Gravity and Strings

508 Extended objects
Fig. 18.1. Open strings cannot carry charge.
Electric–magnetic duality for extended objects. The electric–magnetic dual of a charged
p-brane is, by definition, an object that couples to the electric-magnetic dual of the (p + 1)form
potential A(p+1). Wehave seen several examples, for instance, in Section 16.5.5, in
which we used Poincaréduality to replace the KR 2-form completely by its dual in various
dimensions and also in Sections 16.1.3 and 17.1.1 in which we defined the on-shell duals
of the RR potentials that appear in the N = 2A, B, d = 10 supergravity actions, not being
able to replace the original potentials completely by their duals.
In all (massless) cases, the dual is a ( ˜p + 1)-form Ã( ˜p+1) with ˜p = d − p − 4, whose
field strength is, by definition, the Hodge dual of ˜F ˜p+2 = ⋆ F(p+2). The electric-magnetic
dual of a p-brane is therefore a ˜p-brane with ˜p = p in general, which is electrically charged
with respect to Ã( ˜p+1). Only in even dimensions do objects with p = (d − 4)/2 have duals
of the same dimension, and then we can have p-brane dyons.
Anew feature with respect to point-particles is that there can be self-dual p-branes,
charged with respect to a self-dual potential: point-particles couple only to vectors, which
are dual to vectors in d = 4, but, in d = 4, self- or anti-self-duality is consistent only with
aEuclidean signature. However, self-dual 2-forms in d = 6 and 4-forms in d = 10 can
be consistently defined. They occur in N = (1, 0), d = 6 SUGRA (Section 13.4.1) and in
N = 2B, d = 10 SUEGRA (Chapter 17), and are associated with chiral theories.
Dirac charge quantization. The charge of a p-brane moving in the background A(p+1) field
sourced by a dual ˜p-brane is quantized as in the case of point-particles [718, 881, 882]. 5
Let us consider the quantum propagation of a charged p-brane moving along a closed
path6 so its worldvolume is topologically a (p + 1)-sphere S (p+1) .The interesting term in
the path integral is the WZ term which, using Stokes’ theorem, can be written in the form
(−1) (p+1)
qp dA(p+1) = (−1) (p+1)
qp F(p+2), (18.42)
B (p+2)
B (p+2)
where B (p+2) is one of the many possible (p + 2)-dimensional capping surfaces with
∂B (p+2) = S (p+1) .Toavoid having any ambiguities in the path integral, the difference
must be an integer
between choosing two different capping surfaces B (p+2)
1
and B (p+2)
2
5 The derivation of the Dirac quantization condition we are about to explain is different from those used in
Section 8.7.2 for point-particles, which can also be generalized to charged p-branes.
6 This will be the analog of a point-particle moving along a closed path encircling a Dirac string.