5.7 Supertubes

All reviewed solitonic configurations carry charge under the p + 1-dimensional gauge potential they minimally couple to. In this section, I want to consider an example where this is not the case. This phenomena may occur when a collection of lower-dimensional branes finds it energetically favourable to expand into higher-dimensional ones. The stability of these is due to either an external force, typically provided by non-trivial fluxes in the background, or presence of angular momentum preventing the brane from collapse. A IIA superstring blown-up to a tubular D2-brane [200], a collection of D0-branes turning into a fuzzy 2-sphere [395Jump To The Next Citation Point] or wrapping D-branes with quantised non-trivial world volume gauge fields in AdSm × Sn [419] are examples of the first kind, whereas giant gravitons [386Jump To The Next Citation Point], to be reviewed in Section 5.9, are examples of the second.

Supertubes are tubular D2-branes of arbitrary cross-section in a Minkowski vacuum spacetime supported against collapse by the angular momentum generated by a non-trivial Poynting vector on the D2-brane world volume due to non-trivial electric and magnetic Born–Infeld (BI) fields. They were discovered in [381] and its arbitrary cross-section reported in [380Jump To The Next Citation Point], generalising some particular non-circular cross-sections discussed in [30, 32]. Their stability is definitely not due to an external force, since these states exist in Minkowski spacetime. Furthermore, supertubes can be supersymmetric, preserving 1/4 of the vacuum supersymmetry. At first, the presence of non-trivial angular momentum may appear to be in conflict with supersymmetry, since the latter requires a time-independent energy density. This point, and its connection with the expansion of lower-dimensional branes, will become clearer once I have reviewed the construction of these configurations.

Let me briefly review the arbitrary cross-section supertube from [380Jump To The Next Citation Point]. Consider a D2-brane with world volume coordinates σμ = {t,z,σ } in the type IIA Minkowski vacuum

2 2 2 ⃗ ⃗ ds10 = − dT + dZ + dY ⋅ dY , (357 )
where ⃗Y = {Y i} are Cartesian coordinates on ℝ8. We are interested in describing a tubular D2-brane of arbitrary cross-section extending along the Z direction. To do so, consider the set of bosonic configurations
⃗ T = t, Z = z, Y = ⃗y(σ), F = E dt ∧ dz + B (σ)dz ∧ dσ. (358 )
The static gauge guarantees the tubular nature of the configuration, whereas the arbitrary embedding functions ⃗Y = ⃗y(σ) describe its cross-section. Notice the Poynting vector will not vanish, due to the choice of electric and magnetic components, i.e., the world volume electromagnetic field will indeed carry angular momentum.

To study the preservation of supersymmetry, one solves Eq. (214View Equation). Given the ansatz (358View Equation) and the flat background (357View Equation), this condition reduces to [380Jump To The Next Citation Point]

( ∘-------------------) y′iΓ iΓ ♯ (Γ TZΓ ♯ + E ) 𝜖 + B Γ T Γ ♯ − (1 − E2 )|⃗y ′|2 + B2 𝜖 = 0, (359 )
where the prime denotes differentiation with respect to σ. For generic curves, that is, without imposing extra constraints on the embedding functions ⃗Y = ⃗y(σ), supersymmetry requires both to set |E | = 1 and to impose the projection conditions
Γ TZΓ ♮𝜖 = − sgn (E) 𝜖, Γ TΓ ♮ 𝜖 = sgn (B) 𝜖 (360 )
on the constant background Killing spinors 𝜖. These conditions have solutions, preserving 1/4 of the vacuum supersymmetry, if B(σ ) is a constant-sign, but otherwise completely arbitrary, function of σ. Notice the two projections 360View Equation correspond to string charge along the Z-direction and to D0-brane charge, respectively.

In order to improve our understanding on the arbitrariness of the cross-section, it is instructive to compute the charges carried by supertubes and its energy momentum tensor, to confirm the absence of any pull (tension) along the different spacelike directions where the tube is embedded in 10 dimensions. First, the conjugate momentum Pi and the conjugate variable to the electric field, Π, are

′ Pi = ∂-ℒD2-= ∘------BEy--i--------= sgn(ΠB )y′i, (361 ) ∂Y˙i (1 − E2)|⃗y′|2 + B2 ′2 ′2 Π (σ) = ∂-ℒD2-= ∘------E-|⃗y-|---------= sgn(E )|⃗y-|-, (362 ) ∂E (1 − E2)|⃗y′|2 + B2 |B |
where in the last step the supersymmetry condition |E | = 1 was imposed. Notice supertubes satisfy the identity
|P⃗|2 = |ΠB |. (363 )
Second, the fundamental string qF1 and D0-brane qD0 charges are
∫ ∫ qF1 = dσ Π , qD0 = dσ B . (364 )
Finally, the supertube energy-momentum tensor [380Jump To The Next Citation Point]
| mn ---2-------δSD2--|| ∘ --------------[ −1](μν) m n T (x) = √ −-detg-δg (x)|| = − − det(𝒢 + F ) (𝒢 + F) ∂μX ∂νX , (365 ) mn gmn= ηmn
with Xm = {T, Z,Y i}, has only non-zero components
𝒯 TT = |Π | + |B |, 𝒯 ZZ = − |Π |, 𝒯 Ti = sgn (ΠB )y′. (366 ) i
Some comments are in order:
  1. As expected, the linear momentum density (361View Equation) carried by the tube is responsible for the off-diagonal components Ti 𝒯.
  2. The absence of non-trivial components 𝒯 ij confirms the absence of tension along the cross-section, providing a more technical explanation of why an arbitrary shape is stable.
  3. The tube tension − 𝒯 ZZ = |Π | in the Z-direction is only due to the string density, since D0-branes behave like dust.
  4. The expanded D2-brane does not contribute to the tension in any direction.

Integrating the energy momentum tensor along the cross-section, one obtains the net energy of the supertube per unit length in the Z-direction

∫ T T ℰ = dσ 𝒯 = |qF1| + |qD0|, (367 )
matching the expected energy bound from supersymmetry considerations.

Let me make sure the notion of supersymmetry is properly tied with the expansion mechanism. Supertubes involve a uniform electric field along the tube and some magnetic flux. Using the language and intuition of previous Sections, the former can be interpreted as “dissolved” IIA superstrings and the latter as “dissolved” D0-branes, that have expanded into a tubular D2-brane. Their charges are the ones appearing in the supersymmetry algebra allowing the energy to be minimised. Notice the expanded D2-brane couples locally to the RR gauge potential C3 under which the string and D0-brane constituents are neutral. This is precisely the point made at the beginning of the section: supertubes do not carry D2-brane charge.36 When the number of constituents is large, one may expect an effective description in terms of the higher-dimensional D2-brane in which the original physical charges become fluxes of various types.

The energy bound (367View Equation) suggests supertubes are marginal bound states of D0s and fundamental strings (Fs). This was further confirmed by studying the spectrum of BPS excitations around the circular shape supertube by quantising the linearised perturbations of the DBI action [123Jump To The Next Citation Point, 29Jump To The Next Citation Point]. The quantisation of the space of configurations with fixed angular momentum J [123, 29] allowed one to compute the entropy associated with states carrying these charges

∘ -------------- S = 2π 2(q q − J ). (368 ) D0 F 1
This entropy reproduces the microscopic conjecture made in [364] where the Bekenstein–Hawking entropy was computed using a stretched horizon. These considerations do support the idea that supertubes are typical D0-F bound states.

Supergravity description and fuzzball considerations:
The fact that world volume quantisation reproduces the entropy of a macroscopic configuration and the presence of arbitrary profiles, at the classical level, suggests that supersymmetric supertubes may provide a window to understand the origin of gravitational entropy in a regime of parameters where gravity is reliable. This is precisely one of the goals of the fuzzball programme [363, 361Jump To The Next Citation Point].37 A first step towards this connection was provided by the supergravity realisation of supertubes given in [205Jump To The Next Citation Point]. These are smooth configurations described in terms of harmonic functions whose sources allow arbitrary profiles, thus matching the arbitrary cross-section feature in the world volume description [380].

The notion of supertube is more general than the one described above. Different encarnations of the same stabilising mechanism provide U-dual descriptions of the famous string theory D1-D5 system. To make this connection more apparent, consider supertubes with arbitrary cross-sections in ℝ4 and with an S1 tubular direction, allowing the remaining 4-spacelike directions to be a 4-torus. These supertubes are U-dual to D1-D5 bound states with angular momentum J [361Jump To The Next Citation Point], or to winding undulating strings [362] obtained from the original work [129, 158]. It was pointed out in [361Jump To The Next Citation Point] that in the D1-D5 frame, the actual supertubes correspond to KK monopoles wrapping the 4-torus, the circle also shared by D1 and D5-branes and the arbitrary profile in ℝ438. Smoothness of these solutions is then due to the KK monopole smoothness.

Since the U-dual D1-D5 description involves an AdS3 × S3 near horizon, supertubes were interpreted in the dual CFT: the maximal angular momentum configuration corresponding to the circular profile is global AdS3, whereas non-circular profile configurations are chiral excitations above this vacuum [361Jump To The Next Citation Point].

Interestingly, geometric quantisation of the classical moduli space of these D1-D5 smooth configurations was carried in [435], using the covariant methods originally developed in [156, 503]. The Hilbert space so obtained produced a degeneracy of states that was compatible with the entropy of the extremal black hole in the limit of large charges, i.e., ∘ ---------- S = 2π 2(qD0qF 1). Further work on the quantisation of supergravity configurations in AdS3 × S3 and its relation to chiral bosons can be found in [183]. The conceptual framework described above corresponds to a particular case of the one illustrated in Figure 7View Image.

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