Abstract
Perturbations of a class of semiclassical spiky strings in three dimensional Anti-de Sitter (AdS) spacetime, are investigated using the well-known Jacobi equations for small, normal deformations of an embedded timelike surface. We show that the equation for the perturbation scalar which governs the behaviour of such small deformations, is a special case of the well-known Darboux-Treibich-Verdier (DTV) equation. The eigenvalues and eigensolutions of the DTV equation for our case are obtained by solving certain continued fractions numerically. These solutions are thereafter utilised to further demonstrate that there do exist finite perturbations of the AdS spiky strings. Our results therefore establish that the spiky string configurations in AdS3 are indeed stable against small fluctuations. Comments on future possibilities of work are included in conclusion.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.J. Burden and L.J. Tassie, Some exotic mesons and glueballs from the string model, Phys. Lett. B 110 (1982) 64.
C.J. Burden and L.J. Tassie, Rotating strings, glueballs and exotic mesons, Austral. J. Phys. 35 (1982) 223 [INSPIRE].
C.J. Burden and L.J. Tassie, Additional rigidly rotating solutions in the string model of hadrons, Austral. J. Phys. 37 (1984) 1 [INSPIRE].
C.J. Burden, Gravitational radiation from a particular class of cosmic strings, Phys. Lett. B 164 (1985) 277.
F. Embacher, Rigidly rotating cosmic strings, Phys. Rev. D 46 (1992) 3659 [Erratum ibid. D 47 (1993) 4803].
H.J. de Vega and I.L. Egusquiza, Planetoid string solutions in 3 + 1 axisymmetric space-times, Phys. Rev. D 54 (1996) 7513 [hep-th/9607056] [INSPIRE].
V.P. Frolov, S. Hendy and J.P. De Villiers, Rigidly rotating strings in stationary axisymmetric space-times, Class. Quant. Grav. 14 (1997) 1099 [hep-th/9612199] [INSPIRE].
S. Kar and S. Mahapatra, Planetoid strings: Solutions and perturbations, Class. Quant. Grav. 15 (1998) 1421 [hep-th/9701173] [INSPIRE].
C.J. Burden, Comment on “Stationary rotating strings as relativistic particle mechanics, Phys. Rev. D 78 (2008) 128301.
K. Ogawa, H. Ishihara, H. Kozaki, H. Nakano and S. Saito, Stationary Rotating Strings as Relativistic Particle Mechanics, Phys. Rev. D 78 (2008) 023525 [arXiv:0803.4072] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
M. Kruczenski, Spiky strings and single trace operators in gauge theories, JHEP 08 (2005) 014 [hep-th/0410226] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, A Relation between approaches to integrability in superconformal Yang-Mills theory, JHEP 10 (2003) 017 [hep-th/0308089] [INSPIRE].
L. Dolan, C.R. Nappi and E. Witten, Yangian symmetry in D = 4 superconformal Yang-Mills theory, in proceedings of 3rd International Symposium on Quantum theory and symmetries (QTS3), Cincinnati, U.S.A., September 10–14, 2003, pp. 300–315 (2004) [DOI:https://doi.org/10.1142/9789812702340_0036] [hep-th/0401243] [INSPIRE].
G. Mandal, N.V. Suryanarayana and S.R. Wadia, Aspects of semiclassical strings in AdS 5, Phys. Lett. B 543 (2002) 81 [hep-th/0206103] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
S. Frolov and A.A. Tseytlin, Multispin string solutions in AdS 5 × S 5, Nucl. Phys. B 668 (2003) 77 [hep-th/0304255] [INSPIRE].
R. Ishizeki and M. Kruczenski, Single spike solutions for strings on S 2 and S 3, Phys. Rev. D 76 (2007) 126006 [arXiv:0705.2429] [INSPIRE].
R. Ishizeki, M. Kruczenski, A. Tirziu and A.A. Tseytlin, Spiky strings in AdS 3 × S 1 and their AdS-pp-wave limits, Phys. Rev. D 79 (2009) 026006 [arXiv:0812.2431] [INSPIRE].
S. Biswas and K.L. Panigrahi, Spiky Strings on I-brane, JHEP 08 (2012) 044 [arXiv:1206.2539] [INSPIRE].
A. Banerjee, K.L. Panigrahi and P.M. Pradhan, Spiky strings on AdS 3 × S 3 with NS-NS flux, Phys. Rev. D 90 (2014) 106006 [arXiv:1405.5497] [INSPIRE].
A. Banerjee, S. Bhattacharya and K.L. Panigrahi, Spiky strings in ϰ-deformed AdS, JHEP 06 (2015) 057 [arXiv:1503.07447] [INSPIRE].
S. Bhattacharya, S. Kar and K.L. Panigrahi, Perturbations of spiky strings in flat spacetimes, JHEP 01 (2017) 116 [arXiv:1610.09180] [INSPIRE].
S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [INSPIRE].
J. Garriga and A. Vilenkin, Black holes from nucleating strings, Phys. Rev. D 47 (1993) 3265 [hep-ph/9208212].
J. Guven, Perturbations of a topological defect as a theory of coupled scalar fields in curved space interacting with an external vector potential, Phys. Rev. D 48 (1993) 5562 [gr-qc/9304033].
V. Frolov and A.L. Larsen, Propagation of perturbations along strings, Nucl. Phys. B 414 (1994) 129 [hep-th/9303001].
R. Capovilla and J. Guven, Geometry of deformations of relativistic membranes, Phys. Rev. D 51 (1995) 6736 [gr-qc/9411060].
V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings, J. Phys. A 48 (2015) 475401 [arXiv:1507.01883] [INSPIRE].
A. Jevicki and K. Jin, Solitons and AdS String Solutions, Int. J. Mod. Phys. A 23 (2008) 2289 [arXiv:0804.0412] [INSPIRE].
C. de Sparre, Sur l’equation \( \begin{array}{l}\frac{d^2y}{d{x}^2}+\left[2v\frac{k^2 snxcnx}{dnx}+2{v}_1\frac{snxdnx}{cnx}-2{v}_2\frac{cnxdnx}{snx}\right]\frac{dy}{dx}=\hfill \\ {}\left[\frac{I}{s{n}^2x}\left({n}_3-{v}_2\right)\left({n}_3+{v}_2+1\right)+\frac{d{n}^2x}{c{n}^2x}\left({n}_2-{v}_1\right)\left({n}_2+{v}_1+1\right)+\frac{k^2c{n}^2x}{d{n}^2x}\left({n}_1-v\right)\left({n}_1+v+1\right)\right.\hfill \\ {}\left.+{k}^2s{n}^2x\left(n+v+{v}_1+{v}_2\right)\left(n-v-{v}_1-{v}_2+1\right)+h\right]\hfill \end{array} \), Acta Math. 3 (1883) 105.
C. de Sparre, Sur l’equation \( \begin{array}{l}\frac{d^2y}{d{x}^2}+\left[2v\frac{k^2 snxcnx}{dnx}+2{v}_1\frac{snxdnx}{cnx}-2{v}_2\frac{cnxdnx}{snx}\right]\frac{dy}{dx}=\hfill \\ {}\left[\frac{I}{s{n}^2x}\left({n}_3-{v}_2\right)\left({n}_3+{v}_2+1\right)+\frac{d{n}^2x}{c{n}^2x}\left({n}_2-{v}_1\right)\left({n}_2+{v}_1+1\right)+\frac{k^2c{n}^2x}{d{n}^2x}\left({n}_1-v\right)\left({n}_1+v+1\right)\right.\hfill \\ {}\left.+{k}^2s{n}^2x\left(n+v+{v}_1+{v}_2\right)\left(n-v-{v}_1-{v}_2+1\right)+h\right]y\hfill \end{array} \), Acta Math. 3 (1883) 289.
V.B. Matveev and A.O. Smirnov, On the Link Between the Sparre Equation and Darboux-Treibich-Verdier Equation, Lett. Math. Phys. 76 (2006) 283.
Y-M. Chiang, A. Ching and C-Y. Tsang, Symmetries of the Darboux equation, to appear in Kumamoto J. Math.. [arXiv:1509.03995].
S. Frolov, A. Tirziu and A.A. Tseytlin, Logarithmic corrections to higher twist scaling at strong coupling from AdS/CFT, Nucl. Phys. B 766 (2007) 232 [hep-th/0611269] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1804.07544
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bhattacharya, S., Kar, S. & Panigrahi, K.L. Perturbations of spiky strings in AdS3. J. High Energ. Phys. 2018, 89 (2018). https://doi.org/10.1007/JHEP06(2018)089
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2018)089