four-momentum components $p=(p^0,p^1,p^2,p^3)$ in that basis, and require the finite bound

$E_{\mathrm{a}\mathrm{p}\mathrm{p}}:={\sup }_{p\in {\mathcal{K}}_{\mathrm{a}\mathrm{p}\mathrm{p}}}{\max }_{0\le \nu \le 3}\mathrm{\mid }{p}^{\nu }\mathrm{\mid }\le \epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}\mathrm{.}$

Here $p^0$ is the excitation energy and $p^1,p^2,p^3$ are its spatial momentum components; all have mass dimension one. An application without specified excitations may take ${\mathcal{K}}_{\mathrm{a}\mathrm{p}\mathrm{p}}=\{0\}$. Physical use of this adopted low-energy regime requires background values in ${\mathcal{U}}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ and the supplied excitation bound. This does not impose a hard Fourier-support condition on static vortex profiles. The domain, cutoff and margin are supplied model-validity inputs, not a proof of microscopic accuracy or a bound on omitted terms. In particular, small $\mu$ alone establishes no such error bound.

The displayed functional remains a specified classical two-derivative model for all smooth fields with $\mathrm{\mid }{A}_D\mathrm{\mid }<r$, including fields outside this physical applicability regime; no physical accuracy outside that regime is claimed. Additional hard restrictions on a variational class must be stated separately and checked against any trial profiles or interpolations.

For a specified leading linear-superpotential truncation, explicitly replace $u(A_D)$ by $u_m+c{A}_D$ in this constant-metric model. Then the perturbation is an $F$-type FI deformation of the local $\mathcal{N}=2$ abelian theory, related to a D-type FI deformation by rotation of its auxiliary triplet. This extra truncation admits the critical BPS square completion. The generic holomorphic $u(A_D)$ model above is softly broken to $\mathcal{N}=1$ and is not asserted to have exact BPS vortices; the full physical solution and a globally valid metric are separate inputs. Neither a small $\mu$ nor a holomorphic superpotential theorem alone controls all non-holomorphic metric or higher-derivative corrections.

This is a one-singularity effective data package. It includes neither a simultaneously local electric description of mutually nonlocal dyons nor a claim that the local action extends over the full moduli space. A finite-energy vortex is not part of this Definition; its existence, when needed, is a separate hypothesis.

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

Bilal, §5.3.

Pure $N=2$ SU(2) Coulomb-branch conventions: Pure $N=2$ SU(2) Coulomb-branch and period data.

Independent pure ultraviolet vector-multiplet datum: Independent pure $N=2$ SU(2) ultraviolet vector-multiplet datum.