Source-proposed pure SU(2) $N=2$ Seiberg–Witten low-energy data

Adopt the source-proposed low-energy solution data for pure four-dimensional rigid $\mathcal{N}=2$ Yang--Mills with global gauge group $SU(2)$, one adjoint vector multiplet, rank one, no matter hypermultiplets, and the coupling, units, connection-normalized scalar, and prepotential convention of the independent pure ultraviolet vector-multiplet datum. 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 a positive dynamical scale $\mathrm{\Lambda }$ of mass dimension one. The gauge-invariant coordinate $u$ has mass dimension two; set $w=u\mathrm{/}{\mathrm{\Lambda }}^2$.

The absence of microscopic hypermultiplets is an explicit model input in that independent ultraviolet postulate. The charged monopole hypermultiplet in a local infrared frame below is emergent; it does not change the ultraviolet matter content.

On the regular $w$-plane with singular values $w=1,-1,\mathrm{\infty }$ removed, adopt the family of genus-one curves, where $x,y,w$ are dimensionless:

${\mathrm{\Sigma }}_w:y^2=(x-1)(x+1)(x-w)\mathrm{.}$

and the meromorphic period differential

${\lambda }_{\mathrm{S}\mathrm{W}}=\frac{\sqrt{2}\mathrm{\Lambda }}{2\pi }\frac{ydx}{x^2-1}\mathrm{.}$

Within this proposed solution, take the quantum vacuum-moduli data to be the entire $u$-plane, with a ${\mathbb{Z}}_2$ symmetry $u\mapsto -u$ and singular values $u={\mathrm{\Lambda }}^2,-{\mathrm{\Lambda }}^2,\mathrm{\infty }$. This is part of the adopted solution ansatz, not a consequence of a constructed continuum theory.

For a chosen local cycle basis ${\gamma }_m,{\gamma }_e$ on ${\mathrm{\Sigma }}_w$ with intersection number ${\gamma }_m\cdot {\gamma }_e=1$, define $a_D={\oint }_{{\gamma }_m}{\lambda }_{\mathrm{S}\mathrm{W}}$ and $a={\oint }_{{\gamma }_e}{\lambda }_{\mathrm{S}\mathrm{W}}$. These periods have mass dimension one, are multivalued under continuation around the punctures, and are used in the local prepotential convention $a_D=\mathrm{\partial }\mathcal{F}\mathrm{/}\mathrm{\partial }a$ and $\tau ={\mathrm{\partial }}^2\mathcal{F}\mathrm{/}\mathrm{\partial }{a}^2$ with $\mathrm{Im}\tau >0$. In the source's counterclockwise-loop convention, the period column $(a_D,a{)}^T$ has monodromy matrices

$M_{\mathrm{\infty }}=\left(\begin{array}{cc}-1& 2\\ 0& -1\end{array}\right),M_{+}=\left(\begin{array}{cc}1& 0\\ -2& 1\end{array}\right),M_{-}=\left(\begin{array}{cc}-1& 2\\ -2& 3\end{array}\right)\mathrm{.}$

Adopt the source's formal charge-label lattice ${\mathbb{Z}}^2$ and central-charge pairing ${\mathcal{Z}}_{\mathrm{S}\mathrm{W}}=n_{e}a+n_m{a}_D$ for labels $(n_e,n_m)\in {\mathbb{Z}}^2$. In the supersymmetry convention of this page, the scalar central charge is normalized as $z=\sqrt{2}{\mathcal{Z}}_{\mathrm{S}\mathrm{W}}$. The mass inequality and its saturation statement are separate theorem deductions. No claim is made that every lattice label is an allowed line operator or is represented by a stable particle.

Adopt the singularity interpretation that the $w=1$ point has a light monopole with $a_D=0$, while the $w=-1$ point has a light dyon with $a-a_D=0$ in the chosen frame. These are source-proposed physical inputs tested by their monodromies; they are not deduced here from a constructed continuum theory.

In this source convention the monopole point is $u_m:={\mathrm{\Lambda }}^2$. In the chosen local monopole frame, adopt the source identification of the dual scalar $A_D$ with the period branch $a_D(u)$. This branch satisfies $a_D(u_m)=0$ and has nonzero derivative $a_D^{\mathrm{\prime }}(u_m)=c_0\mathrm{\ne }0$, as expressed by its source-given linear vanishing near $u_m$; the source also treats $u$ as a local coordinate in this patch. Adopt the local holomorphic inverse $u(A_D)$, defined by $a_D(u(A_D))=A_D$, with $u(0)=u_m$ and $c:=u^{\mathrm{\prime }}(0)=c_0^{-1}\mathrm{\ne }0$. These are source-proposed local model inputs, not deductions from a constructed continuum theory or assertions of global single-valuedness. The local monopole superpotential uses this inverse in $\mu u(A_D)$.

In the local monopole patch, adopt the dual $U(1{)}_D$ field content and superpotential described in the preceding effective-model definition. The source two-derivative vector metric is positive and generally field dependent, determined by the local dual coupling; a regulated matching scale is part of any effective action with the light monopole retained. That definition explicitly freezes this metric at $b=g_D^{-2}>0$ for the subsequent classical calculations. Metric freezing, and the further replacement of $u(A_D)$ by its linear Taylor term where explicitly requested, are additional approximations, not exact identities of the full period solution. No bound on their discarded corrections is adopted here. A small $N=1$-breaking mass parameter $\mu$ of mass dimension one adds the local superpotential term $\mu u(A_D)$ to $\sqrt{2}{A}_{D}M\tilde{M}$. The local description applies only near this singularity, with the supplied cutoff ${\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$, margin $\epsilon$, background domain ${\mathcal{U}}_{\mathrm{E}\mathrm{F}\mathrm{T}}$ and excitation bound of that definition; small $\mu$ means $v<\epsilon {\mathrm{\Lambda }}_{\mathrm{E}\mathrm{F}\mathrm{T}}$. These are adopted applicability inputs, not a bound on discarded corrections. This local description does not simultaneously describe the mutually nonlocal dyon at the other singularity.

Here exact low-energy solution means the adopted two-derivative Wilsonian data of this source-defined model on its regular patches. It does not mean a proof of a nonperturbative UV path integral, a global single-valued prepotential, or an identity for arbitrary gauge groups, matter contents, or $N=2$ theories.

Seiberg and Witten, §5.7, §§5.1–5.6 and §6.

Bilal, §§5.3 and 6.1–6.2.

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

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

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