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
To study the preservation of supersymmetry, one solves Eq. (214). Given the ansatz (358
) and the flat
background (357
), this condition reduces to [380
]
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
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 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 [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 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 7
.
http://www.livingreviews.org/lrr-2012-3 |
Living Rev. Relativity 15, (2012), 3
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