Abstract
Despite the rich and fruitful history of the integrability approach to string theory on the AdS3 × S3 × T4 background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. In other integrable superstring backgrounds there is compelling evidence that this formulation captures all wrapping effects exactly and describes the full planar spectrum. This great success motivates us to study the new proposed QSC and develop methods to extract from it concrete predictions for spectral data. The AdS3 × S3 × T4 case presents a significant novel feature and challenge compared to its higher-dimensional analogues — massless modes. It has been conjectured that these manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of \( \mathcal{N} \) = 4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for unprotected string excitations in the planar limit with finite quantum numbers and RR flux. We explain how to extract a systematic expansion around the analogue of the weak ’t Hooft coupling limit in \( \mathcal{N} \) = 4 SYM and also obtain high-precision numerical results. These concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
L. Eberhardt and M.R. Gaberdiel, String theory on AdS3 and the symmetric orbifold of Liouville theory, Nucl. Phys. B 948 (2019) 114774 [arXiv:1903.00421] [INSPIRE].
N. Seiberg and E. Witten, The D1 / D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
O. Ohlsson Sax, A. Sfondrini and B. Stefański, Integrability and the Conformal Field Theory of the Higgs branch, JHEP 06 (2015) 103 [arXiv:1411.3676] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, R) WZW model. I.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
A. Babichenko, B. Stefański Jr. and K. Zarembo, Integrability and the AdS3/CFT2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
A. Sfondrini, Towards integrability for AdS3/CFT2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [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, A. Sfondrini and B. Stefański, Towards the All-Loop Worldsheet S Matrix for AdS3 × S3 × T4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, The complete AdS3 × S3 × T4 worldsheet S matrix, JHEP 10 (2014) 066 [arXiv:1406.0453] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefański, On the spectrum of AdS3 × S3 × T4 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 AdS3 × S3 × T4, J. Phys. A 50 (2017) 024004 [arXiv:1607.00914] [INSPIRE].
S. Ekhammar and D. Volin, Monodromy bootstrap for SU(2|2) quantum spectral curves: from Hubbard model to AdS3/CFT2, JHEP 03 (2022) 192 [arXiv:2109.06164] [INSPIRE].
A. Cavaglià et al., Quantum Spectral Curve for AdS3/CFT2: a proposal, JHEP 12 (2021) 048 [arXiv:2109.05500] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
N. Gromov, Introduction to the Spectrum of N = 4 SYM and the Quantum Spectral Curve, arXiv:1708.03648 [INSPIRE].
M.C. Abbott and I. Aniceto, Massless Lüscher terms and the limitations of the AdS3 asymptotic Bethe ansatz, Phys. Rev. D 93 (2016) 106006 [arXiv:1512.08761] [INSPIRE].
M.C. Abbott and I. Aniceto, Integrable field theories with an interacting massless sector, Phys. Rev. D 103 (2021) 086017 [arXiv:2002.12060] [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].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum spectral curve for arbitrary state/operator in AdS5/CFT4, JHEP 09 (2015) 187 [arXiv:1405.4857] [INSPIRE].
A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum Spectral Curve of the \( \mathcal{N} \) = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
D. Bombardelli et al., The full Quantum Spectral Curve for AdS4/CFT3, JHEP 09 (2017) 140 [arXiv:1701.00473] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4, JHEP 06 (2016) 036 [arXiv:1504.06640] [INSPIRE].
C. Marboe and D. Volin, Quantum spectral curve as a tool for a perturbative quantum field theory, Nucl. Phys. B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Pomeron Eigenvalue at Three Loops in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 115 (2015) 251601 [arXiv:1507.04010] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum spectral curve and structure constants in \( \mathcal{N} \) = 4 SYM: cusps in the ladder limit, JHEP 10 (2018) 060 [arXiv:1802.04237] [INSPIRE].
S. Giombi and S. Komatsu, More Exact Results in the Wilson Loop Defect CFT: Bulk-Defect OPE, Nonplanar Corrections and Quantum Spectral Curve, J. Phys. A 52 (2019) 125401 [arXiv:1811.02369] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Separation of variables in AdS/CFT: functional approach for the fishnet CFT, JHEP 06 (2021) 131 [arXiv:2103.15800] [INSPIRE].
C. Bercini, A. Homrich and P. Vieira, Structure Constants in \( \mathcal{N} \) = 4 SYM and Separation of Variables, arXiv:2210.04923 [INSPIRE].
B. Basso, A. Georgoudis and A.K. Sueiro, Structure Constants of Short Operators in Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 130 (2023) 131603 [arXiv:2207.01315] [INSPIRE].
A. Cavaglià, N. Gromov, J. Julius and M. Preti, Integrability and conformal bootstrap: One dimensional defect conformal field theory, Phys. Rev. D 105 (2022) L021902 [arXiv:2107.08510] [INSPIRE].
A. Cavaglià, N. Gromov, J. Julius and M. Preti, Bootstrability in defect CFT: integrated correlators and sharper bounds, JHEP 05 (2022) 164 [arXiv:2203.09556] [INSPIRE].
S. Caron-Huot, F. Coronado, A.-K. Trinh and Z. Zahraee, Bootstrapping \( \mathcal{N} \) = 4 sYM correlators using integrability, JHEP 02 (2023) 083 [arXiv:2207.01615] [INSPIRE].
B. Eden, D. Plat and A. Sfondrini, Integrable bootstrap for AdS3/CFT2 correlation functions, JHEP 08 (2021) 049 [arXiv:2102.08365] [INSPIRE].
M. Fabri, Hexagonalization in AdS3 × S3 × T4: Mirror corrections, Phys. Rev. D 106 (2022) 126008 [arXiv:2209.01959] [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
S. Frolov and A. Sfondrini, Mirror thermodynamic Bethe ansatz for AdS3/CFT2, JHEP 03 (2022) 138 [arXiv:2112.08898] [INSPIRE].
S. Frolov and A. Sfondrini, New dressing factors for AdS3/CFT2, JHEP 04 (2022) 162 [arXiv:2112.08896] [INSPIRE].
O. Ohlsson Sax and B. Stefański, Closed strings and moduli in AdS3/CFT2, JHEP 05 (2018) 101 [arXiv:1804.02023] [INSPIRE].
R. Borsato et al., The all-loop integrable spin-chain for strings on AdS3 × S3 × T4: the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].
V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Classical limit of Quantum Sigma-Models from Bethe Ansatz, PoS SOLVAY (2006) 005 [hep-th/0703137] [INSPIRE].
D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo, Exploring the spectrum of planar AdS4/CFT3 at finite coupling, JHEP 04 (2018) 117 [arXiv:1803.04748] [INSPIRE].
C. Marboe and D. Volin, The full spectrum of AdS5/CFT4 I: Representation theory and one-loop Q-system, J. Phys. A 51 (2018) 165401 [arXiv:1701.03704] [INSPIRE].
C. Marboe and D. Volin, The full spectrum of AdS5/CFT4 II: Weak coupling expansion via the quantum spectral curve, J. Phys. A 54 (2021) 055201 [arXiv:1812.09238] [INSPIRE].
L. Apolo et al., Deforming symmetric product orbifolds: a tale of moduli and higher spin currents, JHEP 08 (2022) 159 [arXiv:2204.07590] [INSPIRE].
A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [hep-th/0404092] [INSPIRE].
A. Loewy and Y. Oz, Large spin strings in AdS3, Phys. Lett. B 557 (2003) 253 [hep-th/0212147] [INSPIRE].
J.R. David and A. Sadhukhan, Spinning strings and minimal surfaces in AdS3 with mixed 3-form fluxes, JHEP 10 (2014) 049 [arXiv:1405.2687] [INSPIRE].
A. Banerjee and A. Sadhukhan, Multi-spike strings in AdS3 with mixed three-form fluxes, JHEP 05 (2016) 083 [arXiv:1512.01816] [INSPIRE].
L.F. Alday and J.M. Maldacena, Comments on operators with large spin, JHEP 11 (2007) 019 [arXiv:0708.0672] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk, G. Sizov and S. Valatka, Quantum spectral curve at work: from small spin to strong coupling in \( \mathcal{N} \) = 4 SYM, JHEP 07 (2014) 156 [arXiv:1402.0871] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS3/CFT2 Correspondence and Integrability, JHEP 11 (2012) 133 [Erratum ibid. 04 (2013) 003] [arXiv:1209.4049] [INSPIRE].
B. Hoare and A.A. Tseytlin, On string theory on AdS3 × S3 × T4 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 AdS3 × S3 × T4 superstring theory with mixed 3-form flux, Nucl. Phys. B 873 (2013) 395 [arXiv:1304.4099] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefański Jr., The complete worldsheet S matrix of superstrings on AdS3 × S3 × T4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
S. Leurent and D. Volin, Multiple zeta functions and double wrapping in planar N = 4 SYM, Nucl. Phys. B 875 (2013) 757 [arXiv:1302.1135] [INSPIRE].
Acknowledgments
We thank G. Arutyunov, A. Banerjee, B. Basso, A. Belin, A. Bissi, A. Castro, M. Gaberdiel, R. Gopakumar, J. Julius, V. Kazakov, P. Kravchuk, D. le Plat, F. Levkovich-Maslyuk, S. Majumder, J. Maldacena, A. Manenti, D. Martelli, M. Preti, A. Pribytok, N. Primi, A. Sfondrini, N. Sokolova, R. Tateo, A. Tseytlin and M. Wilhelm for discussions, and especially B. Stefański jr, A. Torrielli and D. Volin for discussions and ongoing collaborations.
We thank the Simons Foundation for the Varna Workshop organized by the International Center for Mathematical Sciences in Sofia, and the Kavli Institute For Theoretical Physics, Santa Barbara for hosting the program Integrability in String, Field, and Condensed Matter Theory, for warm hospitality and creating an excellent scientific environment in the final stages of this work.
AC was partially supported by the INFN project SFT and the EU network GATIS+, and was funded by the ERC project EXACTC during part of this work.
SE was supported by the Knut and Alice Wallenberg Foundation under grant “Exact Results in Gauge and String Theories” Dnr KAW 2015.0083.
N.G. and P. R were supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme — 60 — (grant agreement No. 865075) EXACTC. P. R is grateful to DESY Hamburg for warm hospitality during the final stages of this work.
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Cavaglià, A., Ekhammar, S., Gromov, N. et al. Exploring the Quantum Spectral Curve for AdS3/CFT2. J. High Energ. Phys. 2023, 89 (2023). https://doi.org/10.1007/JHEP12(2023)089
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DOI: https://doi.org/10.1007/JHEP12(2023)089