We have so far considered the situations where we put the six-dimensional theory on a two-dimensional space, with coordinates ${x}_{5}$ and ${x}_{6}$, say. We can take a limit where the ${x}_{5}$ direction is much larger than the ${x}_{6}$ direction. Then we can ﬁrst compactify along the ${x}_{6}$ direction and consider an intermediate ﬁve-dimensional theory, see Fig. 6.9. This is believed to give the maximally supersymmetric 5d Yang-Mills theory with gauge group $SU\left(2\right)$.

A string wrapped around the ${x}_{6}$ direction gives rise to a massive electric particle, and a string not wrapped around the ${x}_{6}$ direction becomes a massive magnetic string. This agrees with a basic feature of the 5d Yang-Mills theory where $SU\left(2\right)$ is broken to $U\left(1\right)$: First, we have massive W-bosons which are electric. Second, the standard monopole solutions of 4d gauge theory can be regarded as a solution in 5d gauge theory, by declaring that there is no dependence of the ﬁelds on the additional ﬁfth direction. Then the solutions should be now regarded as representing a magnetic-monopole-string.

By imposing periodic boundary condition in the ${x}^{5}$ direction, we have the situation of Fig. 6.10. We are compactifying the maximally supersymmetric Yang-Mills in ﬁve dimensions on ${S}^{1}$. We therefore should obtain 4d $\mathcal{\mathcal{N}}=4$ super Yang-Mills theory.

The ultraviolet curve $C$ itself is now a torus ${T}^{2}$. Let the complex structure of this ${T}^{2}$ be $\tau $. The Seiberg-Witten curve $\Sigma $ consists of two parallel copies of this torus embedded in $X$, separated by $2\lambda $ in the $x$ direction, where $\lambda $ is now a constant.

We can consider a cycle ${L}_{n,m}$ in $C$, wrapping $n$ times in the ${x}_{6}$ direction and $m$ times in the ${x}_{5}$ directions. Then we can consider a ring-shaped membrane over this cycle, which gives rise to particles of masses

$${M}_{n,m}=2|na+m{a}_{D}|$$ | (6.4.1) |

where

$$a={\int}_{A}\lambda ,\phantom{\rule{1em}{0ex}}{a}_{D}={\int}_{B}\lambda =\tau a.$$ | (6.4.2) |

The particles with $\left(n,m\right)=\left(1,0\right)$ are W-bosons, and the particles with $\left(n,m\right)=\left(0,1\right)$ are monopoles. The peculiar feature of this theory is that the monopoles and the W-bosons both come from ring-shaped membranes. In fact, from the 6d point of view, the distinction of the two directions of the torus is completely arbitrary. Then this theory with a given value of $\tau ={\tau}_{0}$, and the theory with another value of $\tau =-1\u2215{\tau}_{0}$ are the same after the exchange of the W-bosons and the monopoles.

Indeed they match the property of the $\mathcal{\mathcal{N}}=4$ supersymmetric $SU\left(2\right)$ Yang-Mills. This theory is conformal and has an exactly marginal coupling $\tau $. In the semi-classical region, the ratio of the mass of the monopole to that of the W-boson is $\left|\tau \right|$. The $\mathcal{\mathcal{N}}=4$ supersymmetric $SU\left(2\right)$ Yang-Mills has four Weyl fermions in the adjoint representation. The semiclassical quantization of the monopole solution in this situation, as was recalled brieﬂy in Sec. 1.3, makes the monopole states into a massive $\mathcal{\mathcal{N}}=4$ vector multiplet. This makes it possible to exchange it with the W-boson, which is also in a massive $\mathcal{\mathcal{N}}=4$ vector multiplet. In general we expect that there is a massive $\mathcal{\mathcal{N}}=4$ vector multiplet with mass $|na+m{a}_{D}|$, for any coprime pair of integers $\left(n,m\right)$. This should arise from a semi-classical quantization of charge-$m$ monopole background. This is the celebrated conjecture of Sen [53].

The curve of the pure $\mathcal{\mathcal{N}}=2$ $SU\left(2\right)$ theory

$$\frac{{\Lambda}^{2}}{z}+{\Lambda}^{2}z={x}^{2}-u$$ | (6.4.3) |

and the curve of the $\mathcal{\mathcal{N}}=2$ $SU\left(2\right)$ theory with one ﬂavor

$$\frac{2\Lambda \left(x-\mu \right)}{z}+{\Lambda}^{2}z={x}^{2}-u$$ | (6.4.4) |

can be given a similar interpretation. The point is to take ${x}_{5}=log\left|z\right|$ and ${x}_{6}=Arg\phantom{\rule{0.3em}{0ex}}z$, and compactify along the ${x}_{6}$ direction ﬁrst, see Fig. 6.11.

Let us ﬁrst consider the pure theory. The term on the left hand side, ${\Lambda}^{2}\u2215z$, should be regarded as a boundary condition ‘terminating’ the ﬁfth direction ${x}_{5}$, although ${x}_{5}=log\left|z\right|$ formally extends to $-\infty $. The bulk of the ﬁve dimensional theory is maximally supersymmetric. The resulting four-dimensional theory is $\mathcal{\mathcal{N}}=2$, and therefore the boundary breaks half of the supersymmetry, without doing much other than that. A boundary condition which preserves half of the original supersymmetry is called a half-BPS boundary condition. Then we see that the term ${\Lambda}^{2}\u2215z$ represents a half-BPS boundary condition of the 5d theory.

The term ${\Lambda}^{2}z$ is obtained by the ﬂip ${x}_{5}\leftrightarrow -{x}_{5}$, and therefore should represent the same boundary condition. In the end, we see that the system is a compactiﬁcation of the maximally supersymmetric $SU\left(2\right)$ Yang-Mills on a segment, terminated by two boundary conditions breaking half of the supersymmetry, realizing 4d pure $SU\left(2\right)$ Yang-Mills.

Next, let us consider the one-ﬂavor theory. The term ${\Lambda}^{2}z$ is the same as the pure case, so it should give the same half-BPS boundary condition. The boundary condition at $z\sim 0$ is diﬀerent: now we have a term of the form $\Lambda \left(x-\mu \right)\u2215z$. This should mean that one hypermultiplet with the mass $\mu $ in the doublet of $SU\left(2\right)$ lives on this boundary, coupling to the bulk ﬁve-dimensional gauge multiplets.

From this interpretation, it is easy to get the 6d realization of $SU\left(2\right)$ theory with ${N}_{f}=2,3,4$ ﬂavors, namely the theory with ${N}_{f}=2,3,4$ hypermultiplets in the doublet representation. In terms of $\mathcal{\mathcal{N}}=1$ chiral multiplets, we have $\left({Q}_{i}^{a},{\stackrel{\u0303}{Q}}_{a}^{i}\right)$ for $a=1,2$ and $i=1,\dots ,{N}_{f}$, with the superpotential

$$\sum _{i}\left({Q}_{i}\Phi {\stackrel{\u0303}{Q}}^{i}+{\mu}_{i}{Q}_{i}{\stackrel{\u0303}{Q}}^{i}\right)$$ | (6.4.5) |

where ${\mu}_{i}$ are mass terms.

Let us start with the ${N}_{f}=2$ theory. We know how to introduce one hypermultiplet in the doublet at the boundary on the side $z=0$. To do the same on the side $z=\infty $, we just a change of variables $z\leftrightarrow 1\u2215z$. We end up with the setup shown on the left-hand side of Fig. 6.12, with the curve given by

$$\frac{2\Lambda \left(x-{\mu}_{1}\right)}{z}+2\Lambda \left(x-{\mu}_{2}\right)z={x}^{2}-u$$ | (6.4.6) |

with the Seiberg-Witten diﬀerential $\lambda =xdz\u2215z$.

The same curve can be rewritten using another variable ${z}^{\prime}=\left(x-{\mu}_{2}\right)z\u2215\left(2\Lambda \right)$:

$$\frac{\left(x-{\mu}_{1}\right)\left(x-{\mu}_{2}\right)}{{z}^{\prime}}+4{\Lambda}^{2}{z}^{\prime}={x}^{2}-u.$$ | (6.4.7) |

But now we can consider ${x}_{5}^{\prime}=log\left|{z}^{\prime}\right|$, ${x}_{6}^{\prime}=Arg\phantom{\rule{0.3em}{0ex}}{z}^{\prime}$ to reduce ﬁrst to a theory on ${C}^{\prime}$ parameterized by ${z}^{\prime}$, and then to a ﬁve-dimensional theory on a segment parameterized by $\left|{z}^{\prime}\right|$. In this interpretation, the boundary condition on the ${z}^{\prime}=\infty $ side is the same one in the pure $SU\left(2\right)$ case. Therefore, the boundary condition on the ${z}^{\prime}=0$ side given by the term $\left(x-{\mu}_{1}\right)\left(x-{\mu}_{2}\right)\u2215z$ should be the half-BPS condition such that two hypermultiplets in the doublet of $SU\left(2\right)$ live on the boundary.

The description of the system is not complete until we give the one-form $\lambda $ describing the variable tension. In (6.4.6) it is $2\pi i\lambda =xdz\u2215z$ and in (6.4.7) it is $2\pi i{\lambda}^{\prime}=xd{z}^{\prime}\u2215{z}^{\prime}$. Both are obtained by integrating $dx\wedge dlogz=dx\wedge dlog{z}^{\prime}$, see (6.2.1). The two diﬀerentials are not quite equal, however:

$${\lambda}^{\prime}-\lambda =\frac{1}{2\pi i}xdlog\frac{{z}^{\prime}}{z}=\frac{1}{2\pi i}xdlog\left(x-{\mu}_{2}\right).$$ | (6.4.8) |

The diﬀerence is independent of $u$, and its non-zero residue is at ${\mu}_{2}$ at $x={\mu}_{2}$. This means that, given a cycle $L$ on the Seiberg-Witten curve $\Sigma $, we have

$${\oint}_{L}{\lambda}^{\prime}-{\oint}_{L}\lambda =k{\mu}_{2}$$ | (6.4.9) |

where $k$ is an integer. Recall that the BPS mass formula is governed by the expansion

$${\oint}_{L}\lambda =na+m{a}_{D}+{f}_{1}{\mu}_{1}+{f}_{2}{\mu}_{2}$$ | (6.4.10) |

where ${f}_{1,2}$ are ﬂavor charges, see (2.3.9). Therefore, the choice between the two Seiberg-Witten diﬀerentials $\lambda $ and ${\lambda}^{\prime}$ aﬀects the mapping of the ﬂavor charge ${f}_{2}$ and the cycle $L$, but not much else. In general, a change in the Seiberg-Witten diﬀerential by a form which is independent of $u$ and whose residues are integral linear combinations of the hypermultiplet masses are safe. We will encounter them repeatedly later.

Now that we have a boundary condition representing the existence of two doublet hypermultiplets, it is easy to guess the curve of the ${N}_{f}=3$ theory and ${N}_{f}=4$ theory. We just have to combine various boundary conditions which we already found, as in Fig. 6.13. For the ${N}_{f}=3$ theory we ﬁnd

$$\frac{\left(x-{\mu}_{1}\right)\left(x-{\mu}_{2}\right)}{z}+2\Lambda \left(x-{\mu}_{3}\right)z={x}^{2}-u,$$ | (6.4.11) |

and for the ${N}_{f}=4$ theory we ﬁnd

$$f\cdot \frac{\left(x-{\mu}_{1}\right)\left(x-{\mu}_{2}\right)}{z}+{f}^{\prime}\cdot \left(x-{\mu}_{3}\right)\left(x-{\mu}_{4}\right)z={x}^{2}-u$$ | (6.4.12) |

where we put complex numbers $f$ and ${f}^{\prime}$. One of them can be eliminated by a rescaling of $z$.

Our next task is to check that the curves thus obtained via the 6d construction have the correct properties to describe the respective four-dimensional theories. Before proceeding, we need to learn more about the Higgs branch of $\mathcal{\mathcal{N}}=2$ theories in general.