supercurves

Physics Letters B 538 (2002) 366–374

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Supercurves

David Mateos, Selena Ng, Paul K. Townsend

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA,United Kingdom

Received 7 May 2002; accepted 21 May 2002

Editor: P.V. Landshoff

Abstract

The TST-dual of the general 1/4-supersymmetric D2-brane supertube is identified as a 1/4-supersymmetric IIA ‘supercurve’:a string with arbitrary transverse displacement travelling at the speed of light. A simple proof is given of the classical upperbound on the angular momentum, which is also recovered as the semi-classical limit of a quantum bound. The classical bound issaturated by a ‘superhelix’, while the quantum bound is saturated by a bosonic oscillator state in a uniqueSO(8) representation. 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

A supertube is a 1/4-supersymmetric tubular D2-brane configuration of IIA superstring theory that issupported against collapse by the momentum gen-erated by crossed electric and magnetic Born–Infeld(BI) fields on the D2-brane [1,2]. As originally de-scribed, the tube was assumed to have a circularcross-section, in which case the stability is easilyunderstood as a result of the centrifugal force dueto an angular momentum 2-form on the plane ofthe circle. The T-dual (along the axis of the cylin-der) of this rotationally-invariant supertube is a 1/4-supersymmetric helical IIB D-string [3] supportedfrom collapse by the combination of angular momen-tum and momentum along the D-string. The TST-dualis an analogous IIA helical string [1].

E-mail addresses: [email protected] (D. Mateos),[email protected] (S. Ng),[email protected] (P.K. Townsend).

Following [4], in which it was shown that a super-tube cross-section could be elliptical rather than circu-lar, we have recently shown that 1/4 supersymmetryactually allows a supertube to have a cross-section thatis anarbitrary curve inE

8 [5]. Here we show that thisrather surprising fact becomes less surprising once oneconsiders the TST-dual configuration, which we call a‘supercurve’: it turns out that a periodic supercurve isequivalent to a IIA string carrying a transverse wave ofarbitrary profile at the speed of light, and it has beenknown for some time that such configurations preserve1/4 supersymmetry [6,7].

Although a periodic ‘supercurve’ is equivalent to awave on a string, this is not the description of it thatemerges directly from a TST-duality of the supertube.The two features of the supertube that are essential toits stability, and 1/4 supersymmetry, are (i) that theBI electric field take the value that would be ‘critical’in the absence of a magnetic field, and (ii) that theBI magnetic field not change sign. For the TST-dualIIA string, (i) becomes a condition of uniform motion

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D. Mateos et al. / Physics Letters B 538 (2002) 366–374 367

of the string at the speed of light along the T-dualdirection, while (ii) becomes the condition that theshape of the curve be monotonic in this direction.One might think that a string that moves uniformlyin one direction at the speed of light must have avanishing mass density. However, because a Nambu–Goto string is boost-invariant, only the orthogonalvelocity component of a string element is physical andthis ensures that its physical velocity is subluminal.Here one can see that an inconsistency could arise ifthe monotonicity condition were violated because thiswould allow the string to close, but if a closed string inE

9 moves at the speed of light in a fixed direction thenits centre of mass is effectively a massless particle, andthis is possible only if the string has a vanishing mass-density.

It was shown in [7], in the IIB D-string context, thatthe total angular momentum of a 1/4-supersymmetricmomentum-carrying string is maximized, for fixedlinear momentum, when the transverse profile is acircle in a plane inE

8. Here we provide a muchsimpler proof of this result. It turns out that there isa sequence of four bounds on the angular momentum(given 1/4 supersymmetry), according to whether theangular-momentum 2-formL has rank 2, 4, 6 or 8.The ‘rank 2k’ bound is saturated by curves inE9

for which the projection ontoE8 is a closed curvethat has projections to circles of common radii onk

orthogonal planes. These could be called ‘generalized’superhelices; thek = 1 case is the IIA version ofthe IIB D-string ‘superhelix’ described in [3]. As thek = 1 bound is the weakest one, it yields the maximumangular momentum for fixed linear momentum; thisbound is the same as the one obtained for a D-string in[7] and for a supertube in [1,2,5].

One of the motivations for the present work was togain an understanding of the implications of quantummechanics for supertubes. Given the freedom of anarbitrary cross-sectional shape, one may wonder, forexample, whether there is a unique quantum stateassociated to each classical shape. The TST-dualsupercurve provides a simpler context in which topose such questions because any IIA superstringconfiguration corresponds to a state of theperturbativequantum IIA superstring. We analyze the angularmomentum of the 1/4-supersymmetric ‘BPS’ states ofIIA superstring theory and derive a quantum versionof the classical angular momentum bound, which is

recovered as a semi-classical limit. It turns out that(for fixed linear momentum) there is a unique one-string state of the perturbative IIA superstring theoryfor which the quantum bound is saturated, as onemight expect from the fact that the classical bound issaturated by a unique wave profile.

2. IIA supercurves

Let ξa = (t, σ ) be the string worldsheet coordinatesand Xµ = (X0,X) be Cartesian coordinates for theD = 10 Minkowski spacetime. We shall fix the timereparametrization invariance by the choice

(2.1)X0 = t .

In this gauge, the induced worldsheet metricgab issuch that

(2.2)√−detg =

√(1− X 2

)(X′)2 + (

X · X′)2,

where the overdot and prime indicate differentiationwith respect tot andσ , respectively. Let us now writeX = (Z,Y ) and set

(2.3)Z = 1, Y = 0,

in which case

(2.4)√−detg = |Z′|.If we interpret theZ-direction as a T-dual directionthen configurations of this type are precisely what isobtained by a TST-duality transformation of the D2-brane supertube. The conditionZ = 1, which meansthat the string moves with the speed of light in theZ-direction, arises from the condition that the BI electricfield on the supertube take the value that would becritical in the absence of the magnetic field. Also,the condition that the BI magnetic field not changesign translates to the same condition onZ′; we shallsee below that this is needed for preservation ofsupersymmetry.

The condition for a bosonic IIA superstring config-uration to preserve some fraction of supersymmetry ofthe IIA Minkowski vacuum is that the equation

(2.5)Γ ε = ε

368 D. Mateos et al. / Physics Letters B 538 (2002) 366–374

admits non-zero constant spinor solutionsε, whereΓis the ‘kappa-symmetry’ matrix

(2.6)Γ = 1√−detgXµX′ νΓµνΓ�.

As usual,Γ� denotes the product of all ten Diracmatrices, anticommuting with eachΓµ and squaringto the identity. Note thatΓ 2 = 1. For configurationssatisfying (2.3) we have

(2.7)

Γ = sgn(Z′)ΓTZΓ� − 1

|Z′|Y′ · Γ YΓT Γ�(1−ΓTZ).

Let us first consider the case in whichY ′/Z′ is aconstant 8-vector. As long asZ′ is everywhere non-zero, this corresponds to an infinite straight string atan angle arctan(|Y ′|/Z′) to theZ-axis. Although thisstring moves at the speed of light in theZ-direction,only the orthogonal component is physical and this hasthe subluminal magnitude

(2.8)v = |Y ′|√(Z′)2 + |Y ′|2 .

This case therefore corresponds to an orthogonallyboosted infinite straight string, which we would expectto preserve 1/2 supersymmetry. It does so because inthis caseΓ is a constant matrix and Eq. (2.5) admits16 linearly independent solutions forε.

If Y ′/Z′ is not a constant 8-vector, butZ′ isnowhere zero, then Eq. (2.5) is solved by spinorssatisfying

(2.9)ΓTZΓ�ε = sgn(Z′)ε, ΓTZε = ε.

These two conditions are those of a 1/4-supersym-metric superposition of a string and linear momentumalong theZ-axis, although the string now describes anarbitrary curve in the ‘transverse’E

8 space. These arethe IIA supercurves that are TST-dual to the generalsupertube, although one might wish to restrict the term‘supertube’ to apply only to those cases for whichthe cross-section is a closed non-intersecting curve inE

8. For such cases the supercurve will (i) be periodicand (ii) have non-zero angular momentum; it can beinterpreted as a 1/2-supersymmetric string parallel totheZ-axis that has expanded to a 1/4-supersymmetricsupercurve as a result of its angular momentum.There is an entirely analogous result for D-strings that

generalizes the super D-helix of [3], which is the T-dual of the circularly-symmetric supertube.

3. Hamiltonian analysis

So far we have found a class of IIA string con-figurations that preserve 1/4 supersymmetry, but weshould still check that they solve the Nambu–Gotoequations. We will do this by considering how thesesolutions look in phase space. This will also permit asimple demonstration that a supercurve is equivalentto a wave-carrying string. The (bosonic) phase spaceLagrangian density is

(3.1)L= P · X − 1

2e[P 2 + (X′)2] − sP · X′,

wheree and s are Lagrange multipliers that imposethe Hamiltonian and reparametrization constraints,respectively. In theX0 = t gauge this becomes

(3.2)L= PZDtZ + P · DtY −H,

where Dt is the reparametrization-covariant timederivative

(3.3)Dt = ∂t − s∂σ ,

andH is the (bosonic) Hamiltonian density. Using thereparametrization constraint,

(3.4)PZZ′ + P · Y ′ = 0,

we can write the Hamiltonian density as

(3.5)H =√(PZ ∓ Z′)2 + |P ∓ Y ′|2.

For this Hamiltonian density, thePZ andP equationsof motion are

DtZ =H−1(PZ ∓ Z′),(3.6)DtY = H−1(P ∓ Y ′).

It is obvious from (3.5) that, in theZ′ = 1 gauge,the energy is minimized for fixedPZ when

(3.7)P = ±Y ′.

It may be verified that such configurations solve bothEq. (3.6) and the constraint (3.4) provided that

(3.8)|DtZ| = 1, DtY = 0.


For s = 0 this implies |Z| = 1 and Y = 0, so weconclude that the supercurve configurations describedearlier indeed solve the Nambu–Goto equations, andthat they are equivalent to phase space configurationssatisfying (3.7). Given (3.7), they are also solved byZ = 0 but for a different value ofs and with Y nownon-zero, such thatY = ±Y ′ in theZ′ = 1 gauge. Thisis the more conventional description of a supercurveas a string carrying a wave of arbitrary profile thatmoves at the speed of light in one direction along thestring. The two descriptions are equivalent because inboth cases thecovariant time derivatives ofZ andY

are the same, and for either description a case-by-caseanalysis of the signs leads to the conclusion that

(3.9)H = |Z′| + |PZ |.The first term is the energy due to the string massdensity, and the second term is the energy due to themomentum-carrying wave on it.1

The Hamiltonian density is the time–time compo-nent of the string stress-energy tensor density

(3.10)τab = −√−detg gab,

wheregab is the inverse of the induced worldsheetmetric. Observe that this is divergence-free in thegaugeZ′ = 1 as a consequence of theX0 and Z

equations of motion. For the supercurve one finds inthis gauge that

(3.11)τab =(

H −1−1 0

).

The off-diagonal components correspond to the pres-ence of the linear momentum density carried by thewave. The vanishing of the string tension−τσσ pro-vides one explanation of why an arbitrary fixed waveprofile in E

8 is stable.

4. Angular momentum bound

Let us suppose thatY (σ ) is non-zero and periodicwith period 2π . The angular-momentum 2-form per

1 Note that the reparametrization constraint implies thatPZ = 0whenY ′ = 0, as expected since a string withY ′ = 0 is parallel totheZ-axis.

period has components

(4.1)Lij = 1

2π

∮dσ (YiPj − YjPi).

Given (3.7), this becomes

(4.2)Lij = ± 1

2π

∮dσ

(YiY

′j − YjY

′i

),

which is proportional to the area of the closed curveformed by the projection ofY (σ ) over one periodonto the ij -plane. Recall that (3.7) was derived byminimizing the energy in theZ′ = 1 gauge for fixedPZ . We learn from the reparametrization constraintthat the momentum per period in theZ-direction is

(4.3)|"PZ| = 1

2π

∮dσ |Y ′|2

in theZ′ = 1 gauge. The integrand is the square of thelength-density of the curveY (σ ), so we might expect,by some variant of the isoperimetric inequality, thatthe total squared angular momentum per period

(4.4)J 2 = 1

2LijL

ij

will be bounded by some multiple of"PZ . We shallnow establish this bound.

Specifically, we seek to maximizeJ 2 subject tothe constraint (4.3). This is equivalent to maximizing,without constraint, the functional

(4.5)

F [Y ;λ] = J 2[Y ] + 2λ

[∮dσ |Y ′|2 − 2π |"PZ|

]

with respect to the functionsY and the Lagrangemultiplier variableλ. Variation with respect to thefunctionsYi yields second-order differential equationswhich are trivially once-integrated. Setting the inte-gration constants to zero by a translation inE

8 (underwhichJ is invariant) we arrive at the equation

(4.6)Y ′i = 1

λJLij Y

j .

Contracting withY ′i and integrating over a period we

deduce that

(4.7)λ = J

|"PZ| .

By a rotation inE8 (under whichJ is invariant)

we can skew-diagonalizeL; let %p, p = 1, . . . ,4,


be the skew-eigenvalues.L is then diagonal (witheigenvalues±i%p) in the complex basis

(4.8)Wp = Y2p−1 + iY2p,

and the solution of (4.6) is then

(4.9)Wp = RpeiNpσ .

HereRp are arbitrary constants, and the integers

(4.10)Np = |"PZ|J 2

%p

count the number of times per period that the stringwinds around the origin of the complexWp-plane.Note that the overall phase of each of the four complexfunctionsWp in (4.9) may be brought to any desiredvalue by a transformation in the residualSO(2)4

subgroup ofSO(8) that preserves the skew-diagonalform of L.

Given (4.9),L may be computed and one can thusverify that it is skew-diagonal. This calculation alsoreveals that

(4.11)%p = NpR2p.

If %p �= 0 then this together with (4.10) shows that

(4.12)R2p = J 2

2π |"PZ| .

If %p = 0 then (4.10) implies thatNp = 0 too, sothat Rp is undetermined andWp is constant; by atranslation in the complexWp-plane we may stillchooseRp to be given by the formula (4.12). Thus,the solution (4.9) becomes

(4.13)Wp =√

J 2

2π |"PZ| eiNpσ .

Defining

(4.14)N2 ≡4∑

p=1

N2p

and making use of (4.10), we see that the angularmomentum of this solution saturates the bound

(4.15)J � N−1|"PZ|for fixed linear momentum per period|"PZ| andwinding numbersNp .

The geometry of the periodic supercurve withmaximal angular momentum per period thereforedepends in an essential way on the rank of the angular-momentum 2-form; through (4.10) this is determinedby the number of non-zero winding numbersNp ,which, of course, may be specified independently ofthe linear momentum. If only|"PZ | is specified,then the absolute bound on the angular momentumbecomes simply

(4.16)J � |"PZ|,which is the bound discussed in [7] in the ST-dualcontext of the D-string and which is saturated when(say)N1 = 1 andN2 = N3 = N4 = 0. In this caseLhas rank 2 and the supercurve is a helix (of fixed pitchin theZ′ = 1 gauge) in anE3 subspace ofE9; in fact,it is the IIA dual of the superhelix of [3].

5. Light-front gauge

With a view to simplifying the quantum treatmentof supercurves within the context of type II superstringtheory, we now reobtain some of the above results inthe light-front gauge. We define

(5.1)X± = X0 ± Z√2

, P± = P0 ± PZ√2

and set

(5.2)X+ = t, P− = −p+

for positive constantp+. The Hamiltonian constraintmay now be solved for the light-front HamiltoniandensityH = −P+, while the reparametrization con-straint implies that

(5.3)(X−)′ = P · Y ′

p+ .

The bosonic Lagrangian density becomes

(5.4)

Lbos= P · Y − 1

2p+(|P |2 + |Y ′|2) − ∂t

(p+X−)

,

from which one sees that

(5.5)P = p+Y .


Using (5.3), we may rewrite the light-front gaugebosonic Hamiltonian density as

(5.6)Hbos= 1

2p+∣∣P ∓ Y ′∣∣2 ± ∂σX

−.

For periodicY the integral ofHbos over one period isminimized, for fixed increase"X− of X− over oneperiod, when

(5.7)P = ±Y ′.

Given this relation one can show that

(5.8)(X−)′ = ±|Y ′|2p+ , X− = |Y ′|2

(p+)2,

and hence that

(5.9)"X− = ± 1

p+

∮dσ |Y ′|2,

where the integral is over one period. As(X−)′ =−√

2Z′, a string with zeroY ′ also has zeroZ′ and istherefore pointlike. Conversely, ifY ′ is non-zero thenso is"Z, so we have a periodic wave along an infinitestring in theZ-direction.

To determine the fraction of supersymmetry pre-served by solutions satisfying (5.7), we return to thesupersymmetry preservation condition (2.5). Imposingthe conditions

(5.10)Γ+ε = 0, Γ�ε = ±ε,

which are equivalent to (2.9), we find that (2.5) be-comes an identity when (5.7) is satisfied, by virtue ofthe relations (5.3), (5.8) and (5.5). Thus, as expected,1/4 supersymmetry is preserved.

The full IIA superstring action in the light-frontgauge of course includes the Green–Schwarz world-sheet fermionsS± in the 8s and 8c spinor represen-tations ofSO(8). The phase-space Lagrangian densitygeneralizing (5.4) is (omitting total derivatives)

(5.11)L = P · Y − i(S+)T S+ − i(S−)T S− −H,

where the Hamiltonian density is

H = 1

2p+[|P |2 + |Y ′|2 − i(S+)T (S+)′

(5.12)+ i(S−)T (S−)′].

The angular-momentum 2-form generalizing thebosonic expression of (4.1) is

Lij = 1

2π

∮dσ

[YiPj − YjPi − 1

4(S+)T γij S

+

(5.13)− 1

4(S−)T γij S

−],

whereγij is the antisymmetrized product of a pair ofSO(8) Dirac matrices.

The phase space action is invariant under thesupersymmetry transformations generated by

(5.14)Q± = 1

2π√

2p+

∮dσ (P ± Y ′) · γS∓.

We note here for future use that

K = − 1

2π

∮dσ

[P · Y ′ − i

2(S+)T (S+)′

(5.15)− i

2(S−)T (S−)′

]

is invariant under these transformations. WhenP +Y′ = 0, the functionalK becomes the supersymmetricextension of the right-hand side of (4.3), and henceequal to the quantity|"PZ| held fixed in derivingthe classical angular momentum bound. The fermionsmake no difference classically but we shall needto consider them in the following discussion of thequantum-mechanical angular momentum bound.

6. Quantum mechanical bound

The worldsheet fields can be expressed as Fourierseries, the coefficients of which become annihilationand creation operators in the quantum theory. Inparticular, for zero total transverse momentum, wehave the oscillator expansions

P ∓ Y ′ = 2∞∑n=1

√n

[a±n e±inσ + (

a±n

)†e∓inσ

],

(6.1)S± = s± +∞∑n=1

[S±n e±inσ + (

S±n

)†e∓inσ

],

where a±n and S±

n are, respectively, the boson andfermion oscillator annihilation operators. It follows


that

(6.2)

Q± =√

2

p+∞∑n=1

√n

[a±n · γ (

S±n

)† + (a±n

)† · γ S±n

],

and also that

(6.3)K =∞∑n=1

n(NB

n +NFn

),

whereNBn andNF

n are, respectively, the boson andfermion number operators for the oscillators of thenthFourier mode.

One-string states are built by the action of thecreation operators on the oscillator vacuum

(6.4)|0〉 = |0〉+ ⊗ |0〉−,

which is annihilated by bothQ+ andQ−. However,because of the fermion zero modess±, the one-stringvacuum is actually the tensor product of an8v ⊕8c SO(8) multiplet of |0〉+ vacua with an8v ⊕ 8s

multiplet of |0〉− vacua. The bosonic vacua are thusin the

(6.5)1 ⊕ 8 ⊕ 28 ⊕ 35 ⊕ 56

tensor representation ofSO(8). Initially, at least, wewill consider only theSO(8) singlet because we canthen ignore the effects of the fermion zero modes onthe angular momentum. Thus we may take the one-string vacuum to be the oscillator vacuum (6.4). Theone-string states of the form

(6.6)|ψ〉 = |ψ〉+ ⊗ |0〉−with |ψ〉+ �= |0〉+ are annihilated byQ− but not byQ+, and hence preserve 1/4 supersymmetry. These,and the analogous states annihilated byQ+ (but notbyQ−) are, as is well-known, the 1/4-supersymmetric‘BPS’ states of the quantum IIA superstring theory (inlight-front gauge).

Henceforth we shall need to consider only the ‘+’oscillators so we omit the ‘+’ superscript. A basis forthe |ψ〉+ factor in (6.6) is

(6.7)∞⊗n=1

|B〉n ⊗ |F 〉n,

where

|B〉n =8∏

i=1

[(ain

)†]Bin |0〉Bn ,

(6.8)|F 〉n =8∏

α=1

[(Sαn

)†]Fαn |0〉Fn .

HereBin andFα

n are non-negative integers, and|0〉Bnand|0〉Fn are the ground states of thenth bosonic andfermionic oscillators, respectively. Note that each ofthe states (6.7) is an eigenstate of the bosonic andfermionic number operators for thenth oscillator witheigenvalues

(6.9)Bn =8∑

i=1

Nin, Fn =

8∑α=1

Fαn .

Bosonic BPS states are the subset of these states forwhich

∑∞n=1Fn is even (since we chose to construct

them on a bosonic one-string vacuum).Given the oscillator expansion of the worldsheet

fields, the angular momentum 2-formLij is now theoperator

(6.10)Lij =∞∑n=1

[2i

(a†n

)[i (an)j ] − 1

2S†nγij Sn

]+ · · · ,

where we have omitted the zero mode contribution,and the ellipsis indicates terms involving the ‘−’ oscil-lators that are irrelevant to matrix elements involvingonly the BPS states. SinceLij commute with the os-cillator number operators, states of the form (6.8) forfixed Bn andFn can be assembled intoSO(8) multi-plets. Each of these multiplets may be characterized byits Dynkin labelsw = (w1,w2,w3,w4), in terms ofwhich the quadratic Casimir ofSO(8) takes the form[8]

(6.11)J 2 =4∑

a,b=1

Gabwa(wb + 2

),

where

(6.12)Gab =

1 1 1/2 1/21 2 1 1

1/2 1 1 1/21/2 1 1/2 1

.

Note that the formula (6.11) is independent ofn,whereas the expression forK is such that the higher


the level of an oscillator, the larger is its contributionto K.

We seek states that maximizeJ 2 for fixed K, orequivalently states that minimizeK for fixed J 2. Tothis end let us writeK in terms of the separate bosonand fermion contributions asK = KB +KF . Supposefor the moment thatKF = 0; in this case it is clearthat the angular momentum will be maximized forfixed K whenBn = 0 for n > 1, and henceB1 = K,because we may otherwise reduceK without changingJ 2 by the substitutionn → 1. This argument failsfor the Fermi oscillators becauseFn � 8, for all n,due to the Pauli exclusion principle. Thus, once weinclude Fermi oscillators it is no longer obvious thatmaximizingJ 2 for fixedK requires all oscillators withn > 1 to be in their ground states. Nevertheless, itseems to be true for small values ofK and in thesecasesall fermion oscillators are in their ground states.An obvious conjecture is that this will remain truefor all K. We will examine the consequences of thisconjecture below but we should first stress that it isnot needed in the semi-classical limit of largeK, forthe following reason. IfK is large then eitherKB orKF , or both, must be large. But ifKF is large thenmost of the fermion oscillators must be in levels withn � 1 because of the exclusion principle, so a largenumber of fermions would make a large contributionto K for correspondingly small contribution toJ 2 (ascompared to boson oscillators). Thus, for largeK,the angular momentum will be maximized in somestate for whichKB � KF , and this means that thefermion oscillators contribute negligibly to the angularmomentum in the semi-classical limit; we may thenassume, for simplicity, that all fermion oscillators arein their ground states. Note that the effect of choosinga non-singletSO(8) irrep in the decomposition (6.5)also has a negligible effect in the limit of largeJ , thusjustifying our simplifying assumption that the one-string vacuum is anSO(8) singlet.

Since the state with maximalJ for fixedK has allbosonic oscillators withn > 1 in their ground states,the assumption that all fermion oscillators are in theirground states leaves only BPS states of the form

(6.13)∏i

[(ai

1

)†]K |0〉+ ⊗ |0〉−.

These states transform in the[(8v)K ]sym representa-

tion of SO(8). Within this reducible representation, the

SO(8) irrep corresponding to the tracelessKth ranktensor has the largest value ofJ 2; this irrep has Dynkinlabels

(6.14)w = (K,0,0,0).

Using (6.11) we deduce thatJ 2 = K2 + 6K for thisrepresentation, and hence the bound

(6.15)J 2 � K2 + 6K,

with equality for the representation (6.14). Dependingon the validity of the conjecture above, and the effectof choosing a non-singlet representation in (6.5), thisbound may require adjustment for smallK, but it iscertainly valid in the semi-classical limit of largeK,for which it reduces toJ � K. We have derived thisbound in the light-front gauge but the result can ofcourse be applied in any gauge; in particular in the‘physical’ gaugeX0 = Z′ = 1, in which K equals|"PZ|. We thus recover the classical bound (4.16).

7. Conclusions

This Letter was motivated by a desire to better un-derstand the properties of D2-brane supertubes of gen-eral cross section by considering the TST-dual IIA su-perstring configuration. This is a 1/4-supersymmetricIIA string carrying both momentum and angular mo-mentum that we have called a ‘supercurve’; it includesas a special case the ‘superhelix’ described for D-strings in [3]. In fact, IIA supercurves just provide analternative description of a IIA string carrying a left orright moving wave of arbitrary profile. As it has longbeen known that such configurations preserve 1/4 su-persymmetry, this may help demystify the result of [5]that 1/4 supersymmetry allows a supertube to have anarbitrary cross-section inE8.

The main focus of this Letter has been on the an-gular momentum carried by supercurves. The angularmomentum of a supertube is subject to an upper boundthat is saturated by supertubes of planar and circu-lar cross-section [1,2,5]. The latter have superhelicesas their TST duals, so one would expect superhelicesto saturate an upper bound on the angular momentum(for fixed linear momentum). A classical bound of thistype was derived in [7] in a slightly different context.We have given a much simpler proof of it, and a re-finement to the case in which one fixes the rank of


the angular momentum 2-form in addition to the linearmomentum.

We have shown how the classical bound on theangular momentum can be recovered from the semi-classical limit of a quantum bound derived within thecontext of perturbative IIA superstring theory. Thereis a uniqueSO(8) representation for which the angularmomentum is maximal. We would certainly expectthat many of the properties of quantum supercurvescontinue to hold for quantum supertubes, and that wasanother motivation for their study. Our results lead usto expect that a quantum supertube of maximal angularmomentum will again be associated to a uniqueSO(8)representation.

Acknowledgements

We are grateful to Jerome Gauntlett, Jaume Gomis,Joaquim Gomis and Daniel Waldram for helpful dis-cussions. D.M. is supported by a PPARC fellowship.

S.N. is supported by the British Federation of WomenGraduates and the Australian Federation of UniversityWomen (Queensland).

References

[1] D. Mateos, P.K. Townsend, Phys. Rev. Lett. 87 (2001) 011602,hep-th/0103030.

[2] R. Emparan, D. Mateos, P.K. Townsend, J. High EnergyPhys. 07 (2001) 011, hep-th/0106012.

[3] J.-H. Cho, P. Oh, Phys. Rev. D 64 (2001) 106010, hep-th/0105095.

[4] D. Bak, A. Karch, Nucl. Phys. B 626 (2002) 165, hep-th/0110039.

[5] D. Mateos, S. Ng, P.K. Townsend, J. High Energy Phys. 03(2002) 016, hep-th/0112054.

[6] A. Dabholkar, J.P. Gauntlett, J.A. Harvey, D. Waldram, Nucl.Phys. B 474 (1996) 85, hep-th/9511053.

[7] O. Lunin, S.D. Mathur, Nucl. Phys. B 610 (2001) 49, hep-th/0105136.

[8] R. Slansky, Phys. Rep. 79 (1981) 1.

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