Localization of Matters on a String-like Defect

Ichiro Oda

REVIEW

Localization of Matters on a String-like Defect

Ichiro Oda

High Energy Physics - Theory · Submitted 26 Jun 2000 · 28 references

Pith's verdict: A clean, correct zero-mode calculation with one genuinely new result — spin-1 localization on a codimension-2 defect — built on a background whose stability and deficit-angle consistency are never checked.

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arXiv IDhep-th/0006203 (version 2)

Pith IDM5M5JFJR, Pith's permanent record of this paper and its review

CitationIchiro Oda. "Localization of Matters on a String-like Defect." arXiv:hep-th/0006203 (2000). https://pith.science/paper/M5M5JFJR

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Editor's note

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The thing worth knowing: Oda's spin-1 result is real, and it is the reason to read this. On a domain wall in AdS5 a vector field does not localize; here, on a codimension-2 string-like defect, it does, because the angular part of the metric carries its own exponential e^{-c_1 r} and that makes I_1 converge. The algebra holds up — I went through I_0, I_1, I_{1/2}, I_{3/2}, and inequalities (36) and (44) follow from the integrals plus c^2 in Eq. (24). The general solutions (17)-(20) are also a genuine step past Gregory's sourceless case (21), and the author is honest that for p=4 the c-piece drops out of I_1 and everything rides on c_1. That kind of self-observation is a good sign about the author's care, not just the result's convenience. What is soft: this is a zero-mode normalizability check and nothing more. There is no linearized stability analysis of (23), no KK spectrum, no massive modes. That is standard for the genre, so I would call it a limitation rather than a flaw — but the paper should say so out loud instead of letting the word 'localized' do more work than it can bear. The one place I would push harder is the boundary conditions (27). The paper calls them 'not directly relevant' and moves on. They are not free: they tie the deficit angle to c_1 and R_0, and the localization window (44) constrains c_1. Nobody checks that the resulting delta lands in (0, 2 pi), or that t_theta = beta e^{cr} + alpha corresponds to a source with positive energy density. The stress-test's specific formula for delta is not something I verified, but the gap it points at is real. The fermion result — increasing warp factor needed — is acknowledged in the abstract and discussion, so I will not count it against them. Who it is for: people building with two large extra dimensions, and anyone who needs a gravity-only vector localization mechanism. Worth a referee's time; the spin-1 claim should be checked and the boundary-condition consistency demanded.

Referee report

Major objections (2)

[§4.2, Eq. (44)] The vector localization window is derived correctly, but the paper does not check that this window is compatible with a physical string source. The deficit-angle boundary conditions (27) are explicitly set aside as 'not directly relevant to the present analysis for the localization,' but the window constrains c_1 and hence the deficit angle; the paper should verify that the window yields a sensible deficit angle and energy conditions, or state explicitly that this is assumed. Without this, the headline result is conditional on an unverified physicality assumption, and the reader cannot tell whether the window corresponds to a realizable cosmic string.
[§3, Eq. (23)] The stability of the background (23) is not analyzed. The zero-mode integrals are computed for a fixed background; if the background is unstable to linearized perturbations, the localization analysis is formal and does not describe a physically realized configuration. The paper should acknowledge this limitation and cite the stability and fluctuation analyses already in the literature (for example, the Csaki et al. paper [17] and the Gherghetta-Shaposhnikov paper [19] already cited). This is a standard limitation of the genre, but it should be stated rather than left implicit.

Minor objections (8)

[§4.3–4.4] The fermion localization requires c<0, while the boson localization requires c>0. The paper should make explicit that these are different backgrounds, so no single defect localizes all fields without additional interactions. The abstract's phrasing ('provided that additional interactions are not introduced') is ambiguous and should be clarified to say that fermions localize on the c<0 background and that localizing them on the c>0 background requires additional interactions.
[§4.3, Eqs. (56)–(59)] The gauge condition Ψ_θ = 0 is imposed without justification. A sentence explaining why this is consistent with the equations of motion, or how the residual gauge freedom is fixed, would help the reader.
[§3, Eqs. (25)–(26)] The n≥3 result is asserted ('we have checked') without derivation. A brief sketch or appendix showing how the e^B terms force the solution (25) would strengthen the paper.
[Abstract and §1] 'Three-half' should be '3/2' or 'three-halves'; 'spin one-half and three-half fields' is awkward phrasing.
[§4 intro] 'Variuos' is a typo for 'various'.
[Eq. (27)] The notation for the boundary conditions is ambiguous (the prime and the evaluation points at ε and 0). A clearer presentation would help the reader follow the deficit-angle construction.
[§4.1] The statement that 'normalizability of the ground state wave function is equivalent to the condition that the coupling constant is nonvanishing' is cryptic; a sentence of explanation would help.
[§4.1–4.2] The paper could note the overlap region where spin 0 and spin 1 both localize: for p≥4 the vector window (44) is a subset of the scalar window (36), so there is a range of t_θ where both fields are localized simultaneously. Making this explicit would strengthen the physical interpretation.

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The paper

The paper claims

On a defect with two extra dimensions, the spin-1 field localizes where it cannot on a five-dimensional domain wall, while fermions need an increasing warp factor and therefore extra interactions.

The reading

The paper asks whether the now-standard story of how Standard Model fields get trapped on a brane in five dimensions survives when the defect has two transverse dimensions instead of one. Oda first writes down warped solutions to Einstein's equations in general $D$ dimensions, with metric $ds^2 = e^{-cr}ds_4^2 + dr^2 + R_0^2 e^{-c_1 r}dtheta^2$, then goes through spins 0, 1/2, 1, 3/2 and 2 one at a time, computing the zero-mode wavefunction in the extra dimensions and checking whether its normalization integral converges. The result that carries the paper is the spin-1 case: on an AdS$_5$ domain wall a massless vector is never normalizable, but here the angular part of the metric carries its own exponential factor $e^{-c_1 r}$, and that factor is enough to make the vector integral finite for a decreasing warp factor and a suitable range of the angular stress $t_theta$. Fermions run the other way: spin-1/2 and spin-3/2 modes converge only for an increasing warp factor, so gravity alone will not trap them and some further interaction is required. If the analysis is right, the codimension-2 defect reproduces the five-dimensional localization table entry for entry except at spin 1, and that single exception is the entry that decides whether gauge fields need a separate trapping mechanism.

What carries the argument

The load-bearing device is the pair of zero-mode normalization integrals built from the two warp factors: with $P(r)=e^{-cr}$ and $Q(r)=R_0^2e^{-c_1r}$, the scalar and graviton require $I_0 = int dr, P^{p/2-1}Q^{1/2}$ finite, the vector requires $I_1 = int dr, P^{p/2-2}Q^{1/2}$ finite, and the fermions require $I_{1/2}=I_{3/2}=int dr, P^{p/2-1/2}Q^{1/2}alpha(r)^2$ finite after substituting the profiles $alpha propto P^{-p/4}Q^{-1/4}$ and $u propto P^{-(p-2)/4}Q^{-1/4}$ obtained from the Dirac and Rarita-Schwinger equations. Each integral reduces to $int_0^infty dr, e^{-lambda r}$ with a spin-dependent exponent, so localization becomes a sign condition on a linear combination of $c$ and $c_1$. The second ingredient is the background solution itself, in which $c_1$ is tied to the angular stress through $c_1 = c - 8kappa_D^2 t_theta/(pc)$ and $c^2 = (-8Lambda + 8kappa_D^2 t_theta)/(p(p+1))$; this is what lets the angular exponential differ from the radial one and is the sole reason the vector case departs from the domain wall.

Extended reading notes

Core claim

The paper's central claim is that a string-like defect with codimension 2 localizes a massless U(1) vector field on its world-volume, in contrast with the five-dimensional domain wall where no localized vector mode exists. The argument is a normalizability test: for each spin the author extracts the zero-mode profile $chi(r)$ in the transverse coordinates and requires $int_0^infty dr,sqrt{-g},|chi|^2$ to converge, which for the vector means requiring $I_1 = int_0^infty dr, P^{p/2-2}Q^{1/2}$ to be finite with $P=e^{-cr}$ and $Q=R_0^2e^{-c_1r}$. Because the angular metric component carries the extra exponential $e^{-c_1 r}$, $I_1$ converges for $c>0$ provided the angular stress satisfies $Lambda/kappa_D^2 < t_theta < -(p-3)Lambda/(4kappa_D^2)$; in the domain-wall limit the angular factor is trivial and the same integral diverges for any $c$, which is exactly the known no-go. The paper also reports that spin-0 and spin-2 modes share one localization condition (the two integrals are identical), and that spin-1/2 and spin-3/2 modes converge only for $c<0$, the increasing warp factor, so the fermions are not localized by gravity alone.

Load-bearing premise

The argument assumes that the specific warped background with its chosen source profile is a physically realized and stable configuration, and that finiteness of the zero-mode normalization integral is the correct test for localization; neither the stability of the background nor the realizability of the required range of the angular stress is checked.

Editorial extensions

If this is right

In a codimension-2 warped compactification with a decreasing warp factor, gauge fields need no separate confinement mechanism such as a bulk condensate; the warped angular geometry does the work.
The localization table for a string-like defect matches the five-dimensional domain wall for spins 0, 1/2, 3/2 and 2, with spin 1 as the single exception, so results derived for domain-wall phenomenology transfer except where vectors are concerned.
Trapping spin-1/2 and spin-3/2 fields requires interactions beyond gravity, which the author identifies as the motivation for embedding the model in six-dimensional supergravity.
Graviton and scalar zero modes obey identical localization conditions because $I_2 = I_0$, so any tuning that localizes the scalar also localizes the graviton.
The allowed window for the angular stress $t_theta$ is bounded on both sides, so vector localization is a constraint on the defect's matter content rather than an automatic feature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Substituting the background relation $c^2 = (-8Lambda + 8kappa_D^2 t_theta)/(p(p+1))$ into the vector window $Lambda/kappa_D^2 < t_theta < -(p-3)Lambda/(4kappa_D^2)$ converts it into a bounded range for the warp slope $c$ itself, not merely a sign condition; the paper states the inequalities in terms of $t_theta$ and never performs this substitution, so the practical size of the allowed parameter
Since the vector result depends entirely on the mismatch between the radial and angular warp slopes, and $c_1 - c$ is proportional to $t_theta/c$, the effect vanishes continuously as the two slopes are brought together; the domain wall is not a different theory so much as the degenerate limit of this one.
The paper never examines the stability of the background it uses, so a Kaluza-Klein tachyon in the full fluctuation operator could invalidate the zero-mode counting even where the normalization integrals converge; a Sturm-Liouville analysis of the transverse fluctuation operator would settle whether the localized modes are the lightest ones.
Because the localization window is narrow once the equation of state is imposed, the vector result may describe a fine-tuned corner of parameter space rather than a generic property of string-like defects, which is a testable distinction for anyone building a codimension-2 phenomenology.

Original abstract

After presenting string-like solutions with a warp factor to Einstein's equations, we study localization of various spin fields on a string-like defect in a general space-time dimension from the viewpoint of field theory. It is shown that spin 0 and 2 fields are localized on a defect with the exponentially decreasing warp factor. Spin 1 field can be also localized on a defect with the exponentially decreasing warp factor. On the other hand, spin one-half and three-half fields can be localized on a defect with the exponentially increasing warp factor, provided that additional interactions are not introduced. Thus, some mechanism of localization must be invoked for these fermionic fields. These results are very similar to those of a domain wall in five space-time dimensions except the case of spin 1 field.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,830 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.93 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

After presenting string-like solutions with a warp factor to Einstein's equations, we study localization of various spin fields on a string-like defect in a general space-time dimension from the viewpoint of field theory. It is shown that spin 0 and 2 fields are localized on a defect with the exponentially decreasing warp factor. Spin 1 field can be also localized on a defect with the exponentially decreasing warp factor. On the other hand, spin one-half and three-half fields can be localized on a defect with the exponentially increasing warp factor, provided that additional interactions are not introduced. Thus, some mechanism of localization must be invoked for these fermionic fields. These results are very similar to those of a domain wall in five space-time dimensions except the case of spin 1 field.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	After presenting string-like solutions with a warp factor to Einstein's equations, we study localization of various spin fields on a string-like defect in a general space-time dimension from the viewpoint of field theory.	contribution 0.99	0.91	0.07	asserted 0.61
s1	It is shown that spin 0 and 2 fields are localized on a defect with the exponentially decreasing warp factor.	result 1.00	0.91	0.22	asserted 1.00
s2	Spin 1 field can be also localized on a defect with the exponentially decreasing warp factor.	result 1.00	0.89	0.12	asserted 0.81
s3	On the other hand, spin one-half and three-half fields can be localized on a defect with the exponentially increasing warp factor, provided that additional interactions are not introduced.	result 0.99	0.87	0.22	asserted 0.64
s4	Thus, some mechanism of localization must be invoked for these fermionic fields.	interpretation 0.95	0.57	0.17	asserted 0.86
s5	These results are very similar to those of a domain wall in five space-time dimensions except the case of spin 1 field.	interpretation 0.75	0.73	0.18	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v2

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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (28)

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C. Csaki, J. Erlich, T.J. Hollowood and Y. Shirman, hep-th/00010 33

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T. Gherghetta and M. Shaposhnikov, hep-th/0004014

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M. Green, J. Schwarz and E. Witten, String Theory, vol.2 , Cambridge University Press (1988)

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[2]

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L. Randall and R. Sundrum, Phys.Rev.Lett. 83 (1999) 4690, hep-th/9906064

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[8]

A. Pomarol, hep-ph/9911294

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[9]

Y. Grossman and M. Neubert, Phys.Lett. B474 (2000) 361, hep-ph/9912408

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[10]

S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, he p-ph/9912498

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B. Bajc and G. Gabadadze, Phys.Lett. B474 (2000) 282, hep-th/9912232

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M. Cvetic and J. Wang, Phys.Rev. D61 (2000) 124020, hep-th/9912187

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I. Olasagasti and A. Vilenkin, hep-th/0003300

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P. Berglund, T. Hubsch and D. Minic, hep-th/0005162

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Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The End of the Road for Bulk Fields in Warped Randall-Sundrum Braneworlds
gr-qc 2026-01 reject novelty 6.0 of 10

In arbitrary-dimensional warped braneworlds, the derived local conditions leave only the free scalar and the NED model L(F)=b√F as consistent, localizable bulk fields.
Domain wall brane in squared curvature gravity
hep-th 2011-04 conditional novelty 5.0 of 10

An exact, smooth thick-brane solution in R + gamma R^2 gravity localizes the graviton zero mode and left-handed fermions, with massive fermion states trapped as resonances or bound states.
Fermions on Thick Branes in the Background of Sine-Gordon Kinks
hep-th 2008-04 conditional novelty 4.0 of 10

For a sine-Gordon thick brane, a sin(phi) scalar-fermion coupling produces an effective potential with multiple wells that can trap the right-chiral fermion zero mode for certain ranges of the kink parameter q, unlike...
Localization and Mass Spectrum of Matters on Weyl Thick Branes
hep-th 2008-03 conditional novelty 4.0 of 10

On a Weyl-geometry thick brane, scalars, vectors, and one chirality of fermions acquire normalizable zero modes, with the fermion bound-state count set by a free coupling constant.
Localization of Matters on Pure Geometrical Thick Branes
hep-th 2007-08 conditional novelty 4.0 of 10

On Weyl-geometry thick branes, scalar zero modes are normalizable, vector zero modes are not, and a special Yukawa coupling localizes exactly one chiral fermion mode.

Show 2 more citing papers

Large and infinite extra dimensions
hep-ph 2001-04 accept novelty 3.0 of 10

A pedagogical survey of brane-world models in which extra dimensions are large or infinite, covering how matter is trapped on a brane and what observable signatures large, warped or infinite extra dimensions would produce.
Introduction to Extra Dimensions and Thick Braneworlds
hep-th 2017-07 accept novelty 1.0 of 10

The paper surveys extra-dimension and thick-braneworld models, organizing known solutions and localization results into a single pedagogical review.

How the literature uses it

39 arXiv papers cite this in 149 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 75 sentences without a confident label

Contests it 2 papers

Therefore, these results correct the discussion performed in the reference [ref] for the $p$-form fields and generalize the results found in Refs. [ref] , for the scalar and the vector fields, to include its dual fields.

Consistency Conditions for $p$-Form Fields Localization on Codimension two Braneworlds · Application · contest 0.72
As a direct consequence, we conclude that the localization mechanisms proposed in Refs. [ref] are not compatible with the full set of consistency conditions derived here.

The End of the Road for Bulk Fields in Warped Randall-Sundrum Braneworlds · Conclusion · contest 0.91

Builds on it 8 papers

For this case, we reviewed the results obtained in Refs. [ref] for the scalar and the vector fields, and after generalized for a $p$-form in Ref. [ref] .

Consistency Conditions for $p$-Form Fields Localization on Codimension two Braneworlds · Application · 0.90
In a subsequent study, Ref. [ref] extended the application of condition ([ref]) to additional models and found that * Vector Field: * Higher-dimensional models: In contrast to the four-dimensional case, the zero modes of gauge fields in higher-dimensional braneworlds can lead to convergent integrals [ref] .

The End of the Road for Bulk Fields in Warped Randall-Sundrum Braneworlds · Generalization of Duff et al. consistency condition · 0.92
The above expression generalizes the one found in [ref] and in particular for $q=0$ and $q=1$ we recover the previous results.

Antisymmetric Tensor Fields in Codimension Two Brane-World · Localization of $q-$Form Zero Modes · 0.98

Uses its method or tool 1 paper

Following the usual approach, we shall use the spinor representation [ref] [equation]

Fermions in a warped resolved conifold · Fermion Localization · 0.93

Compares with it 5 papers

Therefore, we claim the resolution parameter alters slightly the KK modes near the brane and provides the same KK modes as for the exterior string-like model [ref] .

Gauge vector field localization on 3-brane placed in a warped transverse resolved conifold · Massive modes · 0.88
([ref]) as [equation] Note that equation ([ref]) is similar to that found in string-like geometries [ref] , regardless the cigar term $β(ρ,k)$.

Gravity localization in a string-cigar braneworld · Gravity localization · 0.99
[ref] regardless the factor $(α(r,a)β(r,a))^1/2$.

Scalar field localization on 3-branes placed at a warped resolved conifold · Massless modes · 1.00

Cites it as background 25 papers

In this background, the authors of Refs. [ref] discussed the localization of the free scalar and the free vector fields.

Consistency Conditions for $p$-Form Fields Localization on Codimension two Braneworlds · Conclusions · 0.83
It is well known that the massless scalar field [ref] and the graviton [ref] are localized on branes of different types, and that the spin 1 Abelian vector fields can not be localized on the Randall-Sundrum(RS) model in five dimensions but can be localized in some higher-dimensional cases [ref] .

Localization of Matters on Pure Geometrical Thick Branes · Introduction · 0.91
In five-dimensional spacetime, free scalar fields can typically be localized on branes [ref] , while fermion fields generally require Yukawa couplings with background scalar fields to achieve localization on branes [ref] .

Localization of matter fields on a chameleon brane · Introduction · 0.80

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [17]

C. Csaki, J. Erlich, T.J. Hollowood and Y. Shirman, hep-th/00010 33

View on Pith

[2] [19]

T. Gherghetta and M. Shaposhnikov, hep-th/0004014

View on Pith Pith review arXiv

[3] [1]

M. Green, J. Schwarz and E. Witten, String Theory, vol.2 , Cambridge University Press (1988)

View on Pith 1988

[4] [2]

J. Polchinski, String Theory, Cambridge University Press (1998)

View on Pith 1998

[5] [3]

L. Randall and R. Sundrum, Phys.Rev.Lett. 83 (1999) 4690, hep-th/9906064

View on Pith Pith review arXiv 1999

[6] [4]

J. Polchinski, ”Tasi Lectures on D-branes”, hep-th/9611050

View on Pith Pith review arXiv

[7] [5]

V.A. Rubakov and M.E. Shaposhnikov, Phys.Lett. B125 (1983) 136

View on Pith 1983

[8] [6]

G. Dvali and M. Shifman, Phys.Lett. B396 (1997) 64, hep-th/9612128

View on Pith Pith review arXiv 1997

[9] [7]

H. Davoudias, J.L. Hewett and T.G. Rizzo, Phys.Lett. B473 (2000) 43, hep-ph/9911262

View on Pith Pith review arXiv 2000

[10] [8]

A. Pomarol, hep-ph/9911294

View on Pith Pith review arXiv

[11] [9]

Y. Grossman and M. Neubert, Phys.Lett. B474 (2000) 361, hep-ph/9912408

View on Pith Pith review arXiv 2000

[12] [10]

S. Chang, J. Hisano, H. Nakano, N. Okada and M. Yamaguchi, he p-ph/9912498

View on Pith

[13] [11]

B. Bajc and G. Gabadadze, Phys.Lett. B474 (2000) 282, hep-th/9912232

View on Pith Pith review arXiv 2000

[14] [12]

N. Arkani-Hamed, L. Hall, D. Smith and N. Weiner, hep-ph/99124 53

View on Pith

[15] [13]

A. Chodos and E. Poppitz, Phys.Lett. B471 (1999) 119, hep-th/9909199

View on Pith Pith review arXiv 1999

[16] [14]

A.G. Cohen and D.B. Kaplan, Phys.Lett. B470 (1999) 52, hep-th/9910132

View on Pith Pith review arXiv 1999

[17] [15]

R. Gregory, Phys.Rev.Lett. 84 (2000) 2564, hep-th/9911015

View on Pith Pith review arXiv 2000

[18] [16]

M. Cvetic and J. Wang, Phys.Rev. D61 (2000) 124020, hep-th/9912187

View on Pith Pith review arXiv 2000

[19] [18]

I. Olasagasti and A. Vilenkin, hep-th/0003300

View on Pith Pith review arXiv

[20] [20]

P. Berglund, T. Hubsch and D. Minic, hep-th/0005162

View on Pith Pith review arXiv

[21] [21]

M. Chaichian, M. Gogberashvili, and A.B. Kobakhidze, hep-th/00 05167

View on Pith

[22] [22]

A. Chodos, E. Poppitz and D. Tsimpis, hep-th/0006093

View on Pith Pith review arXiv

[23] [23]

C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, San Francisco: Freeman (1973)

View on Pith 1973

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