Gravity and Strings

564 The extended objects of string theory
available would be utterly hopeless. Thus, we will be pragmatic, focusing on the simplest
families of solutions and the general rules. More information can be obtained from reviews
such as [417, 588, 858].
The solutions we have studied so far describe p-branes at rest, in their lower energy states
in which none of their worldvolume fields is excited and has a non-trivial configuration.
We have checked this by matching the solutions with p-brane sources in Sections 18.2.3
and 19.2.1. Excited worldvolume fields describe, first of all, deformations of the p-brane
in the target spacetime when they involve the embedding coordinates X µ (ξ) (as happens
in all the supersymmetric cases). These deformations can be seen, in certain cases, as other
branes that end on or intersect the original brane. The converse relation is always true: a
brane that ends on or intersects another brane always corresponds to an excitation of the
worldvolume fields of the latter. The dimension of the intersection, which behaves as a dynamical
solitonic object in the worldvolume of the host brane, determines the nature of the
intersecting brane. That dimension is associated with the rank of the excited worldvolume
differential-form fields: k-brane intersections to (k + 1)-form worldvolume fields. There
are three main examples of this correspondence.
1. By definition, open strings end on Dp-branes and their endpoints are seen as pointparticles
electrically charged w.r.t. the BI vector field (BIons). Worldvolume halfsupersymmetric
solutions of the Dp-brane action in flat spacetime describing a
point-like electric charge were found in [209, 434] (see also [502, 646]. There is
always an excited embedding scalar that corresponds to a spike sticking out of the
Dp-brane. The energy of this solitonic worldvolume solution per unit length of the
spike is precisely equal to the fundamental-string tension. Furthermore, perturbations
along the spike have Dirichlet boundary conditions and one concludes that this
solution represents a fundamental string attached to the Dp-brane.
Since these are supersymmetric worldvolume solutions, it is not surprising that there
are solutions describing several parallel (or antiparallel) spikes in equilibrium.
The BI vector field can be dualized into a (p − 2)-form potential, which is another
BI vector for the D3-brane [450, 451, 469]. The D3 electric–magnetic self-duality is
related to type-IIB S duality and the dual BIons turn out to describe D-strings ending
on a D3-brane. 23 The dual (p − 3)-BIons of other Dp-branes are related to this one
by T duality: a D(p − 2) ending on a Dp with a (p − 3)-dimensional intersection
associated with the (p − 2)-form dual of the BI vector field.
These intersections can be written in the form F1 ⊥ Dp(0) and D(p − 2) ⊥ Dp(p −
3).
2. In [567] a supersymmetric worldvolume solution of the M5 equations of motion
in flat spacetime [15, 80, 119, 570, 757] in which two embedding scalars and
the self-dual 2-form 24 were excited was found. It corresponds to the intersection
23 If the D-string ended on N coincident D3-branes, whose worldvolume field theory, contains a non-Abelian
SU(N) BI vector field (in fact, it is a non-linear generalization of N = 4, d = 4 super-Yang–Mills theory),
the intersection would be seen as an SU(N) magnetic monopole in the worldvolume [321, 469].
24 The bosonic worldvolume fields of the string/M-theory extended objects can be found in Table 19.6.