Abstract
We study the exact spectrum of the AdS4/CFT3 duality put forward by Aharony, Bergman, Jafferis and Maldacena (ABJM). We derive thermodynamic Bethe ansatz (TBA) equations for the planar ABJM theory, starting from “mirror” asymptotic Bethe equations which we conjecture. We also propose generalization of the TBA equations for excited states. The recently proposed Y-system is completely consistent with the TBA equations for a large subsector of the theory, but should be modified in general. We find the general asymptotic infinite length solution of the Y-system, and also several solutions to all wrapping orders in the strong coupling scaling limit. To make a comparison with results obtained from string theory, we assume that the all-loop Bethe ansatz of N.G. and P. Vieira is the valid worldsheet theory description in the asymptotic regime. In this case we find complete agreement, to all orders in wrappings, between the solution of our Y-system and generic quasi-classical string spectrum in AdS3 × S1.
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J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [SPIRES].
O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [SPIRES].
J.A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [SPIRES].
J.A. Minahan and K. Zarembo, The Bethe ansatz for superconformal Chern-Simons, JHEP 09 (2008) 040 [arXiv:0806.3951] [SPIRES].
D. Gaiotto, S. Giombi and X. Yin, Spin chains in N = 6 superconformal Chern-Simons-matter theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [SPIRES].
D. Bak and S.-J. Rey, Integrable spin chain in superconformal Chern-Simons theory, JHEP 10 (2008) 053 [arXiv:0807.2063] [SPIRES].
C. Kristjansen, M. Orselli and K. Zoubos, Non-planar ABJM theory and integrability, JHEP 03 (2009) 037 [arXiv:0811.2150] [SPIRES].
B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory, J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [SPIRES].
J.A. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, JHEP 03 (2009) 057 [arXiv:0901.1142] [SPIRES].
D. Bak, H. Min and S.-J. Rey, Generalized dynamical spin chain and 4-loop integrability in N = 6 superconformal Chern-Simons theory, Nucl. Phys. B 827 (2010) 381 [arXiv:0904.4677] [SPIRES].
B. Chen and J.-B. Wu, Semi-classical strings in AdS 4 ∗ CP 3, JHEP 09 (2008) 096 [arXiv:0807.0802] [SPIRES].
J.A. Minahan, O. Ohlsson Sax and C. Sieg, Magnon dispersion to four loops in the ABJM and ABJ models, arXiv:0908.2463 [SPIRES].
J.A. Minahan, O. Ohlsson Sax and C. Sieg, Anomalous dimensions at four loops in N = 6 superconformal Chern-Simons theories, arXiv:0912.3460 [SPIRES].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [SPIRES].
B. Stefanski, jr, Green-Schwarz action for type IIA strings on AdS 4 × CP 3, Nucl. Phys. B 808 (2009) 80 [arXiv:0806.4948] [SPIRES].
G. Arutyunov and S. Frolov, Superstrings on AdS 4 × CP 3 as a coset σ-model, JHEP 09 (2008) 129 [arXiv:0806.4940] [SPIRES].
J. Gomis, D. Sorokin and L. Wulff, The complete AdS 4 × CP 3 superspace for the type IIA superstring and D-branes, JHEP 03 (2009) 015 [arXiv:0811.1566] [SPIRES].
D. Astolfi, V.G.M. Puletti, G. Grignani, T. Harmark and M. Orselli, Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS 4 xCP 3, Nucl. Phys. B 810 (2009) 150 [arXiv:0807.1527] [SPIRES].
P. Sundin, The AdS 4 × CP 3 string and its Bethe equations in the near plane wave limit, JHEP 02 (2009) 046 [arXiv:0811.2775] [SPIRES].
P. Sundin, On the worldsheet theory of the type IIA AdS 4 × CP 3 superstring, JHEP 04 (2010) 014 [arXiv:0909.0697] [SPIRES].
K. Zarembo, Worldsheet spectrum in AdS 4/CFT 3 correspondence, JHEP 04 (2009) 135 [arXiv:0903.1747] [SPIRES].
C. Kalousios, C. Vergu and A. Volovich, Factorized tree-level scattering in AdS 4 xCP 3, JHEP 09 (2009) 049 [arXiv:0905.4702] [SPIRES].
M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [SPIRES].
N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys. 12 (2008) 945 [hep-th/0511082] [SPIRES].
N. Beisert, The analytic Bethe Ansatz for a chain with centrally extended SU(2—2) symmetry, J. Stat. Mech. 0701 (2007) P017 [nlin/0610017] [SPIRES].
R.A. Janik, The AdS 5 × S 5 superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D 73 (2006) 086006 [hep-th/0603038] [SPIRES].
N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [SPIRES].
V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 05 (2004) 024 [hep-th/0402207] [SPIRES].
V.A. Kazakov and K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 10 (2004) 060 [hep-th/0410105] [SPIRES].
N. Beisert, V.A. Kazakov and K. Sakai, Algebraic curve for the SO(6) sector of AdS/CFT, Commun. Math. Phys. 263 (2006) 611 [hep-th/0410253] [SPIRES].
S. Schäfer-Nameki, The algebraic curve of 1-loop planar N = 4 SYM, Nucl. Phys. B 714 (2005) 3 [hep-th/0412254] [SPIRES].
N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS 5 × S 5, Commun. Math. Phys. 263 (2006) 659 [hep-th/0502226] [SPIRES].
N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe Ansaetze for gauge theory and strings, Nucl. Phys. B 727 (2005) 1 [hep-th/0504190] [SPIRES].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].
N. Gromov and P. Vieira, The AdS4/CFT3 algebraic curve, JHEP 02 (2009) 040 [arXiv:0807.0437] [SPIRES]
N. Gromov and P. Vieira, The all loop AdS4/CFT3 Bethe Ansatz, JHEP 01 (2009) 016 [arXiv:0807.0777] [SPIRES].
C. Ahn and R.I. Nepomechie, N=6 super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations, JHEP 09 (2008) 010 [arXiv:0807.1924] [SPIRES].
A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS 3/CFT 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [SPIRES].
J. Ambjorn, 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] [SPIRES].
R.A. Janik and T. Lukowski, Wrapping interactions at strong coupling – the giant magnon, Phys. Rev. D 76 (2007) 126008 [arXiv:0708.2208] [SPIRES].
M.P. Heller, R.A. Janik and T. Lukowski, A new derivation of Lüscher F-term and fluctuations around the giant magnon, JHEP 06 (2008) 036 [arXiv:0801.4463] [SPIRES].
Z. Bajnok and R.A. Janik, Four-loop perturbative Konishi from strings and finite size effects for multiparticle states, Nucl. Phys. B 807 (2009) 625 [arXiv:0807.0399] [SPIRES].
C.-N. Yang and C.P. Yang, One-dimensional chain of anisotropic spin-spin interactions. I: proof of Bethe’s hypothesis for ground state in a finite system, Phys. Rev. 150 (1966) 321 [SPIRES].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [SPIRES].
N. Dorey, Magnon bound states and the AdS/CFT correspondence, J. Phys. A 39 (2006) 13119 [hep-th/0604175] [SPIRES].
M. Takahashi, Thermodynamics of one-dimensional solvable models, Cambridge University Press, Cambridge U.K. (1999).
F.H.L. Essler, H.Frahm, F.Göhmann, A. Klümper and V. Korepin, The one-dimensional Hubbard model, Cambridge University Press, Cambridge U.K. (2005).
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Quantum field theories in finite volume: Excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [SPIRES].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [SPIRES].
D. Fioravanti, A. Mariottini, E. Quattrini and F. Ravanini, Excited state Destri-De Vega equation for sine-Gordon and restricted sine-Gordon models, Phys. Lett. B 390 (1997) 243 [hep-th/9608091] [SPIRES].
A.G. Bytsko and J. Teschner, Quantization of models with non-compact quantum group symmetry: modular XXZ magnet and lattice sinh-Gordon model, J. Phys. A 39 (2006) 12927 [hep-th/0602093] [SPIRES].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [SPIRES].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of anomalous dimensions of planar N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [SPIRES].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Anomalous dimension with wrapping at four loops in N = 4 SYM, Nucl. Phys. B 805 (2008) 231 [arXiv:0806.2095] [SPIRES].
V.N. Velizhanin, Leading transcedentality contributions to the four-loop universal anomalous dimension in N = 4 SYM, Phys. Lett. B 676 (2009) 112 [arXiv:0811.0607] [SPIRES].
G. Arutyunov and S. Frolov, On string S-matrix, bound states and TBA, JHEP 12 (2007) 024 [arXiv:0710.1568] [SPIRES].
F. Fiamberti, A. Santambrogio and C. Sieg, Five-loop anomalous dimension at critical wrapping order in N = 4 SYM, JHEP 03 (2010) 103 [arXiv:0908.0234] [SPIRES].
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] [SPIRES].
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] [SPIRES].
G. Arutyunov and S. Frolov, The dressing factor and crossing equations, J. Phys. A 42 (2009) 425401 [arXiv:0904.4575] [SPIRES].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [SPIRES].
S. Frolov and R. Suzuki, Temperature quantization from the TBA equations, Phys. Lett. B 679 (2009) 60 [arXiv:0906.0499] [SPIRES].
G. Arutyunov and S. Frolov, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [SPIRES].
N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of planar \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory: Konishi dimension at any coupling, Phys. Rev. Lett. 104 (2010) 211601 [arXiv:0906.4240] [SPIRES].
K. Konishi, Anomalous supersymmetry transformation of some composite operators in SQCD, Phys. Lett. B 135 (1984) 439 [SPIRES].
M. Bianchi, S. Kovacs, G. Rossi and Y.S. Stanev, Properties of the Konishi multiplet in N = 4 SYM theory, JHEP 05 (2001) 042 [hep-th/0104016] [SPIRES].
B. Eden, C. Jarczak, E. Sokatchev and Y.S. Stanev, Operator mixing in N = 4 SYM: the Konishi anomaly revisited, Nucl. Phys. B 722 (2005) 119 [hep-th/0501077] [SPIRES].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [SPIRES].
N. Gromov, Y-system and quasi-classical strings, JHEP 01 (2010) 112 [arXiv:0910.3608] [SPIRES].
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] [SPIRES].
R. Roiban and A.A. Tseytlin, Quantum strings in AdS 5 × S 5 : strong-coupling corrections to dimension of Konishi operator, JHEP 11 (2009) 013 [arXiv:0906.4294] [SPIRES].
A.A. Tseytlin, Quantum strings in AdS5 x S5 and AdS/CFT duality, Int. J. Mod. Phys. A 25 (2010) 319 [arXiv:0907.3238] [SPIRES].
D. Bombardelli and D. Fioravanti, Finite-Size Corrections of the \( \mathbb{C}{\mathbb{P}^3} \) Giant Magnons: the Lúscher terms, JHEP 07 (2009) 034 [arXiv:0810.0704] [SPIRES].
I. Shenderovich, Giant magnons in AdS 4/CFT 3 : dispersion, quantization and finite–size corrections, arXiv:0807.2861 [SPIRES].
N. Berkovits, Super-Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [SPIRES].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [SPIRES].
O. Bergman and S. Hirano, Anomalous radius shift in AdS 4/CFT 3, JHEP 07 (2009) 016 [arXiv:0902.1743] [SPIRES].
L. Mazzucato and B.C. Vallilo, On the non-renormalization of the AdS Radius, JHEP 09 (2009) 056 [arXiv:0906.4572] [SPIRES].
G. Bonelli, P.A. Grassi and H. Safaai, Exploring pure spinor string theory on \( Ad{S_4} \times \mathbb{C}{\mathbb{P}^3} \), JHEP 10 (2008) 085 [arXiv:0808.1051] [SPIRES].
N. Beisert, J.A. Minahan, M. Staudacher and K. Zarembo, Stringing spins and spinning strings, JHEP 09 (2003) 010 [hep-th/0306139] [SPIRES].
Z. Tsuboi, Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r + 1|s + 1), J. Phys. A 30 (1997) 7975 [arXiv:0911.5387].
N. Beisert, The analytic Bethe ansatz for a chain with centrally extended su(2|2) symmetry, J. Stat. Mech. 0701 (2007) P017 [nlin/0610017] [SPIRES].
V. Kazakov, A.S. Sorin and A. Zabrodin, Supersymmetric Bethe ansatz and Baxter equations from discrete Hirota dynamics, Nucl. Phys. B 790 (2008) 345 [hep-th/0703147] [SPIRES].
H. Saleur and B. Pozsgay, Scattering and duality in the 2 dimensional OSP(2—2) gross neveu and σ-models, JHEP 02 (2010) 008 [arXiv:0910.0637] [SPIRES].
B. Vicedo, Finite-g Strings, arXiv:0810.3402 [SPIRES].
T. Lukowski and O.O. Sax, Finite size giant magnons in the SU(2) x SU(2) sector of AdS 4 xCP 3, JHEP 12 (2008) 073 [arXiv:0810.1246] [SPIRES].
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 [SPIRES].
P. Dorey, Exact S matrices, hep-th/9810026 [SPIRES].
N. Beisert, C. Kristjansen and M. Staudacher, The dilatation operator of N = 4 super Yang-Mills theory, Nucl. Phys. B 664 (2003) 131 [hep-th/0303060] [SPIRES].
N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [SPIRES].
N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [SPIRES].
M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [SPIRES].
N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS 5 × S 5 strings, JHEP 11 (2006) 070 [hep-th/0609044] [SPIRES].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [SPIRES].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Quantum field theories in finite volume: excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [SPIRES].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [SPIRES].
P. Dorey and R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575 [hep-th/9706140] [SPIRES].
B.I. Zwiebel, Two-loop integrability of planar N = 6 superconformal Chern-Simons theory, J. Phys. A 42 (2009) 495402 [arXiv:0901.0411] [SPIRES].
J.A. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, JHEP 03 (2009) 057 [arXiv:0901.1142] [SPIRES].
C. Ahn and R.I. Nepomechie, Two-loop test of the N = 6 Chern-Simons theory S-matrix, JHEP 03 (2009) 144 [arXiv:0901.3334] [SPIRES].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [SPIRES].
G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [SPIRES].
N. Gromov and V. Mikhaylov, Comment on the scaling function in AdS 4 × CP 3, JHEP 04 (2009) 083 [arXiv:0807.4897] [SPIRES].
N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP 04 (2008) 046 [arXiv:0709.3487] [SPIRES].
T. McLoughlin and R. Roiban, Spinning strings at one-loop in AdS 4 xCP 3, JHEP 12 (2008) 101 [arXiv:0807.3965] [SPIRES].
L.F. Alday, G. Arutyunov and D. Bykov, Semiclassical quantization of spinning strings in AdS 4 xCP 3, JHEP 11 (2008) 089 [arXiv:0807.4400] [SPIRES].
C. Krishnan, AdS4/CFT3 at one loop, JHEP 09 (2008) 092 [arXiv:0807.4561] [SPIRES].
T. McLoughlin, R. Roiban and A.A. Tseytlin, Quantum spinning strings in AdS 4 xCP 3 : testing the Bethe Ansatz proposal, JHEP 11 (2008) 069 [arXiv:0809.4038] [SPIRES].
V. Mikhaylov, On the fermionic frequencies of circular strings, arXiv:1002.1831 [SPIRES].
S. Frolov and A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS 5 × S 5, JHEP 06 (2002) 007 [hep-th/0204226] [SPIRES].
D. Bombardelli, D. Fioravanti and R. Tateo, TBA and Y-system for planar AdS 4/CFT 3, Nucl. Phys. B 834 (2010) 543 [arXiv:0912.4715] [SPIRES].
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Gromov, N., Levkovich-Maslyuk, F. Y-system, TBA and Quasi-Classical Strings in AdS 4 × CP3 . J. High Energ. Phys. 2010, 88 (2010). https://doi.org/10.1007/JHEP06(2010)088
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DOI: https://doi.org/10.1007/JHEP06(2010)088