Self-duality of the 6d theory

6.3 Self-duality of the 6d theory

Now we found that a single type of objects, the membrane of M-theory or equivalently the string of 6d $𝒩 = (2, 0)$ 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.

Figure 6.6: Charged things in 4d and 6d.

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_{worldline} A

(6.3.1)

which creates a nonzero electric ﬁeld $F_{t r} \neq 0$ where

F = d A

(6.3.2)

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

d F = d ⋆ F = 0

(6.3.3)

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

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

\int_{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 $G_{t s r}$ , where $r$ is again the radial direction. The equations of motion are

d G = d ⋆ G = 0

(6.3.5)

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

{(⋆ G)}_{μ ν ρ} = 𝜖_{μ ν ρ α β γ} G^{α β γ} .

(6.3.6)

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

d G = 0, G = ⋆ G .

(6.3.7)

Then a worldsheet extending along the directions $t$ and $s$ has both nonzero electric ﬁeld $G_{t s r}$ and nonzero magnetic ﬁeld $G_{𝜃 ϕ ψ}$ at the same time.

Figure 6.7: Electric and magnetic particles from a single type of objects in 6d

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 $G_{A B C}$ , where $A, \dots$ are indices for six-dimensional spacetime. We can extract four-dimensional two-forms by considering

F_{μ ν} : = G_{6 μ ν}, F_{D μ ν} : = G_{5 μ ν} .

(6.3.8)

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

F_{D} = ⋆_{4} F .

(6.3.9)

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

Now, the string wrapped around $x_{6}$ has nonzero $G_{6 t r}$ and $G_{5 𝜃 ϕ}$ , and therefore it has nonzero $F_{t r}$ . Therefore this becomes an electric particle in four dimensions. Similarly, the string wrapped around $x_{5}$ has nonzero $G_{5 t r}$ and $G_{6 𝜃 ϕ}$ . Therefore it has nonzero $F_{𝜃 ϕ}$ , meaning that it is a magnetic particle in four dimensions.

Figure 6.8: The boundaries of the membranes for a W-boson and a monopole intersect at two points.

In the concrete situation of the pure $S U (2)$ 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.

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