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  and its arbitrary cross-section reported in , 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 . Consider a D2-brane with world volume coordinates in the type IIA Minkowski vacuum
To study the preservation of supersymmetry, one solves Eq. (214). Given the ansatz (358) and the flat background (357), this condition reduces to 
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 and the conjugate variable to the electric field, , are
Integrating the energy momentum tensor along the cross-section, one obtains the net energy of the supertube per unit length in the -direction
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 5.6.2 – 5.6.3, 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 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 (367) 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 [123, 29]. The quantisation of the space of configurations with fixed angular momentum [123, 29] allowed one to compute the entropy associated with states carrying these charges 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.
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 and with an S 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 , or to winding undulating strings  obtained from the original work [129, 158]. It was pointed out in  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 38. 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 .
Interestingly, geometric quantisation of the classical moduli space of these D1-D5 smooth configurations was carried in , 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., . Further work on the quantisation of supergravity configurations in AdS3 × S3 and its relation to chiral bosons can be found in . The conceptual framework described above corresponds to a particular case of the one illustrated in Figure 7.
Living Rev. Relativity 15, (2012), 3
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