# Supertubes

###### Abstract

It is shown that a IIA superstring carrying D0-brane charge can be ‘blown-up’, in a Minkowski vacuum background, to a (1/4)-supersymmetric tubular D2-brane, supported against collapse by the angular momentum generated by crossed electric and magnetic Born-Infeld fields. This ‘supertube’ can be viewed as a worldvolume realization of the sigma-model Q-lump.

[

DAMTP-2001-21

hep-th/0103030

]

## I Introduction

M-theory can be viewed as an extension of string theory to a ‘democratic’ theory of branes, with dualities that allow the whole theory to be constructed from any brane. One aspect of this is that a given brane may often be considered as a partially collapsed version of another, higher-dimensional, brane. Conversely, there may be circumstances in which the lower-dimensional brane is ‘blown-up’ into the higher-dimensional one. One example of this is the observation that a IIA superstring can be blown-up to a tubular D2-brane by placing it in an appropriate (non-supersymmetric) background [1]; the background fields impose an external force that prevents the collapse of the D2-brane. Another example is the observation that a similar non-trivial background can blow-up a collection of D0-branes into a fuzzy 2-sphere [2]; this can again be considered as a D2-brane prevented from collapse by an external force [3]. Another way to support a brane against collapse to a lower-dimensional one is to allow it to carry angular momentum; the case of branes on spheres, first analyzed in [4], has recently found an application to ‘giant gravitons’ [5]; these partially preserve the supersymmetry of an supergravity background [6], a fact that is presumably related to the appearance of angular momentum in the anticommutator of -supersymmetry charges.

Here we shall show that a cylindrical, or tubular, D2-brane in a Minkowski vacuum spacetime can be supported against collapse by the angular momentum generated by crossed electric and magnetic Born-Infeld (BI) fields. The construction involves a uniform electric field along the tube, and a constant magnetic flux. The electric field can be interpreted as a ‘dissolved’ IIA superstring, so the tube is a IIA superstring that has been ‘blown-up’ to a tubular D2-brane. The magnetic flux can be interpreted as a ‘dissolved’ D0-brane charge (per unit length). There are thus similarities to some of the previously proposed stabilization mechanisms but also some differences. Firstly, the background in our case is trivial; there is no external force. Secondly, the tubular D2-brane configuration presented here is supersymmetric; it preserves 1/4 of the supersymmetry of the IIA Minkowski vacuum, hence the term ‘supertube’.

At first sight it would appear unlikely that a brane configuration stabilized by angular momentum could be supersymmetric. For a start, supersymmetry requires a time-independent energy density, which is certainly non-generic for configurations with non-zero angular momentum. Even if this can be arranged, there is still the fact that a supersymmetric configuration in Minkowski space minimizes the energy subject to fixed values of the central charges appearing in the supersymmetry algebra, but angular momentum is not one of these charges. However, although these considerations may make supersymmetric stabilization by angular momentum unlikely, they do not make it impossible. Indeed, in the somewhat different context of field theory solitons there are known examples of supersymmetric solitons that are stabilized by angular momentum. One such example is the -lump of certain massive supersymmetric sigma-models [7], and the D2-brane supertube is essentially its worldvolume realization.

The -lump saturates an energy bound of the form , where is a topological (lump) charge and is a Noether charge [8]. When viewed as a string-like solution of the maximally supersymmetric sigma-model, it preserves 1/4 of the supersymmetry of the sigma-model vacuum [9]. The simplest -lump-string has cylindrical symmetry [10], but for large its energy density is not concentrated on the axis of symmetry but rather in a hollow tube of radius . Therefore the -lump-string tube can be interpreted, for large , as a lump-string that has been ‘blown-up’ into a cylindrical tube of kink-2-brane [11]. This tube is prevented from collapse by a centrifugal force, generated by an angular momentum proportional to the charge . Because the effective action for the kink 2-brane is a Dirac-Born-Infeld (DBI) action [9], there should be a worldvolume description of the ‘blown-up’ sigma-model lump-string as a 1/4-supersymmetric solution of the DBI equations. Any such solution will also solve the DBI equations of the IIA D2-brane (in a IIA Minkowski vacuum) and must preserve at least two supersymmetries (this being 1/4 of the eight supersymmetries of the sigma-model vacuum). This solution is precisely the D2-brane ‘supertube’ to be discussed below; it actually preserves eight of the thirty-two supersymmetries of the IIA vacuum and is therefore still 1/4-supersymmetric.

## Ii Energetics of the Supertube

The D2-brane Lagrangian, for unit surface tension, is

(1) |

where is the induced worldvolume 3-metric and is the BI 2-form field strength. We shall choose spacetime coordinates such that the Minkowski metric is

(2) |

with . If we take the worldvolume coordinates to be with , then we may fix the worldvolume diffeomorphisms for a D2-brane of cylindrical topology by the ‘physical’ gauge choice

(3) |

For a cylindrical D2-brane of (possibly varying, but time-independent) radius at a fixed position in (and with the -axis as the axis of symmetry) the induced metric is

(4) |

where and . We will allow for a time-independent electric field in the -direction, and a time-independent magnetic field , so the BI 2-form field strength is

(5) |

Under these conditions the Lagrangian becomes

(6) |

The corresponding Hamiltonian density is defined as

(7) |

where is the ‘electric displacement’ subject to the Gauss law constraint .

Let us now focus on the D2-brane supertube, for which is constant. In this case, the relation between the electric field and the electric displacement takes the form

(8) |

so the Hamiltonian density becomes

(9) |

where it should be noted that for BI 2-vector potential . The Gauss law constraint implies that is -independent. In addition, the equation of motion for forces to be -independent when is constant, as we are now assuming. Under these conditions is -independent, and its integral over the circle parametrized by yields a constant energy per unit length, namely the tube tension

(10) |

This is a function of and a functional of and . For an appropriate choice of units, the integrals

(11) |

are, respectively, the conserved IIA string charge and the D0-brane charge per unit length carried by the tube. The total D0-brane charge is also conserved, so imposing periodic boundary conditions on the tube, with period , implies conservation of , and also that is quantized in multiples of some unit proportional to . For fixed values of these charges, the tube tension is bounded from below:

(12) |

with equality if and only if

(13) |

The crossed electric and magnetic fields generate a Poynting 2-vector-density with

(14) |

as its only non-zero component. The integral of over yields an angular momentum per unit length along the axis of the cylinder. It is this angular momentum that supports the tube at the constant radius . Substituting (13) into (8) we see that

(15) |

This would be the ‘critical’ electric field if the magnetic field were absent, as is shown by the fact that when .

## Iii Supersymmetry

We now aim to show that the tubular D2-brane configuration just described is 1/4-supersymmetric; but as we also aim to relate it to some previously discussed D2-brane configurations we shall now drop the assumption that the worldvolume fields are independent of and . The number of supersymmetries preserved by any D2-brane configuration is the number of independent Killing spinors of the background for which

(16) |

where is the matrix appearing in the ‘-symmetry’ transformation of the worldvolume spinors [12]. Introducing as the constant matrix with unit square which anticommutes with all ten spacetime Dirac matrices, and as the induced worldvolume Dirac matrices, we have [13]

(17) |

For the D2-brane configuration of interest here

(18) |

where and are the constant Minkowski spacetime Dirac matrices (with ). For the spacetime coordinates that we have chosen, any Killing spinor can be written as

(19) |

where is a constant 32-component spinor of . The condition for preservation of supersymmetry can now be written as

(20) | |||||

It is clear that in order to satisfy this equation for all values of (equal to for our gauge choice) both terms on the right hand side must vanish independently. From the vanishing of the second term we recover the condition that , and we further deduce that must satisfy

(21) |

The vanishing of the first term leads to and

(22) |

For the supertube configuration and is constant, so the constraint above becomes simply

(23) |

The two conditions (21) and (23) are compatible and imply preservation of 1/4 supersymmetry; the minimal energy tubular D2-brane configuration is a supertube. Note that the constraints (21) and (23) are those associated with, respectively, a IIA superstring (in the -direction) and D0-brane charge; in particular, there is no trace of the D2-brane in these conditions! The physical reason for this is that a cylindrical D2-brane carries no net D2-brane charge.

When , the vanishing of the first term on the right hand side of (20) implies that

(24) |

for some constant , and the constraint on the spinor becomes

(25) |

The Gauss law now implies that

(26) |

for some constants and . This is just the ‘BIon in a magnetic background’ solution of [9] representing a IIA string ending on a bound state of D2-branes and D0-branes. To see this more clearly, first note that the constraint (25) indeed corresponds to that of a D2-D0 bound state in the -plane. Secondly, we can invert (26) to find

(27) |

which shows that is a harmonic function on the D2-brane 2-dimensional worldspace, as expected for the BIon [14, 15]. Finally, using (27) we can rewrite the BI field strength (5) as

(28) |

This corresponds to a radial Coulomb-like electric field on the D2-brane worldspace, as expected from the charge at the endpoint of the string, and to a constant density of D0-brane charge per unit worldspace area, precisely as in the solution of [9].

## Iv Discussion

By definition, a Dp-brane is a sink for IIA string charge. However, the D0-brane, being point-like, is special because the IIA charge has nowhere to go and must exit on another IIA superstring. Thus D0-branes can appear as ‘beads’ on a IIA superstring, breaking the 1/2 supersymmetry of the string to 1/4 supersymmetry. Of course, quantum mechanics will ensure that the ground state of such a superstring is one for which the D0-brane charge is uniformly distributed along the string. This ‘charged’ IIA superstring will have a tension exactly equal to the D2-brane supertube discussed above, . What distinguishes them is the angular momentum; the charged superstring has zero angular momentum while the supertube has angular momentum per unit length equal to . In principle, can be specified independently of the string and D0-brane charges, so given and we might expect there to be some supersymmetric string/tube configuration with arbitrary . As long as it is not difficult to see what this will be: a supertube with angular momentum per unit length , together with a charged superstring along its central axis. Because is quantized in the same units as , it is always possible for excess string and D0-charge to ‘condense’ out of a tubular D2-brane to leave behind a supertube supported from collapse by any given less than . On the other hand, it is unclear what the ground state could be when . It is conceivable that is an upper bound on the angular momentum of a supersymmetric IIA superstring configuration with charges and , and that supersymmetry is spontaneously broken when .

There is some similarity here to the status of angular momentum in the context of black holes of supergravity [16, 17, 18]. The black hole mass is determined entirely by its charge, if it is supersymmetric. For a given charge there is a one-parameter family of supersymmetric black hole spacetimes, parametrized by an angular momentum, but there is a critical value of the angular momentum beyond which the physics is qualitatively different. There is also a similarity to the suggestion [6] that the ‘giant graviton’ ground state is not supersymmetric above a critical value of the angular momentum, although the supersymmetry of relevance in that case is rather than Poincaré.

Finally, we wish to comment on some M-theory configurations dual to the D2-brane supertube. The first one involves the M2-brane: the D2-brane action is equivalent to the action for the supermembrane, the equivalence involving an exchange of the BI 1-form potential for a periodically-identified scalar field (with unit period) representing position in the 11th dimension [13]. Specifically, one has

(29) |

For the D2-brane supertube this yields

(30) |

This implies, firstly, that the M2-brane is wound times around the 11th dimension, as expected given the identification of with IIA string charge. Secondly, it implies that must be an integer, obviously equal to the number of IIA strings dissolved in the original D2-brane. Thirdly, it implies that there is a wave at the speed of light in the 11th dimension, as expected from the D0-brane charge; the momentum of this wave is proportional to . Note that if the dimension is periodically identified with period then one can take to arrive at a new (1/4)-supersymmetric IIA configuration in which a helical string rotates, producing a net momentum along the axis of the helix [19].

Another dual M-theory configuration consists of an M5-brane with topology , carrying dissolved membrane charges oriented along orthogonal planes in . To see this, let the be parametrized by , and let the two planes span the directions 1-2 and 3-4. Reducing to the IIA theory by compactifying the 4-direction, and T-dualizing along 2 and 3 leads to the D2-brane supertube.

## Acknowledgments

We thank Barak Kol for correspondence. D.M. is supported by a PPARC fellowship.

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- [19] We owe this observation to Barak Kol and Juan Maldacena; note that T-duality along the axis of the helix interchanges the string with the wave yielding a IIB version of the rotating helical string.