Abstract
We compute the tree-level bosonic S matrix in light-cone gauge for superstrings on pure-NSNS AdS3 × S3 × S3 × S1. We show that it is proportional to the identity and that it takes the same form as for AdS3 × S3 × T4 and for flat space. Based on this, we make a conjecture for the exact worldsheet S matrix and derive the mirror thermodynamic Bethe ansatz (TBA) equations describing the spectrum. Despite a non-trivial vacuum energy, they can be solved in closed form and coincide with a simple set of Bethe ansatz equations — again much like AdS3 × S3 × T4 and flat space. This suggests that the model may have an integrable spin-chain interpretation. Finally, as a check of our proposal, we compute the spectrum from the worldsheet CFT in the case of highest-weight representations of the underlying Kač-Moody algebras, and show that the mirror-TBA prediction matches it on the nose.
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].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [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].
S. Elitzur, O. Feinerman, A. Giveon and D. Tsabar, String theory on AdS 3 × S 3 × S 3 × S 1, Phys. Lett. B 449 (1999) 180 [hep-th/9811245] [INSPIRE].
A. Sevrin, W. Troost and A. Van Proeyen, Superconformal Algebras in Two-Dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].
O. Ohlsson Sax and B. Stefański, Closed strings and moduli in AdS 3 /CFT 2, JHEP 05 (2018) 101 [arXiv:1804.02023] [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].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS 3 and the SL(2, ℝ) WZW model. Part 2. Euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [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].
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 /CFT 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].
N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].
L. Eberhardt and K. Ferreira, The plane-wave spectrum from the worldsheet, JHEP 10 (2018) 109 [arXiv:1805.12155] [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].
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].
O. Ohlsson Sax, A. Sfondrini and B. Stefański, Jr., Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [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].
J.M. Nieto and R. Ruiz, One-loop quantization of rigid spinning strings in AdS 3 × S 3 × T 4 with mixed flux, JHEP 07 (2018) 141 [arXiv:1804.10477] [INSPIRE].
A. Babichenko, B. Stefański, Jr. and K. Zarembo, Integrability and the AdS 3 /CFT 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 /CFT 2 Correspondence and Integrability, JHEP 11 (2012) 133 [Erratum ibid. 1304 (2013) 003] [arXiv:1209.4049] [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].
A. Sfondrini, Towards integrability for AdS 3 /CFT 2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [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, A. Sfondrini, B. Stefański, 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].
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].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, 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. Stefański, Jr., The complete AdS 3 × S 3 × T 4 worldsheet S matrix, JHEP 10 (2014) 66 [arXiv:1406.0453] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefański, 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].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, Jr., The AdS 3 × S 3 × S 3 × S 1 worldsheet S matrix, J. Phys. A 48 (2015) 415401 [arXiv:1506.00218] [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].
P. Sundin and L. Wulff, Worldsheet scattering in AdS 3 /CFT 2, JHEP 07 (2013) 007 [arXiv:1302.5349] [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].
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].
P. Sundin and L. Wulff, The AdS n × S n × T 10 − 2n BMN string at two loops, JHEP 11 (2015) 154 [arXiv:1508.04313] [INSPIRE].
P. Sundin and L. Wulff, The complete one-loop BMN S-matrix in AdS 3 × S 3 × T 4, JHEP 06 (2016) 062 [arXiv:1605.01632] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr. and A. Torrielli, Dressing phases of AdS 3 /CFT 2, Phys. Rev. D 88 (2013) 066004 [arXiv:1306.2512] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr. and A. Torrielli, 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].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, Jr., 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.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
M. Baggio and A. Sfondrini, Strings on NS-NS Backgrounds as Integrable Deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].
A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 07 (2018) 109 [arXiv:1806.00422] [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].
T. Dray and G. ’t Hooft, The Gravitational Shock Wave of a Massless Particle, Nucl. Phys. B 253 (1985) 173 [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the Simplest Theory of Quantum Gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [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].
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].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [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].
L.F. Alday, G. Arutyunov and S. Frolov, New integrable system of 2dim fermions from strings on AdS 5 × S 5, JHEP 01 (2006) 078 [hep-th/0508140] [INSPIRE].
O. Ohlsson Sax and B. Stefański, Jr., Integrability, spin-chains and the AdS3/CFT2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [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].
A. Dei, M.R. Gaberdiel and A. Sfondrini, The plane-wave limit of AdS3 × S3 × S3 × S1, JHEP 08 (2018) 097 [arXiv:1805.09154] [INSPIRE].
T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, Worldsheet scattering in AdS 5 × S 5, JHEP 03 (2007) 094 [hep-th/0611169] [INSPIRE].
G. Arutyunov and S. Frolov, On String S-matrix, Bound States and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [INSPIRE].
M. Baggio, O. Ohlsson Sax, A. Sfondrini, B. Stefański, Jr. and A. Torrielli, Protected string spectrum in AdS 3 /CFT 2 from worldsheet integrability, JHEP 04 (2017) 091 [arXiv:1701.03501] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel, R. Gopakumar and W. Li, BPS spectrum on AdS 3 × S 3 × S 3 × S 1, JHEP 03 (2017) 124 [arXiv:1701.03552] [INSPIRE].
M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli and H. Walsh, On \( T\overline{T} \) deformations and supersymmetry, arXiv:1811.00533 [INSPIRE].
M. Staudacher, The Factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].
S.J. van Tongeren, Introduction to the thermodynamic Bethe ansatz, arXiv:1606.02951 [INSPIRE].
G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].
S.J. van Tongeren, Yang-Baxter deformations, AdS/CFT and twist-noncommutative gauge theory, Nucl. Phys. B 904 (2016) 148 [arXiv:1506.01023] [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and W. Li, A holographic dual for string theory on AdS 3 × S 3 × S 3 × S 1, JHEP 08 (2017) 111 [arXiv:1707.02705] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS 3 at ∥ = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS 3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, arXiv:1812.01007 [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].
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 /CFT 2, JHEP 01 (2011) 015 [arXiv:1007.2720] [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. Basso, F. Coronado, S. Komatsu, H.T. Lam, P. Vieira and D.-l. Zhong, Asymptotic Four Point Functions, arXiv:1701.04462 [INSPIRE].
T. Fleury and S. Komatsu, Hexagonalization of Correlation Functions II: Two-Particle Contributions, JHEP 02 (2018) 177 [arXiv:1711.05327] [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, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles: Nonplanar Integrability in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 121 (2018) 231602 [arXiv:1711.05326] [INSPIRE].
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling handles. Part II. Stratification and data analysis, JHEP 11 (2018) 095 [arXiv:1809.09145] [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].
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].
B. Eden, Y. Jiang, M. de Leeuw, T. Meier, D. le Plat and A. Sfondrini, Positivity of hexagon perturbation theory, JHEP 11 (2018) 097 [arXiv:1806.06051] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, \( \mathrm{T}\overline{\mathrm{T}} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, A solvable irrelevant deformation of AdS 3 /CFT 2, JHEP 12 (2017) 155 [arXiv:1707.05800] [INSPIRE].
L. Eberhardt and K. Ferreira, Long strings and chiral primaries in the hybrid formalism, arXiv:1810.08621 [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: 1812.08195
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 S matrix, mirror TBA and spectrum for the stringy AdS3 × S3 × S3 × S1 WZW model. J. High Energ. Phys. 2019, 72 (2019). https://doi.org/10.1007/JHEP02(2019)072
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)072