One might say strings with variable tension is slightly weird. One way to realize this variation of the tension in a natural manner is to consider that the extra-dimensional space $C$ which have two dimensions is further embedded in a four-dimensional ambient space $X$, and there are two sheets of $\Sigma $ covering $C$ separated in the additional directions of $X$. We then furthermore suppose that there is a membrane extending along two spatial directions plus one temporal direction, which can have ends on the sheets of $\Sigma $. The situation is depicted in Fig. 6.2. Let $z$ be the coordinate of $C$, and $X$ has complex coordinates $\left(z,x\right)$. Then two sheets of $\Sigma $ deﬁne two functions ${x}_{1}\left(z\right)$ and ${x}_{2}\left(z\right)$. Then, a membrane with constant tension $\left|dx\right|\wedge |dlogz|$ , suspended between two sheets, can be regarded as a string with variable string whose tension at a given value of $z$ is given by

$$\text{(tensionat}z\text{)}\ge \left|{\int}_{{x}_{2}\left(z\right)}^{{x}_{1}\left(z\right)}dx\wedge dlogz\right|=\left|{x}_{1}\frac{dz}{z}-{x}_{2}\frac{dz}{z}\right|.$$ | (6.2.1) |

Denoting ${\lambda}_{i}\left(z\right)={x}_{i}dz\u2215z$, we ﬁnd that

In M-theory, there are indeed higher-dimensional objects with these properties. We consider an eleven dimensional spacetime of the form

$${\mathbb{R}}^{3,1}\times X\times {\mathbb{R}}^{3}.$$ | (6.2.3) |

M-theory has six-dimensional objects called M5-branes. We put one M5-brane on

$${\mathbb{R}}^{3,1}\times \Sigma \times \left\{0\right\}$$ | (6.2.4) |

where $\Sigma \subset X$ is the curve, and $0$ is the origin of the additional ${\mathbb{R}}^{3}$. This gives a four-dimensional theory. M-theory also has three-dimensional objects called M2-branes, which can have ends on M5-branes. We can take one M2-brane on

$${\mathbb{R}}^{0,1}\times \text{disc}\times \left\{0\right\}$$ | (6.2.5) |

where ${\mathbb{R}}^{0,1}\subset {\mathbb{R}}^{3,1}$ is the worldline of a particle in the four-dimensional spacetime, and the $\text{disc}\subset X$ has its boundary on $\Sigma $ as depicted in Fig. 6.2. For more details on this point, the reader should start from the original paper [52].

It is also useful to regard the intermediate situation when we regard the system as a six-dimensional one on ${\mathbb{R}}^{3,1}\times C$. This six-dimensional theory is known as the 6d $\mathcal{\mathcal{N}}=\left(2,0\right)$ theory of type $SU\left(2\right)$.

Let us apply this higher-dimensional idea to the curve (4.3.1) of the pure $SU\left(2\right)$ theory concretely. For easy reference we reproduce the curve here:

$$\Sigma :\phantom{\rule{2em}{0ex}}{\Lambda}^{2}z+\frac{{\Lambda}^{2}}{z}={x}^{2}-u.$$ | (6.2.6) |

We consider $\Sigma $ to be embedded in a four-dimensional space $X$. Given a point $z$ on $C$, we ﬁnd two $x$ coordinates by solving the quadratic equation above, as depicted on the left hand side of Fig. 6.3. Let the solutions be $\pm x\left(z\right)$. As the point $z$ moves on $C$, they form two sheets of the curve $\Sigma $, see the right hand side of Fig. 6.3. The coordinate $x$ always appears as a way to describe the one-form on $C$ giving the tension, so it is convenient to multiply them always by $dz\u2215z$, and say that two sheets have coordinates $\pm \lambda =x\left(z\right)dz\u2215z$. We use this convention from now on.

We can now consider a ring-shaped membrane suspended between the two sheets over the $A$ cycle, see Fig. 6.3. Note that the tension as a string on $C$ is $2\lambda $, and the mass is given by

$$M\ge \left|2{\int}_{A}\lambda \right|=\left|2a\right|.$$ | (6.2.7) |

We can minimize the tension by solving (6.1.6), which give rise to a conﬁguration with the mass

$$M=\left|2a\right|.$$ | (6.2.8) |

Note that this has the correct mass to be a W-boson, which has electric charge $n=2$ in our normalization, which is for the triplets of $SU\left(2\right)$. It is also to be noted that there is no way to have a membrane whose mass is given by

$${M}^{\prime}=\left|a\right|,$$ | (6.2.9) |

because there is simply no way to suspend the membrane to have just one ends over the $A$-cycle. Therefore, this higher-dimensional reasoning has more explanatory power than just regarding the curve $\Sigma $ as an auxiliary object producing the holomorphic functions $a\left(u\right)$ and ${a}_{D}\left(u\right)$ with the correct monodromy properties. This procedure knows that there is no dynamical particle with electric charge $n=1$ in this system.

Next, we can consider a disc-shaped membrane suspended between the sheets of $\Sigma $ so that they have endpoints over the branch points ${z}_{+}$, ${z}_{-}$ of $C$, see Fig. 6.4. By a similar reasoning as above, the mass of this membrane is

$$M=2\left|{\int}_{{z}_{-}}^{{z}_{+}}\lambda \right|=\left|{\int}_{B}\lambda \right|=\left|{a}_{D}\right|.$$ | (6.2.10) |

This is a correct mass formula for the monopole, whose magnetic charge is $m=1$. In terms of a variable-tension string on $C$, it is to be noted that this corresponds to an open string, ending at the points where the tension $2\lambda $ becomes zero.

We can also connect the two branch points ${z}_{\pm}$ by going around the phase direction of $z$, as shown in Fig. 6.5. As shown there, the membrane is topologically the sum of the two conﬁgurations considered so far, and we ﬁnd that the mass of this conﬁguration is

$$M=|2a+{a}_{D}|.$$ | (6.2.11) |

This is the correct mass formula for the dyon, with the electric charge $n$ and the magnetic $m$ given by $\left(n,m\right)=\left(2,1\right)$. By going around $n$ times when we connect the branch points, we see that there are dyons with mass $|2na+{a}_{D}|$ for integral $n$. We also see there is no way to connect the branch points to have dyons with mass $|\left(2n+1\right)a+{a}_{D}|$, which is compatible with the ﬁeld theory analysis in Sec. 1.3.