Gravity and Strings

18.1 Generalities 507
The conservation law for the current d ⋆ j = 0 suggests the following definition for the
charge qp associated with A(p+1):
qp ≡
B (d−p−1)
⋆
j, (18.38)
where, by definition, B (d−p−1) is a capping surface whose boundary is (topologically)
∂B (d−p−1) = S (d−p−2) . More precisely, this is the charge contained in the capping surface.
The total charge would be calculated by integrating over a capping surface whose boundary
is the (d − p − 2)-sphere at infinity S (d−p−2)
∞ .Asusual this definition is invariant under
smooth deformations of the capping surface in source-free ( j = 0) regions.
Using the generalization of the Gauss law Eq. (18.37) and Stokes’ theorem, we obtain
the usual definition
qp = (−1) d+p
S (d−p−2)
⋆F(p+2), (18.39)
which is also invariant under smooth deformations of the (d − p − 2)-dimensional surface
in source-free regions.
Given a p-brane sweeping out a (p + 1)-dimensional worldvolume W (p+1) ,wecan immediately
construct a conserved current for it that generalizes Eq. (8.49) for the current of
a point-like charged object:
j µ1···µp+1(x) = qp
= qp
W (p+1)
W (p+1)
dX µ1 ∧ ···∧dX µp+1 δ(d) (x − X (ξ))
√ |g(X)|
d p+1 ξ ∂(X µ1 ···X µp+1)
∂(ξ µ1 ···ξ µp+1)
δ (d) (x − X (ξ))
√ .
|g(X)|
(18.40)
The charge associated with this current is qp,except when the p-brane worldvolume has
boundaries, since, in that case, we can continuously deform the surface of integration and
contract it to a point without meeting the p-brane source, obtaining zero charge. This is
easy to visualize for a string of finite length in a four-dimensional target space: the string
charge with respect to a 2-form potential is calculated through a closed line integral around
the string that can be slid off the string and contracted to a point as shown in Figure 18.1. In
asense this explains why the KR 2-form does not appear in the open-string spectrum. It is
clear now that qp = 0 only when the p-brane has compact topology or extends to infinity.
A brane with boundaries can also carry charge if the boundaries are attached to another
object just as open strings are attached to D-branes. This case, which is more complicated
(and interesting), will be studied in Section 19.6 since it depends strongly on the theory we
are considering.
On substituting the above p-brane current into the interaction term Eq. (18.34), we obtain
the standard form of the WZ term for p-branes which appears in the p-brane worldvolume
action,
A(p+1). (18.41)
(−1) (p+1) qp
W (p+1)