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 one explicit scale, cycle, and charge 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.

Choose a symplectic cycle basis $(A,B)$ at a weak-coupling base point $u_0>\Lambda^2$ and define

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

The $A$ orientation is fixed by $a\sim\sqrt{2u}>0$ at $u_0$. This asymptotic condition removes the otherwise ambiguous overall sign and scale of the differential.

Exact periods and coupling

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

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

They can be expressed in complete elliptic integrals after mapping the four branch points, including infinity, to a standard modulus. The effective coupling is

$$\tau(u)=\frac{da_D/du}{da/du}.$$

Because it is the elliptic period ratio, $\operatorname{Im}\tau>0$ on a regular patch. Analytic continuation of the elliptic integrals reproduces the integral monodromies below.

At large $u$, expanding the periods yields

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

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

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

The two charges have nonzero Dirac pairing,

$$\langle\gamma_m,\gamma_d\rangle=-1.$$

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

With based loops and multiplication chosen so that the positive singularity acts first in the written product,

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

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.

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. 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 fixed, within the stated assumptions, by:

• 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

At the dyon point, show that the period combination $a_d=a_D-a$ is invariant under $M_d$.

Solution

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 $\gamma_d=(1,-1)$.

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.