Let us construct holomorphic functions $a$ and ${a}_{D}$ satisfying these monodromies explicitly. Note that a holomorphic function is uniquely determined by its singularities. Therefore, if we ﬁnd a candidate with the correct properties around the singularities and at inﬁnity of its domain of deﬁnition, it is necessarily the correct answer itself, assuming that we identiﬁed the singularities correctly. Therefore, it suﬃces to construct a candidate and then check that it satisﬁes the conditions.

We ﬁrst introduce two auxiliary complex variables $x$ and $z$, and then we consider an equation

$$\Sigma :\phantom{\rule{1em}{0ex}}{\Lambda}^{2}z+\frac{{\Lambda}^{2}}{z}={x}^{2}-u.$$ | (4.3.1) |

We consider this equation as deﬁning a complex one-dimensional subspace of a complex two-dimensional
space of $x$ and
$z$.^{7}
As the equation changes as we change $u$,
the shape of this subspace also changes. This complex one-dimensional object is called the
Seiberg-Witten curve.^{8}
A diﬀerential

$$\lambda =x\frac{dz}{z},$$ | (4.3.2) |

called the Seiberg-Witten diﬀerential, plays an important role
later.^{9}

The space parametrized by $z$ is important in itself. We add the point at $z=\infty $ to the complex plane of $z$, or equivalently, we regard $z$ to be the complex coordinate of a sphere. We denote this sphere by $C$, and call it the ultraviolet curve of this system. The variable $x$ as a function of $z$ has four square-root branch points, see Fig. 4.6.

Then the curve $\Sigma $ is a two-sheeted cover of $C$,

$$\Sigma \stackrel{2:1}{\to}C,$$ | (4.3.3) |

see Fig. 4.7. We then draw two one-dimensional cycles $A$, $B$ on the curves as shown in the ﬁgures, and we declare that

$$a=\frac{1}{2\pi i}{\oint}_{A}\lambda ,\phantom{\rule{2em}{0ex}}{a}_{D}=\frac{1}{2\pi i}{\oint}_{B}\lambda .$$ | (4.3.4) |

Let us check that the functions $a\left(u\right)$ and ${a}_{D}\left(u\right)$ thus deﬁned satisfy physically expected properties. First, let us compute $\tau \left(a\right)$:

$$\tau \left(a\right)=\frac{\partial {a}_{D}}{\partial a}=\frac{\partial {a}_{D}\u2215\partial u}{\partial a\u2215\partial u}.$$ | (4.3.5) |

The $u$ derivatives can be computed in the following way:

$$\begin{array}{lll}\hfill \frac{\partial a}{\partial u}& ={\int}_{A}\frac{\partial}{\partial u}\lambda ={\int}_{A}\frac{dz}{xz}\phantom{\rule{2em}{0ex}}& \hfill \text{(4.3.6)}\\ \hfill \frac{\partial {a}_{D}}{\partial u}& ={\int}_{B}\frac{\partial}{\partial u}\lambda ={\int}_{B}\frac{dz}{xz}\phantom{\rule{2em}{0ex}}& \hfill \text{(4.3.7)}\end{array}$$where the $u$ derivative within the integral is taken at ﬁxed $z$. The diﬀerential $\omega =dz\u2215\left(xz\right)$ is ﬁnite on $\Sigma $, even at apparently dangerous points $z=0$, $z=\infty $ or at $x=0$. For example, when $x=0$, $z\sim c+{c}^{\prime}{x}^{2}$ for some constants $c$ and ${c}^{\prime}$. Then $dz\u2215\left(xz\right)\sim \left(2{c}^{\prime}\u2215c\right)dx$.

Given an open path on the curve $\Sigma $ from a ﬁxed point ${P}_{0}$, we ﬁnd a map from the endpoint of the path to another complex plane

$$t={\int}_{{P}_{0}}^{P}\omega .$$ | (4.3.8) |

As shown in Fig. 4.8, the curve $\Sigma $ is mapped to a parallelogram in the complex plane, bounded by the lines which are the images of the cycles $A$ and $B$. Now, any holomorphic mapping such as (4.3.8) preserves the angles. Therefore, the image of the cycle $B$ is always to the left of the image of the cycle $A$. Then

$$\tau \left(a\right)=\frac{\partial {a}_{D}\u2215\partial u}{\partial a\u2215\partial u}=\frac{{\int}_{B}dz\u2215\left(xz\right)}{{\int}_{A}dz\u2215\left(xz\right)}$$ | (4.3.9) |

takes the values to the left of the real axis, and therefore

$$Im\tau \left(a\right)>0,$$ | (4.3.10) |

which guarantees that the coupling squared ${g}^{2}\left(a\right)$ is always positive. This complex number $\tau \left(a\right)$ is called the period or the complex structure of the torus.

Let us check the curve (4.3.1) reproduces the monodromy we determined from physical considerations. Write the curve $\Sigma $ as

$$z+\frac{1}{z}=\frac{{x}^{2}}{{\Lambda}^{2}}-\frac{u}{{\Lambda}^{2}}.$$ | (4.3.11) |

From this we see that when $\left|u\right|\gg {\Lambda}^{2}$, we ﬁnd two branch points ${z}_{\pm}$ of the function $x\left(z\right)$ around

$${z}_{+}\sim -u\u2215{\Lambda}^{2},\phantom{\rule{2em}{0ex}}{z}_{-}\sim -{\Lambda}^{2}\u2215u.$$ | (4.3.12) |

We also have branch points at $z=0$ and $z=\infty $, and we take the branch cuts to run from $z=0$ to $z={z}_{-}$, and from $z={z}_{+}$ to $z=\infty $.

We put the $A$-cycle at $\left|z\right|=1$. Then the integral over it is very easy: $x\simeq \sqrt{u}$ around $\left|z\right|=1$, and therefore

$$a=\frac{1}{2\pi i}\oint x\frac{dz}{z}\simeq \sqrt{u}.$$ | (4.3.13) |

As for the $B$-cycle integral, the dominant contribution comes when the variable $z$ is not very close to the branch points. The variable $x$ can be again approximated by $\sqrt{u}\simeq a$, and therefore

$${a}_{D}=\frac{2}{2\pi i}{\int}_{{z}_{+}}^{{z}_{-}}x\frac{dz}{z}\simeq \frac{2\cdot 2}{2\pi i}{\int}_{u\u2215{\Lambda}^{2}}^{1}a\frac{dz}{z}\simeq -\frac{8a}{2\pi i}log\frac{a}{\Lambda}.$$ | (4.3.14) |

From these two equations we ﬁnd that $a$ and ${a}_{D}$ deﬁned via the curve $\Sigma $ have the correct monodromy around $u\sim \infty $,

$${M}_{\infty}=\left(\begin{array}{cc}\hfill -1\hfill & \hfill 4\hfill \\ \hfill 0\hfill & \hfill -1\hfill \end{array}\right).$$ | (4.3.15) |

By a more careful computation, we can explicitly ﬁnd corrections to (4.3.14), or to its derivative $\tau \left(a\right)$. From the form of the curve (4.3.1), it is clear that the corrections can be expanded in powers of ${\Lambda}^{2}$, but in fact they are given by powers of ${\Lambda}^{4}$. We ﬁnd

$$\tau \left(a\right)=-\frac{8}{2\pi i}log\frac{a}{\Lambda}+\sum _{k=0}^{\infty}{c}_{k}{\left(\frac{\Lambda}{a}\right)}^{4k}$$ | (4.3.16) |

where ${c}_{k}$ are dimensionless rational numbers. We now know the terms hidden as $\cdots \phantom{\rule{0.3em}{0ex}}$ in (4.1.8). This expansion can be understood for example by introducing $\stackrel{\u0303}{z}={\Lambda}^{2}z$. Then the curve is $\stackrel{\u0303}{z}+{\Lambda}^{4}\u2215\stackrel{\u0303}{z}={x}^{2}-u$, and we can compute $a$, ${\xe3}_{D}$ by considering ${\Lambda}^{4}\u2215\stackrel{\u0303}{z}$ as a perturbation to the leading-order form of the curve $\stackrel{\u0303}{z}={x}^{2}-u$. An eﬃcient method to compute ${c}_{k}$ from the contour integral can be found e.g. in [43].

Let us interpret these corrections in the powers of ${\Lambda}^{4}$. From (4.1.9), we know that the term ${\Lambda}^{4k}$ carries the phase ${e}^{ik{\mathit{\theta}}_{UV}}$ where ${\mathit{\theta}}_{UV}$ is the theta angle. It corresponds to a conﬁguration with instanton number $k$, as we learned in (3.2.6). This expansion explicitly demonstrates that the only perturbative correction to the low-energy coupling $\tau \left(a\right)$ is from the one-loop level, and there are non-perturbative corrections from the instantons. An honest path-integral computation in the instanton background should reproduce the coeﬃcients ${c}_{k}$. For the one-instanton contribution ${c}_{1}$ this was done in [44]. It was later extended to all $k$ in [7, 45]. Summarizing, we see that various quantities are given by a combination of a one-loop logarithmic contribution plus instanton corrections. It is now known that they agree to all orders in the instanton expansion, thanks to the developments starting from [7].

Let us next study the monodromy around the strongly-coupled singularities. Taking a look at (4.3.11) again, it is clear that when $z+1\u2215z=\pm 2$ we have a rather special situation. When $u=2{\Lambda}^{2}$, the two branch points collide at ${z}_{\pm}=-1$, and when $u=-2{\Lambda}^{2}$, they collide at ${z}_{\pm}=+1$. These are the singularities $u=\pm {u}_{0}$ introduced in Fig. 4.4.

Let us study the behavior close to $u={u}_{0}=2{\Lambda}^{2}$ as an example. We let $u=2{\Lambda}^{2}+\delta u$. Then the branch points are at

$${z}_{\pm}-1\propto \pm \sqrt{\delta u}.$$ | (4.3.17) |

The close up of the branch points ${z}_{\pm}$ and the cycles $A$, $B$ are shown in Fig. 4.9.

When we slowly change the value of $u$ around $u=2{\Lambda}^{2}$, two branch points $z={z}_{\pm}$ are exchanged. This modiﬁes the cycle $A$ as shown in the ﬁgure, which is equivalent to the original cycle $A$ minus the cycle $B$. The cycle $B$ is clearly unchanged. Therefore we have

$$a\to a-{a}_{D},\phantom{\rule{2em}{0ex}}{a}_{D}\to {a}_{D}$$ | (4.3.18) |

or equivalently, the monodromy is

$${M}_{+}=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right),$$ | (4.3.19) |

reproducing (4.2.9).

Let us study the physics at $u={u}_{0}=2{\Lambda}^{2}$. We perform the $S$ transformation (1.2.19)

$${a}^{\prime}=-{a}_{D},\phantom{\rule{1em}{0ex}}{a}_{D}^{\prime}=a$$ | (4.3.20) |

exchanging the electric and magnetic charges. These are given as functions of $u$ by

$${a}^{\prime}=c\left(u-{u}_{0}\right),\phantom{\rule{1em}{0ex}}{a}_{D}^{\prime}=\frac{{a}^{\prime}}{2\pi i}log{c}^{\prime}\left(u-{u}_{0}\right)$$ | (4.3.21) |

where $c$ and ${c}^{\prime}$ are two constants, from which we ﬁnd

$${\tau}_{D}\left({a}^{\prime}\right)=\frac{\partial {a}_{D}^{\prime}}{\partial {a}^{\prime}}\sim +\frac{log{a}^{\prime}}{2\pi i}.$$ | (4.3.22) |

Note that ${a}^{\prime}$ sets the energy scale of the system. The result shows the same behavior as the running of the coupling of an $\mathcal{\mathcal{N}}=2$ supersymmetric $U\left(1\right)$ gauge theory with one charged hypermultiplet, consisting of $\mathcal{\mathcal{N}}=1$ chiral multiplets $\left(Q,\stackrel{\u0303}{Q}\right)$. The superpotential coupling is then

$$\int {d}^{2}\mathit{\theta}Q{a}^{\prime}\stackrel{\u0303}{Q}.$$ | (4.3.23) |

Writing

$${\tau}_{D}=\frac{4\pi i}{{g}_{D}^{2}}+\frac{{\mathit{\theta}}_{D}}{2\pi},$$ | (4.3.24) |

we ﬁnd

$${g}_{D}\to 0$$ | (4.3.25) |

as we approach $u\to {u}_{0}$. The mass of the quantum of $Q$ is given by the BPS mass formula to be

$$\text{massofquantumof}Q\text{}=\left|{a}^{\prime}\right|=\left|{a}_{D}\right|.$$ | (4.3.26) |

Therefore, we identify the charged chiral multiplet $Q$ as the second quantized version of the monopole in the original theory. The monopoles, which were very heavy in the weakly coupled region, are now very light.

The behavior at $u=-{u}_{0}$ is easily given by applying the discrete R-symmetry (4.2.5). As we map by ${T}^{2}$, we ﬁnd that the very light particles now have electric charge $n=2$ and magnetic charge $m=1$, i.e. they are dyons. From these reasons, the point $u={u}_{0}=2\Lambda $ is often called the monopole point, and the point $u=-{u}_{0}=-2\Lambda $ the dyon point.