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

worldlineA (6.3.1)

which creates a nonzero electric field Ftr0 where

F = dA (6.3.2)

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

dF = dF = 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

worldsheetB. (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 Gtsr, where r is again the radial direction. The equations of motion are

dG = dG = 0 (6.3.5)

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

(G)μνρ = 𝜖μνραβγGαβγ. (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

dG = 0,G = G. (6.3.7)

Then a worldsheet extending along the directions t and s has both nonzero electric field Gtsr and nonzero magnetic field G𝜃ϕψ 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 x5,6, and consider strings wrapped along each of the directions, as shown in Fig. 6.7. Denote the 6d three-form field-strength by GABC, where A, are indices for six-dimensional spacetime. We can extract four-dimensional two-forms by considering

Fμν := G6μν,FDμν := G5μν. (6.3.8)

The 6d self-duality G = 6G translates to the equality

FD = 4F. (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 x6 has nonzero G6tr and G5𝜃ϕ, and therefore it has nonzero Ftr. Therefore this becomes an electric particle in four dimensions. Similarly, the string wrapped around x5 has nonzero G5tr and G6𝜃ϕ. Therefore it has nonzero F𝜃ϕ, 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 SU(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.