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REVIEW 4 major objections 5 minor 9 references

Strings, Topological Change and Dark Matter

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims dark matter consists of 'dark charges' at the ends of Kalb-Ramond strings on a 3-sphere brane, and that this turns the standard halo density profile into an exact Einstein-Maxwell solution.

desk verdict A speculative string-theory dark matter model whose one solid math result is reverse-engineered and whose one-sign charge asymmetry is stipulation, not physics. read the letter →

arxiv 1909.01077 v1 pith:6TXQWGGL submitted 2019-08-12 physics.gen-ph gr-qchep-th

Gerald E. Marsh This is my paper · ORCID
classification physics.gen-phgr-qchep-th
keywords darkmatterKalb-RamondstringschargechargeddustEinstein-MaxwellequationsLane-Emdenequationtopologicalchange3-spherebrane
topics Dark Matter
open problems Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes that dark matter is made of 'dark charges'—the endpoints of open Kalb-Ramond strings that terminate on a 3-sphere brane representing the early universe. Because each endpoint behaves like a point charge under a Maxwell-type field, and because the model arranges for only one sign of dark charge to end up inside the observable space, the dark matter acts as charged dust in which electrostatic repulsion exactly balances gravitational attraction ($|q|=\sqrt{G}\,m$). Feeding this into the Einstein-Maxwell equations gives an exact density profile $\rho(r)=\rho_0 a^2/(a^2+r^2)$, the same form used to fit dark-matter halos as an approximate isothermal Lane-Emden solution. If the scenario is right, the standard halo profile is not a fitting function but a consequence of string-endpoint dark charge. The paper also argues that quantum tunneling could carry the universe from $\mathbb{S}^3$ to a negatively curved space, matching recent curvature data.

What carries the argument

The load-bearing identity is the extremal charged-dust condition $|q|=\sqrt{G}\,m$, which makes the mutual repulsion of same-sign dark charges exactly balance gravitational attraction. The metric is the static Majumdar-Papapetrou form $ds^2=f^2dt^2-f^{-2}(dr^2+r^2d\Omega^2)$, and Bonnor's equation relating $f$ to the density is the machine that converts the guessed $f(r)=\sqrt{(4\pi\rho_0/3)(a^2+r^2)}$ into the exact profile $\rho(r)=\rho_0 a^2/(a^2+r^2)$. Two supporting mechanisms carry the rest: Kalb-Ramond string endpoints supply the charges, and small boundary components around each charge make the Kalb-Ramond potential single-valued on $\mathbb{S}^3$, so that a global $B_{\mu\nu}$ can exist.

What would settle it

A first-principles computation of open-string pair creation on a brane during inflation would settle the sign asymmetry: if the first-created endpoint has equal probability of either sign, the single-sign population and the charged-dust equilibrium are not guaranteed. Observationally, a dark-matter halo whose inner density follows $\rho_0 a^2/(a^2+r^2)$ but whose outer envelope falls off faster than $r^{-2}$ would also contradict the exact-solution claim, since the exact solution has divergent total mass.

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Extended reading notes

Core claim

The central claim is that the dark matter in galactic halos can be identified with the charged endpoints of Kalb-Ramond strings terminating on a brane. In this reading the endpoint charges are not ordinary electric charges; they are 'dark charges' whose fields satisfy Maxwell's equations, in line with the original Kalb-Ramond interpretation of an electromagnetic-type interaction between string endpoints. The constructive result is that for the Majumdar-Papapetrou metric of charged dust in equilibrium, Bonnor's equation admits the exact solution $f(r)=\sqrt{(4\pi\rho_0/3)(a^2+r^2)}$, which yields the density $\rho(r)=\rho_0 a^2/(a^2+r^2)$. This is precisely the profile usually obtained as an approximate solution of the isothermal Lane-Emden equation, so the model converts a fitting formula into an exact Einstein-Maxwell solution. The paper further claims that a single sign of dark charge can dominate inside $\mathbb{S}^3$ if string creation is asymmetric in time and sign, and that quantum topological change allows the transition from $\mathbb{S}^3$ to an open, negatively curved universe.

Load-bearing premise

The model collapses if string creation is not asymmetric in favor of one sign of dark charge: Section 4 assumes one end of a charged string is always created first, and the first end always carries the same sign, so only one sign accumulates inside $\mathbb{S}^3$.

Editorial extensions

If this is right

  • Dark-matter halos with the profile $\rho(r)=\rho_0 a^2/(a^2+r^2)$ are exact solutions of the coupled Einstein-Maxwell equations for charged dust, so flat rotation curves can be read as an equilibrium between gravity and dark-charge repulsion.
  • Dark matter in this model has a fixed charge-to-mass ratio $|q|=\sqrt{G}\,m$; if the dark charge is one elementary unit, the dark-matter mass is forced to roughly $3.6\times10^{-9}$ kg, near the reduced Planck mass.
  • The observable universe should contain a net excess of one sign of dark charge, because inflation separated the two sign populations and sent the opposite sign effectively to infinity.
  • The early universe can begin as $\mathbb{S}^3$ and undergo a quantum topological transition to an open negative-curvature space, consistent with recent indications of negative spatial curvature.
  • Because the exact profile has infinite total mass, isolated halos cannot extend forever; the profile is adequate for galaxies in clusters, but an isolated halo with a finite outer edge would require a modification of the model.

Reading between the lines

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

  • The paper leaves implicit that if dark halos are same-sign charged dust, the dark charge should exert its own repulsive pressure; comparing the shapes of merging-cluster dark-matter halos with purely gravitational simulations would bound the dark charge-to-mass ratio.
  • If the first-created endpoint sign is a local accident rather than a global rule, different causal patches could end up with opposite net dark charge, predicting compensated regions or dark-charge domain walls in large-scale structure.
  • The coincidence between the minimal dark-charge mass and the reduced Planck mass opens a search channel the paper does not develop: compact dark objects of roughly Planck mass could be looked for through gravitational lensing or timing, rather than through particle-scattering searches.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes that dark matter consists of dark charges at the endpoints of open Kalb-Ramond strings terminating on a 3-sphere brane in the early universe. Sections 1-3 supply string-theory background, a two-ball model of S^3, and a homology argument for a single-valued Kalb-Ramond potential when endpoint charges are isolated by boundary components. Section 4 stipulates that only one sign of dark charge appears in the interior of S^3, with the opposite sign carried to infinity by inflation during a GUT phase transition. Section 5 sketches a quantum-cosmological topological transition from a positively curved to a negatively curved spatial section. The quantitative core is in Sections 6-7: Eq. (6.3) gives the standard pseudo-isothermal dark-matter halo profile rho = rho0 r0^2/(r0^2+r^2) as an approximate Lane-Emden solution, and Eq. (7.6) presents a function f(r) which, substituted into Bonnor's equation (7.3) of Majumdar-Papapetrou charged-dust theory, yields Eq. (7.7), an exact Einstein-Maxwell density profile of the same algebraic form. The author concludes that dark matter may be such charged dust with charge-to-mass ratio sqrt(G).

Significance. The algebraic core of the paper is sound: the function f(r) of Eq. (7.6) does satisfy Bonnor's equation (7.3) and produces the density profile (7.7) with the stated normalization, and the paper is commendably explicit about its assumptions and about the infinite-total-mass limitation of the solution. The observation that an exact Majumdar-Papapetrou solution reproduces the pseudo-isothermal profile is mildly interesting but is an inverse construction, not a derivation. As a dark-matter model, the paper does not support its central claim: the one-sign charge asymmetry of Section 4 is assumed without mechanism and conflicts with the compact S^3 geometry; the empirical agreement is engineered through the free choice of f(r), a, and rho0; and the model introduces long-range dark self-interactions of gravitational strength whose compatibility with the collisionless behavior inferred from the Bullet Cluster observations (cited in the paper itself) is never examined. These gaps, rather than any disagreement with current consensus, are what prevent the manuscript from establishing its conclusions.

major comments (4)
  1. [Section 4 and Fig. 3.1] The one-sign asymmetry that the entire equilibrium rests on is stipulated in Section 4: "it will be assumed here that one end of a charged string is created first and the other a very short time later," and "It is also assumed that because the string charge has a vector character that the first end always has the same sign dark charge." No mechanism from string theory, GUT cosmology, or elsewhere is given, and the assumption is load-bearing because the charged-dust equilibrium of Section 7 requires a region containing charges of a single sign with |q| = sqrt(G) m so that electrostatic repulsion can balance gravity. The assumption is also in tension with the paper's own geometry: in the two-ball model of S^3 used in Figs. 2.2 and 3.1, the point at infinity is a point of S^3, and a Kalb-Ramond string carries charges of opposite sign at its two ends, so the Section 4 scenario places the compensating opposite-sign charges on S^3 at infinity. The paper never shows that the long-range dark fields of those distant charges are negligible throughout the interior, and never addresses the Gauss-law constraint on a closed spatial manifold, according to which the total charge must vanish; the asymptotically flat Majumdar-Papapetrou solution of Section 7 is therefore not obviously realizable inside the proposed compact spatial section. Since this premise is the only thing that prevents the dark charges from neutralizing or from losing the repulsive support of the equilibrium, the central claim is unsupported at its foundation.
  2. [Section 7, Eqs. (7.3)-(7.7)] The agreement between the exact solution and the empirical halo profile is by construction. On page 22, the paper states that "The question addressed here is whether it is possible to find a function f(r) that would result in a radially unlimited density distribution matching that given in Eq. (6.3) for dark matter," and then exhibits f(r) in Eq. (7.6) precisely to reproduce Eq. (7.7). Both a and rho0 are free parameters, and the target profile (6.3) itself comes from truncating the Chandrasekhar series solution of the isothermal Lane-Emden equation to the first two terms. The "exact" match of Eq. (7.7) with Eq. (6.3) is therefore a consistency check of an inverse construction, not a prediction of the dark-charge model, and it carries no independent evidence for the physical identification of dark matter with string-endpoint charges; the claimed surprise that the profiles coincide is an artifact of having chosen f(r) to force that coincidence. The paper acknowledges that both profiles share the unattractive feature of infinite total mass, which further limits the empirical significance of the match.
  3. [Section 6, p. 19, and Section 7, p. 21] Section 6 cites the Bullet Cluster results as supporting the standard picture that dark matter "only interacts gravitationally," but the proposed model gives dark matter a long-range dark Maxwell force whose strength, at the extremal condition |q| = sqrt(G) m, is equal to the gravitational force. Dark matter with self-interactions of gravitational strength would be expected to behave very differently from collisionless cold dark matter in halo dynamics and in cluster mergers, including the very Bullet Cluster separation of dark matter and baryons cited in the Introduction. The paper provides no estimate of the effective self-interaction cross-section of the dark charges and no discussion of whether the long-range force is compatible with the observational constraints it invokes; this is a testable consequence of the model that is left unexamined.
  4. [Sections 5-7] There is no dynamical bridge between the early-universe scenario and the local static solution of Section 7. Section 4 describes inflation sweeping opposite-sign charges to infinity, and Section 5 describes a quantum topological transition from S^3 to a negatively curved spatial section, but nothing in Sections 5-7 explains why the surviving same-sign dark charges would assemble into spherically symmetric, static Majumdar-Papapetrou equilibria with the scale radius a and central density rho0 of Eq. (7.7), nor how the equilibrium is reached starting from the GUT-era string population. The scale radius and central density are free parameters with no derivation from the string or cosmological inputs, so the model does not connect its microscopic ingredients to the observed halo parameters.
minor comments (5)
  1. [Section 6, Eq. (6.3)] The definition of r0 below Eq. (6.3) is typeset as r0 = 6K/4piG rho0; as written the units are inconsistent, and the text likely intends r0^2 = 6K/(4piG rho0), with the factor 6 versus 9 issue already noted by the author.
  2. [Section 6, Eqs. (6.1)-(6.3)] The derivation of Eq. (6.3) from the isothermal Lane-Emden equation should be stated more precisely: the two-term truncation of Chandrasekhar's power series yields the quadratic core of the profile, not the full algebraic form rho0 r0^2/(r0^2+r^2), which is the empirical modified isothermal profile.
  3. [Section 7, p. 21] The numerical example giving a minimal mass of about 3.6e-9 kg from the electron charge should be shown explicitly; the value depends on the electrostatic unit convention, and the quoted agreement with the reduced Planck mass is not transparent as written.
  4. [Section 5, Eq. (5.2)] The notation k(t)^{1/2} below Eq. (5.2) is confusing when k passes through zero and changes sign; the transition should be stated in terms of the complexified metric of Eq. (5.6), which is the object actually used by Martin et al.
  5. [Throughout] Several equations are garbled in the typeset text (for example, Eqs. (1.8)-(1.10) and (7.6)), the name "Friedmann-Lemaître-Robinson-Walker" in Section 2 should be "Friedmann-Lemaître-Robertson-Walker", and the fractional exponents in Eqs. (7.4)-(7.6) should be re-checked against the author's earlier paper.

Circularity Check

2 steps flagged · score 6.0 of 10

Halo density 'match' is reverse-engineered via f(r); one-sign dark-charge asymmetry is assumed rather than derived.

  1. fitted input called prediction [Section 7, Eqs. (7.3)-(7.7), target Eq. (6.3) in Section 6]
    "The question addressed here is whether it is possible to find a function f(r) that would result in a radially unlimited density distribution matching that given in Eq. (6.3) for dark matter. Indeed, one can. Substitution of [Eq. (7.6)] into Eq (7.3) yields rho(r)=rho0 a^2/(a^2+r^2), where a is now a free constant. This has the same form as Eq. (6.3) except that now the equality is exact and r(r) is derived from a solution of the Einstein-Maxwell field equations."

    Eq. (6.3) is the target halo profile the paper aims to reproduce. Instead of deriving f(r) from string theory or from an independent physical principle, Eq. (7.6) is selected precisely so that Bonnor's equation (7.3) returns that same profile. The output rho(r)=rho0 a^2/(a^2+r^2) therefore reduces to the input Eq. (6.3) by construction, with rho0 and a left as free parameters. The phrase 'derived from a solution of the Einstein-Maxwell field equations' describes an inverse-engineering ansatz, not a prediction; the paper itself quotes Lemos and Zanchin that the procedure is 'an art of correct guessing.' The claimed match is guaranteed by the choice of f(r).

  2. self definitional [Section 4 (and Summary), one-sign dark-charge assumption]
    "In what follows, it is necessary that only one sign of the dark charge at the ends of Kalb-Ramond strings appear within S^3. That there exists at least one possible scenario for this to occur will be shown here. ... To resolve this creation enigma, it will be assumed here that one end of a charged string is created first and the other a very short time later. It is also assumed that because the string charge has a vector character that the first end always has the same sign dark charge."

    The section announces it will exhibit a scenario producing only one sign of dark charge inside S^3, but the 'scenario' consists of assuming that the first-created string end always has the same sign dark charge. The required property is thus the premise, not a derived consequence of string theory, GUT, or inflation. The Summary later calls this 'an argument ... given,' but Section 4 labels it an assumption. The charged-dust equilibrium of Section 7 and the resulting density profile depend on this stipulated asymmetry, so a load-bearing part of the model is question-begging.

full rationale

The central quantitative claim, that the proposed dark matter gives the halo density profile rho(r)=rho0 a^2/(a^2+r^2), is not independently predicted. In Section 7 the paper explicitly asks for an f(r) that reproduces the empirical profile Eq. (6.3), then chooses such an f(r) and substitutes it into Bonnor's equation to recover the same profile exactly. This is a fitted input called a prediction: the equality is constructed, and rho0 and a are free parameters. Separately, the one-sign dark-charge asymmetry needed for the charged-dust equilibrium is stipulated in Section 4 as an assumption about string creation, so the model's physical support is partly question-begging. The self-citation to G. E. Marsh 2013 in Section 6 is present, but Section 7 reproduces the calculation, so the citation itself is not the load-bearing source and does not add to the circularity score. The topological-change and Kalb-Ramond material is independent background, but it does not determine the density profile or the charge asymmetry. Overall, the paper contains partial circularity: the halo-profile 'match' is by construction, and the one-sign premise is assumed rather than derived.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim rests on several ad hoc assumptions: the existence and Maxwell-like behavior of dark charges, the one-sign charge asymmetry, the S^3 brane setup, and quantum topological change. The density profile itself is reproduced by choosing f(r) to fit the target, with the scale radius and central density left free. No independent evidence is provided for the invented dark charge entity.

free parameters (3)
  • a (halo scale radius) = varies per galaxy; not predicted
    In Eq. (7.6) and Eq. (7.7), a is stated to be a free constant setting the length scale of the density profile. The model does not derive it from string theory or cosmology.
  • rho0 (central dark matter density) = varies per galaxy; not predicted
    rho0 appears in the f(r) ansatz and in the resulting density profile, but no mechanism predicts its value for a given galaxy. It is an empirical scale.
  • minimal dark charge q0 = 10^-19 C (assumed)
    Section 7 chooses the minimal dark charge to be equivalent to one electron charge to estimate a minimal mass near the reduced Planck mass. This choice is not derived from the string model.
assumptions (5)
  • ad hoc to paper Dark charges and their fields obey Maxwell's equations.
    Section 1 states: 'These fields will nevertheless be assumed to obey the Maxwell equations.' This is not implied by string theory and is introduced specifically to use the charged dust formalism.
  • ad hoc to paper Only one sign of dark charge appears within S^3.
    Section 4 assumes one string end is created first, always with the same sign dark charge, and inflation separates the opposite sign to infinity. This asymmetry is load-bearing for the charged dust equilibrium.
  • ad hoc to paper The very early universe is an S^3 brane with Kalb-Ramond strings terminating on it.
    Section 2 chooses the brane to be S^3 to allow spin structure and topological change. There is no independent evidence that the early universe was an S^3 brane in the required string theory sense.
  • domain assumption Quantum topological change from S^3 to a negatively curved space is possible.
    Section 5 relies on Martin et al. and De Lorenci et al. for semiclassical Wheeler-De Witt topological transitions, and notes that the Green function computation is not possible. This is a contested, non-standard assumption.
  • domain assumption Dark matter halo density profiles are approximately described by the isothermal Lane-Emden solution.
    Section 6 takes the standard empirical profile Eq. (6.3) as the target. This profile is an approximation to the isothermal Lane-Emden equation and is not exact for real halos.
invented entities (1)
  • dark charge and associated dark Maxwell field
    purpose: To act as dark matter at string endpoints and to produce a charged dust density profile matching dark matter halos.
    Section 1 defines dark charges as charges at string endpoints that obey Maxwell equations but are not electromagnetic. No mass, coupling, or detection signature is predicted, so there is no falsifiable handle outside the paper.

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Pith. "Pith review of Strings, Topological Change and Dark Matter." pith.science (2026). https://pith.science/paper/6TXQWGGL

@misc{pith2026190901077,
  author       = {Pith},
  title        = {Pith review of: Strings, Topological Change and Dark Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6TXQWGGL}},
  note         = {Machine review of arXiv:1909.01077}
}
read the original abstract

Dark matter, first postulated by Jacobus Kapteyn in 1922 and later by Fritz Zwicky in 1933, has remained an enigma ever since proof of its existence was confirmed in 1970 by Vera Rubin and Kent Ford by plotting the rotation curve for the Andromeda galaxy. Here, some concepts from string theory and topological change in quantum cosmology are used to formulate a new model for dark matter. The density profiles of dark matter halos are often modeled as an approximate solution to the Lane-Emden equation. Using the model proposed here for dark matter, coupled with previous work showing that the approximate solution to the Lane-Emden equation can be an exact solution of the Einstein-Maxwell equations, provides a new insight into the possible nature of dark matter.

Figures

Figures reproduced from arXiv: 1909.01077 by the authors.

Figure 2.1
Figure 2.1. The two-ball model of 𝕊𝟑 given by the union of the surface of two 3-balls given by ℎ: 𝜕𝐵 E " → 𝜕𝐵 F " so that 𝕊𝟑 = 𝐵 E " ∪° 𝐵 F " . The point q is on the surface (boundary) of 𝐵 E " and h(q) is on the surface of 𝐵 F " . If 𝒫 ⊂ 𝐵 F " is set equal to ¥, as will later be the case, then 𝐵 F " = {𝒫¥} ∪ (ℝ𝟑 − Int 𝐵 E " ), where “Int” means interior. {𝒫¥} ∪ (ℝ𝟑 − Int 𝐵 E " ) is a topological ball with center at infinity. T… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figures from the paper (1 more)
Figure 5.1
Figure 5.1. Figure 5.1: Observed magnitude versus redshift plotted for well-measures distant Type Ia supernovae. [Adapted from S. Perlmutter, "Supernovae, Dark Energy, and the Accelerating Universe", Physics Today, April 2003.] 6. A String Model for Dark Matter† The density profiles of dark…

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Distribution of Dark Matter in the Spiral Galaxy NGC 3198

    STRINGS, TOPOLOGICAL CHANGE AND DARK MATTER Gerald E. Marsh Argonne National Laboratory (Retired) gemarsh@uchicago.edu ABSTRACT Dark matter, first postulated by Jacobus Kapteyn in 1922 and later by Fritz Zwicky in 1933, has remained an enigma ever since proof of its existence was confirmed in 1970 by Vera Rubin and Kent Ford by plotting the rotation curve...

    work page 1985

  2. [2]

    Every FLRW model is a 4-dimensional submanifold (hypersurface) in this 5-dimensional space

    D3-branes and Friedmann-Lemaître-Robinson-Walker cosmological models It has been shown by Lachièze-Rey9 that all FLRW pseudo-Riemannian manifolds can be embedded in a flat 5-dimensional Minkowski manifold with Lorentzian signature. Every FLRW model is a 4-dimensional submanifold (hypersurface) in this 5-dimensional space. In what follows, the 3-dimensiona...

    work page 1904

  3. [3]

    Figure A1

    Figure A1 shows a charged Kalb-Ramond string terminating on a 2-sphere. Figure A1. A charged Kalb-Ramond string terminating on a 2-sphere with its dark end charges isolated by boundary components. The closed curve C1 is homologous to zero (because it lies on a 2-sphere), but C2 is not. In Fig. A1, the bulk is 3-space. Without the string, the homology is g...

    work page 2004

  4. [4]

    Isothermal spheres and charged dust

    The hydrostatic balance equation may then be integrated to yield 𝜌=𝜌Y 𝑒𝑥𝑝(−Φ/𝐾), † Much of this section, in the context of electric charge, originally appeared in: G. E. Marsh, “Isothermal spheres and charged dust”, J. Phys. Astron. 2, (2013). 20 (6.1) where F is the gravitational potential. F/K must then be a solution of the isothermal Lane-Emden equatio...

    work page 2013

  5. [5]

    Supernovae, Dark Energy, and the Accelerating Universe

    Evolution of the universe and topological change The problem with considering 𝕊𝟑 as a model for the very early universe, is that it is now known that the universe is not closed and is either flat or hyperbolic should the matter density be below the critical value even by a small amount. In 1967, Geroch20 showed that changes in the topology of spacelike se...

    work page 1967

  6. [6]

    A note on flat metric connections with antisymmetric torsion

    pp. 189-195. 4 For additional discussion see the 2002 PhD dissertation by Terry Glenn Pilling (North Dakota State University of Agriculture and Applied Science). 5 I. Agricola, “A note on flat metric connections with antisymmetric torsion”, Diff. Geom. and Appl. 28, (2010), pp. 480-487. 6 E. Zaslow, Mirror Symmetry in T. Gowers, Ed. The Princeton Companio...

    work page 2010

  7. [104]

    On the embedding of spacetime in higher-dimensional spaces with torsion

    8 M. Kalb and P. Ramond, “Classical direct interstring action”, Phys. Rev. D 9, 2273 (1974). See also R. Ademollo, et al., “Theory of an Interacting String and Dual Resonance Model”, CERN Report No. TH-1702 (1973). 9 M. Lachièze-Rey, Astron. Astrophys. 364, 894-900 (2000) 10 C. Romero, et al., “On the embedding of spacetime in higher dimensional spaces wi...

    work page Pith review arXiv 1974

  8. [523]

    Marsh, An Introduction to the Standard Model of Particle Physics for the Non-Specialist (World Scientific, New Jersey 2018), p

    7 G.E. Marsh, An Introduction to the Standard Model of Particle Physics for the Non-Specialist (World Scientific, New Jersey 2018), p

    work page 2018

Show all 9 references
  1. [2009]

    Einstein-Cartan Theory

    11 A. Trautman, “Einstein-Cartan Theory”, Encyclopedia of Mathematical Physics, Edited by J.P. Francoise, GL. Naber and Tsou S.T. (Elsevier, Oxford 2006), Vol. 2, pp. 189-195. 12 J.W. Milnor and J.D. Stasheff, Characteristic Classes (Princeton University Press, New Jersey 1974...

    1988 arXiv

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