31 May 2001

Physics Letters B 508 (2001) 381–384www.elsevier.nl/locate/npe

Hodge duality on the brane✩

M.J. Duff, James T. LiuRandall Laboratory, Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA

Received 21 March 2001; accepted 16 April 2001Editor: M. Cvetic

Abstract

It has been claimed that whereas scalars can be bound to a Randall–Sundrum brane, higherp-form potentials cannot, incontradiction with the Hodge duality between 0-form and 3-form potentials in the five-dimensional bulk. Here we show thata 3-form in the bulk correctly yields a 2-form on the brane, in complete agreement with both bulk and brane duality. Wealso emphasize that the phenomenon of photon screening in the Randall–Sundrum geometry is ruled out by the bulk Einsteinequation. 2001 Published by Elsevier Science B.V.

It is well known that in the Randall–Sundrum brane world picture [1], a massless scalar field in the(d + 1)-dimensional bulk results in a massless scalar being bound to thed-dimensional brane. In recent work [2], however,it is claimed that masslessp-form fields withp � (d − 2)/2 in the bulk yield no massless fields on the brane. Inparticular, only scalars are bound to the brane ind = 4. The argument runs as follows. The action for a masslessp-form potentialA[p] with (p + 1)-form field strengthF[p+1] = dA[p] in the(d + 1)-dimensional bulk is given by

(1)Sbulk =∫

dd+1x

[− 1

2(p + 1)!√

−g gM1N1gM2N2 · · · gMp+1Np+1FM1M2...Mp+1FN1N2...Np+1

].

Denoting the coordinatesxM = (xµ, z), the Kaluza–Klein ansatz for the metric is

(2)gMN dxM dxN = e−2k|z|gµν(x) dxµ dxν + dz2.

If the p-form ansatz is

(3)Aµ1µ2...µp (x, z) = Aµ1µ2...µp (x),

with all other components vanishing (so that the field strength reduces straightforwardly,F[p+1] = F[p+1]), thenthe resulting brane action is

(4)Sbrane=∫

ddx

[− 1

2(p + 1)!√−g gµ1ν1gµ2ν2 · · ·gµp+1νp+1Fµ1µ2...µp+1Fν1ν2...νp+1

]∫dz e2(p+1− d

2 )k|z|.

✩ Research supported in part by DOE Grant DE-FG02-95ER40899 Task G.E-mail address: jimliu@umich.edu (J.T. Liu).

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00520-2

382 M.J. Duff, J.T. Liu / Physics Letters B 508 (2001) 381–384

The criterion for being bound to the brane is the convergence of thez integral, or

(5)p <d − 2

2.

So the ansatz (3) would imply in particular that only 0-forms can be bound to the brane ind = 4.However, this argument is in contradiction with the well-known result that, in the absence of topological

obstructions [3], ap-form potentialA[p] in the bulk is dual to a(d −p −1)-form potentialA[d−p−1] with (d −p)-

form field strengthF [d−p] = d A[d−p−1]:

(6)√

−g gM1N1gM2N2 · · · gMd−pNd−p F N1N2...Nd−p = 1

(p + 1)!εM1M2...Md−pN1N2...Np+1FN1N2...Np+1.

Indeed, it is easy to see that the Kaluza–Klein ansatz (3) cannot be chosen for bothA[p] and A[d−p−1]simultaneously since it would not be compatible with (6).

The resolution of the paradox is to keep the ansatz (3) forp < (d − 2)/2 but modify it forp � (d − 2)/2. First,for p > d/2 we write

(7)Aµ1µ2...µp−1z(x, z) = e−2(p− d2 )k|z|Aµ1µ2...µp−1(x),

with other components vanishing, so that

(8)Fµ1µ2...µpz = e−2(p− d2 )k|z|Fµ1µ2...µp .

This is now perfectly compatible with (6). One finds

(9)√−g gµ1ν1gµ2ν2 · · ·gµd−pνd−p Fν1ν2...νd−p = 1

p! εµ1µ2...µd−pν1ν2...νpFν1ν2...νp ,

whereF[p] andF[d−p] result from ansatzes (3) and (7), respectively. So duality in the bulk implies duality on thebrane as it must for consistency. The brane action now describes a(p − 1)-form potential:

(10)Sbrane=∫

ddx

[− 1

2 · p!√−g gµ1ν1gµ2ν2 · · ·gµpνpFµ1µ2...µpFν1ν2...νp

]∫dz e−2(p− d

2 )k|z|.

The criterion for being bound to the brane, or convergence of thez integral, is now

(11)p >d

2.

In particular ford = 4, ap-form with p > 2 in the bulk yields a(p − 1)-form on the brane. This now resolvesthe original duality paradox. Thus a 3-form which is dual to a scalar in the five-dimensional bulk yields a 2-formwhich is dual to a scalar on the four-dimensional brane.

For the intermediate valuesp = d/2 andp = (d − 2)/2, neither thep-form nor the(p − 1)-form is bound tothe brane. Note in particular therefore that photons are not bound to the brane ind = 4 if they are described byMaxwell’s equations in the bulk. In this respect, we agree with [2,4].1

A few comments on the Kaluza–Klein reduction are now in order. Firstly, the ansatzes (3) and (7) are notarbitrary, but must be chosen to respect the bulkp-form equation of motion,

(12)∇M1FM1M2...Mp+1 = 0.

1 In the supersymmetric Randall–Sundrum brane world [5–7], the graviphotons on thed = 4 brane have a different origin in terms ofodd-dimensional self-duality equations in the bulk [8–10]. Similarly ind > 4, higherp-forms may be bound if their bulk origin is not givenby (1).

M.J. Duff, J.T. Liu / Physics Letters B 508 (2001) 381–384 383

In the Randall–Sundrum background, (2), this becomes

∇µ1Fµ1µ2...µp+1 + e−2(p+1− d2 )k|z|∂z

(e2(p− d

2 )k|z|Fzµ2µ3...µp+1

) = 0,

(13)∇µ1Fµ1µ2...µpz = 0.

Kaluza–Klein reduction of ap-form potential generically yields bothp-form and(p − 1)-form potentials on thebrane. A natural ansatz in the warped background would be to choose

(14)A[p](x, z) = A[p](x) + A[p−1](x) ∧ df (z),

wheref (z) is a priori an arbitrary function ofz. Note that there is a freedom of gauge choice in making theansatz (14). One may transform to a gaugeAµ1µ2...µp−1z = 0, whereupon (14) becomes

(15)A[p](x, z) = A[p](x) + (−1)pf (z)F[p](x).

This is the form of the ansatz used in [8,10]. Either way, substituting the resulting field strength

(16)F[p+1](x, z) = F[p+1](x) + F[p](x) ∧ df (z)

into (13), one then obtains thed-dimensional equations of motion,

(17)d ∗d F[p+1] = 0, d ∗d F[p] = 0,

along with the constraintf ′(z) = e−2(p− d2 )k|z|. This is the origin of the ansatzes (3) and (7).

While the ansatzes (14) may always be made, the localization argument presented above indicates that at mostonly one of eitherA[p] or A[p−1] is bound to the brane — the former forp < (d − 2)/2 and the latter forp > d/2.For intermediate values ofp, there is no binding, yielding the result that a(d/2)-form field strength on the branecannot originate from a bulk action of the form (1) (but may arise instead from odd-dimensional self-dualityequations). This is in contrast to ordinary Kaluza–Klein reduction on a circle, where both potentials survive in themassless sector.

Of course, (12) is not the only equation of motion we must consider. There is also the bulk Einstein equationwith cosmological constantΛ = −d(d − 1)k2:

RMN − 1

2gMNR

(18)

= −1

2gMNΛ + 1

2 · p![gM1N1gM2N2 · · · gMpNp FMM1M2...Mp FNN1N2...Np

− 1

2(p + 1)gMN gM1N1gM2N2 · · · gMp+1Np+1FM1M2...Mp+1FN1N2...Np+1

].

Substituting the ansatze (2) and (3) yields

(19)

Rµν − 1

2gµνR = 1

2 · p!e2pk|z|

[gµ1ν1gµ2ν2 · · ·gµpνpFµµ1µ2...µpFνν1ν2...νp

− 1

2(p + 1)gµνg

µ1ν1gµ2ν2 · · ·gµp+1νp+1Fµ1µ2...µp+1Fν1ν2...νp+1

]from theµν components, and

(20)R = 1

2(p + 1)!e2pk|z|gµ1ν1gµ2ν2 · · ·gµp+1νp+1Fµ1µ2...µp+1Fν1ν2...νp+1

from thezz component. Since consistency of the Einstein equation requires that thez-dependence cancel, this rulesout all butp = 0. Alternatively, it is straightforward to see by taking the trace of the first equation that (19) and (20)are inconsistent unlessp = 0.

384 M.J. Duff, J.T. Liu / Physics Letters B 508 (2001) 381–384

Similarly, substituting the ansatzes (2) and (7) yields

(21)

Rµν − 1

2gµνR = 1

2(p − 1)!e2(d−p−1)k|z|

[gµ1ν1gµ2ν2 · · ·gµpνpFµµ1µ2...µp−1Fνν1ν2...νp−1

− 1

2pgµνg

µ1ν1gµ2ν2 · · ·gµpνpFµ1µ2...µpFν1ν2...νp

]and

(22)R = − 1

2 · p!e2(d−p−1)k|z|gµ1ν1gµ2ν2 · · ·gµpνpFµ1µ2...µpFν1ν2...νp ,

which rules out all butp = (d − 1). So when Kaluza–Klein consistency of the coupled Einsteinp-form fieldequations is imposed, the onlyp-forms that can be bound to the brane are the 0-form and its (brane) dual the(d −2)-form. This result could not have been found merely by substituting the ansatz into the action and demandingconvergence of thez-integral. This highlights the well-known dangers of substituting a Kaluza–Klein ansatz intoan action rather than the equations of motion [11]. Note that ind � 4, the Einstein equation imposes no furtherrestrictions. In particular ind = 4, both 0-forms and 2-forms are still bound to the brane. Whereas ind = 6, forexample, only 0-forms and the dual 4-forms are bound to the brane; the 1-form allowed by (5) and 3-form allowedby (11) are ruled out.

This also has interesting consequences for the phenomenon of “screening” discussed in [2]. There, thep-formsfor which thez-integral diverged were interpreted as validd-dimensional modes, albeit modes whose charge hasbeen screened. However, we see that with the ansatzes (3) and (7), there are no solutions of the coupled Einsteinp-form field equations except forp = 0 andp = (d − 1), for which the integral converges. One may thus take oneof two attitudes. If one is content to take a test particle approach with no gravitational dynamics, then the screeningphenomenon may occur. However, if one takes the view that the combined gravitational andp-form dynamicsmust be taken into account, then if screening occurs at all, it cannot occur in the way described in [2] where theansatz (3) was employed.

Acknowledgements

We have enjoyed useful discussions with Hong Lü, Eva Silverstein, and Lisa Randall. We are especially indebtedto Chris Pope for pointing out the extra restrictions imposed by the Einstein equation. A somewhat differentapproach to the duality problem has recently been presented in [12].

References

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