Let us close this section by very brieﬂy mentioning the Seiberg-Witten solution of various other gauge theories. First, let us copy the solutions for $SU\left(N\right)$ and $SO\left(2N\right)$ with ${N}_{f}$ ﬂavors. The Seiberg-Witten curve for $SU\left(N\right)$ with ${N}_{f}$ ﬂavors was

$$\frac{{\Lambda}^{N}}{z}+z{\Lambda}^{N-{N}_{f}}\prod _{i=1}^{{N}_{f}}\left(x-{\mu}_{i}\right)={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N-1}x+{u}_{N}$$ | (11.6.1) |

and the Seiberg-Witten curve for $SO\left(2N\right)$ with ${N}_{f}$ ﬂavors was

Here, for simplicity, we dropped the ﬂavor terms multiplying $1\u2215z$ on the left hand sides. For the full expressions, see (10) and (21), respectively.

Without derivations, we present here the Seiberg-Witten curves for other classical gauge groups. The Seiberg-Witten curve for $SO\left(2N+1\right)$ with ${N}_{f}$ ﬂavors is

and the Seiberg-Witten curve for $Sp\left(N\right)$ with ${N}_{f}$ ﬂavors is

$$\begin{array}{cc}& \frac{{\Lambda}^{2N+2}}{{z}^{1\u22152}}+2c+{\Lambda}^{2N+2-2{N}_{f}}{z}^{1\u22152}\prod _{i=1}^{{N}_{f}}\left({x}^{2}-{\mu}_{i}{\phantom{\rule{0.0pt}{0ex}}}^{2}\right)\\ & ={x}^{2}\left({x}^{2N}+{u}_{2}{x}^{2N-2}+{u}_{4}{x}^{2N-4}+\cdots +{u}_{2N}\right)& \text{(11.6.4)}\end{array}$$where ${c}^{2}={\Lambda}^{4N+4-2{N}_{f}}{\prod}_{i=1}^{{N}_{f}}\left(-{\mu}_{i}^{2}\right)$. The diﬀerential is always just $\lambda =xdz\u2215z$. We again dropped the ﬂavor terms multiplying $1\u2215z$ on the left hand sides.

The curves so far can be always written as

$$\frac{F\left(x\right)}{z}+z\stackrel{\u0303}{F}\left(x\right)=P\left(x\right)$$ | (11.6.5) |

for some polynomials $F\left(x\right)$, $\stackrel{\u0303}{F}\left(x\right)$ and $P\left(x\right)$. In the older literature, it is more common to ﬁnd the curve and the diﬀerential in the form

$${y}^{2}=P{\left(x\right)}^{2}-4F\left(x\right)\stackrel{\u0303}{F}\left(x\right),\phantom{\rule{2em}{0ex}}\lambda =\frac{x}{2}dlog\frac{P\left(x\right)-y}{P\left(x\right)+y}.$$ | (11.6.6) |

To relate (11.6.5) and (11.6.6), note that the equation (11.6.6) implies that the combination

$${\zeta}_{\pm}=\frac{1}{2}\left(P\left(x\right)\pm y\right)$$ | (11.6.7) |

satisﬁes

$${\zeta}_{+}+{\zeta}_{-}=P\left(x\right),\phantom{\rule{2em}{0ex}}{\zeta}_{+}{\zeta}_{-}=F\left(x\right)\stackrel{\u0303}{F}\left(x\right).$$ | (11.6.8) |

Comparing with (11.6.5), we ﬁnd

$${\zeta}_{+}=\frac{F\left(x\right)}{z},\phantom{\rule{2em}{0ex}}{\zeta}_{-}=z\stackrel{\u0303}{F}\left(x\right).$$ | (11.6.9) |

This also explains the diﬀerential given in (11.6.6).

This older form is mathematically easier to deal with in certain situations, because the branch cut of the function $y\left(x\right)$ is at most of order 2. Mathematically, such Riemann surfaces are called hyperelliptic, and have a few special properties compared to more general Riemann surfaces. That said, the forms we use in the rest of the lecture note is much more physical and usually more useful.

A good summary of the curves for classical gauge groups listed above can be found e.g. in [67]. For exceptional gauge groups, the situation is more complicated. Although one can write the Seiberg-Witten curve, it is more natural to study the Seiberg-Witten geometry, which is a complex 3-dimensional space, ﬁbered over the ultravioletcurve $C$. A very nice presentation for the pure theory can be found in [68]. With matter hypermultiplets, a modern reference is [69].