Supergravity Supertubes - PDF Free Download

Preprint typeset in JHEP style. - PAPER VERSION DAMTP-2001-49 hep-th/0106012 Supergravity Supertubes arxiv:hep-th/0106012v1 1 Jun 2001 David Mateos and Paul K. Townsend Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences Wilberforce Road, Cambridge CB3 0WA, United Kingdom E-mail: D.Mateos@damtp.cam.ac.uk, P.K.Townsend@damtp.cam.ac.uk Abstract: We find the asymptotic supergravity solution sourced by a supertube: a 1/4-supersymmetric D0-charged IIA superstring that has been blown up to a cylindrical D2-brane by angular momentum. It has no net D2-brane charge but does have a D2-brane dipole moment. It also has closed timelike curves in the dipole region, which leads to a ghost-induced instability of D2-supertube probes. We comment on similarities to the enhançon and the heterotic dyonic instanton. Keywords: D-branes, Supersymmetry and Duality, Brane Dynamics in Gauge Theories.

Contents 1. Introduction 1 2. The Supergravity Supertube 3 3. The D2-brane Dipole, Multi-tubes and CTCs 5 4. Brane Probes: Supersymmetry 8 5. Brane Probes: Energetics 11 6. Discussion 14 1. Introduction We have recently shown that a cylindrical D2-brane can be supported against collapse by the angular momentum generated by electric and magnetic Born-Infeld fields [1]. These fields can be interpreted as some number q s of dissolved IIA superstrings and some number per unit length q 0 of dissolved D0-branes. The angular momentum is proportional to q 0 q s and the energy is minimized for a cylinder radius proportional to q0 q s. Somewhat surprisingly, this configuration preserves 1/4 of the supersymmetry of the IIA Minkowski vacuum, hence the name supertube. The gravitational back-reaction was not considered in [1], but when this is taken into account the supertube becomes a source for the IIA supergravity fields that govern the low-energy limit of the closed string sector of IIA superstring theory. In the limit of weak string coupling this source is a singular (distributional) one, but it will be non-singular at non-zero coupling and one might expect it to generate a non-singular solution of IIA supergravity. Such a solution must solve the source-free IIA supergravity equations asymptotically and it must have the same string and D0-brane charges as the D2-brane supertube, as well as the same angular momentum. It must also carry a D2-brane dipole moment, and preserve 1/4 of the supersymmetry of the IIA vacuum. The aim of this paper is to exhibit this IIA supergravity solution, which we call the supergravity supertube, and to study its properties. 1

Since the D2-brane supertube is a supersymmetric solution one might expect the force between parallel 1 tubes to vanish, allowing them to be superposed. This is certainly the case for the D0-charged superstrings to which the supertube reduces in the limit of zero angular momentum, and it has been argued in the context of matrix theory [2] that it is also true of supertubes. As any force between parallel supertubes is transmitted by the IIA supergravity fields, it would seem necessary to consider the supergravity supertube solution to verify this, and this is one motivation for the present work. In fact, we shall exhibit multi-tube solutions representing a number of parallel supertubes with arbitrary locations and radii, which implies the existence of a no-force condition between parallel supertubes. By considering D0 and IIA string probes in this background we also establish a no-force condition between supertubes and strings and D0-branes. Ideally, we would wish to have a full IIA supergravity/supertube configuration, rather than just the asymptotic supergravity tube solution, since we might expect the asymptotic solution to become unphysical in the region where the source should be. This would be evidence that this region should be excised, to be replaced by a different and hopefully physical solution of the IIA supergravity field equations with a D2-supertube source. This expectation is indeed realized: the Killing vector field associated with the angular momentum becomes timelike at a radius determined by the D2-brane dipole field, thus allowing timelike curves that enter this region to close, forming closed timelike curves (CTCs). This global violation of causality cannot cause unphysical behaviour of any local probe (as we verify for a D0-brane probe) but it can and does cause unphysical behaviour for D2-brane probes. The pathology is very similar to one described in [3] but is caused by the appearance of a ghost-excitation on the D2-brane worldvolume; we interpret this as evidence that the dipole region containing CTCs must be excised from the spacetime. The angular momentum of the D2-brane supertube is not a free parameter but is instead determined by the other charges 2. Not surprisingly, this is not reflected in the asymptotic supergravity solution, for which the angular momentum can be freely specified. Thus, the matching of the asymptotic supergravity tube solution with excised dipole region onto a full spacetime that covers this region should fix this free parameter. Precisely this phenomenon was recently shown to occur in the context of a heterotic dyonic instanton, which provides a non-singular completion of a 1/4-supersymmetric rotating brane solution of N=1 D=10 supergravity [4]. We explore this further in a 1 Or concentric; however, there is no topological distinction between parallel and concentric supertubes in a ten-dimensional spacetime. 2 This becomes an upper bound on the angular momentum if one considers a combined chargedsuperstring/supertube configuration since the charged superstring carries no angular momentum. 2

concluding section. As noted in [1], the D2-brane supertube is T-dual to a helical rotating IIB D-string (the S-dual of which is T-dual to a helical rotating IIA string). These 1/4-supersymmetric rotating helical strings have since been studied in two recent papers [5, 6] which appeared during the completion of the work described here. We should mention that a IIA supergravity solution has been previously proposed as describing the supergravity fields of a supertube [7], but as it has no angular momentum this identification cannot be correct. Also, a supergravity solution representing the asymptotic fields of a IIB rotating helical D-string was presented in [6]; given that it preserves the same fraction of supersymmetry it should be T-dual to the solution studied here, but we have not investigated this. 2. The Supergravity Supertube Our starting point will be a solution of D=11 supergravity found in [8] that describes the intersection of two rotating M5-branes and an M2-brane, with an M-wave along the string intersection. Although the generic solution of this type preserves only 1/8 of the 32 supersymmetries of the M-theory vacuum, the special case in which we set to zero the M5-brane charges preserves 1/4 supersymmetry. The metric and 4-form field strength of this solution are (in our conventions) [ ] ds 2 11 = U 2/3 dt 2 + dz 2 + K (dt + dz) 2 + 2 (dt + dz) A + dx 2 + U 1/3 d y d y, F 4 = dt du 1 dx dz (dt + dz) dx d (U 1 A). (2.1) Here y = {y i } are Cartesian coordinates on 8. The membrane extends along the x- and z-directions, and the wave propagates along the z-direction. The functions U and K and the 1-form A depend only on y. U and K are harmonic on 8 and are associated with the membrane and the wave, respectively. A determines the angular momentum of the solution, and its field strength da satisifes the source-free Maxwell-like equation d 8 da = 0, (2.2) where 8 is the Hodge dual operator on 8. The reduction to ten dimensions of this solution along the z-direction yields a 1/4-supersymmetric IIA supergravity solution: ds 2 10 = U 1 V 1/2 (dt A) 2 + U 1 V 1/2 dx 2 + V 1/2 d y d y, B 2 = U 1 (dt A) dx + dt dx, C 1 = V 1 (dt A) + dt, C 3 = U 1 dt dx A, e φ = U 1/2 V 3/4, (2.3) 3

where V = 1 + K, and B 2 and C p are the Neveu-Schwarz and Ramond-Ramond potentials, respectively, with gauge-invariant field strengths H 3 = db 2, F 2 = dc 1, G 4 = dc 3 db 2 C 1. (2.4) We have chosen a gauge for the potentials B 2, C 1 and C 3 such that they all vanish at infinity when U and V are chosen such that the metric at infinity is the D=10 Minkowski metric in canonical Cartesian coordinates. We note here for future use that G 4 = U 1 V 1 (dt A) dx da. (2.5) The Killing spinors of the solution (2.3) follow from those in eleven dimensions given in [8], but we have verified them directly in D=10 with the conventions of [9]. They take a simple form when the metric is written as with the orthonormal 1-forms ds 2 10 = et e t + e x e x + e i e i, (2.6) e t = U 1/2 V 1/4 (dt A), e x = U 1/2 V 1/4 dx, e i = V 1/4 dy i. (2.7) In this basis, the Killing spinors become ǫ = U 1/4 V 1/8 ǫ 0, (2.8) where ǫ 0 is a constant 32-component spinor subject to the constraints Γ tx Γ ǫ 0 = ǫ 0, Γ t Γ ǫ 0 = ǫ 0. (2.9) These two constraints preserve 1/4 supersymmetry and are those associated with a IIA string charge aligned with the x-axis and D0-brane charge, as required for a supertube, although we still must specify the functions U and V and the 1-form A before we can make this identification. The simplest transverse asymptotically flat solutions are obtained by setting U = 1 + Q s 6Ωλ, V = 1 + Q 0 6 6Ωλ, A = 1 L ij y j 6 2 Ωλ 8 dyi, (2.10) where λ = y is the radial distance in 8 and Ω is the volume of the unit 7-sphere. The constant Q s is a IIA string charge, while Q 0 is a D0-brane charge per unit length; this can be seen from their asymptotic contributions to their respective field strengths H 3 and F 2. (The signs of these charges are flipped by taking t t and/or x x.) 4

Similarly, the constants L ij = L ji are the components of the angular momentum 2-form, in units for which the Einstein equation in Einstein frame is G µν = (1/2)T µν. Evaluation of the ADM integral for brane tension [10] (in the same units) shows that the tube tension is 3 τ = Q s + Q 0, (2.11) exactly as for the D2-brane supertube (no factors of the string coupling constant appear in this formula because the asymptotic value of the dilaton vanishes for our chosen solution). We have still to specify the rank of the angular-momentum 2-form L. In previous studies of rotating intersecting branes, including [8], L has been taken to have rank 4 or 8. To describe the supertube we must take L to have rank 2, as will become clear below. Choosing 8 coordinates such that ds 2 ( 8 ) = dr 2 + r 2 dϕ 2 + dρ 2 + ρ 2 dω 2 5, (2.12) where dω 2 5 is the SO(6)-invariant metric on the unit 5-sphere, we may write the oneform A for rank-2 L as Jr 2 A = 2Ω (r 2 + ρ 2 4 dϕ, (2.13) ) where J is a constant angular momentum (the one non-zero skew-eigenvalue of L). 3. The D2-brane Dipole, Multi-tubes and CTCs We have now found a 1/4-supersymmetric solution of IIA supergravity that carries all the charges required for its interpretation as the asymptotic solution generated by a supertube source. However, we would also expect the fields of the D2-brane supertube to carry a non-zero D2-brane dipole moment determined by the size of the tube. There is certainly a local D2-brane charge distribution because the electric components of G 4 are non-zero. Specifically, for A given by (2.13), we have [ ] Jr 2 G 4 = d dt dx dϕ +..., (3.1) 2Ω(r 2 + ρ 2 ) 4 3 This result shows that the energy required to dissociate the D2-supertube into IIA strings and D0-branes equals the energy of a (non-supersymmetric) rotating D2-brane with vanishing Born-Infeld fields, and hence that the supertube is a genuine bound state of IIA strings and D0-branes with a D2-brane. To see this consider a (non-supersymmetric) configuration of a rotating tubular D2- brane with zero Born-Infeld fields, and suppose that the string and D0-brane charges needed for the supertube are initially on distant IIA strings and D0-branes. The strings and D0-branes are attracted by the D2-brane, so the energy is lowered as they approach it and dissolve into it; the bound state thus produced is a supertube with binding energy equal to the energy of the initial D2-brane. 5

where the dots stand for subleading terms in an expansion in 1/λ for λ. The integral of G 4 over any 6-sphere at infinity vanishes, showing that there is no D2-brane net charge. However, the expression (3.1) has the right form to be interpreted as the dipole field sourced by a cylindrical D2-brane aligned with the x-axis; the scale of the dipole moment is set by the angular momentum: µ 2 J. (3.2) In turn, the dipole moment determines the size of the source as r 2 D2 µ 2, (3.3) where r D2 is the radius of the cylinder. This can be seen as follows. The dipole moment of a spherical D2-brane of radius R scales as µ R 3 [11]. For a cylindrical D2-brane of length a this must be replaced by µ ar 2. In the present situation a is infinite, so µ 2 in equations (3.2) and (3.3) is actually a dipole moment per unit length. Thus, we conclude that r D2 J, (3.4) in agreement with the worldvolume analysis of [1], where the radius of the supertube was found to be exactly J. However, there is potential discrepancy with the results of [1]: the radius of the supertube is also equal to Q 0 Q s because the angular momentum is determined by the charges; instead, J is a free parameter in the supergravity solution. Clearly, only the supergravity solution with J = ± Q 0 Q s can be identified as the asymptotic limit of a full supergravity supertube solution of IIA superstring theory. We will comment further on this point in the final section. Having completed our construction of the asymptotic supergravity supertube solution, and its identification as the asymptotic fields produced by a D2-supertube source, we are now in a position to answer the question raised in the introduction concerning the force between parallel supertubes. Multi-tube solutions representing N parallel tubes with arbitrary locations and charges are easily constructed by generalizing (2.10) to U = 1 + 1 6Ω V = 1 + 1 6Ω A = 1 2Ω N n=1 N n=1 N n=1 Q (n) s [(u u n ) 2 + (v v n ) 2 + ρ 2 ] 3, Q (n) 0 [(u u n ) 2 + (v v n ) 2 + ρ 2 ] 3, J (n) [(u u n ) 2 + (v v n ) 2 + ρ 2 ] 4 [(u u n )dv (v v n )du]. (3.5) 6

Here u and v are Cartesian coordinates on the 2-plane selected by the angular momentum. The n-th tube is located at ρ = 0, u = u n, v = v n, and it carries string and D0 charges Q s (n) and Q (n) 0, respectively, and angular momentum J (n). The total charges and angular momentum are the sums of those carried by each tube. This shows that there is no force between sufficiently separated stationary parallel supertubes. By sufficient separation we mean that the dipole regions surrounding the individual centres of the metric should not overlap, because these dipole regions should be replaced by a full solution that includes a (non-singular) source. One argument for excision of the dipole region of the supergravity supertube spacetime is that timelike paths that enter this region can be closed to yield closed timelike curves, leading to a global violation of causality. This happens because the Killing vector field l ϕ, which is associated to rotations on the 2-plane selected by the angular momentum, is not everywhere spacelike. Its norm squared is ( l 2 = g ϕϕ = V 1 2 r 2 U 1 V 1 f 2), (3.6) where Thus l 2 0 provided that f = Jr2 2Ωλ 8. (3.7) r 3 J (r 2 /λ 2 ) ( Q0 + 6Ωλ 6 )( Q s + 6Ωλ 6 ). (3.8) This inequality is certainly satisfied if r > 3 J / Q 0 Q s, but for sufficiently small λ it is violated. When this happens the closed orbits of l are timelike; they are therefore closed timelike curves. These CTCs are also naked in the sense that there is no event horizon to prevent them from being deformed to pass through any point of the spacetime. Neither can they be removed by a change of frame, since the causal structure is unchanged by any non-singular conformal rescaling of the metric. It should be appreciated, however, that the surface defined by l 2 = 0 is not singular; it is not even a coordinate singularity. In particular, the metric signature does not change on this surface despite the fact that for l 2 < 0 there are two commuting timelike Killing vector fields. In fact, the only physical singularity of the metric is at λ = 0, the origin of 8. Thus, the local geometry is perfectly physical everywhere away from λ = 0 even though causality is violated globally. The largest value of r at which l becomes null occurs when ρ = 0, and for large Q 0 and Q s this happens when J r Q0 Q s. (3.9) 7

If we now set J = Q 0 Q s, (3.10) which is the value found in [1] for the D2-brane supertube, then we conclude that l becomes null when r Q 0 Q s. (3.11) If we identify this radius as that of the supertube source then we have agreement with the worldvolume analysis [1] and with the estimate (3.4) based on the dipole moment. It is reasonable that we should find this agreement with the worldvolume analysis when Q 0 and Q s are large because this is the condition expected for validity of the supergravity approximation. The CTCs of the supergravity supertube metric have a very simple origin in eleven dimensions. The IIA supergravity supertube lifts to the solution (2.1) of elevendimensional supergravity from which we started, with U and V as in (2.10) and A as in (2.13). If the eleventh coordinate z is not periodically identified then this elevendimensional spacetime has no CTCs, but CTCs are created by the identifications needed for compactification on a circle. To see this, consider an orbit of the vector field ξ = ϕ + β z, for real constant β. When z is periodically identified this curve will be closed for some dense set of values of β. It will also be timelike if ξ 2 < 0, which will happen if β 2 + 2V 2 f β + UV 2 r 2 < 0. (3.12) This inequality can be satisfied for real β only if r 2 < U 1 V 1 f 2, but this is precisely the condition that l be timelike. Thus, there will be closed timelike orbits of ξ in the D=11 spacetime precisely when the closed orbits of l become timelike in the D=10 spacetime. However, from the eleven-dimensional perspective, these CTCs arise from periodic identification and can therefore be removed by passing to the universal covering space. An analogous phenomenon was described in [12], where it was shown that the CTCs of over-rotating supersymmetric five-dimensional black holes are removed by lifting the solution to ten dimensions and passing to the universal covering space. 4. Brane Probes: Supersymmetry We shall now examine the behaviour of brane probes in the supertube spacetime. Specifically, we shall consider D0-branes, strings and D2-branes. We shall first show that the 1/4 supersymmetry of the background is preserved by stationary D0-brane, IIA-string and D2-supertube probes (when suitably aligned). The D=10 spacetime metric can be written as ds 2 10 = et e t + e x e x + e r e r + e ϕ e ϕ + e a e a, (4.1) 8

for orthonormal 1-forms e t = U 1/2 V 1/4 (dt A), e x = U 1/2 V 1/4 dx, e r = V 1/4 dr, e ϕ = V 1/4 r dϕ, e a = V 1/4 dρ a. (4.2) Here {ρ a } are Cartesian coordinates on the 6 space orthogonal to the 2-plane selected by the angular momentum. Let Γ t, Γ x, Γ r, Γ ϕ and {Γ a } be the ten constant tangentspace Dirac matrices associated to the above basis, and let Γ be the constant matrix of unit square which anticommutes with all them. The Killing spinors in this basis take the form ( ǫ = U 1/4 V 1/8 M + ǫ 0, M ± exp ± 1 ) 2 ϕ Γ rϕ, (4.3) where ǫ 0 is a constant 32-component spinor subject to the constraints (2.9). The number of supersymmetries preserved by any brane configuration in a given spacetime is the number of independent Killing spinors ǫ of the background for which Γǫ = ǫ, (4.4) where Γ is the matrix appearing in the κ-symmetry transformation of the worldvolume spinors, its particular form depending on the background and on the type of brane. For a stationary D0-brane in the gauge in which worldline time is identified with t we have Γ D0 = Γ t Γ. (4.5) For a IIA string in the same temporal gauge and, additionally, with the string coordinate identified with x, we have Γ string = Γ tx Γ. (4.6) These two Γ-matrices commute and the Killing spinors of the background are simultanteous eigenstates of Γ D0 and Γ string with unit eigenvalue, so the inclusion of these D0-brane and IIA-string probes does not break any supersymmetries that are preserved by the background. In the absence of a D2-dipole, this result is to be expected. Somewhat surprisingly, it continues to be true in the presence of the D2-dipole field, even though planar D2-branes exert an attractive force on D0-branes and IIA strings. Now we consider a probe consisting of a supertube itself, that is, a D2-brane of cylindrical topology with string and D0-brane charges. The analysis is very similar to that of [1] except that the Born-Infeld field strength F must now be replaced by the 9

background-covariant field strength F = F B 2. As in [1] we choose F to have the form F = E dt dx + B dx dϕ. (4.7) This yields 4 F = E dt dx + B dx dϕ, (4.8) where E = E + U 1 1, B = B + U 1 f. (4.9) In the orthonormal basis (4.2) we have F = Ē et e x + B e x e ϕ, (4.10) where Ē = UE, B = U 1/2 V 1/2 r 1 (B fe). (4.11) In terms of these variables, Γ takes the same form as in flat space: Γ = 1 ( Γ txϕ + Ē Γ ϕγ + B ) Γ t Γ, (4.12) det(g + F) where g is the induced metric and det(g + F) = 1 Ē2 + B 2. (4.13) The condition for supersymmetry (4.4) thus becomes [ M + B Γt Γ ] [ det(g + F) ǫ 0 + M γ ϕ Γ Γtx Γ + Ē] ǫ 0 = 0. (4.14) Since this equation must be satisfied for all values of ϕ, both terms must vanish independently. Given the constraints (2.9) satisfied by ǫ 0, the vanishing of the second term requires that Ē = 1. Equivalently, E = U 1. (4.15) The first term in (4.14) then vanishes identically if B 0, which using (4.15) is equivalent to B 0. In the B = 0 case we have 1/2 supersymmetry, so we shall assume that B > 0. (4.16) 4 We need not distinguish between spacetime forms and their pullbacks here because there is no difference in the physical gauge, as described in detail in [1]. 10

5. Brane Probes: Energetics We now turn to the energetics of probes in the supergravity supertube background. Our aim is to uncover any effects of the global causality violation due to the CTCs in the dipole region. We would not expect this to have any effect on a local probe such as a D0-brane, and we shall begin by verifying this. The action for a D0-brane of unit mass is S D0 = e φ det g C 1. (5.1) In the gauge where the worldline time is identified with t we find the Lagrangian density L = V 1 1 2f ϕ UV 1/2 (g ϕϕ ϕ 2 + v 2 ) + V 1 (1 f ϕ) 1, (5.2) where the overdot indicates differentiation with respect to t, f is given by (3.7), and v 2 g ij Ẋ i Ẋ j, X i = {x, r, ρ a }. (5.3) Note that v 2 0. The Lagrangian (5.2) appears to lead to unphysical behaviour when g ϕϕ becomes negative. However, an expansion in powers of the velocities yields L = 1 + 1 2 Ur2 ϕ 2 + 1 2 UV 1/2 v 2 +. (5.4) The first term is minus the (unit) positive mass of the D0-brane. As expected from the supersymmetry of a stationary D0-brane, all the velocity-independent potential terms cancel. In addition, the kinetic energy is positive-definite regardless of the sign of g ϕϕ (and this remains true to all orders in the velocity). There is therefore nothing to prevent a D0-brane from entering the dipole region, and its dynamics in this region is perfectly physical, at least locally. The same applies to aligned IIA superstrings; again this is not surprising because this is a probe that is local in ϕ. We now turn to the supertube itself. This is a more significant test of the geometry because one could imagine building up the source of the supergravity solution by accretion of concentric D2-brane supertubes. We begin by considering a general tubular D2-brane aligned with the background. For our purposes we may assume that the worldvolume scalar fields r and ρ a are uniform in x, but we shall also assume, initially, that they are time-independent (as will be the case for a supertube, since this is a D2-brane tube at a minimum of the potential energy). The action S D2 = e φ det(g + F) (C 3 + C 1 F) (5.5) 11

in the physical gauge (as used in [1]) yields the Lagrangian density L = U 1/2 V 1 V r 2 (U 2 E 2 ) + U 1 (B fe) 2 + V 1 (B fe) B. (5.6) The variable conjugate to E is Π δl δe = δl δe = The Hamiltonian density is defined as U1/2 r 2 E + U 1/2 V 1 (B fe)f V r2 (U 2 E 2 ) + U 1 (B fe) 2 V 1 f. (5.7) H ΠE L = ΠE L + Π(1 U 1 ). (5.8) For supersymmetric configurations we have L = B and E = U 1, so 5 H = Π + B (5.9) for a supertube. Neither Π nor B is invariant under background gauge transformations. However, their worldspace integrals are gauge invariant 6. Given the assumption of x- independence and a unit period for ϕ, we can identify q s = dϕπ as the IIA string charge and q 0 = dϕb as the D0-brane charge per unit length of the supertube probe. For constant Π and B we therefore have The probe supertube tension is then seen to be Π = q s, B = q 0. (5.10) τ probe = q s + q 0, (5.11) exactly as in the case of a Minkowski background. Setting E = U 1 in (5.7) we deduce that ΠB = r 2 (5.12) at the supertube radius, and hence that again as in a Minkowski background. r = q s q 0, (5.13) 5 Note that both B (see equation (4.16)) and Π (from its definition (5.7)) are positive for supersymmetric configurations. 6 We should consider only time-independent gauge transformations because the Hamiltonian is not expected to be invariant under time-dependent gauge transformations. We should also consider only x-independent gauge transformations to be consistent with our assumption of x-independence. 12

Thus, supersymmetry places no restriction on the possible radius of a probe supertube in a supergravity supertube background. However, the supertube minimizes the energy for given q s and q 0 (and hence implies stability) only if there are no ghost excitations of the D2-brane, that is, excitations corresponding in the quantum theory to negative norm states. However, there is a negative norm state for sufficiently small λ, and there is some choice of q 0 and q s such that this occurs whenever g ϕϕ < 0. To establish this, we now allow for time-dependent r and ρ a, and we expand the Lagrangian density in powers of v 2 = ṙ 2 + δ ab ρ a ρ b. (5.14) We find L = L 0 + 1 2 Mv2 +, (5.15) where L 0 is the Lagrangian density of (5.6), and M = U1/2 V 1/4 (B 2 + 2U 1 fb + U 1 V r 2 ) det(g + F), (5.16) which reduces to M = UV 1/2 B 1 ( B 2 + 2U 1 fb + U 1 V r 2) (5.17) for a supertube. In both of these expressions for M, the denominator is positive, and so is the numerator as long as g ϕϕ > 0, but the factor B 2 + 2U 1 fb + U 1 V r 2 (5.18) becomes negative for some choices of B and r (and hence q 0 and q s for a supertube) whenever g ϕϕ < 0. Note that the supertube does not become tachyonic in the dipole region; its tension continues to be given by (5.11). A tachyon instability leads to runaway behaviour in which the kinetic energy increases at the expense of potential energy. The instability here is instead due to the possibility of negative kinetic energy, characteristic of a ghost 7. Clearly, this should not happen in any physical completion of the asymptotic solution to include a supertube source, so this is the clearest indication that the dipole region 7 As representations of the Poincaré group, a tachyon is a particle with spacelike energy-momentum vector, and hence negative mass-squared (corresponding in the quantum theory to an excitation about a vacuum that is a local maximum of the energy rather than a local minimum). A ghost is a particle with non-spacelike energy-momentum vector but negative energy; at the level of the particle Lagrangian (or, more generally, brane Lagrangian, as discussed here) this corresponds to negative mass. 13

containing the CTCs is unphysical and must be excised. So what will actually happen to a supertube probe that approaches the source region in the complete solution? Presumably, it will be absorbed by the source and become part of it. Since one must go beyond the test-brane approximation to see this, it may be that the instability due to ghost excitations of the supertube will be replaced by a breakdown of the test-brane approximation. 6. Discussion We have found the asymptotic IIA supergravity solution corresponding to a source provided by the D2-brane supertube found in [1]. We have also found multi-tube solutions, which shows that parallel supertubes exert no force on each other, and we have confirmed this by showing that when (suitably aligned) stationary D0-brane, IIAstring and D2-supertube probes are introduced into the background they preserve all supersymmetries of the background. This provides confirmation from supergravity of the matrix model results of [2] for multi-supertubes. The asymptotic supergravity supertube solution can be continued into the origin of the transverse 8-dimensional space without singularity, except at the origin itself, but acausal behaviour is encountered at a radius set by a D2-dipole field, which can be identified with the region occupied by the supertube source. Although this global violation of causality has no effect on D0-brane or IIA string probes, it causes an instability of D2-supertube probes due to the fact that the supertube tension no longer minimizes the energy for fixed D0 and string charges. This is made possible by the appearance of a negative-norm state (a ghost) on a supertube at a radius for which the rotational Killing vector field l becomes timelike. The global violation of causality that this produces (due to the occurrence of CTCs) is thus manifested by a local pathology on supertube probes (which is possible because these probes are themselves non-local). Athough it would require a departure from the test-brane approximation, the supergravity supertube solution could be built up, in principle, by bringing in supertube probes from infinity. Thus unphysical behaviour of supertube probes constitutes evidence of a breakdown of the validity of the asymptotic solution in the dipole region, which must therefore be excised as unphysical. It is straightforward to modify the IIA supergravity supertube solution to one that provides the asymptotic fields for a supertube in a Kaluza-Klein vacuum spacetime of the form (1,5) K3. Although at weak string coupling the supertube source is distributional, and hence singular, one might expect it to be non-singular at strong coupling. To examine this possibility we should look for solutions of the D=6 supergravity / Yang-Mills theory that governs the low-energy limit of the S-dual T 4 -compactified 14

heterotic string theory. The distributional D2-brane of the IIA theory now appears as a non-singular 2-brane with a magnetic monopole core. Conceivably, a tubelike configuration of this monopole 2-brane can be supported against collapse by angular momentum in just such a way that its effective worldvolume description is as a D=6 supertube (that is, a d2-brane supertube of iia little string theory). In this case, one might hope to find a full non-singular supergravity / super Yang-Mills (SYM) solution that is asymptotic to the IIA supergravity tube (after the redefinition of fields required by the IIA/heterotic duality) but completed in the interior by a solution with the SYM-supertube source. There is no known SYM-supertube solution of D=6 SYM theory, but if it exists it will likely have an angular momentum that is fixed by the other charges, exactly as required for the supergravity solution representing a supertube. Precisely this phenomenon was found to occur in the analogous case in which an asymptotic rotating brane solution of D=10 supergravity studied in [12] was completed in the interior by a dyonic instanton [14] solution of the SYM theory. The resulting supergravity/sym solution is the heterotic dyonic instanton of [4]. The main difference between the two cases is that the supergravity supertube, and presumably also its heterotic completion, has an angular momentum 2-form of rank 2, whereas the supergravity solution of [12] has an angular momentum 2-form of rank 4, as does its completion to the heterotic dyonic instanton, although this completion is possible for only one value of the angular momentum. A further aspect of this analogy is that the rotating brane solution of D=10 supergravity studied in [12] can (when viewed as a solution of N=1 supergravity) be dimensionally reduced to yield a rotating D=5 heterotic black hole found in [13]. These black holes have CTCs in just the same way that the CTCs of the D=10 supergravity supertube may be removed in D=11. The CTCs are hidden by an event horizon for small angular momentum, but they become naked at a critical value of the angular momentum; this is presumably the value at which the completion to a heterotic dyonic instanton is possible, but this has not been investigated. In summary, it appears that the supergravity supertube spacetime studied here will eventually take its place in a more general theory of supergravity solutions for supersymmetric sources supported by angular momentum. Acknowledgments We thank Jerome Gauntlett and Carlos Herdeiro for discussions. D.M. is supported by a PPARC fellowship. 15

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