Now we found that a single type of objects, the membrane of M-theory or equivalently the string of 6d $\mathcal{\mathcal{N}}=\left(2,0\right)$ theory, gives rise to both electrically charged objects such as W-bosons and magnetically charged objects such as monopoles, see Fig. 6.3 and Fig. 6.4. To get a handle of this property, let us ﬁrst recall basic features of charged particles in four dimensions, see Fig. 6.6.

In a ﬁrst-quantized framework, an electric particle sitting at the origin of the space, extending along the time direction $t$, couples to the electromagnetic potential via

$${\int}_{\text{worldline}}A$$ | (6.3.1) |

which creates a nonzero electric ﬁeld ${F}_{tr}\ne 0$ where

$$F=dA$$ | (6.3.2) |

and $r$ is the radial direction. The equations of motion are

$$dF=d\star F=0$$ | (6.3.3) |

outside of the worldline. Note that in four dimensional Lorentzian space, we have ${\star}^{2}=-1$ acting on two-forms. Therefore we cannot impose the condition $\star F=F$.

Let us consider a theory described by a two-form $B$ in six dimensions, to which a string couples via the term

$${\int}_{\text{worldsheet}}B.$$ | (6.3.4) |

Let us say that the string extends along the spatial direction $s$ and the time direction $t$. This conﬁguration creates a nonzero electric ﬁeld ${G}_{tsr}$, where $r$ is again the radial direction. The equations of motion are

$$dG=d\star G=0$$ | (6.3.5) |

outside of the worldsheet. Here $\star $ is the six-dimensional Hodge star operation, given by

$${\left(\star G\right)}_{\mu \nu \rho}={\mathit{\epsilon}}_{\mu \nu \rho \alpha \beta \gamma}{G}^{\alpha \beta \gamma}.$$ | (6.3.6) |

In six dimensions with Lorentizan signature, ${\star}^{2}=1$ acting on three-forms, so we can demand the equations of motion of the form

$$dG=0,\phantom{\rule{1em}{0ex}}G=\star G.$$ | (6.3.7) |

Then a worldsheet extending along the directions $t$ and $s$ has both nonzero electric ﬁeld ${G}_{tsr}$ and nonzero magnetic ﬁeld ${G}_{\mathit{\theta}\varphi \psi}$ at the same time.

Now, let us put this theory on a two-torus with coordinates ${x}_{5,6}$, and consider strings wrapped along each of the directions, as shown in Fig. 6.7. Denote the 6d three-form ﬁeld-strength by ${G}_{ABC}$, where $A,\dots $ are indices for six-dimensional spacetime. We can extract four-dimensional two-forms by considering

$${F}_{\mu \nu}:={G}_{6\mu \nu},\phantom{\rule{2em}{0ex}}{F}_{D\phantom{\rule{1em}{0ex}}\mu \nu}:={G}_{5\mu \nu}.$$ | (6.3.8) |

The 6d self-duality $G={\star}_{6}G$ translates to the equality

$${F}_{D}={\star}_{4}F.$$ | (6.3.9) |

Therefore, the single self-dual two-form ﬁeld in 6d gives rise to a single $U\left(1\right)$ ﬁeld strength.

Now, the string wrapped around ${x}_{6}$ has nonzero ${G}_{6tr}$ and ${G}_{5\mathit{\theta}\varphi}$, and therefore it has nonzero ${F}_{tr}$. Therefore this becomes an electric particle in four dimensions. Similarly, the string wrapped around ${x}_{5}$ has nonzero ${G}_{5tr}$ and ${G}_{6\mathit{\theta}\varphi}$. Therefore it has nonzero ${F}_{\mathit{\theta}\varphi}$, meaning that it is a magnetic particle in four dimensions.

In the concrete situation of the pure $SU\left(2\right)$ theory, W-bosons and monopoles arise from the membranes as shown in Fig. 6.8. We see that the boundaries of the membrane for a W-boson and the boundary of the membrane for a monopole intersect at two points. In general, the Dirac pairing as particles in the four-dimensional spacetime can be found in this way by counting the number of intersections, once signs given by the orientation are included.