5.6 Branes within branes
The existence of Wess–Zumino couplings of the form
suggests that on-shell non-trivial magnetic flux configurations can source the electric components of the
corresponding RR potentials. Thus, one may speculate with the existence of D
-Dp and
D
-Dp bound states realised as on-shell solutions in the higher dimensional D-brane effective action.
In this section, I will review the conditions the magnetic fluxes must satisfy to describe such
supersymmetric bound states.
The analysis below should be viewed as a further application of the techniques described previously, and
not as a proper derivation for the existence of such bound states in string theory. The latter can be a rather
subtle quantum mechanical question, which typically involves non-abelian phenomena [496
, 185
]. For
general discussions on D-brane bound states, see [447, 424
, 425], on marginal D0-D0 bound states [445],
on D0-D4 bound states [446, 486] while for D0-D6, see [470]. D0-D6 bound states in the presence of
-fields, which can be supersymmetric [391], were considered in [501]. There exist more general analysis
for the existence of supersymmetric D-branes with non-trivial gauge fields in backgrounds with non-trivial
NS-NS 2-forms in [372].
5.6.1 Dp-D(p + 4) systems
These are bound states at threshold corresponding to the brane array
Motivated by the Wess–Zumino coupling
, one considers the ansatz on the D
-brane
effective action
Let me first discuss when such configurations preserve supersymmetry. Consider type IIA
,
even though there is an analogous analysis for type IIB.
reduces to
where I already used the static gauge and the absence of excited transverse scalars, so that
. For
the same reason,
, involving a
determinant.
Given our experience with previous systems, it is convenient to impose the supersymmetry projection
conditions on the constant Killing spinors that are appropriate for the system at hand. These are
Notice that commutativity of both projectors is guaranteed due to the dimensionality of both constituents,
which is what selects the Dp-D
nature of the bound state in the first place. Inserting these into the
kappa symmetry preserving condition, the latter reduces to
where
. Requiring the last term in Eq. (338) to vanish is equivalent to the self-duality
condition
When the latter holds, Eq. (338) is trivially satisfied. Eq. (339) is the famous instanton equation in four
dimensions.
The Hamiltonian analysis done in [225] again confirms its BPS nature.
5.6.2 Dp-D(p + 2) systems
These are non-threshold bound states corresponding to the brane array
Motivated by the Wess–Zumino coupling
, one considers the ansatz on the D
-brane
effective action
Since there is a single non-trivial magnetic component, I will denote it by
to ease the notation.
The DBI determinant reduces to
whereas the kappa symmetry preserving condition in type IIA is
for
. This is solved by the supersymmetry projection
for any
, for the magnetic flux satisfying
To interpret the solution physically, assume the world space of the D
-brane is of
the form
. This will quantise the magnetic flux threading the 2-torus according to
To derive this expression, I used the fact that the 2-torus has area
and I rescaled the magnetic field
according to
, since it is in the latter units that it appears in brane effective actions. Since the
energy density satisfies
, flux quantisation allows us to write the latter as
matching the non-threshold nature of the bound state
where the last term stands for the energy of
Dp-branes.
5.6.3 F-Dp systems
These are non-threshold bound states corresponding to the brane array
Following previous considerations, one looks for bosonic configurations with the ansatz
Given the absence of transverse scalar excitations,
and
, where
. The kappa symmetry preserving condition reduces to
This is solved by the supersymmetry projection condition
whenever
To physically interpret the solution, compute its energy density
where I already used that
. These configurations are T-dual to a system of D0-branes
moving on a compact space. In this T-dual picture, it is clear that the momentum along the compact
direction is quantised in units of
. Thus, the electric flux along the T-dual circle must also be
quantised, leading to the condition
where the world volume of the Dp-brane is assumed to be
. In this way, one can rewrite the energy
for the F-Dp system as
which corresponds to the energy of a non-threshold bound state made of a Dp-brane and
fundamental
strings (
).