Abstract
Via an appropriate field redefinition of the fermions, we find a set of conditions under which light cone gauge fixed world sheet theories of strings on two different backgrounds are related by a double Wick rotation. These conditions take the form of a set of transformation laws for the background fields, complementing a set of transformation laws for the metric and B field we found previously with a set for the dilaton and RR fields, and are compatible with the supergravity equations of motion. Our results prove that at least to second order in fermions, the AdS5 × S5 mirror model which plays an important role in the field of integrability in AdS/CFT, represents a string on ‘mirror AdS5 × S5’, the background that follows from our transformations. We discuss analogous solutions for AdS3 × S3 × T4 and AdS2 × S2 × T6. The main ingredient in our derivation is the light cone gauge fixed action for a string on an (almost) completely generic background, which we explicitly derive to second order in fermions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
R.R. Metsaev and A.A. Tseytlin, Type IIB superstring action in AdS5 × S5 background, Nucl. Phys. B 533 (1998) 109 [hep-th/9805028] [INSPIRE].
J. Ambjørn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].
G. Arutyunov and S. Frolov, On String S-matrix, Bound States and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS5 × S5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
G. Arutyunov and S. Frolov, String hypothesis for the AdS5 × S5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS5 × S5 Mirror Model, JHEP 05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
D. Bombardelli, D. Fioravanti and R. Tateo, Thermodynamic Bethe Ansatz for planar AdS/CFT: A proposal, J. Phys. A 42 (2009) 375401 [arXiv:0902.3930] [INSPIRE].
N. Gromov, V. Kazakov, A. Kozak and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar N = 4 Supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N}=4 \) Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, arXiv:1411.4758 [INSPIRE].
G. Arutyunov and S.J. van Tongeren, AdS5 × S5 mirror model as a string σ-model, Phys. Rev. Lett. 113 (2014) 261605 [arXiv:1406.2304] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS5 × S5) η superstring, Theor. Math. Phys. 182 (2015) 23 [arXiv:1403.6104] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS5 × S5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS5 × S5 superstring, JHEP 10 (2014) 132 [arXiv:1406.6286] [INSPIRE].
G. Arutyunov, R. Borsato and S. Frolov, S-matrix for strings on η-deformed AdS5 × S5, JHEP 04 (2014) 002 [arXiv:1312.3542] [INSPIRE].
B. Hoare, R. Roiban and A.A. Tseytlin, On deformations of AdS n × Sn supercosets, JHEP 06 (2014) 002 [arXiv:1403.5517] [INSPIRE].
T. Kameyama and K. Yoshida, Anisotropic Landau-Lifshitz σ-models from q-deformed AdS5 × S5 superstrings, JHEP 08 (2014) 110 [arXiv:1405.4467] [INSPIRE].
M. Khouchen and J. Kluson, Giant Magnon on Deformed AdS3 × S3, Phys. Rev. D 90 (2014) 066001 [arXiv:1405.5017] [INSPIRE].
C. Ahn and P. Bozhilov, Finite-size giant magnons on η-deformed AdS5 × S5, Phys. Lett. B 737 (2014) 293 [arXiv:1406.0628] [INSPIRE].
G. Arutyunov and D. Medina-Rincon, Deformed Neumann model from spinning strings on (AdS5 × S5) η , JHEP 1410 (2014) 50 [arXiv:1406.2536] [INSPIRE].
A. Banerjee and K.L. Panigrahi, On the rotating and oscillating strings in (AdS3 × S3) κ , JHEP 09 (2014) 048 [arXiv:1406.3642] [INSPIRE].
T. Kameyama and K. Yoshida, A new coordinate system for q-deformed AdS5 × S5 and classical string solutions, J. Phys. A 48 (2015) 075401 [arXiv:1408.2189] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS 5 × S 5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
K. Sfetsos and D.C. Thompson, Spacetimes for λ-deformations, JHEP 12 (2014) 164 [arXiv:1410.1886] [INSPIRE].
O. Lunin, R. Roiban and A.A. Tseytlin, Supergravity backgrounds for deformations of AdS n × Sn supercoset string models, Nucl. Phys. B 891 (2015) 106 [arXiv:1411.1066] [INSPIRE].
B. Hoare, Towards a two-parameter q-deformation of AdS3 × S3 × M4 superstrings, Nucl. Phys. B 891 (2015) 259 [arXiv:1411.1266] [INSPIRE].
T. Matsumoto and K. Yoshida, Yang-Baxter deformations and string dualities, JHEP 03 (2015) 137 [arXiv:1412.3658] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, Finite-size Effects from Giant Magnons, Nucl. Phys. B 778 (2007) 1 [hep-th/0606126] [INSPIRE].
L. Wulff, The type-II superstring to order θ 4, JHEP 07 (2013) 123 [arXiv:1304.6422] [INSPIRE].
A.A. Tseytlin, On dilaton dependence of type-II superstring action, Class. Quant. Grav. 13 (1996) L81 [hep-th/9601109] [INSPIRE].
M. Cvetič, H. Lü, C.N. Pope and K.S. Stelle, T duality in the Green-Schwarz formalism and the massless/massive IIA duality map, Nucl. Phys. B 573 (2000) 149 [hep-th/9907202] [INSPIRE].
J.H. Schwarz, Covariant Field Equations of Chiral N = 2 D = 10 Supergravity, Nucl. Phys. B 226 (1983) 269 [INSPIRE].
M. Kruczenski and A.A. Tseytlin, Semiclassical relativistic strings in S5 and long coherent operators in N = 4 SYM theory, JHEP 09 (2004) 038 [hep-th/0406189] [INSPIRE].
D. Tong, String Theory, arXiv:0908.0333 [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
M.J. Duff, H. Lü and C.N. Pope, Supersymmetry without supersymmetry, Phys. Lett. B 409 (1997) 136 [hep-th/9704186] [INSPIRE].
M.J. Duff, H. Lü and C.N. Pope, AdS5 × S5 untwisted, Nucl. Phys. B 532 (1998) 181 [hep-th/9803061] [INSPIRE].
C.M. Hull, Timelike T duality, de Sitter space, large-N gauge theories and topological field theory, JHEP 07 (1998) 021 [hep-th/9806146] [INSPIRE].
T. Klose, Review of AdS/CFT Integrability, Chapter IV.3: N = 6 Chern-Simons and Strings on AdS4 × CP3, Lett. Math. Phys. 99 (2012) 401 [arXiv:1012.3999] [INSPIRE].
A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS3 /CFT2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
P. Sundin and L. Wulff, One- and two-loop checks for the AdS3 × S3 × T4 superstring with mixed flux, J. Phys. A 48 (2015) 105402 [arXiv:1411.4662] [INSPIRE].
G. Arutyunov and S. Frolov, Superstrings on AdS4 × CP3 as a Coset σ-model, JHEP 09 (2008) 129 [arXiv:0806.4940] [INSPIRE].
B. Stefański Jr., Green-Schwarz action for Type IIA strings on AdS4 × CP3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Towards the All-Loop Worldsheet S Matrix for AdS3 × S3 × T4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
B. Hoare, A. Pittelli and A. Torrielli, Integrable S-matrices, massive and massless modes and the AdS2 × S2 superstring, JHEP 11 (2014) 051 [arXiv:1407.0303] [INSPIRE].
A. Sfondrini, Towards integrability for AdS3 /CFT2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].
K. Zarembo, Worldsheet spectrum in AdS4 /CFT3 correspondence, JHEP 04 (2009) 135 [arXiv:0903.1747] [INSPIRE].
T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, Worldsheet scattering in AdS5 × S5, JHEP 03 (2007) 094 [hep-th/0611169] [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: 1412.5137
Correspondent fellow at Steklov Mathematical Institute, Moscow, Russia. (Gleb Arutyunov)
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Arutyunov, G., van Tongeren, S.J. Double Wick rotating Green-Schwarz strings. J. High Energ. Phys. 2015, 27 (2015). https://doi.org/10.1007/JHEP05(2015)027
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2015)027