Recall the BPS mass formula of the pure theory of a particle with electric charge $n$ and magnetic charge $m$,

$$M\ge |na+m{a}_{D}|=\left|{\int}_{L}\lambda \right|$$ | (6.1.1) |

where $L$ is a cycle on the curve which goes around $n$ times along the $A$ direction and $m$ times along the $B$ direction. The basic idea we employ is to take this equation seriously: we regard the four-dimensional particle as arising from a string wrapped on the cycle $L$. Then $\lambda $ is something like the tension of the string. In this section we introduce a factor of $2\pi i$ in the deﬁnition of $\lambda $, to lighten the equations.

To make this idea more concrete, suppose a six-dimensional theory which has strings as its
excitation^{10} ,
and assume this theory is on a two-dimensional space
$C$ times the four-dimensional
Minkowski space ${\mathbb{R}}^{1,3}$.
Further assume that the tension of the string depends on these extra-dimensional
directions. Namely, let us assume that there is a locally-holomorphic one-form
$\lambda $
such that the tension of an inﬁnitesimal segment of a string, parameterized by
$s$, is
given by

$$\left|\lambda \right|:=\left|\frac{\lambda}{ds}\right|ds,$$ | (6.1.2) |

see the left hand side of Fig. 6.1. There, the two-dimensional space is taken to be a torus for deﬁniteness.

A string looks like a particle from the point of view of the uncompactiﬁed four dimensions, and its mass is given by the integral of its variable tension:

$$M={\int}_{L}\left|\lambda \right|$$ | (6.1.3) |

The right hand side can be bounded below using an integral version of the triangle inequality:

$${\int}_{L}\left|\lambda \right|\ge \left|{\int}_{L}\lambda \right|.$$ | (6.1.4) |

The inequality can be visualized by considering the curve in the complex plane deﬁned by

$$f\left(s\right)={\int}_{{P}_{0}}^{s}\lambda $$ | (6.1.5) |

parameterized by $s$, where ${P}_{0}$ is a ﬁxed point on the cycle $L$. Then the left hand side of (6.1.4) is the length of the parameterized curve $f\left(s\right)$, while the right hand side is the distance between the end-points of $f\left(s\right)$, see the right hand side of Fig. 6.1. Then clearly the former is longer than the latter, and the equality is attained only when the line $f\left(s\right)$ itself is a straight line. Or equivalently

$$Arg\phantom{\rule{0.3em}{0ex}}\frac{\lambda}{ds}=\text{constant}.$$ | (6.1.6) |

When the cycle $L$ is topologically trivial, the image of the function $f\left(s\right)$ is itself a loop, and the right hand side of (6.1.4) is zero. When the cycle $L$ is nontrivial, the image of the function $f\left(s\right)$ can be an open segment. As $\lambda $ is holomorphic, the diﬀerence between the two ends of the segment only depends on the topology of the cycle $L$. Say $L$ wraps the $A$-cycle $n$ times and the $B$-cycle $m$ times. Combining (6.1.3) and (6.1.4), we ﬁnd

$$M\ge |na+m{a}_{D}|$$ | (6.1.7) |

where $a$, ${a}_{D}$ are deﬁned by the relations

$$a={\int}_{A}\lambda ,\phantom{\rule{2em}{0ex}}{a}_{D}={\int}_{B}\lambda .$$ | (6.1.8) |

This reproduces the BPS mass formula (6.1.1). We learned furthermore that the inequality is saturated only when (6.1.6) is satisﬁed. Therefore we regard (6.1.6) as the BPS equation for the string excitation.