×

Quantum spectral curve for arbitrary state/operator in \(\mathrm{AdS}_{5}/\mathrm{CFT}_{4}\). (English) Zbl 1388.81214

Summary: We give a derivation of quantum spectral curve (QSC) – a finite set of Riemann-Hilbert equations for exact spectrum of planar \( \mathcal{N}=4 \) SYM theory proposed in our recent paper [“Quantum spectral curve for planar \(\mathcal{N}=4\) super-Yang-Mills theory”, Phys. Rev. Lett. 112, No. 1, Article ID 011602, 5 p. (2014; doi:10.1103/PhysRevLett.112.011602)]. We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system – a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
83C47 Methods of quantum field theory in general relativity and gravitational theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys.99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
[2] 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] [INSPIRE].
[3] N. Gromov, V. Kazakov and P. Vieira, Exact spectrum of \[planarN=4 \mathcal{N}=4\] supersymmetric Yang-Mills theory: konishi dimension at any coupling, Phys. Rev. Lett.104 (2010) 211601 [arXiv:0906.4240] [INSPIRE].
[4] S. Frolov, Konishi operator at intermediate coupling, J. Phys.A 44 (2011) 065401 [arXiv:1006.5032] [INSPIRE]. · Zbl 1208.81160
[5] F. Levkovich-Maslyuk, Numerical results for the exact spectrum of planar AdS4/CFT3, JHEP05 (2012) 142 [arXiv:1110.5869] [INSPIRE].
[6] 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] [INSPIRE]. · Zbl 1186.81102
[7] 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]. · Zbl 1176.81091
[8] G. Arutyunov and S. Frolov, Thermodynamic Bethe ansatz for the AdS5 × S5mirror model, JHEP05 (2009) 068 [arXiv:0903.0141] [INSPIRE].
[9] A. Cavaglia, D. Fioravanti and R. Tateo, Extended Y-system for the AdS5/CF T4correspondence, Nucl. Phys.B 843 (2011) 302 [arXiv:1005.3016] [INSPIRE]. · Zbl 1207.81105
[10] N. Gromov, V. Kazakov and Z. Tsuboi, PSU(2, 2|4) character of quasiclassical AdS/CFT, JHEP07 (2010) 097 [arXiv:1002.3981] [INSPIRE]. · Zbl 1290.81053
[11] N. Gromov, V. Kazakov, S. Leurent and Z. Tsuboi, Wronskian solution for AdS/CFT Y-system, JHEP01 (2011) 155 [arXiv:1010.2720] [INSPIRE]. · Zbl 1214.81214
[12] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Solving the AdS/CFT Y-system, JHEP07 (2012) 023 [arXiv:1110.0562] [INSPIRE].
[13] S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett.109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].
[14] 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]. · Zbl 1331.81276
[15] N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for PlanarN \[\mathcal{N} = super\]-Yang-Mills Theory, Phys. Rev. Lett.112 (2014) 011602 [arXiv:1305.1939] [INSPIRE].
[16] D. Volin, Quantum spectral curve for AdS5/CF T4spectral problem, talk given at Integrability in gauge and string theory, August 19-23, Utrecht, The Netherlands (2013).
[17] D. Volin and C. Marboe, Quantum spectral curve as a tool for a perturbative quantum field theory, Nucl. Phys.B 899 (2015) 810 [arXiv:1411.4758] [INSPIRE]. · Zbl 1331.81209
[18] N. Gromov, F. Levkovich-Maslyuk, G. Sizov and S. Valatka, Quantum spectral curve at work: from small spin to strong coupling \[inN=4 \mathcal{N}=4\] SYM, JHEP07 (2014) 156 [arXiv:1402.0871] [INSPIRE].
[19] A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, Quantum spectral curve of \[theN=6 \mathcal{N}=6\] supersymmetric Chern-Simons theory, Phys. Rev. Lett.113 (2014) 021601 [arXiv:1403.1859] [INSPIRE].
[20] N. Gromov and G. Sizov, Exact slope and interpolating functions in N = 6 supersymmetric Chern-Simons theory, Phys. Rev. Lett.113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].
[21] N. Beisert, V.A. Kazakov, K. Sakai and K. Zarembo, The algebraic curve of classical superstrings on AdS5 × S5, Commun. Math. Phys.263 (2006) 659 [hep-th/0502226] [INSPIRE]. · Zbl 1125.81037
[22] N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021 [hep-th/0610251] [INSPIRE].
[23] N. Beisert and M. Staudacher, Long-range PSU(2, 2|4) Bethe ansatze for gauge theory and strings, Nucl. Phys.B 727 (2005) 1 [hep-th/0504190] [INSPIRE]. · Zbl 1126.81328
[24] J. Balog and A. Hegedus, AdS5 × S5mirror TBA equations from Y-system and discontinuity relations, JHEP08 (2011) 095 [arXiv:1104.4054] [INSPIRE]. · Zbl 1298.81239
[25] D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP08 (2012) 134 [arXiv:1203.1913] [INSPIRE].
[26] N. Drukker, Integrable Wilson loops, JHEP10 (2013) 135 [arXiv:1203.1617] [INSPIRE]. · Zbl 1342.83237
[27] Z. Bajnok et al., The spectrum of tachyons in AdS/CFT, JHEP03 (2014) 055 [arXiv:1312.3900] [INSPIRE].
[28] P. Kulish and N.Y. Reshetikhin, Generalized Heisenberg ferromagnet and the Gross-Neveu model, J. Exp. Theor. Phys.53 (1981) 108.
[29] P. Kulish, Integrable graded magnets, J. Sov. Math.35 (1986) 2648. · Zbl 0606.58054
[30] B. Sutherland, Low-lying eigenstates of the one-dimensional Heisenberg ferromagnet for any magnetization and momentum, Phys. Rev. Lett.74 (1995) 816 [INSPIRE].
[31] 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]. · Zbl 0896.58035
[32] 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] [INSPIRE]. · Zbl 1150.82009
[33] Z. Tsuboi, Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r + 1|s + 1), J. Phys.A 30 (1997) 7975 [arXiv:0911.5386] [INSPIRE]. · Zbl 0940.81025
[34] V. Kazakov, S. Leurent, and D. Volin, T-system on T-hook: grassmannian solution and twisted quantum spectral curve, in preparation. · Zbl 1390.81225
[35] Z. Tsuboi, Wronskian solutions of the T, Q and Y-systems related to infinite dimensional unitarizable modules of the general linear superalgebra gl(M |N ), Nucl. Phys.B 870 (2013) 92 [arXiv:1109.5524] [INSPIRE]. · Zbl 1262.17017
[36] Z. Tsuboi, Analytic Bethe ansatz and functional equations associated with any simple root systems of the Lie superalgebra sl(r + 1|s + 1), PhysicaA 252 (1998) 565 [arXiv:0911.5387] [INSPIRE].
[37] N. Gromov and P. Vieira, Complete 1-loop test of AdS/CFT, JHEP04 (2008) 046 [arXiv:0709.3487] [INSPIRE]. · Zbl 1246.81250
[38] G.P. Pronko and Yu.G. Stroganov, Bethe equations ‘on the wrong side of equator’, J. Phys.A 32 (1999) 2333 [hep-th/9808153] [INSPIRE]. · Zbl 0964.82013
[39] Z. Tsuboi, Solutions of the T-system and Baxter equations for supersymmetric spin chains, Nucl. Phys.B 826 (2010) 399 [arXiv:0906.2039] [INSPIRE]. · Zbl 1203.82029
[40] N. Gromov and V. Kazakov, Review of AdS/CFT integrability, chapter III.7: Hirota dynamics for quantum integrability, Lett. Math. Phys.99 (2012) 321 [arXiv:1012.3996] [INSPIRE]. · Zbl 1242.81111
[41] N. Beisert, R. Hernandez and E. Lopez, A crossing-symmetric phase for AdS5 × S5strings, JHEP11 (2006) 070 [hep-th/0609044] [INSPIRE].
[42] D. Volin, Minimal solution of the AdS/CFT crossing equation, J. Phys.A 42 (2009) 372001 [arXiv:0904.4929] [INSPIRE]. · Zbl 1176.81106
[43] R.A. Janik, The AdS5 × S5superstring worldsheet S-matrix and crossing symmetry, Phys. Rev.D 73 (2006) 086006 [hep-th/0603038] [INSPIRE].
[44] P. Vieira and D. Volin, Review of AdS/CFT integrability. Chapter III.3: the dressing factor, Lett. Math. Phys.99 (2012) 231 [arXiv:1012.3992] [INSPIRE]. · Zbl 1255.81223
[45] M. Staudacher, The factorized S-matrix of CFT/AdS, JHEP05 (2005) 054 [hep-th/0412188] [INSPIRE].
[46] N. Beisert, The SU(2|2) dynamic S-matrix, Adv. Theor. Math. Phys.12 (2008) 945 [hep-th/0511082] [INSPIRE]. · Zbl 1146.81047
[47] A. Santambrogio and D. Zanon, Exact anomalous dimensions of N = 4 Yang-Mills operators with large R charge, Phys. Lett.B 545 (2002) 425 [hep-th/0206079] [INSPIRE]. · Zbl 0998.81059
[48] N. Gromov, Y-system and quasi-classical strings, JHEP01 (2010) 112 [arXiv:0910.3608] [INSPIRE]. · Zbl 1269.81129
[49] N. Gromov and A. Sever, Analytic solution of Bremsstrahlung TBA, JHEP11 (2012) 075 [arXiv:1207.5489] [INSPIRE]. · Zbl 1397.81062
[50] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic solution of Bremsstrahlung TBA II: turning on the sphere angle, JHEP10 (2013) 036 [arXiv:1305.1944] [INSPIRE]. · Zbl 1342.83035
[51] D. Bombardelli, D. Fioravanti and R. Tateo, TBA and Y-system for planar AdS4/CF T3, Nucl. Phys.B 834 (2010) 543 [arXiv:0912.4715] [INSPIRE]. · Zbl 1204.81136
[52] N. Gromov and F. Levkovich-Maslyuk, Y-system, TBA and quasi-classical strings in AdS4 × CP3, JHEP06 (2010) 088 [arXiv:0912.4911] [INSPIRE]. · Zbl 1288.81108
[53] A. Cavaglia, D. Fioravanti and R. Tateo, Discontinuity relations for the AdS4/CF T3correspondence, Nucl. Phys.B 877 (2013) 852 [arXiv:1307.7587] [INSPIRE]. · Zbl 1284.81203
[54] N. Gromov and F. Levkovich-Maslyuk, Y-system and β-deformed N = 4 super-Yang-Mills, J. Phys.A 44 (2011) 015402 [arXiv:1006.5438] [INSPIRE]. · Zbl 1205.81104
[55] G. Arutyunov, M. de Leeuw and S.J. van Tongeren, The exact spectrum and mirror duality of the (AdS5 × S5)ηsuperstring, Theor. Math. Phys.182 (2015) 23 [arXiv:1403.6104] [INSPIRE]. · Zbl 1317.81211
[56] A. Babichenko, B. Stefanski, Jr. and K. Zarembo, Integrability and the AdS3/CF T2correspondence, JHEP03 (2010) 058 [arXiv:0912.1723] [INSPIRE]. · Zbl 1271.81118
[57] R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski, Towards the all-loop worldsheet S matrix for AdS3 × S3 × T4, Phys. Rev. Lett.113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
[58] S. Negro and F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys.B 875 (2013) 166 [arXiv:1306.1476] [INSPIRE]. · Zbl 1282.81161
[59] N. Gromov, V. Kazakov and P. Vieira, Finite volume spectrum of 2D field theories from Hirota dynamics, JHEP12 (2009) 060 [arXiv:0812.5091] [INSPIRE].
[60] V. Kazakov and S. Leurent, Finite size spectrum of SU(N ) principal chiral field from discrete Hirota dynamics, arXiv:1007.1770 [INSPIRE]. · Zbl 1332.81216
[61] S.E. Derkachov, Baxter’s Q-operator for the homogeneous XXX spin chain, J. Phys.A 32 (1999) 5299 [solv-int/9902015] [INSPIRE]. · Zbl 0962.82023
[62] S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Separation of variables for the quantumSL2ℝ \[\text{S}\text{L}\left(2,\mathbb{R}\right)\] spin chain, JHEP07 (2003) 047 [hep-th/0210216] [INSPIRE].
[63] A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky and A.N. Manashov, Baxter Q-operator for graded SL(2|1) spin chain, J. Stat. Mech. (2007) P01005 [hep-th/0610332] [INSPIRE]. · Zbl 1456.81224
[64] R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Baxter operators and hamiltonians for ‘nearly all’ integrable closedgln \[\mathfrak{g}\mathfrak{l}(n)\] spin chains, Nucl. Phys.B 874 (2013) 620 [arXiv:1112.3600] [INSPIRE]. · Zbl 1282.82015
[65] R. Frassek, T. Lukowski, C. Meneghelli and M. Staudacher, Oscillator construction of SU(n|m) Q-operators, Nucl. Phys.B 850 (2011) 175 [arXiv:1012.6021] [INSPIRE]. · Zbl 1215.81047
[66] V. Kazakov and P. Vieira, From characters to quantum (super)spin chains via fusion, JHEP10 (2008) 050 [arXiv:0711.2470] [INSPIRE]. · Zbl 1245.82016
[67] V. Kazakov, S. Leurent and Z. Tsuboi, Baxter’s Q-operators and operatorial Backlund flow for quantum (super)-spin chains, Commun. Math. Phys.311 (2012) 787 [arXiv:1010.4022] [INSPIRE]. · Zbl 1247.82023
[68] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory. 2. Q operator and DDV equation, Commun. Math. Phys.190 (1997) 247 [hep-th/9604044] [INSPIRE]. · Zbl 0908.35114
[69] S. Leurent, D. Serban and D. Volin, Six-loop Konishi anomalous dimension from the Y-system, Phys. Rev. Lett.109 (2012) 241601 [arXiv:1209.0749] [INSPIRE].
[70] P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys.B 482 (1996) 639 [hep-th/9607167] [INSPIRE]. · Zbl 0925.82044
[71] 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] [INSPIRE]. · Zbl 1186.81102
[72] N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys.B 670 (2003) 439 [hep-th/0307042] [INSPIRE]. · Zbl 1058.81581
[73] M. Gunaydin and D. Volin, in preparation.
[74] D. Volin, String hypothesis for gl(n|m) spin chains: a particle/hole democracy, Lett. Math. Phys.102 (2012) 1 [arXiv:1012.3454] [INSPIRE]. · Zbl 1253.82024
[75] H.P. Jakobsen, The full set of unitarizable highest weight modules of basic classical lie superalgebras, Memoirs of American Mathematical Society volume 532, American Mathematical Society, U.S.A. (1994). · Zbl 0811.17002
[76] H. Furutsu and K. Nishiyama, Classification of irreducible super-unitary representations of SU(p, q|n), Commun. Math. Phys.141 (1991) 475. · Zbl 0741.17001
[77] A. Okounkov and G. Olshanski, Shifted Schur functions, St. Petersburg Math. J9 (1996) 239.
[78] A. Molev, Factorial supersymmetric schur functions and super capelli identities, Amer. Math. Soc. Transl. Ser2 (1997) 109. · Zbl 0955.05112
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.