String Theory and M-Theory

6.2 D-branes in type II superstring theories 205
in the absence of charges and currents. Here F is the two-form field strength
describing the electric and magnetic fields. Notice that the above equations
are symmetric under the interchange of F and ⋆F .
More generally, one should include electric and magnetic sources. Electrically
charged particles (or electric monopoles) exist, but magnetic monopoles
have not been observed yet. Most likely, magnetic monopoles exist with
masses much higher than have been probed experimentally. When sources
are included, Maxwell’s equations become
dF = ⋆Jm and d ⋆ F = ⋆Je. (6.47)
In each case J = Jµdx µ is a one-form related to the current and charge
density as
Jµ = (ρ,j), (6.48)
with µ = 0, . . . , 3 in the case of four dimensions. For a point-like electric
charge the charge density is described by a delta function ρ = eδ (3) (r), where
e denotes the electric charge. Similarly, a point-like magnetic source has an
associated magnetic charge, which we denote by g. These charges can be
defined in terms of the field strength
e = ⋆F and g = F, (6.49)
S 2
where the integrations are carried out over a two-sphere surrounding the
charges.
Electric and magnetic charges are not independent. Indeed, as Dirac
pointed out in 1931, the wave function of an electrically charged particle
moving in the field of a magnetic monopole is uniquely defined only if the
electric charge e is related to the magnetic charge g by the Dirac quantization
condition 4
e · g ∈ 2π
The derivation of this result is described in Exercise 6.3.
Generalization to p-branes
S 2
. (6.50)
The preceding considerations can be generalized to p-branes that couple to
(p + 1)-form gauge fields in D dimensions. To determine the possibilities for
stable p-branes, it is worthwhile to consider the types of conserved charges
that they can carry. This entails generalizing the statement that a point
particle (or 0-brane) can carry a charge such that it acts as a source for a
4 For dyons, which carry both electric and magnetic charge, the Dirac quantization rule generalizes
to Witten’s rule: e1g2 − e2g1 = 2πn.