The Pure SU(2) Seiberg–Witten Solution

Pure four-dimensional $\mathcal N=2$ $SU(2)$ Yang–Mills theory is the canonical complete Seiberg–Witten example. Semiclassical running fixes the behavior at infinity; two finite singularities supply a massless monopole and dyon; an elliptic curve and differential reproduce their monodromies and exact periods. The construction is given in Seiberg and Witten 1994, §§2–6, with an explicit pedagogical derivation in Bilal 1996, §§3–6. Every numerical label below belongs to the chapter’s frozen scale, cycle, charge, path, and chamber convention.

Required background. Curves, differentials, and periods fixes the geometric construction. Helpful background. BPS particles and central charges and instantons and zero modes explain two independent physical limits.

Semiclassical data

The adjoint scalar can be diagonalized as

$$\phi\sim\begin{pmatrix}a/2&0\\0&-a/2\end{pmatrix}$$

up to normalization and the Weyl reflection $a\sim-a$. Use the gauge-invariant coordinate

$$u=\langle\operatorname{Tr}\phi^2\rangle$$

normalized so that

$$a(u)\sim\sqrt{2u}$$

at large positive $u$.

The one-loop prepotential is

$$\mathcal F_{\mathrm{1-loop}} =\frac{i}{2\pi}a^2\log\frac{a^2}{\Lambda^2},$$

up to a quadratic polynomial. Hence

$$a_D\sim\frac{i}{\pi}a \left(\log\frac{a^2}{\Lambda^2}+1\right).$$

This display chooses the representative with no extra quadratic term. The curve-and-cycle convention below fixes a different representative, $\Delta\mathcal F=(i/(2\pi))(\log 4-3)a^2$; equivalently, $\Lambda_{\mathrm{UV}}^2=(e^3/4)\Lambda^2$ when $\Lambda$ is the curve scale.

Taking $u$ counterclockwise around infinity sends $a\to-a$ and shifts the logarithm. In the period ordering $(a_D,a)$ this gives

$$M_\infty=\begin{pmatrix}-1&2\\0&-1\end{pmatrix}.$$

One isolated finite singularity cannot factor this monodromy into the required nontrivial integral physics while respecting the discrete symmetry. The exact solution has two.

Curve, differential, and discriminant

Choose

$$y^2=(x-u)(x-\Lambda^2)(x+\Lambda^2)$$

with

$$\lambda_{\mathrm{SW}} =\frac{\sqrt2}{2\pi} \frac{(x-u)\,dx}{y}.$$

The finite branch points collide at

$$u=+\Lambda^2, \qquad u=-\Lambda^2.$$

The polynomial discriminant is proportional to

$$\Lambda^4(u^2-\Lambda^4)^2,$$

so its zero locus has precisely those two finite points for nonzero $\Lambda$. Overall powers and constants in the discriminant do not affect the locus but do matter when comparing algebraic conventions.

Take $\Lambda>0$ and the weak-coupling base point $u_b=2\Lambda^2$. On its fiber use cuts $[-\Lambda^2,+\Lambda^2]$ and $[u_b,\infty]$. Let $A$ encircle the first cut and let $B$ be the oriented lift of a contour enclosing $+\Lambda^2$ and $u_b$, with

$$A\circ B=+1.$$

Define

$$a=\oint_A\lambda_{\mathrm{SW}}, \qquad a_D=\oint_B\lambda_{\mathrm{SW}}.$$

Orient the cycles so $a\sim\sqrt{2u}>0$ and $a_D\in i\mathbb R_{>0}$ on the positive real weak-coupling ray. These conditions remove the otherwise ambiguous signs and scale of the differential. The corresponding $u$-plane meridians and their nonintersecting stems are displayed in the shared rank-one figure linked below.

Exact periods and coupling

The period derivatives are integrals of the holomorphic differential $dx/y$:

$$\frac{da}{du}\propto\oint_A\frac{dx}{y}, \qquad \frac{da_D}{du}\propto\oint_B\frac{dx}{y}.$$

For real $u>\Lambda^2$, define

$$s=u+\Lambda^2, \qquad m=\frac{2\Lambda^2}{s}.$$

With complete elliptic integrals in the parameter convention, the oriented periods are

$$\begin{aligned} a(u) &=\frac{2\sqrt{2s}}{\pi}E(m),\\ a_D(u) &=\frac{2i\sqrt{2s}}{\pi} \bigl[K(1-m)-E(1-m)\bigr]. \end{aligned}$$

The effective coupling is therefore

$$\tau(u)=\frac{da_D/du}{da/du} =i\frac{K(1-m)}{K(m)}.$$

Because it is the elliptic period ratio, $\operatorname{Im}\tau>0$ on this regular ray. At the checksum-frozen base point,

$$\frac{a}{\Lambda}=1.966685301550330\ldots, \qquad \frac{a_D}{i\Lambda}=0.473434436535537\ldots, \qquad \frac{\tau}{i}=0.854584443278744\ldots.$$

The structured figure record recomputes these numbers from independent quadrature before checking monodromies. Analytic continuation of the elliptic integrals reproduces the integral matrices below.

At large $u$, expanding the periods yields

$$a(u)=\sqrt{2u}\left[1-\frac{\Lambda^4}{16u^2} +O\!\left(\frac{\Lambda^8}{u^4}\right)\right],$$

and

$$a_D(u)=\frac{i}{\pi}a \left(\log\frac{a^2}{\Lambda^2}+\log 4-2\right) +O\!\left(\frac{\Lambda^4}{a^3}\right).$$

The logarithmic coefficient is scheme-independent, while the additive constant here is fixed before evaluation by the displayed curve, differential, $A/B$ cycle orientations, and singularity scale. The power corrections reproduce the instanton expansion. The absence of odd powers follows from the anomalous discrete R-symmetry in this scale convention.

The two light charges

Choose charges $\gamma=(p,q)$ with

$$Z_\gamma=pa_D+qa.$$

At $u=+\Lambda^2$, take the vanishing charge to be the monopole

$$\gamma_m=(1,0), \qquad Z_m=a_D.$$

Its monodromy is

$$M_m=\begin{pmatrix}1&0\\-2&1\end{pmatrix}.$$

At $u=-\Lambda^2$, with the declared base paths, take

$$\gamma_d=(1,-1), \qquad Z_d=a_D-a,$$

and

$$M_d=\begin{pmatrix}-1&2\\-2&3\end{pmatrix}.$$

Their coordinate determinant and physical particle pairing are, respectively,

$$\omega(\gamma_m,\gamma_d)=-1, \qquad \langle\gamma_m,\gamma_d\rangle_{\mathrm{part}}=-2.$$

They do not become massless at the same point, so each singularity separately admits a weakly coupled local electric frame. Their nonlocality explains why no single frame describes both singular regions at once.

The global monodromy check

Let $\ell_+$ and $\ell_-$ denote the counterclockwise based meridians fixed by the chapter convention, and define

$$\rho(\ell_\infty)=\rho(\ell_+)\rho(\ell_-).$$

Matrices act on period columns from the left, so the rightmost factor is applied first. With that explicit operator convention,

$$M_mM_d =\begin{pmatrix}-1&2\\0&-1\end{pmatrix} =M_\infty.$$

This check links three independent ingredients: the monopole threshold, the dyon charge, and the ultraviolet beta function. Changing cuts conjugates all matrices; reversing a loop inverts its factor. A product that matches only after changing one matrix in isolation is inconsistent.

The shared rank-one geometry, chamber, and deformation figure shows the base-fiber cycles and based $u$-plane paths that fix these labels. Its machine-readable record contains the exact period fixture, particle-pairing crosswalk, and matrix checks.

Local effective theories

Near the monopole point, use $a_D$ as the electric scalar of a dual $U(1)$ vector multiplet and include a monopole hypermultiplet $(M,\widetilde M)$. The local superpotential in $\mathcal N=1$ notation is

$$W_{\mathrm{local}} =\sqrt2\,A_D M\widetilde M$$

before further deformations. In the frozen orientation,

$$a_D(u) =\frac{i}{2\Lambda}(u-\Lambda^2) +O\!\left(\frac{(u-\Lambda^2)^2}{\Lambda^3}\right), \qquad a(\Lambda^2)=\frac{4\Lambda}{\pi}.$$

The logarithm in the photon-only coupling is exactly the threshold obtained by integrating this hypermultiplet out.

Near the dyon point, use the special coordinate

$$a_d=a_D-a$$

and the duality frame in which the dyon is electric. These are two patches of the same global solution, not two simultaneous local Lagrangians.

Uniqueness and checks

The solution is unique only within the stated ansatz: the quantum moduli space is the $u$-plane; there are exactly two finite singularities and no hidden additional ones; the anomalous discrete symmetry exchanges them; the metric is regular and positive away from the discriminant; and the weak-coupling asymptotics are fixed. Under those hypotheses, the remaining data are:

• the one-complex-dimensional $u$ plane and Weyl-invariant asymptotics;
• the anomalous discrete symmetry exchanging the two finite singularities;
• the monodromy at infinity from the one-loop beta function;
• integral local monodromies from one light hypermultiplet at each point;
• a genus-one curve and differential with the correct dimensions and asymptotics.

Independent checks include instanton coefficients in the large-$a$ expansion, positivity of $\operatorname{Im}\tau$, BPS central charges, and the controlled $\mathcal N=1$ deformation on the final page of this chapter.

What the solution does not say

It does not give one global electric Lagrangian, list the BPS spectrum without a chamber, or prove ordinary QCD confinement. It solves the exact two-derivative Coulomb-branch dynamics of the supersymmetric theory and selected protected consequences.

The same curve can be paired with different genuine-line lattices for different global forms. Those theories share local couplings but differ in extended observables.

Common pitfalls

Calling both finite singularities monopole points. Their light charges differ in the fixed base-point frame; one is a dyon in the convention above.

Changing $\Lambda$ without moving the singularities and periods together. Scale redefinitions change every numerical anchor.

Inferring a chamber-independent BPS tower from monodromy. Monodromy constrains charges and periods; stability requires wall-crossing data.

Exercises

Starting from $\gamma_d=(1,-1)$ and $M_\gamma=\mathbf1+2(J\gamma)\gamma^T$:

1. reconstruct $M_d$ and show that $a_d=a_D-a$ is invariant;
2. multiply it by $M_m$ in the declared based-loop order;
3. give the inverse-transpose action on an arbitrary charge column and verify central-charge invariance.

Solution

The charge formula gives

$$M_d=\begin{pmatrix}-1&2\\-2&3\end{pmatrix}.$$

Under

$$\binom{a_D'}{a'} =\begin{pmatrix}-1&2\\-2&3\end{pmatrix} \binom{a_D}{a},$$

one has

$$a_D'-a' =(-a_D+2a)-(-2a_D+3a) =a_D-a.$$

This is equivalent to $\gamma_d^TM_d=\gamma_d^T$ for $\gamma_d=(1,-1)$.

The ordered product is

$$M_mM_d =\begin{pmatrix}-1&2\\0&-1\end{pmatrix} =M_\infty.$$

For any charge column $\beta$, set $\beta'=M^{-T}\beta$ when $\Pi'=M\Pi$. Then $\beta'^T\Pi'=\beta^T\Pi$, so the transported BPS mass is independent of the chosen period components.

References

• Bilal, Adel. “Duality in $\mathcal N=2$ SUSY $SU(2)$ Yang–Mills Theory: A Pedagogical Introduction to the Work of Seiberg and Witten.” 1996. arXiv:hep-th/9601007.
• Seiberg, Nathan, and Edward Witten. “Electric–Magnetic Duality, Monopole Condensation, and Confinement in $\mathcal N=2$ Supersymmetric Yang–Mills Theory.” Nuclear Physics B 426 (1994): 19–52; erratum 430 (1994): 485–486. arXiv:hep-th/9407087.