### 5.4 BIons

Perhaps one of the most pedagogical examples of brane solitons are BIons. These were first described
in [128, 234] and correspond to on-shell supersymmetric D-brane configurations representing a fundamental
string ending on the D-brane, i.e., the defining property of the D-brane itself. They correspond to the array
of branes
Working in the static gauge describes the vacuum infinite Dp-brane. The static soliton excites a transverse
scalar field and the electric field , while setting the magnetic components of
the gauge field to zero
The gauge invariant character of the scalar ensures its physical observability as a deformation of the flat
world volume geometry described by the global static gauge, whereas the electric field can be understood as
associated to the end of the open string, which is seen as a charged particle from the world volume
perspective. A second way of arguing the necessity for such electric charge is to remember that fundamental
strings are electrically charged under the NS-NS two form. The latter appears in the effective action
through the gauge invariant form . Thus, turning on is equivalent to turning such
charge.

##### Supersymmetry analysis:

Let me analyse the amount of supersymmetry preserved by configurations (289)
in type IIA and type IIB, separately. In both cases, the matrix equals
while the induced gamma matrices are decomposed as
where stands for world space indices. Due to the electric ansatz for the gauge field, the
kappa symmetry matrix has only two contributions. In particular, for type IIA
Summing over world volume time, one obtains
Using the duality relation
one can write the first term on the right-hand side of Eq. (293) as
Using the same duality relation and proceeding in an analogous way, the second term equals
Inserting Eqs. (295) and (296), the kappa symmetry preserving condition can be expressed as
Given the physical interpretation of the sought soliton, one imposes the following two supersymmetry
projection conditions
corresponding to having a type IIA Dp-brane along directions and a fundamental string along
the transverse direction . Since both Clifford valued matrices commute, the dimensionality of the
subspace of solutions is eight, as corresponds to preserving of the bulk supersymmetry.
Plugging these projections into Eq. (297), the kappa symmetry preserving condition reduces to
It is clear that the BPS condition
derived from requiring the coefficient of to vanish, solves Eq. (300). Indeed, the last term in
Eq. (300) vanishes due to antisymmetry, whereas the square root of the determinant equals one, whenever
Eq. (301) holds.
The analysis for type IIB Dp-branes works analogously by appropriately dealing with
the different bulk fermion chiralities, i.e., one should replace by . Thus, the supersymmetry
projection conditions (298) and (299) are replaced by
corresponding to having a type IIB Dp-brane along the directions and a fundamental string
along the transverse direction .
Satisfying the BPS equation (301) does not guarantee the on-shell nature of the configuration. Given
the non-triviality of the gauge field, Gauss’ law must be imposed, where is the conjugate
momentum to the electric field, which reduces to

when Eq. (301) is satisfied. Thus, the transverse scalar must be a harmonic function on the
-dimensional D-brane world space

##### Hamiltonian analysis:

Using the phase space formulation of the D-brane Lagrangian in Eqs. (222) and
(223), I will reproduce the BPS bound (301) and interpret the charges carried by BIons. Working in static
gauge, the world space diffeomorphism constraints are trivially solved for static configurations, i.e.,
, and in the absence of magnetic gauge field excitations, i.e., . The Hamiltonian constraint
can be solved for the energy density [225]
Since , Eq. (306) is equivalent to [225]
There exists an energy bound
being saturated if and only if
This is precisely the relation (301) derived from the solution to the kappa symmetry preserving
condition (214) (the sign is related to the sign of the fundamental string charge). Thus, the total energy
integrated over the D-brane world space satisfies
where is the charge
To interpret this charge as the charge carried by a string, consider the most symmetric solution to
Eq. (305), for Dp-branes with , depending on the radial coordinate in world space , i.e.,
,
where stands for the volume of the unit -sphere. This describes a charge at the origin. Gauss’s
law allows us to express the energy as an integral over a (hyper)sphere of radius surrounding the charge.
Since is constant over this (hyper)sphere, one has
Thus, the energy is infinite since as , but this divergence has its physical origin on the
infinite length of a string of finite and constant tension [128, 234]. See [164] for a discussion of the
D-string case, corresponding to string junctions.