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Long-range \(\mathfrak{psu}(2,2|4)\) Bethe ansätze for gauge theory and strings. (English) Zbl 1126.81328

Summary: We generalize various existing higher-loop Bethe ansätze for simple sectors of the integrable long-range dynamic spin chain describing planar \(\mathcal N=4\) super-Yang-Mills theory to the full \(\mathfrak{psu}(2,2|4)\) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the \(\mathfrak{su}(1|2)\) subsector, corresponding to a long-range \(t\)-\(J\) model, we are able to derive, up to three loops, the S-matrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the \(\mathfrak{su}(1|2)\) and \(\mathfrak{su}(1|2)\) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on \(\text{AdS}_5\times S^5\) is also described by a \(\mathfrak{su}(1|2)\) spin chain. We propose asymptotic all-order Bethe equations for this putative “string chain”, which differ in a systematic fashion from the gauge theory equations.

MSC:

81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T27 Continuum limits in quantum field theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] Bethe, H., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der Linearen Atomkette, Z. Phys., 71, 205 (1931) · JFM 57.1587.01
[2] Minahan, J. A.; Zarembo, K., The Bethe-ansatz for \(N = 4\) super-Yang-Mills, JHEP, 0303, 013 (2003)
[3] Beisert, N.; Staudacher, M., The \(N = 4\) SYM integrable super spin chain, Nucl. Phys. B, 670, 439 (2003) · Zbl 1058.81581
[4] Beisert, N.; Kristjansen, C.; Staudacher, M., The dilatation operator of \(N = 4\) conformal super-Yang-Mills theory, Nucl. Phys. B, 664, 131 (2003) · Zbl 1051.81044
[5] Lipatov, L. N., Evolution equations in QCD, (Boffi, S.; Ciofi Degli Atti, C.; Giannini, M., Perspectives in Hadronic Physics, Proceedings of the Conference, ICTP. Perspectives in Hadronic Physics, Proceedings of the Conference, ICTP, Trieste, Italy, 12-16 May 1997 (1998), World Scientific: World Scientific Singapore) · Zbl 1365.81115
[6] Belitsky, A. V.; Braun, V. M.; Gorsky, A. S.; Korchemsky, G. P., Integrability in QCD and beyond, Int. J. Mod. Phys. A, 19, 4715 (2004) · Zbl 1059.81164
[7] Bena, I.; Polchinski, J.; Roiban, R., Hidden symmetries of the \(AdS_5 \times S^5\) superstring, Phys. Rev. D, 69, 046002 (2004)
[8] Arutyunov, G.; Russo, J.; Tseytlin, A. A., Spinning strings in \(AdS_5 \times S^5\): New integrable system relations, Phys. Rev. D, 69, 086009 (2004)
[9] Kazakov, V. A.; Marshakov, A.; Minahan, J. A.; Zarembo, K., Classical/quantum integrability in AdS/CFT, JHEP, 0405, 024 (2004)
[10] Kazakov, V. A.; Zarembo, K., Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP, 0410, 060 (2004)
[11] Beisert, N.; Kazakov, V. A.; Sakai, K., Algebraic curve for the SO(6) sector of AdS/CFT · Zbl 1125.81036
[12] Beisert, N.; Kazakov, V.; Sakai, K.; Zarembo, K., The algebraic curve of classical superstrings on \(AdS_5 \times S^5\) · Zbl 1125.81037
[13] Schäfer-Nameki, S., The algebraic curve of 1-loop planar \(N = 4\) SYM, Nucl. Phys. B, 714, 3 (2005) · Zbl 1194.81234
[14] Beisert, N.; Kazakov, V. A.; Sakai, K.; Zarembo, K., Complete spectrum of long operators in \(N = 4\) SYM at one loop, JHEP, 0507, 030 (2005)
[15] Arutyunov, G.; Frolov, S.; Staudacher, M., Bethe ansatz for quantum strings, JHEP, 0410, 016 (2004)
[16] Beisert, N., Spin chain for quantum strings, Fortschr. Phys., 53, 852 (2005) · Zbl 1069.81052
[17] Arutyunov, G.; Zamaklar, M., Linking Bäcklund and monodromy charges for strings on \(AdS_5 \times S^5\), JHEP, 0507, 026 (2005)
[18] Staudacher, M., The factorized S-matrix of CFT/AdS, JHEP, 0505, 054 (2005)
[19] Berenstein, D.; Maldacena, J. M.; Nastase, H., Strings in flat space and pp waves from \(N = 4\) super-Yang-Mills, JHEP, 0204, 013 (2002)
[20] Frolov, S.; Tseytlin, A. A., Multi-spin string solutions in \(AdS_5 \times S^5\), Nucl. Phys. B, 668, 77 (2003) · Zbl 1031.81051
[21] Frolov, S.; Tseytlin, A. A., Semiclassical quantization of rotating superstring in \(AdS_5 \times S^5\), JHEP, 0206, 007 (2002)
[22] Minahan, J. A., Circular semiclassical string solutions on \(AdS_5 \times S^5\), Nucl. Phys. B, 648, 203 (2003)
[23] Russo, R.; Tanzini, A., The duality between IIB string theory on pp-wave and \(N = 4\) SYM: A status report, Class. Quantum Grav., 21, S1265 (2004)
[24] Zarembo, K., Semiclassical Bethe ansatz and AdS/CFT, C. R. Phys., 5, 1081 (2004) · Zbl 1068.81057
[25] Beisert, N., The dilatation operator of \(N = 4\) super-Yang-Mills theory and integrability, Phys. Rep., 405, 1 (2005)
[26] Beisert, N.; Frolov, S.; Staudacher, M.; Tseytlin, A. A., Precision spectroscopy of AdS/CFT, JHEP, 0310, 037 (2003)
[27] Beisert, N., The su(2/3) dynamic spin chain, Nucl. Phys. B, 682, 487 (2004) · Zbl 1036.82513
[28] Serban, D.; Staudacher, M., Planar \(N = 4\) gauge theory and the Inozemtsev long range spin chain, JHEP, 0406, 001 (2004)
[29] Beisert, N.; Dippel, V.; Staudacher, M., A novel long range spin chain and planar \(N = 4\) super-Yang-Mills, JHEP, 0407, 075 (2004)
[30] Berkovits, N., Quantum consistency of the superstring in \(AdS_5 \times S^5\) background, JHEP, 0503, 041 (2005)
[31] Callan, C. G.; Lee, H. K.; McLoughlin, T.; Schwarz, J. H.; Swanson, I.; Wu, X., Quantizing string theory in \(AdS_5 \times S^5\): Beyond the pp-wave, Nucl. Phys. B, 673, 3 (2003) · Zbl 1058.81643
[32] Minahan, J. A., The SU(2) sector in AdS/CFT, Fortschr. Phys., 53, 828 (2005) · Zbl 1069.81055
[33] Beisert, N., The complete one-loop dilatation operator of \(N = 4\) super-Yang-Mills theory, Nucl. Phys. B, 676, 3 (2004) · Zbl 1097.81575
[34] Beisert, N., BMN operators and superconformal symmetry, Nucl. Phys. B, 659, 79 (2003) · Zbl 1087.81518
[35] Kotikov, A. V.; Lipatov, L. N.; Velizhanin, V. N., Anomalous dimensions of Wilson operators in \(N = 4\) SYM theory, Phys. Lett. B, 557, 114 (2003) · Zbl 1009.81040
[36] Moch, S.; Vermaseren, J. A.M.; Vogt, A., The three-loop splitting functions in QCD: The non-singlet case, Nucl. Phys. B, 688, 101 (2004) · Zbl 1149.81371
[37] Kotikov, A. V.; Lipatov, L. N.; Onishchenko, A. I.; Velizhanin, V. N., Three-loop universal anomalous dimension of the Wilson operators in \(N = 4\) supersymmetric Yang-Mills theory · Zbl 1247.81486
[38] Eden, B.; Jarczak, C.; Sokatchev, E.; Stanev, Y. S., Operator mixing in \(N = 4\) SYM: The Konishi anomaly revisited, Nucl. Phys. B, 722, 119 (2005) · Zbl 1128.81324
[39] Eden, B., A two-loop test for the factorised S-matrix of planar \(N = 4\) · Zbl 1109.81353
[40] Parnachev, A.; Ryzhov, A. V., Strings in the near plane wave background and AdS/CFT, JHEP, 0210, 066 (2002)
[41] McLoughlin, T.; Swanson, I., N-impurity superstring spectra near the pp-wave limit, Nucl. Phys. B, 702, 86 (2004) · Zbl 1198.81159
[42] Fischbacher, T.; Klose, T.; Plefka, J., Planar plane-wave matrix theory at the four loop order: Integrability without BMN scaling, JHEP, 0502, 039 (2005)
[43] Park, I. Y.; Tirziu, A.; Tseytlin, A. A., Spinning strings in \(AdS_5 \times S^5\): One-loop correction to energy in SL(2) sector, JHEP, 0503, 013 (2005)
[44] Hernández, R.; López, E.; Periáñez, A.; Sierra, G., Finite size effects in ferromagnetic spin chains and quantum corrections to classical strings, JHEP, 0506, 011 (2005)
[45] Gubser, S. S.; Klebanov, I. R.; Polyakov, A. M., Gauge theory correlators from non-critical string theory, Phys. Lett. B, 428, 105 (1998) · Zbl 1355.81126
[46] Zhang, F. C.; Rice, T. M., Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B, 37, 3759 (1988)
[47] Förster, D., Staggered spin and statistics in the supersymmetric \(t-J\) model, Phys. Rev. Lett., 63, 2140 (1989)
[48] Schlottmann, P., Integrable narrow-band model with possible relevance to heavy fermion systems, Phys. Rev. B, 36, 5177 (1987)
[49] Foerster, A.; Karowski, M., Algebraic properties of the Bethe ansatz for an \(spl(2, 1)\) supersymmetric \(t-J\) model, Nucl. Phys. B, 396, 611 (1993)
[50] Sutherland, B., Model for a multicomponent quantum system, Phys. Rev. B, 12, 3795 (1975)
[51] Sutherland, B., A brief history of the quantum soliton with new results on the quantization of the Toda lattice, Rocky Mountain J. Math., 8, 431 (1978)
[52] Yang, C.-N., Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett., 19, 1312 (1967) · Zbl 0152.46301
[53] Andrei, N.; Furuya, K.; Lowenstein, J. H., Solution of the Kondo problem, Rev. Mod. Phys., 55, 331 (1983)
[54] Sutherland, B., (Shastry, B. S.; Jha, S. S.; Singh, V., Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory. Exactly Solvable Problems in Condensed Matter and Relativistic Field Theory, Lecture Notes in Physics, vol. 242 (1985), Springer: Springer Berlin)
[55] Göhmann, F.; Seel, A., A note on the Bethe ansatz solution of the supersymmetric \(t-J\) model, Czech. J. Phys., 53, 1041 (2003)
[56] Klose, T.; Plefka, J., On the integrability of large \(N\) plane-wave matrix theory, Nucl. Phys. B, 679, 127 (2004) · Zbl 1045.81534
[57] Minahan, J. A., Higher loops beyond the \(SU(2)\) sector, JHEP, 0410, 053 (2004)
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