Abstract
Building on [31] we investigate the integrable structure of the Wess-Zumino-Witten (WZW) model describing closed strings on AdS3× S 3× T4. Using the recently-proposed integrable S matrix we show analytically that all wrapping corrections cancel and that the theory has a natural spin-chain interpretation. We construct the integrable spin chain and discuss its relation with the WZW description. Finally we compute the spin-chain spectrum in closed form and show that it matches the WZW prediction on the nose.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [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].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(2, ℝ) WZW model 1.: the spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
A. Pakman, Unitarity of supersymmetric SL(2, ℝ)/U(1) and no ghost theorem for fermionic strings in AdS 3 × N , JHEP 01 (2003) 077 [hep-th/0301110] [INSPIRE].
D. Israel, C. Kounnas and M.P. Petropoulos, Superstrings on NS5 backgrounds, deformed AdS 3 and holography, JHEP 10 (2003) 028 [hep-th/0306053] [INSPIRE].
S. Raju, Counting giant gravitons in AdS 3, Phys. Rev. D 77 (2008) 046012 [arXiv:0709.1171] [INSPIRE].
G. Giribet, A. Pakman and L. Rastelli, Spectral flow in AdS 3 /CF T 2, JHEP 06 (2008) 013 [arXiv:0712.3046] [INSPIRE].
K. Ferreira, M.R. Gaberdiel and J.I. Jottar, Higher spins on AdS 3 from the worldsheet, JHEP 07 (2017) 131 [arXiv:1704.08667] [INSPIRE].
A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3 /CF T 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS 3 /CF T 2 correspondence and integrability, JHEP 11 (2012) 133 [Erratum ibid. 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
A. Sfondrini, Towards integrability for AdS3 /CFT2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 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].
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 et al., 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, 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 et al., Dressing phases of AdS 3 /CFT 2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, On the spectrum of AdS 3 × S 3 × T 4 strings with Ramond-Ramond flux, J. Phys. A 49 (2016) 41LT03 [arXiv:1605.00518] [INSPIRE].
R. Borsato et al., 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].
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].
T. Lloyd et al., 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].
G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, The off-shell symmetry algebra of the light-cone AdS 5 × S 5 superstring, J. Phys. A 40 (2007) 3583 [hep-th/0609157] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS 5 × S 5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [INSPIRE].
M. Baggio and A. Sfondrini, Strings on NS-NS backgrounds as integrable deformations, to be published in Phys. Rev. D Rapid Commun., arXiv:1804.01998 [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( t\overline{t} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Natural tuning: towards a proof of concept, JHEP 09 (2013) 045 [arXiv:1305.6939] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \) -deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, The complete AdS 3× S 3× T 4 worldsheet S matrix, JHEP 10 (2014) 66 [arXiv:1406.0453] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, A spin chain for the symmetric product CFT 2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., Integrability and the conformal field theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [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].
M.C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS 3 asymptotic Bethe ansatz, Phys. Rev. D 93 (2016) 106006 [arXiv:1512.08761] [INSPIRE].
G. Arutyunov and S. Frolov, Integrable Hamiltonian for classical strings on AdS 5 × S 5, JHEP 02 (2005) 059 [hep-th/0411089] [INSPIRE].
G. Arutyunov and S. Frolov, Uniform light-cone gauge for strings in AdS 5 × S 5 : Solving SU(1|1) sector, JHEP 01 (2006) 055 [hep-th/0510208] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, Finite-size effects from giant magnons, Nucl. Phys. B 778 (2007) 1 [hep-th/0606126] [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].
A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s matrices in two-dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 1. Stable particle states, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
M. Lüscher, Volume dependence of the energy spectrum in massive quantum field theories. 2. Scattering states, Commun. Math. Phys. 105 (1986) 153 [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
M. Baggio et al., Protected string spectrum in AdS 3 /CFT 2 from worldsheet integrability, JHEP 04 (2017) 091 [arXiv:1701.03501] [INSPIRE].
G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
Z. Bajnok, Review of AdS/CFT integrability, Chapter III.6: thermodynamic Bethe ansatz, Lett. Math. Phys. 99 (2012) 299 [arXiv:1012.3995] [INSPIRE].
A. Dei, M.R. Gaberdiel and A. Sfondrini, The plane-wave limit of AdS3 × S3 × S3 × S1, arXiv:1805.09154 [INSPIRE].
E. Whittaker and G. Watson, A course of modern analysis, Cambridge University Press, Cambridge U.K. (1996).
D.M. Hofman and J.M. Maldacena, Giant magnons, J. Phys. A 39 (2006) 13095 [hep-th/0604135] [INSPIRE].
J.R. David and A. Sadhukhan, Spinning strings and minimal surfaces in AdS 3 with mixed 3-form fluxes, JHEP 10 (2014) 49 [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. 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].
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].
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. Hernandez and J.M. Nieto, Elliptic solutions in the Neumann-Rosochatius system with mixed flux, Phys. Rev. D 91 (2015) 126006 [arXiv:1502.05203] [INSPIRE].
R. Hernández, J.M. Nieto and R. Ruiz, Pulsating strings with mixed three-form flux, JHEP 04 (2018) 078 [arXiv:1803.03078] [INSPIRE].
J.M. Nieto and R. Ruiz, One-loop quantization of rigid spinning strings in AdS 3 × S 3 × T 4 with mixed flux, arXiv:1804.10477 [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure constants and integrable bootstrap in planar N = 4 SYM theory,arXiv:1505.06745[INSPIRE].
B. Eden and A. Sfondrini, Tessellating cushions: four-point functions in \( \mathcal{N}=4 \) SYM, JHEP 10 (2017) 098 [arXiv:1611.05436] [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of correlation functions, JHEP 01 (2017) 130 [arXiv:1611.05577] [INSPIRE].
B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP 02 (2018) 170 [arXiv:1710.10212] [INSPIRE].
T. Bargheer et al., Handling handles I: nonplanar integrability, arXiv:1711.05326 [INSPIRE].
B. Eden and A. Sfondrini, Three-point functions in \( \mathcal{N}=4 \) SYM: the hexagon proposal at three loops, JHEP 02 (2016) 165 [arXiv:1510.01242] [INSPIRE].
B. Basso, V. Goncalves, S. Komatsu and P. Vieira, Gluing hexagons at three loops, Nucl. Phys. B 907 (2016) 695 [arXiv:1510.01683] [INSPIRE].
B. Basso, V. Goncalves and S. Komatsu, Structure constants at wrapping order, JHEP 05 (2017) 124 [arXiv:1702.02154] [INSPIRE].
J. Teschner, On structure constants and fusion rules in the SL(2, ℂ) / SU(2) WZNW model, Nucl. Phys. B 546 (1999) 390 [hep-th/9712256] [INSPIRE].
J. Teschner, Operator product expansion and factorization in the H+(3) WZNW model, Nucl. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(2, ℝ) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
C.A. Cardona and C.A. Núñez, Three-point functions in superstring theory on AdS 3 × S 3 × T 4, JHEP 06 (2009) 009 [arXiv:0903.2001] [INSPIRE].
C.A. Cardona and I. Kirsch, Worldsheet four-point functions in AdS 3 /CF T 2, JHEP 01 (2011) 015 [arXiv:1007.2720] [INSPIRE].
A. Sfondrini, Latest news from AdS 3 /CFT 2, talk at Integrability in Gauge and String Theory (IGST2017), July 17-21, Paris, France (2017).
G. Giribet et al., Superstrings on AdS 3 at k = 1, arXiv:1803.04420 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS 3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
P. Cooper et al., Looking for integrability on the worldsheet of confining strings, JHEP 04 (2015) 127 [arXiv:1411.0703] [INSPIRE].
A. Mohsen, Fermions on the worldsheet of effective strings via coset construction, Phys. Rev. D 93 (2016) 106007 [arXiv:1603.08178] [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 and B. Stefanski, The AdS 3 × S 3 × S 3 × S 1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS 5 × S 5 superstring action, Phys. Rev. Lett. 112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
S.J. van Tongeren, Integrability of the AdS5 × S5 superstring and its deformations, J. Phys. A 47 (2014) 433001 [arXiv:1310.4854] [INSPIRE].
R. Borsato, Integrable strings for AdS/CFT, Ph.D. thesis, Imperial Coll., London, 2015. arXiv:1605.03173 [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \) -deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
A. Cavaglià et al., Generalised Born-Infeld models, Lax operators and the \( T\overline{T} \) perturbation, arXiv:1806.11515.
C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].
G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [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].
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: 1806.00422
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
Dei, A., Sfondrini, A. Integrable spin chain for stringy Wess-Zumino-Witten models. J. High Energ. Phys. 2018, 109 (2018). https://doi.org/10.1007/JHEP07(2018)109
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2018)109