Seiberg–Witten theory

Definition

Consider four-dimensional $N = 2$ supersymmetric Yang–Mills theory with gauge group $SU(2)$ and dynamical scale $\Lambda > 0$. Its space of vacua contains the Coulomb branch, a copy of the complex plane. Its coordinate $u$ is the vacuum expectation value of $\operatorname{tr}\phi^2$ for the adjoint scalar $\phi$. Seiberg–Witten theory is the following description of the low-energy effective theory on the Coulomb branch.

1. The effective theory is a $U(1)$ gauge theory determined by a holomorphic function $\mathcal{F}$, the prepotential, of a local coordinate $a$ on the Coulomb branch. Set $a_D = \partial\mathcal{F}/\partial a$ and $\tau = da_D / da$; the imaginary part of $\tau$ is positive and is proportional to the inverse squared effective coupling.
2. To each $u$ is attached an elliptic curve, the Seiberg–Witten curve, which in one standard normalization is $E_u \colon y^2 = (x^2 - u)^2 - \Lambda^4$. It is smooth exactly when $u \ne \pm\Lambda^2$.
3. There is a meromorphic differential $\lambda_u$ on $E_u$, the Seiberg–Witten differential, whose derivative in $u$ is a holomorphic differential up to an exact form. For a basis $\alpha, \beta$ of the first homology of $E_u$, one has $a = \oint_\alpha \lambda_u$ and $a_D = \oint_\beta \lambda_u$. So $\tau$ is the period ratio of the elliptic curve.
4. The mass of a state with electric charge $n_e$ and magnetic charge $n_m$ saturating the BPS bound is $\sqrt{2} \abs{n_e a + n_m a_D}$.