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 [380], 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 [380]. Consider a D2-brane with world volume coordinates in the type IIA Minkowski vacuum

where are Cartesian coordinates on . 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 The static gauge guarantees the tubular nature of the configuration, whereas the arbitrary embedding functions 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. (214). Given the ansatz (358) and the flat background (357), this condition reduces to [380]

where the prime denotes differentiation with respect to . For generic curves, that is, without imposing extra constraints on the embedding functions , supersymmetry requires both to set and to impose the projection conditions on the constant background Killing spinors . These conditions have solutions, preserving 1/4 of the vacuum supersymmetry, if is a constant-sign, but otherwise completely arbitrary, function of . Notice the two projections 360 correspond to string charge along the -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 and the conjugate variable to the electric field, , are

where in the last step the supersymmetry condition was imposed. Notice supertubes satisfy the identity Second, the fundamental string and D0-brane charges are Finally, the supertube energy-momentum tensor [380] with , has only non-zero components Some comments are in order:- As expected, the linear momentum density (361) carried by the tube is responsible for the off-diagonal components .
- The absence of non-trivial components confirms the absence of tension along the cross-section, providing a more technical explanation of why an arbitrary shape is stable.
- The tube tension in the -direction is only due to the string density, since D0-branes behave like dust.
- 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 -direction

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 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

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.

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
[361], or to winding undulating strings [362] obtained from the original work [129, 158]. It was
pointed out in [361] 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 AdS_{3} × S^{3} near horizon, supertubes were interpreted in
the dual CFT: the maximal angular momentum configuration corresponding to the circular profile
is global AdS_{3}, whereas non-circular profile configurations are chiral excitations above this
vacuum [361].

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., . Further work on the quantisation of
supergravity configurations in AdS_{3} × S^{3} 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 7.

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: |