#### Perturbations of spiky strings in AdS3

Journal of High Energy Physics
June 2018, 2018:89 | Cite as
Perturbations of spiky strings in AdS3
AuthorsAuthors and affiliations
Soumya BhattacharyaSayan KarKamal L. Panigrahi
Open Access
Regular Article - Theoretical Physics
First Online: 19 June 2018
Received: 26 April 2018
Accepted: 05 June 2018
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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.
Keywords Bosonic Strings Gauge-gravity correspondence
ArXiv ePrint: 1804.07544
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References
[1]
C.J. Burden and L.J. Tassie, Some exotic mesons and glueballs from the string model, Phys. Lett. B 110 (1982) 64.ADSMathSciNetCrossRefGoogle Scholar
[2]
C.J. Burden and L.J. Tassie, Rotating strings, glueballs and exotic mesons, Austral. J. Phys. 35 (1982) 223 [INSPIRE].
[3]
C.J. Burden and L.J. Tassie, Additional rigidly rotating solutions in the string model of hadrons, Austral. J. Phys. 37 (1984) 1 [INSPIRE].ADSCrossRefGoogle Scholar
[4]
C.J. Burden, Gravitational radiation from a particular class of cosmic strings, Phys. Lett. B 164 (1985) 277.Google Scholar
[5]
F. Embacher, Rigidly rotating cosmic strings, Phys. Rev. D 46 (1992) 3659 [Erratum ibid. D 47 (1993) 4803].Google Scholar
[6]
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].ADSGoogle Scholar
[7]
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].ADSCrossRefMATHGoogle Scholar
[8]
S. Kar and S. Mahapatra, Planetoid strings: Solutions and perturbations, Class. Quant. Grav. 15 (1998) 1421 [hep-th/9701173] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
[9]
C.J. Burden, Comment on “Stationary rotating strings as relativistic particle mechanics, Phys. Rev. D 78 (2008) 128301.ADSGoogle Scholar
[10]
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].ADSGoogle Scholar
[11]
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
[12]
M. Kruczenski, Spiky strings and single trace operators in gauge theories, JHEP 08 (2005) 014 [hep-th/0410226] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
[13]
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].ADSCrossRefMATHGoogle Scholar
[14]
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
[15]
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].ADSMathSciNetCrossRefGoogle Scholar
[16]
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].
[17]
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].ADSCrossRefMATHGoogle Scholar
[18]
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].ADSGoogle Scholar
[19]
S. Frolov and A.A. Tseytlin, Multispin string solutions in AdS 5 × S 5, Nucl. Phys. B 668 (2003) 77 [hep-th/0304255] [INSPIRE].
[20]
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].
[21]
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].ADSMATHGoogle Scholar
[22]
S. Biswas and K.L. Panigrahi, Spiky Strings on I-brane, JHEP 08 (2012) 044 [arXiv:1206.2539] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
[23]
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].ADSGoogle Scholar
[24]
A. Banerjee, S. Bhattacharya and K.L. Panigrahi, Spiky strings in ϰ-deformed AdS, JHEP 06 (2015) 057 [arXiv:1503.07447] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
[25]
S. Bhattacharya, S. Kar and K.L. Panigrahi, Perturbations of spiky strings in flat spacetimes, JHEP 01 (2017) 116 [arXiv:1610.09180] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
[26]
S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [INSPIRE].ADSCrossRefGoogle Scholar
[27]
J. Garriga and A. Vilenkin, Black holes from nucleating strings, Phys. Rev. D 47 (1993) 3265 [hep-ph/9208212].
[28]
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].
[29]
V. Frolov and A.L. Larsen, Propagation of perturbations along strings, Nucl. Phys. B 414 (1994) 129 [hep-th/9303001].
[30]
R. Capovilla and J. Guven, Geometry of deformations of relativistic membranes, Phys. Rev. D 51 (1995) 6736 [gr-qc/9411060].
[31]
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].ADSMathSciNetMATHGoogle Scholar
[32]
A. Jevicki and K. Jin, Solitons and AdS String Solutions, Int. J. Mod. Phys. A 23 (2008) 2289 [arXiv:0804.0412] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
[33]
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.MathSciNetCrossRefGoogle Scholar
[34]
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.MathSciNetCrossRefGoogle Scholar
[35]
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.ADSMathSciNetCrossRefMATHGoogle Scholar
[36]
Y-M. Chiang, A. Ching and C-Y. Tsang, Symmetries of the Darboux equation, to appear in Kumamoto J. Math.. [arXiv:1509.03995].
[37]
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].ADSMathSciNetCrossRefMATHGoogle Scholar
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© The Author(s) 2018
Authors and Affiliations
Soumya Bhattacharya1Sayan Kar1Kamal L. Panigrahi1Email author1.Department of Physics and Centre for Theoretical StudiesIndian Institute of Technology KharagpurKharagpurIndia