5.6 Branes within branes

The existence of Wess–Zumino couplings of the form
∫ ∫ 𝒞p+1 ∧ ℱ ∧ ℱ , 𝒞p+1 ∧ ℱ , (332 ) Dp+4 Dp+2
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(p + 4)-Dp and D(p + 2)-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 [496Jump To The Next Citation Point, 185Jump To The Next Citation Point]. For general discussions on D-brane bound states, see [447, 424Jump To The Next Citation Point, 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 B-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

D(p + 4) : 1 .p . . .p + 4 x x Dp : 1 .p x x x x x x . (333 )
Motivated by the Wess–Zumino coupling 𝒞 ∧ ℱ ∧ ℱ, one considers the ansatz on the D(p + 4 )-brane effective action
μ μ i i X = σ , μ = 0, ...,p + 4, X = c, i = p + 5,...,9, Va = Va(σb), a,b = p + 1,...,p + 4. (334 )

Let me first discuss when such configurations preserve supersymmetry. Consider type IIA (p = 2k ), even though there is an analogous analysis for type IIB. Γ κ reduces to

--------1-------------1---- μ1...μ2k+5 ( k+1 Γ κ = ∘ ----------------(2k + 5)!𝜀 Γ μ1...μ2k+5Γ â™¯ −( det(ημν) + Fμν) 2k + 5 + F μ1μ2Γ μ3...μ2k+5Γ k♯ ( 2 ) ( ) ) 1- 2k + 5 4 k+1 + 2 4 2 F μ1μ2F μ3μ4Γ μ5...μ2k+5Γ â™¯ , (335 )
where I already used the static gauge and the absence of excited transverse scalars, so that γμ = Γ μ. For the same reason, det(ημν + F μν) = det (δab + Fab), involving a 4 × 4 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

Γ 0...p+4Γ k♯+1𝜖 = 𝜖, (336 ) k+1 Γ 0...pΓ♯ 𝜖 = 𝜖. (337 )
Notice that commutativity of both projectors is guaranteed due to the dimensionality of both constituents, which is what selects the Dp-D(p + 4) nature of the bound state in the first place. Inserting these into the kappa symmetry preserving condition, the latter reduces to
∘ -------------- ( ) det(δab + Fab)𝜖 = 1 + 1-&tidle;FabFab − 1Γ abΓ â™¯Fab 𝜖, (338 ) 4 2
where &tidle;Fab = 1𝜀abcdF 2 cd. Requiring the last term in Eq. (338View Equation) to vanish is equivalent to the self-duality condition
&tidle;ab ab F = F . (339 )
When the latter holds, Eq. (338View Equation) is trivially satisfied. Eq. (339View Equation) is the famous instanton equation in four dimensions35. 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

D (p + 2) : 1 .p p + 1 p + 2 x x x x (340 ) Dp : 1 .p x x x x x x
Motivated by the Wess–Zumino coupling 𝒞 ∧ ℱ, one considers the ansatz on the D(p + 4)-brane effective action
X μ = σ μ, μ = 0, ...,p + 2, Xi = ci, i = p + 3,...,9, V = V (σb). (341 ) a a
Since there is a single non-trivial magnetic component, I will denote it by Fab ≡ F to ease the notation. The DBI determinant reduces to
2 − det(𝒢 μν + ℱ μν) = 1 + F , (342 )
whereas the kappa symmetry preserving condition in type IIA is
√ -----2- ( k k+1 ) 1 + F 𝜖 = Γ 0...p+2Γ♯ + Γ 0...pΓ♯ F 𝜖 (343 )
for p = 2k. This is solved by the supersymmetry projection
( k k+1) cosα Γ 0...p+2Γ â™¯ + sin αΓ 0...pΓ â™¯ 𝜖 = 𝜖, (344 )
for any α, for the magnetic flux satisfying
F = tan α. (345 )

To interpret the solution physically, assume the world space of the D(p + 2 )-brane is of the form ℝp × T2. This will quantise the magnetic flux threading the 2-torus according to

∫ (2π)2kα ′ 2 F = 2πk =⇒ F = --L-L----. (346 ) T 1 2
To derive this expression, I used the fact that the 2-torus has area L1L2 and I rescaled the magnetic field according to F → 2π α′F, since it is in the latter units that it appears in brane effective actions. Since the energy density satisfies ℰ 2 = T2 (1 + F 2) D(p+2), flux quantisation allows us to write the latter as
( )2 ℰ 2 = T2D(p+2) + T 2Dp --k-- , (347 ) L1L2
matching the non-threshold nature of the bound state
∘ --2---------2-- E = E D(p+2) + E kDp, (348 )
where the last term stands for the energy of k Dp-branes.

5.6.3 F-Dp systems

These are non-threshold bound states corresponding to the brane array

Dp : 1 . . .p x x x x F : x x x x p x x x x . (349 )
Following previous considerations, one looks for bosonic configurations with the ansatz
X μ = σμ , Xi = ci, a F0ρ = F0ρ(σ ). (350 )
Given the absence of transverse scalar excitations, γμ = Γ μ and ∘ -------------- √ ------- − det(𝒢 + ℱ ) = 1 − F 2, where F0 ρ ≡ F. The kappa symmetry preserving condition reduces to
√ ------- ( ) 1 − F 2𝜖 = Γ 0...pΓ k♯ − F Γ 0ρΓ â™¯Γ 0...pΓ k♯ 𝜖 + (1 − F Γ 0ρΓ â™¯)Γ 0...pΓ k♯𝜖. (351 )
This is solved by the supersymmetry projection condition
( k ) cosαΓ 0...pΓ♯ + sinα Γ 0ρΓ â™¯ 𝜖 = 𝜖 , (352 )
whenever
F = − sin α . (353 )

To physically interpret the solution, compute its energy density

ℰ2 = ℰ02+ F 2, (354 )
where I already used that ρ F0ρ = F = E. 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 1∕L. Thus, the electric flux along the T-dual circle must also be quantised, leading to the condition
F = -1--n- , (355 ) 2πα′L
where the world volume of the Dp-brane is assumed to be ℝp × S1. In this way, one can rewrite the energy for the F-Dp system as
∘ ----------(--)2- E = E2Dp + T2f n- , (356 ) L
which corresponds to the energy of a non-threshold bound state made of a Dp-brane and n fundamental strings (Tf).
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