Without further ado, let us introduce the Seiberg-Witten curves. First, the Seiberg-Witten curve for the pure $SU\left(N\right)$ theory is given by

$$\Sigma :\phantom{\rule{2em}{0ex}}\frac{{\Lambda}^{N}}{z}+{\Lambda}^{N}z={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N}$$ | (11.2.1) |

with the diﬀerential $\lambda =xdz\u2215z$ as always. The ultraviolet curve $C$ is just a sphere with the complex coordinate $z$. At each point on the ultraviolet curve $z$, we have $N$ solutions to the equation above. Therefore, $\Sigma $ is an $N$-sheeted cover of $C$.

Let us check that this curve reproduces the semiclassical behavior. We introduce variables ${\underline{a}}_{i}$ via

$${x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N}=\prod _{i}\left(x-{\underline{a}}_{i}\right).$$ | (11.2.2) |

We declare the A-cycle on the ultraviolet curve to be the unit circle $\left|z\right|=1$. As the Seiberg-Witten curve is an $N$-sheeted cover, we can lift this curve to each sheet, which we call the cycle ${A}_{i}$. Assume we are in the regime $\left|{\underline{a}}_{i}\right|\sim E$ independent of $i$, and $E\gg \Lambda $. Then, we can approximately solve (11.2.1) by

$${x}_{i}={\underline{a}}_{i}+O\left(\Lambda \right).$$ | (11.2.3) |

It is more convenient to regard $\lambda =xdz\u2215z$ itself to be the coordinate of the sheets. Then we have

$${\lambda}_{i}={\underline{a}}_{i}\frac{dz}{z}+O\left(\Lambda \right).$$ | (11.2.4) |

The situation is shown in Fig. 11.1. The integral of $\lambda $ on the cycle ${A}_{i}$ is easy to evaluate:

$${a}_{i}:=\frac{1}{2\pi i}{\oint}_{{A}_{i}}\lambda ={\underline{a}}_{i}+O\left(\Lambda \right).$$ | (11.2.5) |

Now we can suspend a ring-shaped membrane suspended between the $i$-th sheet and the $j$-th sheet. The mass of this object is

$$|\frac{1}{2\pi i}{\oint}_{{A}_{i}}\lambda -\frac{1}{2\pi i}{\oint}_{{A}_{j}}\lambda |=\left|\frac{1}{2\pi i}{\oint}_{A}\left({\lambda}_{i}-{\lambda}_{j}\right)\right|=|{a}_{i}-{a}_{j}|.$$ | (11.2.6) |

This reproduces the mass of the W-boson.

To see the monopoles, we need to understand the structure of the branching of the $N$-sheeted cover $\Sigma \to C$. It is convenient to regard the combination $y={\Lambda}^{N}\left(z+1\u2215z\right)$ as one coordinate. Then, the equation (11.2.1) can be thought of determining the intersections of the graph of the polynomial

$$y=P\left(x\right)={x}^{N}+{u}_{2}{x}^{N-2}+\cdots +{u}_{N}$$ | (11.2.7) |

and a horizontal line

$$y={\Lambda}^{N}\left(z+\frac{1}{z}\right)$$ | (11.2.8) |

as shown in Fig. 11.2. Of course the ﬁgure needs to be complexiﬁed, but the reader should be able to get the idea.

As is apparent, two out of $N$ sheets meet at $\left(N-1\right)$ values of $y=\Lambda \left(z+1\u2215z\right)$, each of which becomes a pair ${z}_{i}^{\pm}$ of branch points on the $z$-sphere with ${z}_{i}^{+}{z}_{i}^{-}=1$. Note that the $i$-th sheet and the $\left(i+1\right)$-st sheet meet at this pair of branch points. Then we can suspend a disk-shaped membrane between this pair of branch points, as shown in Fig. 11.3.

In the semiclassical regime when $\left|{\underline{a}}_{i}\right|\sim \left|E\right|\gg \left|\Lambda \right|$, we have

$$\left|{z}_{i}^{+}\right|\sim \frac{{E}^{N}}{{\Lambda}^{N}},\phantom{\rule{2em}{0ex}}\left|{z}_{i}^{-}\right|\sim \frac{{\Lambda}^{N}}{{E}^{N}}.$$ | (11.2.9) |

We call the path connecting ${z}_{i}^{+}$ and ${z}_{i}^{-}$ as ${B}_{i}$. Then

$$\begin{array}{lll}\hfill {M}_{\text{monopole}}& =\left|\frac{1}{2\pi i}{\int}_{{B}_{i}}\left({\lambda}_{i}-{\lambda}_{i+1}\right)\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(11.2.10)}\\ \hfill & \sim \left|\left({a}_{i}-{a}_{i+1}\right)\frac{1}{2\pi i}{\int}_{{\Lambda}^{N}\u2215{E}^{N}}^{{E}^{N}\u2215{\Lambda}^{N}}\frac{dz}{z}\right|\phantom{\rule{2em}{0ex}}& \hfill \text{(11.2.11)}\\ \hfill & \sim |\left({a}_{i}-{a}_{i+1}\right)\frac{2N}{2\pi i}log\frac{E}{\Lambda}|.\phantom{\rule{2em}{0ex}}& \hfill \text{(11.2.12)}\end{array}$$This reproduces the mass of the monopole, by identifying

$$\tau \left(E\right)=\frac{2N}{2\pi i}log\frac{E}{\Lambda}.$$ | (11.2.13) |

This correctly reproduces the one-loop running of the pure $SU\left(N\right)$ theory.

Let us check that our curve satisﬁes the condition that the coupling matrices of the low-energy
$U{\left(1\right)}^{N-1}$ theory
is positive deﬁnite. For this purpose we need to understand the geometry of the Seiberg-Witten curve
$\Sigma $ better. This is
an $N$-sheeted
cover of $C$ with
$2N-2$ branch points
${z}_{i}^{\pm}$ of order 2 and
2 branch points $z=0,\infty $
of order $N$. The
genus $g$ of
$\Sigma $ is then determined by the
Riemann-Hurwitz formula^{15} :

$$\chi \left(\Sigma \right)=N\chi \left(C\right)-\left(2N-2\right)-2\left(N-1\right)$$ | (11.2.14) |

where $\chi \left(\Sigma \right)=2-2g$ and $\chi \left(C\right)=2$ are the Euler number of the respective surfaces. We ﬁnd $g=N-1$. The basis of the 1-cycles consists of $\left(2N-2\right)$ cycles ${A}_{i}$ and ${\stackrel{\u0303}{B}}^{i}$, $i=1,\dots ,N-1$, where the intersections are given by

$${A}_{i}\cdot {A}_{j}=0={\stackrel{\u0303}{B}}^{i}\cdot {\stackrel{\u0303}{B}}^{j},\phantom{\rule{2em}{0ex}}{A}_{i}\cdot {\stackrel{\u0303}{B}}^{j}={\delta}_{i}^{j}.$$ | (11.2.15) |

Here the dot product counts the number of intersections (including signs) of two one-cycles. The resulting set of cycles is shown in Fig. 11.4.

The ﬁgures 11.3 and 11.4 are drawn in a rather diﬀerent manner. The cycles from ${A}_{1}$ to ${A}_{N-1}$ can be directly identiﬁed. We have

$${A}_{N}=-{A}_{1}-{A}_{2}\cdots -{A}_{N-1}$$ | (11.2.16) |

as far as the line integral of holomorphic forms are concerned. Correspondingly, the variables ${a}_{i}$ as deﬁned in (11.2.5) are not linearly independent, and we have

$${a}_{N}=-{a}_{1}-\cdots -{a}_{N-1}.$$ | (11.2.17) |

The combination ${B}_{i}-{B}_{i+1}$ in Fig. 11.3 intersects with ${A}_{i}$ positively and with ${A}_{i+1}$ negatively. Then, we see

$${B}_{i}-{B}_{i+1}={\stackrel{\u0303}{B}}^{i}-{\stackrel{\u0303}{B}}^{i+1}.$$ | (11.2.18) |

Equivalently, ${\stackrel{\u0303}{B}}^{i}$ is a closed one-cycle completing the open path ${B}_{i}$ in a way independent of $i$. Then we deﬁne

$${a}_{D}^{i}:=\frac{1}{2\pi i}{\oint}_{{\stackrel{\u0303}{B}}^{i}}\lambda $$ | (11.2.19) |

on the curve. Let us consider

$${\tau}^{ij}:=\frac{\partial {a}_{D}^{i}}{\partial {a}_{j}}={X}_{D}^{ik}{\left({X}^{-1}\right)}_{k}^{j}\phantom{\rule{2em}{0ex}}$$ | (11.2.20) |

where

$${X}_{i}^{k}:=\frac{\partial {a}_{i}}{\partial {u}_{k}},\phantom{\rule{2em}{0ex}}{X}_{D}^{jk}:=\frac{\partial {a}_{D}^{j}}{\partial {u}_{k}}.$$ | (11.2.21) |

Deﬁning

$${\omega}_{k}=\frac{\partial}{\partial {u}_{k}}\lambda {\left|\right.}_{\text{constant}z\text{}},$$ | (11.2.22) |

we ﬁnd

It can be checked that ${\omega}_{2,3,\dots ,N}$ form a basis of holomorphic non-singular one-forms on $\Sigma $. The matrix ${\tau}^{ij}$ formed this way is known mathematically as the period matrix of $\Sigma $, and is known to satisfy

$${\tau}^{ij}={\tau}^{ji},\phantom{\rule{2em}{0ex}}Im{\tau}^{ij}\phantom{\rule{1em}{0ex}}\text{ispositivede\ufb01nite}.$$ | (11.2.24) |

From the ﬁrst condition, we see that there is locally a function $F\left({a}_{i}\right)$ such that

$${a}_{D}^{i}=\frac{\partial F}{\partial {a}_{i}},\phantom{\rule{2em}{0ex}}{\tau}^{ij}=\frac{{\partial}^{2}F}{\partial {a}_{i}\partial {a}_{j}}.$$ | (11.2.25) |

This justiﬁes that we identify ${a}_{i}$, ${a}_{D}^{i}$ deﬁned this way with the ${a}_{i}$ appearing in the low-energy description of $U{\left(1\right)}^{N-1}$ gauge theory. The inverse gauge coupling matrix is given by $Im{\tau}^{ij}$, whose positive deﬁniteness is guaranteed by the mathematical relation (11.2.24).