Self Gravitating Fundamental Strings

Gary T. Horowitz; Joseph Polchinski

REVIEW

Self Gravitating Fundamental Strings

Gary T. Horowitz, Joseph Polchinski

High Energy Physics - Theory · Submitted 18 Jul 1997 · 11 references

Pith's verdict: The four-dimensional contraction law is solid and useful; the higher-dimensional collapse is an interpretation that the abstract states as a derived result.

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arXiv IDhep-th/9707170 (version 1)

Pith IDG4EUUDW2, Pith's permanent record of this paper and its review

CitationGary T. Horowitz, Joseph Polchinski. "Self Gravitating Fundamental Strings." arXiv:hep-th/9707170 (1997). https://pith.science/paper/G4EUUDW2

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Editor's note

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Two things worth your time here. First, the d=3 result: a string at level N, once g exceeds g_o ~ N^{-3/8}, sits at size l ~ alpha'^{1/2}/(g^2 N^{1/2}) and shrinks smoothly to the string scale by g_c ~ N^{-1/4}. That comes out of a clean mean-field treatment — one thermal-scalar loop in a self-consistent weak background — and the scaling argument is transparent. Second, the dimensional bookkeeping: g_o ~ N^{(d-6)/8} against g_c ~ N^{-1/4} tells you at a glance that d=3 contracts, d=4 turns on right at the black hole scale, d=5 leaves a gap, and d>=6 is out of reach of the original description. That organizing statement is the real contribution and the reason to read the paper. The appendix is not filler either: the checks that the stress tensor satisfies the correct divergence identity and integrates to the string mass are what make a mean-field source believable. Where it is soft: the abstract's higher-dimensional sentence (s3) says the string 'rapidly collapses to a black hole.' The body says 'we interpret this,' and the evidence is that the variational functional (3.18) is unbounded below for d>=5. An unbounded static functional at fixed normalization tells you the ansatz has no stable saddle point; it does not by itself choose collapse over fragmentation or over breakdown of the one-loop truncation. The interpretation is the natural one, but it is an interpretation, and the abstract should read that way. The d=3 contraction law does not depend on it. Smaller points: zeta_0 is left as an O(1) undetermined number, so everything is parametric; the d>6 dual-description paragraph is speculative and the authors flag it as such; and the 'hysteresis' language for d=5 is looser than the mean-field treatment actually supports. One correction to the record: the classifier reports no stated limitations. That is wrong. The paper is explicit about the mean-field step, the weak-field linearization, the undetermined eigenvalue, and footnote 1 on fluctuations and Hawking radiation. They are honest about their approximations; the only place the prose outruns them is that one abstract sentence. For anyone working on the correspondence principle or the Hagedorn transition this is required reading. Send it out.

Referee report

Significance

This is a well-motivated and carefully hedged scaling analysis that extends the correspondence principle in the string-to-black-hole direction. The d=3 result (3.21) is a clean scaling law with a nice internal consistency check at both endpoints: at g_o it reproduces the random-walk size N^{1/4}α'^{1/2}, and at g_c it gives the string scale. The identification of g_o and the dimension-dependent classification (d=3 smooth contraction, d=4 coincidence, d≥5 collapse, d>6 dual description) is a genuinely useful contribution, and the paper is honest about the mean-field nature of the calculation and about the O(1) coefficient ζ_0 being undetermined. The appendix's derivation of ⟨T_μν⟩ with the consistency checks (4.6) and (4.8) is a solid piece of supporting work.

Major objections (1)

[Abstract, sentence s3; §3, paragraph beginning 'For d = 4 and d = 5'] The abstract's third sentence states that in higher dimensions the string 'rapidly collapses to a black hole' as a result, but §3 presents this as an interpretation: 'We interpret this as saying that once the interaction becomes important the long string collapses all the way to a black hole.' The technical content of §3 is that the variational functional I in Eq. (3.18) is unbounded below for d ≥ 5, which establishes the absence of a stable mean-field saddle point; it does not by itself distinguish collapse-to-black-hole from fragmentation, a different saddle, or breakdown of the one-loop truncation. The authors acknowledge the interpretive step, so the remaining issue is that the abstract does not carry the hedge. Since the classifier identifies this as the paper's headline claim, either the abstract should say 'we argue' or 'we interpret,' or §3 should add a sentence explaining why the instability is toward ever-smaller ℓ and hence toward the black hole rather than another resolution. The d=3 result (3.21) is not affected by this point.

Minor objections (6)

[§3, Eq. (3.14)] The quantity defined as the 'dimensionless string coupling' g^2 = β_H^2κ^2/(2π^2(d-2)ω_{d-1}α'^2) has dimensions length^{d-3} once κ^2 ~ g_string^2 α'^{(d-1)/2} is used, so it is dimensionless only for d=3. The results g_o ~ N^{(d-6)/8} in Eq. (1.2) and the d=5 discussion therefore implicitly use string units (α'=1). Please state this convention explicitly or restore the α' factors, so that readers can check the dimensional consistency of (1.2) and (3.21).
[§1, Eq. (1.5)] As printed, the equation is dimensionally inconsistent (the left side has dimensions of inverse length while the right side has dimensions of length if g_c is dimensionless), and the exponent N^{7/6} is not obviously reproduced by substituting Eq. (1.4) together with the d=5 relation l_p ~ g^{1/2}α'^{1/2}, which gives N' ~ N^{9/8}. Please show the intermediate step or correct the exponent.
[§3, Eq. (3.11)] The linearization conditions h_ττ ≪ 1 and β_C^2-β_H^2 ≪ 1 are stated, but in d=3 at g ~ g_c one has g_c^2 M ~ 1, so the potential depth is O(1) precisely where the black hole is said to form. This is expected physically, but a sentence identifying where (3.11) fails would clarify the domain of validity of the contraction law (3.21).
[§3, paragraph after Eq. (3.2)] The claim that the quartic self-intersection interaction is 'less relevant' than the cubic gravitational one and 'can be neglected' would be strengthened by an explicit estimate of the quartic contribution at the scales ℓ ~ m^{-1} relevant to the bound state, rather than a relevance argument alone.
[§2 vs §3] §2 states that the thermal scalar 'has no apparent dynamical significance,' while §3 uses its bound-state wavefunction as the shape of the string and its saddle point as the criterion for collapse. A sentence reconciling these two uses (χ as a static bookkeeping device whose saddle point encodes the typical configuration) would help the reader.
[§3, final paragraph] The conclusion that 'most excited states of strings in higher dimensions never form a black hole for any value of the string coupling' rests on the assumption that the solitonic string decays rapidly into dual strings. The text hedges this ('one might expect,' 'might be quite slow'), but the sentence as written reads as a firm conclusion; please mark it explicitly as conditional on the decay rate.

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The paper

The paper claims

A single highly excited string, made to feel its own gravity as the coupling is raised, shrinks and eventually becomes a black hole — and how it does so depends on how many spatial dimensions it lives in.

The reading

The paper asks a concrete question about a single, isolated, highly excited string: as the string coupling $g$ is slowly turned up, does the string keep its free random-walk size until it is small enough to be a black hole, or do long-range gravitational and dilaton attraction shrink it first? The authors answer that it depends on the number $d$ of noncompact spatial dimensions, because the cubic coupling between the string and the graviton/dilaton background is relevant only below six dimensions. In four spacetime dimensions ($d=3$) the string contracts smoothly, with size $l$ set by $l^2 g^2 N^{1/2} = 1$, so it reaches the string scale exactly where a black hole of the same mass would form. In five and more dimensions the string stays at its random-walk size until a larger critical coupling, then collapses suddenly, producing a hysteresis loop between the string and black hole descriptions. The result matters because it tests the recently proposed correspondence principle from the string side: black holes must hand over not to typical random-walking strings but to long strings constrained to a volume of order the string scale, and the authors argue the entropy counting survives because constrained walks still have entropy of order $N^{1/2}$ times a constant.

What carries the argument

The argument is carried by the thermal scalar: a single complex scalar field $chi$ of winding number one, living in one fewer dimension, whose mass $m^2( beta) = ( beta^2 - beta_H^2)/(4 pi^2 alpha'^2)$ vanishes at the Hagedorn temperature and whose path integral reproduces the critical behaviour of the string density of states. Placing $chi$ in a static metric and dilaton background with the redshift factor $G_{tau tau}$ appearing explicitly (because it measures the proper length of the winding string) turns the problem into a nonlinear Schrödinger equation with an attractive Coulomb self-interaction, Eq. (3.13), whose coupling is the string coupling. A scaling argument on the associated variational functional then decides everything: the negative term scales as $rho^{d-2}$ and the positive gradient term as $rho^2$, so a lowest bound state exists only for $d=3$; for $d >= 5$ the functional is unbounded below, which the authors read as collapse to a black hole, and for $d=4$ the two terms scale alike so a critical coupling separates a stable from an unstable regime.

Extended reading notes

Core claim

The central claim is that the evolution of a highly excited string with increasing coupling is dimension-dependent, and that the crossover couplings can be computed. A free string at level $N$ has mass $M^2 = 4N/alpha'$ and random-walk size $l = N^{1/4} alpha'^{1/2}$; a black hole of that mass has horizon of order the string scale at $g_c ~ N^{-1/4}$. The long-range attraction first matters when the effective dimensionless cubic coupling $g M^{(6-d)/4}$ becomes of order one, giving $g_o ~ N^{(d-6)/8}$. For $d=3$ one finds $g_o < g_c$, and solving the resulting self-consistent bound-state equation gives $l ~ alpha'^{1/2}/(g^2 N^{1/2})$, so the string contracts continuously from the random-walk size to the string scale as $g$ runs from $g_o$ to $g_c$. For $d=4$ the two couplings coincide and the string stays large until it collapses; for $d=5$ one has $g_o > g_c$, so raising and then lowering the coupling traces different paths — a hysteresis. For $d=6$ the interaction is marginal and the string stays large until $g ~ 1$; for $d>6$ the authors note the original description breaks down and suggest passing to a dual weakly coupled theory, where the fate of the solitonic string depends on how fast it decays into the lighter dual degrees of freedom.

Load-bearing premise

The whole result rests on a mean-field step: the single string's stress energy is replaced by its average over all strings of the same mass, and the resulting self-consistent bound-state equation is trusted to describe one string — with the further interpretive step that where the variational functional has no minimum, the physical string collapses into a black hole.

Editorial extensions

If this is right

In four spacetime dimensions the correspondence principle works as originally stated from the string side: a typical excited string shrinks smoothly to the string scale and becomes a black hole at $g_c ~ N^{-1/4}$.
In five and more dimensions the black hole does not revert to a typical random-walking string; it becomes a long string constrained to a volume of order the string scale, so the entropy match must be sought among atypical, compact states.
The entropy count survives this restriction because constrained random walks still have entropy proportional to the number of steps, $N^{1/2}$, with only the numerical coefficient changed — a coefficient the correspondence principle never fixed anyway.
For a given level $N$, the black hole formed at $g_o$ in $d=5$ has mass $M_{bh} ~ N^{7/16}/l_p$, and lowering the coupling back to $g_c$ produces a string at a higher level $N' = N^{7/6}$; the dominant entropy of that black hole comes from these heavier compact strings.
In more than six dimensions the question is deferred to a dual weakly coupled description, where the fundamental string appears as a solitonic string that may or may not survive long enough to collapse.

Reading between the lines

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

The same scaling logic should apply to a string with angular momentum or electric charge, where the centrifugal or Coulomb terms compete with the self-attraction and could shift the crossover coupling; that would be a direct check of the mechanism rather than of the specific numbers.
The hysteresis in five dimensions is a statement about which description is valid, not about two different objects, so it suggests a concrete adiabatic-protocol thought experiment: cycle the coupling and ask whether work is done, which is the kind of question a holographic or free-energy computation could settle.
The instability of the variational functional for $d >= 5$ is a mean-field statement about a single string; if the collapse is real, one would expect it to show up as a runaway in a numerical simulation of a self-attracting polymer chain with a long-range $1/r^{d-2}$ attraction, a check that has not been done in the polymer literature the authors cite.
The authors' remark that the numerical coefficient in the entropy is outside the accuracy of the correspondence principle leaves open whether a sharper treatment of constrained walks could turn that coefficient into a quantitative prediction for black hole entropy in four dimensions.

Original abstract

We study the configuration of a typical highly excited string as one slowly increases the string coupling. The dominant interactions are the long range dilaton and gravitational attraction. In four spacetime dimensions, the string slowly contracts from its initial (large) size until it approaches the string scale where it forms a black hole. In higher dimensions, the string stays large until the coupling reaches a critical value, and then it rapidly collapses to a black hole. The implications for the recently proposed correspondence principle are discussed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,095 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.15 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the configuration of a typical highly excited string as one slowly increases the string coupling. The dominant interactions are the long range dilaton and gravitational attraction. In four spacetime dimensions, the string slowly contracts from its initial (large) size until it approaches the string scale where it forms a black hole. In higher dimensions, the string stays large until the coupling reaches a critical value, and then it rapidly collapses to a black hole. The implications for the recently proposed correspondence principle are discussed.

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	We study the configuration of a typical highly excited string as one slowly increases the string coupling.	contribution 1.00	0.88	0.04	asserted 0.81
s1	The dominant interactions are the long range dilaton and gravitational attraction.	result 0.53	0.48	0.06	asserted 1.00
s2	In four spacetime dimensions, the string slowly contracts from its initial (large) size until it approaches the string scale where it forms a black hole.	result 1.00	0.79	0.24	asserted 1.00
s3	In higher dimensions, the string stays large until the coupling reaches a critical value, and then it rapidly collapses to a black hole.	result 1.00	0.80	0.29	asserted 1.00
s4	The implications for the recently proposed correspondence principle are discussed.	interpretation 1.00	0.42	0.04	unclear 0.92

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.52.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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 (11)

[1]

G. T. Horowitz and J. Polchinski, Phys. Rev. D55 (1997) 6189, hep- th/9612146

View on Pith 1997
[2]

L. Susskind, hep-th/9309145

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[3]

A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99, hep- th/9601029

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[4]

For reviews and further references see J. Maldacena, hep-th /9607235, hep-th/9705078; G. Horowitz, gr-qc/9704072

View on Pith Pith review arXiv
[5]

R. Emparan, hep-th/9704204; H. Sheinblatt, hep-th/9705054; S. Das, hep-th/9705165; S. Mathur, hep-th/9706151

View on Pith Pith review arXiv
[6]

P. Salomonson and B.-S. Skagerstam, Nucl. Phys. B268 (1986) 349; Physica A158 (1989) 499; D. Mitchell and N. Turok, Phys. Rev. Lett. 58 (1987) 1577; Nucl. Phys. B294 (1987) 1138

View on Pith 1986
[7]

B. Sathiapalan, Phys. Rev. D35 (1987) 3227; I. A. Kogan, JETP Lett. 45 (1987) 709; J. J. Atick and E. Witten, Nucl. Phys. B310 (1988) 291

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[8]

D. Lowe and L. Thorlacius, Phys. Rev. D51 (1995) 665

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[9]

R. Hagedorn, Nuovo Cim. Suppl. 3 (1965) 147; K. Huang and S. Weinberg, Phys. Rev. Lett. 25 (1970) 895; S. Fubini and G. Veneziano, Nuovo Cim. A64 (1969) 1640; S. Frautschi, Phys. Rev. D3 (1971) 2821; R. D. Carlitz, Phys. Rev. D5 (1972) 3231

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[10]

R. Brandenberger and C. Vafa, Nucl. Phys. B316 (1989) 391. 19

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[11]

P. J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca (1971); P. G. De Gennes, Scaling Concepts in Polymer Physics, Cornell Univer- sity Press, Ithaca (1984); J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press, Cambridge (1996). 20

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String gas cosmology is a coherent but unfinished string-theory alternative to inflation, with a proposed Hagedorn-phase origin for scale-invariant perturbations and several acknowledged open problems.
Exploring String Theory Solutions: Black hole thermodynamics with $\alpha'$ corrections and type II compactifications
hep-th 2025-05 unverdicted novelty 1.0 of 10

A doctoral thesis restating published results on α'-corrected black holes in heterotic string theory and on type II flux compactifications, with no new standalone results.

How later papers used it

97 later papers cite it

Evidence: substantial. 97 of the citing papers share no author with it, and 29 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 244 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 2 used its method or tool, 1 compared its own results with it and 1 cited it as support.

Uses its method or tool 2 of 9 papers with this label

Based its approach on the formalism for self-gravitating string states near the Hagedorn temperature developed by Horowitz and Polchinski in this paper.

String Theory in a Pinch: Resolving the Gregory-Laflamme Singularity (2024) · 0.99 · paraphrased: the licence does not allow quoting
In the next section, we start with the Horowitz-Polchinski (HP) equations of motion, [ref] , associated with the $SL(2,ℝ)_k/U(1)$ cigar, and show that the order of the equations can be reduced to accommodate the FZZ duality.

A Puncture in the Euclidean Black Hole (2021) · Introduction · 0.82

Compares with it 1 of 12 papers with this label

Showed that its first sub-leading correction, due entirely to zero modes of the massive worldsheet scalars, exactly matches the sub-dominant contribution recently computed within an effective approach based on this paper's string star model.

Semiclassical Quantization of the Superstring and Hagedorn Temperature (2023) · 1.00 · paraphrased: the licence does not allow quoting

Cites it as support 1 of 8 papers with this label

Said that massless interactions, namely the gravitational one, dominating the mass shift for very massive strings is expected on general reasons, citing this paper.

String Mass Shifts (2009) · 0.95 · paraphrased: the licence does not allow quoting

How the literature uses it

96 arXiv papers cite this in 239 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 · 161 sentences without a confident label

Contests it 1 paper

The approach we follow goes beyond those used so far, which assumed some sort of approximation in the description of strings (random-walk, polymers, thermal scalar, …) and led sometimes to results actually being in contradiction (see [ref] ).

Self-interacting fundamental strings and black holes · Introduction · contest 0.72

Builds on it 6 papers

For $d≥6$, where the Horowitz-Polchinski EFT [ref] does not have suitable solutions, the modified effective Lagrangian allows one to study the black hole CFT in an expansion in powers of $d-6$ and $T_H-T$.

On Small Black Holes in String Theory · Introduction · 0.95
This is an extension of the EFT for the winding mode together with the metric around flat space [ref] to more general cases that include curved metric and (possibly) background fluxes such as the Kalb-Ramond flux $H_3 0$.

The black hole/string transition in AdS$_3$ and confining backgrounds · Winding-string condensates in type II string theory · 0.96
In section [ref] we review the solution, slightly extending the work of [ref] by providing the explicit form of the solution, giving the form of the solution in dimensions $D=4,5,6$ and presenting a formula for the classical entropy.

On the black hole/string transition · Introduction · 0.97

Uses its method or tool 9 papers

Our approach is based on the formalism for self-gravitating string states near the Hagedorn temperature developed by Horowitz and Polchinski (HP) in [ref] .

String Theory in a Pinch: Resolving the Gregory-Laflamme Singularity · Introduction · 0.99
The main idea will be to use the correspondence principle formulated in [ref] , following earlier suggestions in [ref] .

How to Expose a Black Hole · Introduction · 0.90
In the next section, we start with the Horowitz-Polchinski (HP) equations of motion, [ref] , associated with the $SL(2,ℝ)_k/U(1)$ cigar, and show that the order of the equations can be reduced to accommodate the FZZ duality.

A Puncture in the Euclidean Black Hole · Introduction · 0.82

Compares with it 12 papers

As $β→β_H$, the solution of the modified equations for all $d≤6$ goes to zero for all $r$, as in [ref] .

On Small Black Holes in String Theory · Introduction and summary · 0.82
The properties of the W star are reminiscent of the Horowitz-Polchinski solution [ref] .

String stars, small black holes, and the supersymmetric index · Properties of the W star · 1.00
As we will show explicitly, the first sub-leading correction, which is entirely due to the zero modes of the massive worldsheet scalars, exactly matches with the sub-dominant contribution recently computed within an effective approach [ref] based on the Horowitz-Polchinski [ref] string star model.

Semiclassical Quantization of the Superstring and Hagedorn Temperature · Introduction · 1.00

Cites it as support 7 papers

To maintain ([ref]) one is forced to speculate that this apparent mismatch is removed by accounting for the self-gravitation of the long string [ref] .

Hagedorn temperature and physics of black holes · Long-string picture of L. Susskind · 0.97
This is indeed the case for the size of a string and a black hole [ref] .[The size of a highly excited string can be estimated by a random walk picture which provides $√(M)$, on the other hand the size of a black hole becomes on the order one at the correspondence point.

Massless radiation from heavy rotating string and Kerr/string correspondence · The correspondence · 0.97
In fact, precisely when $β_BH ≳β_H$ (with $β_BH-β_H ≪ℓ_s$) it is the Horowitz-Polchinski saddle the one that dominates the dynamics of the system [ref] .

The Quantum Gravity Scale and the Swampland · Determining $$ via black holes · 0.97

Cites it as background 30 papers

According to the correspondence principle [ref] , at some particular value of $g_s$, the description of the system changes from that of a black hole to that of a fundamental string.

How to Expose a Black Hole · Review of the correspondence principle · 0.86
In this section, we briefly review the Horowitz-Polchinski model [ref] , which describes self-gravitating strings near the Hagedorn temperature [ref] .

Fluid model of a black hole-string transition · Horowitz-Polchinski model · 0.90
After reduction on the Euclidean time circle, the low-energy effective theory in the $d$ non-compact directions contains the spatial metric $g_ab$, the dilaton $Φ$, the radion $φ$, and the thermal scalar $χ$, with action [ref] [equation] Here $ℛ$ is the Ricci scalar of $g_ab$ and $G_N$ is Newton's constant in $D$ spacetime dimensions.

Love at the String Scale: Tidal Deformability Across the Black Hole-String Transition · Horowitz-Polchinski string stars · 0.94

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.

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G. T. Horowitz and J. Polchinski, Phys. Rev. D55 (1997) 6189, hep- th/9612146

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L. Susskind, hep-th/9309145

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For reviews and further references see J. Maldacena, hep-th /9607235, hep-th/9705078; G. Horowitz, gr-qc/9704072

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R. Emparan, hep-th/9704204; H. Sheinblatt, hep-th/9705054; S. Das, hep-th/9705165; S. Mathur, hep-th/9706151

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B. Sathiapalan, Phys. Rev. D35 (1987) 3227; I. A. Kogan, JETP Lett. 45 (1987) 709; J. J. Atick and E. Witten, Nucl. Phys. B310 (1988) 291

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D. Lowe and L. Thorlacius, Phys. Rev. D51 (1995) 665

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