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].

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 actionLet 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

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 satisfyingTo 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.

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 wheneverTo 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 ().
Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
This work is licensed under a Creative Commons License. E-mail us: |