Abstract
Circular strings pulsating in AdS3 × S3 × T 4 with mixed R-R and NS-NS three-form fluxes can be described by an integrable deformation of the one-dimensional Neumann-Rosochatius mechanical model. In this article we find a general class of pulsating solutions to this integrable system that can be expressed in terms of elliptic functions. In the limit of strings moving in AdS3 with pure NS-NS three-form flux, where the action reduces to the SL(2, ℝ) WZW model, we find agreement with the analysis of the classical solutions of the system performed using spectral flow by Maldacena and Ooguri. We use our elliptic solutions in AdS3 to extend the dispersion relation beyond the limit of pure NS-NS flux.
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References
A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS 3 /CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
K. Zarembo, Strings on Semisymmetric Superspaces, JHEP 05 (2010) 002 [arXiv:1003.0465] [INSPIRE].
K. Zarembo, Algebraic Curves for Integrable String Backgrounds, arXiv:1005.1342 [INSPIRE].
O. Ohlsson Sax and B. Stefanski Jr., Integrability, spin-chains and the AdS 3 /CFT 2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].
N. Rughoonauth, P. Sundin and L. Wulff, Near BMN dynamics of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 07 (2012) 159 [arXiv:1204.4742] [INSPIRE].
O. Ohlsson Sax, B. Stefanski Jr. and A. Torrielli, On the massless modes of the AdS 3 /CFT 2 integrable systems, JHEP 03 (2013) 109 [arXiv:1211.1952] [INSPIRE].
C. Ahn and D. Bombardelli, Exact S-matrices for AdS 3 /CFT 2, Int. J. Mod. Phys. A 28 (2013) 1350168 [arXiv:1211.4512] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic su(1|1)2 S-matrix for AdS 3 /CFT 2, JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations for AdS 3 /CFT 2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, The all-loop integrable spin-chain for strings on AdS 3 × S 3 × T 4 : the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., Towards the All-Loop Worldsheet S Matrix for AdS 3 × S 3 × T 4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete AdS 3 × S 3 × T 4 worldsheet S matrix, JHEP 10 (2014) 066 [arXiv:1406.0453] [INSPIRE].
M. Beccaria, F. Levkovich-Maslyuk, G. Macorini and A.A. Tseytlin, Quantum corrections to spinning superstrings in AdS 3 × S 3 × M 4 : determining the dressing phase, JHEP 04 (2013) 006 [arXiv:1211.6090] [INSPIRE].
M. Beccaria and G. Macorini, Quantum corrections to short folded superstring in AdS 3 × S 3 × M 4, JHEP 03 (2013) 040 [arXiv:1212.5672] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, Dressing phases of AdS 3 /CFT 2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
M.C. Abbott, The AdS 3 × S 3 × S 3 × S 1 Hernández-López phases: a semiclassical derivation, J. Phys. A 46 (2013) 445401 [arXiv:1306.5106] [INSPIRE].
M.C. Abbott and I. Aniceto, Macroscopic (and Microscopic) Massless Modes, Nucl. Phys. B 894 (2015) 75 [arXiv:1412.6380] [INSPIRE].
M.C. Abbott and I. Aniceto, An improved AFS phase for AdS 3 string integrability, Phys. Lett. B 743 (2015) 61 [arXiv:1412.6863] [INSPIRE].
P. Sundin and L. Wulff, Worldsheet scattering in AdS 3 /CFT 2, JHEP 07 (2013) 007 [arXiv:1302.5349] [INSPIRE].
T. Lloyd and B. Stefanski Jr., AdS 3 /CFT 2 , finite-gap equations and massless modes, JHEP 04 (2014) 179 [arXiv:1312.3268] [INSPIRE].
P. Sundin, Worldsheet two- and four-point functions at one loop in AdS 3 /CFT 2, Phys. Lett. B 733 (2014) 134 [arXiv:1403.1449] [INSPIRE].
A. Sfondrini, Towards integrability for AdS 3 /CFT 2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS 3 /CFT 2 Correspondence and Integrability, JHEP 11 (2012) 133 [Erratum JHEP 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
B. Hoare and A.A. Tseytlin, On string theory on AdS 3 × S 3 × T 4 with mixed 3-form flux: tree-level S-matrix, Nucl. Phys. B 873 (2013) 682 [arXiv:1303.1037] [INSPIRE].
B. Hoare and A.A. Tseytlin, Massive S-matrix of AdS 3 × S 3 × T 4 superstring theory with mixed 3-form flux, Nucl. Phys. B 873 (2013) 395 [arXiv:1304.4099] [INSPIRE].
B. Hoare, A. Stepanchuk and A.A. Tseytlin, Giant magnon solution and dispersion relation in string theory in AdS 3 × S 3 × T 4 with mixed flux, Nucl. Phys. B 879 (2014) 318 [arXiv:1311.1794] [INSPIRE].
C. Ahn and P. Bozhilov, String solutions in AdS 3 × S 3 × T 4 with NS-NS B-field, Phys. Rev. D 90 (2014) 066010 [arXiv:1404.7644] [INSPIRE].
J.R. David and A. Sadhukhan, Spinning strings and minimal surfaces in AdS 3 with mixed 3-form fluxes, JHEP 10 (2014) 049 [arXiv:1405.2687] [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. Babichenko, A. Dekel and O. Ohlsson Sax, Finite-gap equations for strings on AdS 3 × S 3 × T 4 with mixed 3-form flux, JHEP 11 (2014) 122 [arXiv:1405.6087] [INSPIRE].
L. Bianchi and B. Hoare, AdS 3 × S 3 × M 4 string S-matrices from unitarity cuts, JHEP 08 (2014) 097 [arXiv:1405.7947] [INSPIRE].
R. Roiban, P. Sundin, A. Tseytlin and L. Wulff, The one-loop worldsheet S-matrix for the AdS n × S n × T 10−2n superstring, JHEP 08 (2014) 160 [arXiv:1407.7883] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete worldsheet S matrix of superstrings on AdS 3 × S 3 × T 4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
P. Sundin and L. Wulff, One- and two-loop checks for the AdS 3 × S 3 × T 4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
A. Stepanchuk, String theory in AdS 3 × S 3 × T 4 with mixed flux: semiclassical and 1-loop phase in the S-matrix, J. Phys. A 48 (2015) 195401 [arXiv:1412.4764] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The AdS 3 × S 3 × S 3 × S 1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].
A. Banerjee, K.L. Panigrahi and M. Samal, A note on oscillating strings in AdS 3 × S 3 with mixed three-form fluxes, JHEP 11 (2015) 133 [arXiv:1508.03430] [INSPIRE].
A. Banerjee and A. Sadhukhan, Multi-spike strings in AdS 3 with mixed three-form fluxes, JHEP 05 (2016) 083 [arXiv:1512.01816] [INSPIRE].
A. Banerjee, S. Biswas and R.R. Nayak, D1 string dynamics in curved backgrounds with fluxes, JHEP 04 (2016) 172 [arXiv:1601.06360] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr., A. Torrielli and O. Ohlsson Sax, On the dressing factors, Bethe equations and Yangian symmetry of strings on AdS 3 × S 3 × T 4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].
S.P. Barik, M. Khouchen, J. Klusoň and K.L. Panigrahi, SL(2, Z) invariant rotating (m, n) strings in AdS 3 × S 3 with mixed flux, Eur. Phys. J. C 77 (2017) 298 [arXiv:1610.03402] [INSPIRE].
S.P. Barik, K.L. Panigrahi and M. Samal, Perturbations of Pulsating Strings, arXiv:1708.05202 [INSPIRE].
R. Hernández and J.M. Nieto, Spinning strings in AdS 3 × S 3 with NS-NS flux, Nucl. Phys. B 888 (2014) 236 [Erratum ibid. B 895 (2015) 303] [arXiv:1407.7475] [INSPIRE].
R. Hernández and J.M. Nieto, Elliptic solutions in the Neumann-Rosochatius system with mixed flux, Phys. Rev. D 91 (2015) 126006 [arXiv:1502.05203] [INSPIRE].
G. Arutyunov, S. Frolov, J. Russo and A.A. Tseytlin, Spinning strings in AdS 5 × S 5 and integrable systems, Nucl. Phys. B 671 (2003) 3 [hep-th/0307191] [INSPIRE].
G. Arutyunov, J. Russo and A.A. Tseytlin, Spinning strings in AdS 5 × S 5 : New integrable system relations, Phys. Rev. D 69 (2004) 086009 [hep-th/0311004] [INSPIRE].
K. Uhlenbeck, Equivariant harmonic maps into spheres, Lect. Notes Math. 949 (1982) 39.
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(2, ℝ) WZW model. I: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
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Hernández, R., Nieto, J.M. & Ruiz, R. Pulsating strings with mixed three-form flux. J. High Energ. Phys. 2018, 78 (2018). https://doi.org/10.1007/JHEP04(2018)078
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DOI: https://doi.org/10.1007/JHEP04(2018)078