Abstract
The hybrid NLIE of AdS5 × S5 is applied to a wider class of states. We find that the Konishi state of the orbifold AdS5 × S5/\( {\mathbb{Z}_S} \) satisfies A 1 NLIE with the source terms which are derived from contour deformation trick. For general states, we construct a deformed contour with which the contour deformation trick yields the correct source terms.
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J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [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 \( \mathcal{N} = 4 \) Supersymmetric Yang-Mills Theory: TBA and excited states, Lett. Math. Phys. 91 (2010) 265 [arXiv:0902.4458] [INSPIRE].
G. Arutyunov and S. Frolov, Thermodynamic Bethe Ansatz for the AdS 5 × S 5 mirror model, JHEP 05 (2009) 068 [arXiv:0903.0141] [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, String hypothesis for the AdS 5 × S 5 mirror, JHEP 03 (2009) 152 [arXiv:0901.1417] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Exact Spectrum of Anomalous Dimensions of Planar \( \mathcal{N} = 4 \) Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 103 (2009) 131601 [arXiv:0901.3753] [INSPIRE].
A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS 5/CFT 4 correspondence, Nucl. Phys. B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE].
J. Balog and A. Hegedus, AdS 5 × S 5 mirror TBA equations from Y-system and discontinuity relations, JHEP 08 (2011) 095 [arXiv:1104.4054] [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].
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] [INSPIRE].
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] [INSPIRE].
S. Frolov, Konishi operator at intermediate coupling, J. Phys. A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE].
F. Levkovich-Maslyuk, Numerical results for the exact spectrum of planar AdS 4/CFT 3, JHEP 05 (2012) 142 [arXiv:1110.5869] [INSPIRE].
S. Frolov, Scaling dimensions from the mirror TBA, arXiv:1201.2317 [INSPIRE].
G. Arutyunov, S. Frolov and R. Suzuki, Exploring the mirror TBA, JHEP 05 (2010) 031 [arXiv:0911.2224] [INSPIRE].
J. Suzuki, Spinons in magnetic chains of arbitrary spins at finite temperatures, J. Phys. A 32 (1999) 2341 [cond-mat/9807076].
R. Suzuki, Hybrid NLIE for the Mirror AdS 5 × S 5, J. Phys. A 44 (2011) 235401 [arXiv:1101.5165] [INSPIRE].
A. Klümper and M.T. Batchelor, An analytic treatment of finite-size corrections in the spin-1 antiferromagnetic XXZ chain, J. Phys. A 23 (1990) L189.
A. Klümper, M.T. Batchelor and P.A. Pearce, Central charges of the 6- and 19- vertex models with twisted boundary conditions, J. Phys. A 24 (1991) 3111.
C. Destri and H. de Vega, New approach to thermal Bethe ansatz, hep-th/9203064 [INSPIRE].
C. Destri and H. de Vega, New thermodynamic Bethe ansatz equations without strings, Phys. Rev. Lett. 69 (1992) 2313 [INSPIRE].
C. Destri and H. De Vega, Unified approach to thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories, Nucl. Phys. B 438 (1995) 413 [hep-th/9407117] [INSPIRE].
R. Roiban, On spin chains and field theories, JHEP 09 (2004) 023 [hep-th/0312218] [INSPIRE].
D. Berenstein and S.A. Cherkis, Deformations of \( \mathcal{N} = 4 \) SYM and integrable spin chain models, Nucl. Phys. B 702 (2004) 49 [hep-th/0405215] [INSPIRE].
K. Ideguchi, Semiclassical strings on AdS 5 × S 5/Z(M) and operators in orbifold field theories, JHEP 09 (2004) 008 [hep-th/0408014] [INSPIRE].
N. Beisert and R. Roiban, Beauty and the twist: the Bethe ansatz for twisted \( \mathcal{N} = 4 \) SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].
N. Beisert and R. Roiban, The Bethe ansatz for Z(S) orbifolds of \( \mathcal{N} = 4 \) super Yang-Mills theory, JHEP 11 (2005) 037 [hep-th/0510209] [INSPIRE].
S. Ananth, S. Kovacs and H. Shimada, Proof of all-order finiteness for planar beta-deformed Yang-Mills, JHEP 01 (2007) 046 [hep-th/0609149] [INSPIRE].
S. Ananth, S. Kovacs and H. Shimada, Proof of ultra-violet finiteness for a planar non-supersymmetric Yang-Mills theory, Nucl. Phys. B 783 (2007) 227 [hep-th/0702020] [INSPIRE].
H. Lin, O. Lunin and J.M. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 10 (2004) 025 [hep-th/0409174] [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].
S. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for superconformal deformations of \( \mathcal{N} = 4 \) super Yang-Mills theory, JHEP 07 (2005) 045 [hep-th/0503192] [INSPIRE].
S. Frolov, R. Roiban and A.A. Tseytlin, Gauge-string duality for (non)supersymmetric deformations of \( \mathcal{N} = 4 \) super Yang-Mills theory, Nucl. Phys. B 731 (2005) 1 [hep-th/0507021] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Finite-size effects in the superconformal beta-deformed \( \mathcal{N} = 4 \) SYM, JHEP 08 (2008) 057 [arXiv:0806.2103] [INSPIRE].
F. Fiamberti, A. Santambrogio, C. Sieg and D. Zanon, Single impurity operators at critical wrapping order in the beta-deformed \( \mathcal{N} = 4 \) SYM, JHEP 08 (2009) 034 [arXiv:0811.4594] [INSPIRE].
J. Minahan and C. Sieg, Four-Loop Anomalous Dimensions in Leigh-Strassler Deformations, arXiv:1112.4787 [INSPIRE].
D.V. Bykov and S. Frolov, Giant magnons in TsT-transformed AdS 5 × S 5, JHEP 07 (2008) 071 [arXiv:0805.1070] [INSPIRE].
J. Gunnesson, Wrapping in maximally supersymmetric and marginally deformed \( \mathcal{N} = 4 \) Yang-Mills, JHEP 04 (2009) 130 [arXiv:0902.1427] [INSPIRE].
M. Beccaria and G.F. De Angelis, On the wrapping correction to single magnon energy in twisted \( \mathcal{N} = 4 \) SYM, Int. J. Mod. Phys. A 24 (2009) 5803 [arXiv:0903.0778] [INSPIRE].
Z. Bajnok, A. Hegedus, R.A. Janik and T. Lukowski, Five loop Konishi from AdS/CFT, Nucl. Phys. B 827 (2010) 426 [arXiv:0906.4062] [INSPIRE].
M. de Leeuw and T. Lukowski, Twist operators in \( \mathcal{N} = 4 \) beta-deformed theory, JHEP 04 (2011) 084 [arXiv:1012.3725] [INSPIRE].
M. Beccaria, F. Levkovich-Maslyuk and G. Macorini, On wrapping corrections to GKP-like operators, JHEP 03 (2011) 001 [arXiv:1012.2054] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Finite-size effect for four-loop Konishi of the β-deformed \( \mathcal{N} = 4 \) SYM, Phys. Lett. B 693 (2010) 380 [arXiv:1006.2209] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, TBA, NLO Lüscher correction and double wrapping in twisted AdS/CFT, JHEP 12 (2011) 059 [arXiv:1108.4914] [INSPIRE].
C. Ahn, D. Bombardelli and M. Kim, Finite-size effects of β-deformed AdS 5/CFT 4 at strong coupling, Phys. Lett. B 710 (2012) 467 [arXiv:1201.2635] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed \( \mathcal{N} = 4 \) super-Yang-Mills, J. Phys. A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE].
G. Arutyunov, M. de Leeuw and S.J. van Tongeren, Twisting the Mirror TBA, JHEP 02 (2011) 025 [arXiv:1009.4118] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, Orbifolded Konishi from the Mirror TBA, J. Phys. A 44 (2011) 325404 [arXiv:1103.5853] [INSPIRE].
M. de Leeuw and S.J. van Tongeren, The spectral problem for strings on twisted AdS 5 × S 5, Nucl. Phys. B 860 (2012) 339 [arXiv:1201.1451] [INSPIRE].
M. Beccaria and G. Macorini, Y-system for \( {\mathbb{Z}_S} \) orbifolds of \( \mathcal{N} = 4 \) SYM, JHEP 06 (2011) 004 [Erratum ibid. 1201 (2012) 112] [arXiv:1104.0883] [INSPIRE].
K. Zoubos, Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].
P. Dorey and R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575 [hep-th/9706140] [INSPIRE].
G. Arutyunov, S. Frolov and S.J. van Tongeren, Bound States in the Mirror TBA, JHEP 02 (2012) 014 [arXiv:1111.0564] [INSPIRE].
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] [INSPIRE].
A. Hegedus, Finite size effects in the SS model: two component nonlinear integral equations, Nucl. Phys. B 679 (2004) 545 [hep-th/0310051] [INSPIRE].
J. Suzuki, Excited states nonlinear integral equations for an integrable anisotropic spin 1 chain, J. Phys. A 37 (2004) 11957 [hep-th/0410243] [INSPIRE].
A. Hegedus, Nonlinear integral equations for finite volume excited state energies of the O(3) and O(4) nonlinear σ-models, J. Phys. A 38 (2005) 5345 [hep-th/0412125] [INSPIRE].
A. Hegedus, Nonlinear integral equations for the finite size effects of RSOS and vertex-models and related quantum field theories, Nucl. Phys. B 732 (2005) 463 [hep-th/0507132] [INSPIRE].
A. Hegedus, F. Ravanini and J. Suzuki, Exact finite size spectrum in super sine-Gordon model, Nucl. Phys. B 763 (2007) 330 [hep-th/0610012] [INSPIRE].
A. Hegedus, Finite size effects and 2-string deviations in the spin-1 XXZ chains, J. Phys. A A 40 (2007) 12007 [arXiv:0706.1411] [INSPIRE].
N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP 12 (2009) 060 [arXiv:0812.5091] [INSPIRE].
J. Caetano, Unified approach to the SU(2) Principal Chiral Field model at Finite Volume, arXiv:1012.2600 [INSPIRE].
A. Klümper and P.A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A 183 (1992) 304.
A. Klümper, Thermodynamics of the anisotropic spin-1/2 Heisenberg chain and related quantum chains, Z. Phys. B 91 (1993) 507 [cond-mat/9306019].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, JHEP 07 (2012) 023 [arXiv:1110.0562] [INSPIRE].
C. Ahn, Z. Bajnok, D. Bombardelli and R.I. Nepomechie, Twisted Bethe equations from a twisted S-matrix, JHEP 02 (2011) 027 [arXiv:1010.3229] [INSPIRE].
N. Beisert, The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2) Symmetry, J. Stat. Mech. 0701 (2007) P01017 [nlin/0610017].
N. Gromov and V. Kazakov, Why Y? Exploiting Hirota integrable dynamics in AdS/CFT, talk presented by V. Kazakov at Conference on Integrability in Gauge and String Theory 2010, Nordita, Stockholm Sweden (2010), http://agenda.albanova.se/contributionDisplay.py?contribId=258&confId=1561.
V. Kazakov and S. Leurent, Finite Size Spectrum of SU(N) Principal Chiral Field from Discrete Hirota Dynamics, arXiv:1007.1770 [INSPIRE].
J. Balog and A. Hegedus, Quasi-local formulation of the mirror TBA, JHEP 05 (2012) 039 [arXiv:1106.2100] [INSPIRE].
J. Balog and A. Hegedus, Hybrid-NLIE for the AdS/CFT spectral problem, arXiv:1202.3244 [INSPIRE].
I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations, Commun. Math. Phys. 188 (1997) 267 [hep-th/9604080] [INSPIRE].
D. Volin, Lecture notes on quantum integrability, downloadable from https://nordita.webex.com/mw0306ld/mywebex/personalroom/personalroom.do?siteurl=nordita&AT=meet&username=Nordita.
G. Arutyunov, M. de Leeuw, R. Suzuki and A. Torrielli, Bound State Transfer Matrix for AdS 5 × S 5 Superstring, JHEP 10 (2009) 025 [arXiv:0906.4783] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and Z. Tsuboi, Wronskian Solution for AdS/CFT Y-system, JHEP 01 (2011) 155 [arXiv:1010.2720] [INSPIRE].
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Suzuki, R. Contour deformation trick in hybrid NLIE. J. High Energ. Phys. 2012, 152 (2012). https://doi.org/10.1007/JHEP07(2012)152
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DOI: https://doi.org/10.1007/JHEP07(2012)152