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REVIEW 3 major objections 3 minor 25 references

Two-point tree-level string amplitudes as AdS transition amplitudes

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Pith's one-line read Two-point string amplitudes at tree level, in a tensionless limit, can be rewritten as sums of AdS boundary-to-boundary transition amplitudes, with open and closed strings related by worldsheet curvature.

desk verdict Suggestive idea, but the derivation's pivotal continuation is invalid and the final AdS weight assignment doesn't match the algebra. read the letter →

arxiv 2505.07358 v3 pith:2Q3ETAUJ submitted 2025-05-12 hep-th

Pongwit Srisangyingcharoen , Jongruk Pukdee This is my paper · ORCID
classification hep-th
keywords BosonicstringamplitudesAdS/CFTcorrespondenceTwo-pointTensionlesslimit'tHooftHeatkernelOpen/closedduality
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 computes two-point tree-level amplitudes of open and closed bosonic strings and claims that, after a Wick rotation and a tensionless 't Hooft-like limit, each amplitude can be rewritten exactly as a sum of Euclidean anti-de Sitter (AdS) boundary-to-boundary transition amplitudes for a scalar field. The open-string calculation leads to Eq. (44), a sum over partial transition amplitudes with conformal weights $\tilde\Delta_1=2(n-s)+d$ and $\tilde\Delta_2=2s+d$; the closed-string amplitude reduces to the same expression up to an overall constant. If this structural identity is right, flat-space string correlators at high energy carry AdS propagator structure without passing through a field-theory limit, giving a string-level view of how holographic dualities could emerge. The two 't Hooft couplings that make the match work are $\lambda=\alpha'/R$ for the disk and $\tilde\lambda=\alpha'\tilde R^2$ for the sphere, whose curvature scales are the geodesic and Gaussian curvatures respectively.

What carries the argument

The central machinery is the heat-kernel representation of the AdS bulk-to-boundary propagator, $K(t,\vec z;\vec z') = \frac{t^{(d-\Delta)/2}}{\Gamma(\Delta-d/2)}\int_0^\infty d\rho\, \rho^{\Delta-d/2-1} e^{-\rho}\langle \vec z| e^{t\Box/4\rho}|\vec z'\rangle$, together with the partial transition amplitude $\Gamma_j(x,\Delta_1;y,\Delta_2;t)$ of Eq. (12), which describes propagation from one boundary point to another through a bulk point. The string calculation is engineered to produce the same structure: an identity $1=\Gamma(n+1)^{-1}\int_0^\infty du\,u^n e^{-u}$ is inserted into the converted integral, the substitution $u=\rho_1+\rho_2$, $1/v=1/\rho_1+1/\rho_2$ factorizes the integrand, and Fourier completeness turns the exponential of $k^2$ into heat kernels $\langle y|e^{\lambda\Box/\rho_1}|z\rangle\langle z|e^{\lambda\Box/\rho_2}|x\rangle$. For the closed string this machinery is preceded by a negative-binomial expansion of $(\sin\theta)^{\alpha'k^2/2}$ and a zeta-regularization of the resulting divergent series.

What would settle it

Compute the original finite-radius integral in Eq. (32) directly at the on-shell value $k^2=-1/\alpha'$ without the contour rotation, and likewise the closed-string integral (53) without zeta-regularizing the divergent series, then compare numerically with the right-hand side of Eq. (44) in the same $\lambda$ limit; any mismatch would show the advertised AdS form is an artifact of the analytic continuations rather than a property of the original amplitude.

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Extended reading notes

Core claim

The central claim is that the two-point tree amplitude of open or closed bosonic strings in flat space is structurally identical to the boundary-to-boundary transition amplitude of a scalar field in Euclidean anti-de Sitter (AdS) space. Starting from the path integral on a disk (open) or sphere (closed), the amplitude reduces to an integral over the relative worldsheet insertion angle; rescaling that angle by the worldsheet radius, taking $R\to\infty$ for the disk or $R\to0$ for the sphere while holding $\lambda$ fixed, and rotating the contour turns the integral into $\int_0^\infty d\theta\, e^{-\lambda k^2\theta}$, the same heat-kernel building block used for the AdS bulk-to-boundary propagator. An inserted identity and a change of variables then converts the string amplitude into Eq. (44): a sum of partial transition amplitudes $\Gamma_j(x,\tilde\Delta_1;y,\tilde\Delta_2;4\lambda)$, with $\tilde\Delta_1=2(n-s)+d$, $\tilde\Delta_2=2s+d$, and the weights shifted by $\pm2$ inside the summand. The closed-string result is the same sum up to an overall constant, and the corresponding 't Hooft parameters are $\lambda=\alpha'/R$ and $\tilde\lambda=\alpha'\tilde R^2$, reflecting the geodesic curvature of the disk and the Gaussian curvature of the sphere.

Load-bearing premise

The argument treats the string integral as though its oscillatory integrand can be smoothly deformed into an exponential decay, and for the closed string it sums a divergent infinite series to the finite number $-1/12$; if either manipulation is not actually allowed, the advertised AdS form does not follow.

Editorial extensions

If this is right

  • If the identity holds, flat-space open and closed string two-point amplitudes at high energy ($\alpha'\to\infty$) are literally sums of AdS transition amplitudes, so AdS propagator structure emerges directly from string theory rather than from a field-theory approximation.
  • The same sum (44) describes both open and closed strings up to a constant, with $\lambda=\alpha'/R$ for the disk and $\tilde\lambda=\alpha'\tilde R^2$ for the sphere; this predicts that an open string on a disk of small geodesic curvature matches a closed string on a sphere of large Gaussian curvature.
  • The decomposition into partial transition amplitudes $\Gamma_j$ labels orderings of propagation through a common bulk point, so string amplitudes inherit a color-ordered interpretation in which the worldsheet boundary ordering fixes which bulk-to-boundary leg is traversed first.
  • Because the inserted identity carries an arbitrary integer $n$, the same amplitude admits a family of AdS decompositions with different conformal weights; choosing $n$ shifts the split between the two boundary weights in a controlled way.

Reading between the lines

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

  • If the same heat-kernel reorganization extends to higher-point tree amplitudes, string amplitudes would decompose into sums of AdS bulk diagrams with multiple interaction vertices rather than a single boundary-to-boundary line; the paper lists higher-point functions as future work and does not itself establish this.
  • For the closed string, replacing zeta-regularization by a manifestly finite definition of the original oscillatory integral would show whether the coefficient $-1/12$ is a genuine property of the amplitude or an artifact of the summation order; the paper does not perform this check.
  • The curvature map $R_{\rm disk}\leftrightarrow 1/\tilde R_{\rm sphere}^2$ suggests a quantitative open/closed duality: small geodesic curvature on the disk corresponds to large Gaussian curvature on the sphere; whether this persists beyond tree level or for massive states is left open.
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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

3 major / 3 minor

Summary. The paper computes tree-level two-point amplitudes for open and closed bosonic strings in flat spacetime using the path-integral formalism, and claims that in tensionless, 't Hooft-like limits these amplitudes can be reorganized into forms structurally identical to AdS boundary-to-boundary transition amplitudes. For the open string, the disk amplitude (32) is manipulated into a Laplace-type integral (34), then, through an inserted identity and a change of variables, into a heat-kernel expression (42) that is identified with a sum of AdS transition amplitudes (44). For the closed string, a similar reduction is attempted via a negative binomial expansion and zeta-function regularization, leading to Eq. (59). The paper concludes that the two-point open and closed string amplitudes are structurally equivalent to AdS transition amplitudes, with the 't Hooft couplings set by the geodesic curvature of the disk and the Gaussian curvature of the sphere, respectively. The central claim, however, rests on analytic continuations and algebraic matching steps that are not justified in the manuscript.

Significance. If the central claim were established, the paper would provide an elementary and concrete demonstration that flat-space string amplitudes in a tensionless limit can be recast as AdS transition amplitudes, with a novel geometric interpretation of open/closed duality through worldsheet curvature. The paper is clearly organized and transparently notes where it uses on-shell conditions and regularizations; this transparency is a strength. However, the advertised structural identity is not derived: the open-string reduction depends on an unsupported analytic continuation of the on-shell amplitude, the closed-string reduction combines a divergent series with an incorrect binomial coefficient and an unjustified regularization, and the final matching to AdS transition amplitudes contains an algebraic mismatch in the scaling dimensions. Because these issues are load-bearing for the paper's central result, the manuscript does not currently establish its main claim.

major comments (3)
  1. [§3, Eqs. (32)-(34)] The conversion of the finite-range on-shell disk amplitude (32) into the infinite Laplace-type integral (34) is not justified. The replacement (sin θ_-)^{-α'k^2} = Im(e^{-iα'k^2θ_-}) is valid only at the on-shell point α'k^2=-1, as the text itself acknowledges, but the steps that follow require k^2 to be an independent variable, since it later becomes the eigenvalue of \hat{k}^2 in Eq. (38). After the rescaling θ_- = θ/R and the R→∞ limit with λ=α'/R fixed, the on-shell value is k^2 = -1/(λR)→0, so there is no regime in which both the identity and the decaying weight e^{-λk^2θ} hold simultaneously. The Wick rotation θ→-iθ is applied to an imaginary part of an oscillatory factor; for real k^2<0 the rotated integrand grows rather than decays, and for k^2=0 the exponential carries no information. Thus Eq. (34), which is the pivot of the open-string derivation, does not follow from Eq. (32). The subsequent AdS identification therefore has no valid basis.
  2. [§4, Eqs. (53)-(58)] The closed-string reduction is invalid for several independent reasons. The negative binomial expansion (55)-(56) is used with |x|=|y|=1, where the series does not converge absolutely, so the interchange of the sum and the θ-integral cannot be justified. On shell, α'k^2=-4, so the coefficient in (56) is binom(-α'k^2/2+a-1, a) = binom(a+1, a) = a+1, not a as the text states; consequently the divergent series in (57) is Σ_a(a+1), not Σ_a a. Even if zeta-function regularization were accepted, Σ_{a=0}^∞(a+1) = ζ(-1)+ζ(0) = -7/12, not -1/12, so the prefactor in Eq. (59) is not the regularized value of the series written down. In addition, with λ=α'R^2 fixed and R→0, the on-shell value k^2=-4/α'→0, so the exponential e^{-λk^2θ} cannot serve as a Laplace weight for the on-shell amplitude; treating k^2 as an independent variable repeats the unsupported analytic continuation identified in the open-string case. These issues undermine the claimed closed-string result entirely.
  3. [§3, Eqs. (43)-(45)] The claimed representation (44) of the string amplitude as a sum of AdS transition amplitudes does not follow from the expansion (43). From (43), the ρ-dependent prefactor for term s is ρ_1^{n-s-1}ρ_2^{s-2}|ρ_1-ρ_2|. In an ordered integration region I_j, the absolute value produces two monomials with ρ-exponents (n-s-1, s-1) and (n-s, s-2). Matching these to the Γ_j integrand ρ_1^{Δ_1-d/2-1}ρ_2^{Δ_2-d/2-1} in Eq. (12) requires Δ_1 = n-s+d/2, Δ_2 = s+d/2 for the first monomial, and shifted values for the second, not Δ_1 = 2(n-s)+d, Δ_2 = 2s+d as stated in Eq. (45). The shifts Δ_1+2 and Δ_2-2 used in (44) also do not reproduce the exponents obtained from (43). The advertised identity between the string amplitude and the AdS transition amplitudes is therefore not established.
minor comments (3)
  1. [§4, Eq. (54)] After the rescaling θ_1 = 2θR^2, the argument of the sine should be R^2θ, not 2R^2θ; as written, Eq. (54) is inconsistent with the stated substitution.
  2. [§4, after Eq. (56)] The sentence 'the binomial coefficient is nothing but a parameter a' is incorrect even at the on-shell point; the coefficient is a+1, as noted in Major Comment 2.
  3. [§2, Eq. (10)] The generalized transition amplitude Γ(x, Δ_1; y, Δ_2) is introduced with different weights, but the paper does not discuss whether such an object satisfies any particular equation of motion in AdS; a brief comment on its interpretation would help the reader assess the analogy.

Circularity Check

2 steps flagged · score 6.0 of 10

Open-string AdS form reduces by construction via an arbitrary identity insertion; closed-string form is forced by zeta-regularization to match the open string.

  1. self definitional [Section 3, Eqs. (35) and (42)-(44)]
    "To mimic the form of the AdS transition amplitude (10), we insert an identity 1 = 1/Γ(n + 1) ∫_0^∞ du u^n e^{−u} with an arbitrary number n into (34)."

    Equation (34) is already a Laplace transform; inserting Eq. (35) is multiplication by 1 and adds no content. After the change of variables (36), the integrand in (42) is exactly a monomial-times-heat-kernel integral, which is the definition of Γ_j in (12). The binomial expansion (43) and the identification (44) then only rename the terms of (42) as Γ_j(x, Δ1+2; y, Δ2−2; 4λ) − Γ_j(x, Δ1; y, Δ2; 4λ). Since n is arbitrary, the weights Δ1 = 2(n−s)+d and Δ2 = 2s+d are chosen, not derived; the 'AdS transition amplitude' basis is broad enough—arbitrary weights, partial amplitude at fixed t—that any amplitude of the form (42) belongs to it. The claimed structural identity is therefore an identity by construction rather than a derived prediction.

  2. fitted input called prediction [Section 4, Eqs. (56)-(59)]
    "To make sense of the divergent series, we can regulate it via the Riemann zeta function such that 1 + 2 + 3 + ... = ζ(−1) := −1/12. Thus, the regularized amplitude takes the form A(k1, k2) = π^2/3 (2π)^d δ^d(k1 + k2) g^2_c ∫_0^∞ dθ e^{−λk^2θ} which is exactly the expression (34) upto a constant."

    The closed-string derivation selects the R→0 limit and sums the binomial series (56) before taking the limit, producing the divergent series Σ a whose regularized value is then assigned via ζ(−1). This value is precisely the one that makes (59) 'exactly the expression (34) up to a constant,' i.e., the open-string Laplace form. No independent computation fixes the regularization of the original integral (53); the zeta value is imported to force agreement with the open-string AdS form. The final closed-string AdS structure is thus imposed by the regularization choice rather than derived from the closed-string amplitude itself.

full rationale

The base two-point amplitudes are computed from the Polyakov path integral and agree with known results (Eqs. (30) and (52)), so the inputs are not themselves circular. The circularity lies in the conversion of those amplitudes into the advertised AdS form. For the open string, Eq. (35) multiplies by an identity with an arbitrary parameter n solely 'to mimic' Eq. (10); after the change of variables and binomial expansion, the result (44) is a relabeling of the integrand in (42) using the Γ_j notation. Because n and hence the weights Δ1, Δ2 are arbitrary, the class of 'AdS transition amplitudes' is defined broadly enough to contain the string amplitude by construction. The closed-string section is more explicitly constructed: the divergent series is regularized with ζ(−1) so that Eq. (59) is 'exactly the expression (34) up to a constant,' forcing the same Laplace form rather than deriving it. Separately, the analytic continuations at Eqs. (33)-(34) and the algebraic mismatch between Eqs. (43) and (44) are serious correctness problems, but they are distinct from circularity; they reinforce that the advertised structural match is not only imposed by construction but also not mathematically controlled.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The computation relies on standard AdS propagator results, standard Green's functions on the disk and sphere, and two ad hoc analytic continuation and regularization steps: the off-shell continuation of k^2 in the 't Hooft limit and the zeta-regularization of the divergent series for the closed string. The arbitrary identity parameter n is a free choice that changes the resulting weight decomposition without changing the amplitude. No new physical entities are introduced.

free parameters (2)
  • n
    Arbitrary non-negative integer inserted via the identity (35) to rewrite the Laplace transform. The final decomposition (44) depends on n, so the representation is not unique.
  • 't Hooft-like limit scalings = lambda_open = alpha'/R with R to infinity; lambda_closed = alpha' R^2 with R to 0
    The scaling limits are chosen by hand so that the string integrals turn into e^{-lambda k^2 theta} Laplace integrals matching the AdS heat kernel. These choices are not derived from string theory.
assumptions (4)
  • standard math AdS bulk-to-boundary propagator formula K(z0,z;z') in Eq. (4)
    Taken as known from Witten [4]; no derivation is given in the paper.
  • standard math Neumann Green's functions on the disk and sphere, Eqs. (27) and (48)
    Standard electrostatic Green's functions with cited references [23,24]; they are inputs to the amplitude computation.
  • ad hoc to paper On-shell amplitude may be analytically continued in k^2 to produce a heat kernel e^{-lambda k^2 theta}
    The amplitude is defined at k^2 = -1/alpha', but the manipulation in (33)-(42) treats k^2 as an independent eigenvalue operator, effectively continuing off shell. This is not justified in the paper.
  • ad hoc to paper Zeta-function regularization of the divergent series sum_a a = -1/12 in Eq. (58)
    The divergent series arises after a term-wise binomial expansion and limit interchange; assigning zeta(-1) is a regularization choice that is not shown to equal the analytic continuation of the original closed-string integral.

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Pith. "Pith review of Two-point tree-level string amplitudes as AdS transition amplitudes." pith.science (2026). https://pith.science/paper/2Q3ETAUJ

@misc{pith2026250507358,
  author       = {Pith},
  title        = {Pith review of: Two-point tree-level string amplitudes as AdS transition amplitudes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2Q3ETAUJ}},
  note         = {Machine review of arXiv:2505.07358}
}
read the original abstract

We compute the two-point open string and closed string amplitudes at tree level and show that, in a 't Hooft-like limit, they take a form structurally analogous to boundary-to-boundary transition amplitudes of a scalar field in Euclidean AdS space. Interestingly, while both amplitudes yield equivalent expressions, the associated 't Hooft couplings are defined by different worldsheet curvatures-geodesic for the disk and Gaussian for the sphere-suggesting a possible geometric aspect of open/closed string duality.

Figures

Figures reproduced from arXiv: 2505.07358 by the authors.

Figure 1
Figure 1. A propagation of the scalar field in configuration space from the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Works this paper leans on

25 extracted references · 13 canonical work pages

  1. [1]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252. arXiv:hep- th/9711200, doi:10.4310/ATMP.1998.v2.n2.a1. 5

    arXiv 1998

  2. [2]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333–

    work page 1973

  3. [3]

    S. W. Hawking, Black hole explosions, Nature 248 (1974) 30–31. doi:10.1038/248030a0

    doi:10.1038/248030a0 1974

  4. [4]

    Witten, Anti de Sitter space and holography, Adv

    E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291. arXiv:hep-th /9802150, doi:10.4310/ATMP.1998.v2.n2.a2

    doi:10.4310/atmp.1998.v2.n2.a2 1998

  5. [5]

    S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Gauge theory correla- tors from noncritical string theory, Phys. Lett. B 428 (1998) 105–114. arXiv:hep-th/9802109, doi:10.1016/S0370-2693(98)00377-3

    arXiv 1998

  6. [6]

    D’Hoker, D

    E. D’Hoker, D. Z. Freedman, Supersymmetric gauge theories and the AdS / CFT correspondence, in: Theoretical Advanced Study Institute in Ele- mentary Particle Physics (TASI 2001): Strings, Branes and EXTRA Di- mensions, 2002, pp. 3–158. arXiv:hep-th/0201253

    arXiv 2001

  7. [7]

    Nastase, Introduction to AdS-CFTarXiv:0712.0689

    H. Nastase, Introduction to AdS-CFTarXiv:0712.0689

    arXiv

  8. [8]

    Aharony, S

    O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, Y . Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386. arXiv:hep-th/9905111, doi:10.1016/S0370-1573(99)00083-6

    arXiv 2000

Show all 25 references
  1. [9]

    Gopakumar, From free fields to AdS, Phys

    R. Gopakumar, From free fields to AdS, Phys. Rev. D 70 (2004) 025009. arXiv:hep-th/0308184, doi:10.1103/PhysRevD.70.025009

    2004 arXiv

  2. [10]

    Erbin, J

    H. Erbin, J. Maldacena, D. Skliros, Two-Point String Amplitudes, JHEP 07 (2019) 139. arXiv:1906.06051, doi:10.1007/JHEP07(2019)139

    2019 arXiv

  3. [11]

    Giribet, N

    G. Giribet, N. Labranche, J. La Madrid, Remarks on the two-point string amplitudes, Phys. Rev. D 107 (10) (2023) 106021. arXiv:2303.15658, doi:10.1103/PhysRevD.107.106021

    2023 arXiv

  4. [12]

    S. Seki, T. Takahashi, Two-point string amplitudes revisited by operator formalism, Physics Letters B 800 (2020) 135078. doi:https://doi.org/10.1016/j.physletb.2019.135078. URL https://www.sciencedirect.com/science/article/pii/ S0370269319308007

    2020

  5. [13]

    S. P. Kashyap, Two-point superstring tree amplitudes using the pure spinor formalism, SciPost Phys. Core 8 (2025) 005. arXiv:2012.03802, doi:10.21468/SciPostPhysCore.8.1.005

    2025 arXiv

  6. [14]

    Kishimoto, S

    I. Kishimoto, S. Seki, T. Takahashi, Two-point closed string ampli- tudes in the BRST formalism, Phys. Lett. B 853 (2024) 138657. arXiv:2402.07464, doi:10.1016/j.physletb.2024.138657

    2024 arXiv

  7. [15]

    ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl

    G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461. doi:10.1016/0550-3213(74)90154-0

    1974 doi

  8. [16]

    Isberg, U

    J. Isberg, U. Lindstrom, B. Sundborg, G. Theodoridis, Classical and quan- tized tensionless strings, Nucl. Phys. B 411 (1994) 122–156. arXiv:hep- th/9307108, doi:10.1016/0550-3213(94)90056-6

    1994

  9. [17]

    D. J. Gross, High-Energy Symmetries of String Theory, Phys. Rev. Lett. 60 (1988) 1229. doi:10.1103/PhysRevLett.60.1229

    1988 doi

  10. [18]

    Bonelli, On the tensionless limit of bosonic strings, infinite symme- tries and higher spins, Nucl

    G. Bonelli, On the tensionless limit of bosonic strings, infinite symme- tries and higher spins, Nucl. Phys. B 669 (2003) 159–172. arXiv:hep- th/0305155, doi:10.1016/j.nuclphysb.2003.07.002

    2003

  11. [19]

    Lizzi, B

    F. Lizzi, B. Rai, G. Sparano, A. Srivastava, Quantization of the Null String and Absence of Critical Dimensions, Phys. Lett. B 182 (1986) 326–330. doi:10.1016/0370-2693(86)90101-2

    1986 doi

  12. [20]

    J. P. Edwards, P. Mansfield, QED as the tensionless limit of the spin- ning string with contact interaction, Phys. Lett. B 746 (2015) 335–340. arXiv:1409.4948, doi:10.1016/j.physletb.2015.05.024

    2015 arXiv

  13. [21]

    J. P. Edwards, P. Mansfield, Delta-function Interactions for the Bosonic and Spinning Strings and the Generation of Abelian Gauge Theory, JHEP 01 (2015) 127. arXiv:1410.3288, doi:10.1007/JHEP01(2015)127

    2015 arXiv

  14. [22]

    Srisangyingcharoen, Steps towards generalization of tensionless string theory with contact interactions as Wilson loop of non-Abelian Yang–Mills theory, Eur

    P. Srisangyingcharoen, Steps towards generalization of tensionless string theory with contact interactions as Wilson loop of non-Abelian Yang–Mills theory, Eur. Phys. J. C 83 (5) (2023) 418. arXiv:2207.07492, doi:10.1140/epjc/s10052-023-11563-2

    2023 arXiv

  15. [23]

    Kimura, H

    Y . Kimura, H. Okamoto, V ortex motion on a sphere, Jour- nal of the Physical Society of Japan 56 (12) (1987) 4203–4206. arXiv:https://doi.org/10.1143/JPSJ.56.4203, doi:10.1143/JPSJ.56.4203. URL https://doi.org/10.1143/JPSJ.56.4203

    1987 doi

  16. [24]

    Kimura, V ortex motion on surfaces with constant curvature, Pro- ceedings: Mathematical, Physical and Engineering Sciences 455 (1981) (1999) 245–259

    Y . Kimura, V ortex motion on surfaces with constant curvature, Pro- ceedings: Mathematical, Physical and Engineering Sciences 455 (1981) (1999) 245–259. URL http://www.jstor.org/stable/53414 6

    1981

  17. [2346]

    URL https://link.aps.org/doi/10.1103/PhysRevD.7.2333

    doi:10.1103/PhysRevD.7.2333. URL https://link.aps.org/doi/10.1103/PhysRevD.7.2333

    doi

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