Algebraic curves for integrable string backgrounds

K. Zarembo

Informatics and Automation, Problems of modern theoretical and mathematical physics: Gauge theories and superstrings, Tome 272 (2011), pp. 286-298.

Résumé

Many Ramond–Ramond backgrounds that arise in the AdS/CFT correspondence are described by integrable sigma-models. The equations of motion for classical spinning strings in these backgrounds can be solved quite generally by the finite-gap integration method via classical Bethe equations and algebraic curves. This construction is reviewed for general $\mathbb Z_4$ cosets and then exemplified by the backgrounds that arise in the AdS$_5$/CFT$_4$, AdS$_4$/CFT$_3$ and AdS$_3$/CFT$_2$ dualities.

Bibliographie

[1] Maldacena J.M., “The large $N$ limit of superconformal field theories and supergravity”, Adv. Theor. and Math. Phys., 2 (1998), 231, arXiv: hep-th/9711200 | MR | Zbl

[2] Gubser S.S., Klebanov I.R., Polyakov A.M., “Gauge theory correlators from non-critical string theory”, Phys. Lett. B, 428 (1998), 105, arXiv: hep-th/9802109 | DOI | MR

[3] Witten E., “Anti de Sitter space and holography”, Adv. Theor. and Math. Phys., 2 (1998), 253, arXiv: hep-th/9802150 | MR | Zbl

[4] Faddeev L.D., How algebraic Bethe ansatz works for integrable model, E-print, 1996, arXiv: hep-th/9605187 | MR

[5] Berenstein D., Maldacena J., Nastase H., “Strings in flat space and pp waves from $\mathcal N=4$ super Yang Mills”, J. High Energy Phys., 2002, no. 4, 013, arXiv: hep-th/0202021 | DOI | MR

[6] Gubser S.S., Klebanov I.R., Polyakov A.M., “A semi-classical limit of the gauge/string correspondence”, Nucl. Phys. B, 636 (2002), 99, arXiv: hep-th/0204051 | DOI | MR | Zbl

[7] Frolov S., Tseytlin A.A., “Multi-spin string solutions in $AdS_5\times S^5$”, Nucl. Phys. B, 668 (2003), 77, arXiv: hep-th/0304255 | DOI | MR | Zbl

[8] Tseytlin A.A., Spinning strings and AdS/CFT duality, E-print, 2003, arXiv: hep-th/0311139 | MR

[9] Plefka J., “Spinning strings and integrable spin chains in the AdS/CFT correspondence”, Living Rev. Relativ., 8 (2005), 9, arXiv: hep-th/0507136 | DOI | Zbl

[10] Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Ser. Sov. Math., Springer, Berlin, 1987 | MR | Zbl

[11] Novikov S., Manakov S.V., Pitaevskii L.P., Zakharov V.E., Theory of solitons: The inverse scattering method, Contemp. Sov. Math., Consultants Bureau, New York, 1984 | MR

[12] Kazakov V.A., Marshakov A., Minahan J.A., Zarembo K., “Classical/quantum integrability in AdS/CFT”, J. High Energy Phys., 2004, no. 5, 024, arXiv: hep-th/0402207 | DOI | MR

[13] Beisert N., Staudacher M., “Long-range $\mathfrak {psu}(2,2|4)$ Bethe ansätze for gauge theory and strings”, Nucl. Phys. B., 727 (2005), 1, arXiv: hep-th/0504190 | DOI | MR | Zbl

[14] Gromov N., Kazakov V., Vieira P., “Exact spectrum of anomalous dimensions of planar $N=4$ supersymmetric Yang–Mills theory”, Phys. Rev. Lett., 103 (2009), 131601, arXiv: 0901.3753 | DOI | MR

[15] Bombardelli D., Fioravanti D., Tateo R., “Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal”, J. Phys. A: Math. and Theor., 42 (2009), 375401, arXiv: 0902.3930 | DOI | MR | Zbl

[16] Gromov N., Kazakov V., Kozak A., Vieira P., “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 | DOI | MR | Zbl

[17] Arutyunov G., Frolov S., “Thermodynamic Bethe ansatz for the $\mathrm {AdS}_5\times \mathrm S^5$ mirror model”, J. High Energy Phys., 2009, no. 5, 068, arXiv: 0903.0141 | DOI | MR

[18] Gromov N., “Integrability in AdS/CFT correspondence: quasi-classical analysis”, J. Phys. A: Math. and Theor., 42 (2009), 254004 | DOI | MR | Zbl

[19] Gromov N., Kazakov V., Tsuboi Z., $\mathrm {PSU}(2,2|4)$ character of quasiclassical AdS/CFT, E-print, 2010, arXiv: 1002.3981 | MR

[20] Alday L.F., Maldacena J., “Null polygonal Wilson loops and minimal surfaces in anti-de-Sitter space”, J. High Energy Phys., 2009, no. 11, 082, arXiv: 0904.0663 | DOI | MR

[21] Alday L.F., Gaiotto D., Maldacena J., Thermodynamic bubble ansatz, E-print, 2009, arXiv: 0911.4708 | MR

[22] Alday L.F., Maldacena J., Sever A., Vieira P., Y-system for scattering amplitudes, E-print, 2010, arXiv: 1002.2459 | MR

[23] Beisert N., Kazakov V.A., Sakai K., Zarembo K., “The algebraic curve of classical superstrings on $AdS_5\times S^5$”, Commun. Math. Phys., 263 (2006), 659, arXiv: hep-th/0502226 | DOI | MR | Zbl

[24] Babichenko A., Stefański B., \textup {Jr.“, Zarembo K.} Integrability and the $AdS_3/CFT_2$ correspondence”, J. High Energy Phys., 2010, no. 3, 058, arXiv: 0912.1723 | DOI | MR | Zbl

[25] Eichenherr H., Forger M., “On the dual symmetry of the non-linear sigma models”, Nucl. Phys. B., 155 (1979), 381 | DOI | MR

[26] Metsaev R.R., Tseytlin A.A., “Type IIB superstring action in $\mathrm {AdS}_5\times S^5$ background”, Nucl. Phys. B, 533 (1998), 109, arXiv: hep-th/9805028 | DOI | MR | Zbl

[27] Berkovits N., Bershadsky M., Hauer T., Zhukov S., Zwiebach B., “Superstring theory on $\mathrm {AdS}_2\times S^2$ as a coset supermanifold”, Nucl. Phys. B, 567 (2000), 61, arXiv: hep-th/9907200 | DOI | MR | Zbl

[28] Roiban R., Siegel W., “Superstrings on $AdS_5\times S^5$ supertwistor space”, J. High Energy Phys., 2000, no. 11, 024, arXiv: hep-th/0010104 | DOI | MR | Zbl

[29] Bena I., Polchinski J., Roiban R., “Hidden symmetries of the $\mathrm {AdS}_5\times \mathrm S^5$ superstring”, Phys. Rev. D, 69 (2004), 046002, arXiv: hep-th/0305116 | DOI | MR

[30] Arutyunov G., Frolov S., “Superstrings on $\mathrm {AdS}_4\times \mathbb {CP}^3$ as a coset sigma-model”, J. High Energy Phys., 2008, no. 9, 129, arXiv: 0806.4940 | DOI | MR | Zbl

[31] Stefański B., Jr., “Green–Schwarz action for type IIA strings on $AdS_4\times CP^3$”, Nucl. Phys. B, 808 (2009), 80, arXiv: 0806.4948 | DOI | MR | Zbl

[32] Rahmfeld J., Rajaraman A., “Green–Schwarz string action on $\mathrm {AdS}_3\times \mathrm S^3$ with Ramond–Ramond charge”, Phys. Rev. D, 60 (1999), 064014, arXiv: hep-th/9809164 | DOI | MR

[33] Park J., Rey S.-J., “Green–Schwarz superstring on $AdS_3\times S^3$”, J. High Energy Phys., 1999, no. 1, 001, arXiv: hep-th/9812062 | MR | Zbl

[34] Metsaev R.R., Tseytlin A.A., “Superparticle and superstring in $AdS_3\times S^3$ Ramond–Ramond background in the light-cone gauge”, J. Math. Phys., 42 (2001), 2987, arXiv: hep-th/0011191 | DOI | MR | Zbl

[35] Chen B., He Y.-L., Zhang P., Song X.-C., “Flat currents of the Green–Schwarz superstrings in $AdS_5\times S^1$ and $AdS_3\times S^3$ backgrounds”, Phys. Rev. D, 71 (2005), 086007, arXiv: hep-th/0503089 | DOI | MR

[36] Adam I., Dekel A., Mazzucato L., Oz Y., “Integrability of type II superstrings on Ramond–Ramond backgrounds in various dimensions”, J. High Energy Phys., 2007, no. 6, 085, arXiv: hep-th/0702083 | DOI | MR

[37] Zhou J.-G., “Super 0-brane and GS superstring actions on $\mathrm {AdS}_2\times S^2$”, Nucl. Phys. B, 559 (1999), 92, arXiv: hep-th/9906013 | DOI | MR

[38] Verlinde H., Superstrings on $AdS_2$ and superconformal matrix quantum mechanics, E-print, 2004, arXiv: hep-th/0403024

[39] Serganova V.V., “Klassifikatsiya prostykh veschestvennykh superalgebr Li i simmetricheskikh superprostranstv”, Funkts. analiz i ego pril., 17:3 (1983), 46 | MR | Zbl

[40] Polyakov A.M., “Conformal fixed points of unidentified gauge theories”, Mod. Phys. Lett. A, 19 (2004), 1649, arXiv: hep-th/0405106 | DOI | MR | Zbl

[41] Zarembo K., Strings on semisymmetric superspaces, E-print, 2010, arXiv: 1003.0465 | MR

[42] Kazakov V.A., Zarembo K., “Classical/quantum integrability in non-compact sector of AdS/CFT”, J. High Energy Phys., 2004, no. 10, 060, arXiv: hep-th/0410105 | DOI | MR

[43] Beisert N., Kazakov V.A., Sakai K., “Algebraic curve for the SO(6) sector of AdS/CFT”, Commun. Math. Phys., 263 (2006), 611, arXiv: hep-th/0410253 | DOI | MR | Zbl

[44] Schäfer-Nameki S., “The algebraic curve of 1-loop planar $\mathcal N=4$ SYM”, Nucl. Phys. B, 714 (2005), 3, arXiv: hep-th/0412254 | DOI | MR | Zbl

[45] Dorey N., Vicedo B., “On the dynamics of finite-gap solutions in classical string theory”, J. High Energy Phys., 2006, no. 7, 014, arXiv: hep-th/0601194 | DOI | MR

[46] Dorey N., Vicedo B., “A symplectic structure for string theory on integrable backgrounds”, J. High Energy Phys., 2007, no. 3, 045, arXiv: hep-th/0606287 | DOI | MR

[47] Zarembo K., “Semiclassical Bethe ansatz and AdS/CFT”, C. r. Phys., 5 (2004), 1081, arXiv: hep-th/0411191 | DOI | MR

[48] Gromov N., Vieira P., “The $AdS_4/CFT_3$ algebraic curve”, J. High Energy Phys., 2009, no. 2, 040, arXiv: 0807.0437 | DOI | MR | Zbl

[49] Beisert N., Kazakov V.A., Sakai K., Zarembo K., “Complete spectrum of long operators in $\mathcal N=4$ SYM at one loop”, J. High Energy Phys., 2005, no. 7, 030, arXiv: hep-th/0503200 | DOI | MR | Zbl

[50] Gromov N., Vieira P., “Complete 1-loop test of AdS/CFT”, J. High Energy Phys., 2008, no. 4, 046, arXiv: 0709.3487 | DOI | MR | Zbl

[51] Kac V.G., “A sketch of Lie superalgebra theory”, Commun. Math. Phys., 53 (1977), 31 | DOI | MR | Zbl

[52] Frappat L., Sorba P., Sciarrino A., Dictionary on Lie superalgebras, E-print, 1996, arXiv: hep-th/9607161 | Zbl

[53] Berkovits N.J., Vafa C., Witten E., “Conformal field theory of AdS background with Ramond–Ramond flux”, J. High Energy Phys., 1999, no. 3, 018, arXiv: hep-th/9902098 | DOI | MR | Zbl

[54] Pesando I., “The GS type IIB superstring action on $AdS_3\times S^3\times T^4$”, J. High Energy Phys., 1999, no. 2, 007, arXiv: hep-th/9809145 | DOI | MR | Zbl