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The Algebraic Curve of Classical Superstrings on AdS 5×S 5

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Abstract

We investigate the monodromy of the Lax connection for classical IIB superstrings on AdS 5×S 5. For any solution of the equations of motion we derive a spectral curve of degree 4+4. The curve consists purely of conserved quantities, all gauge degrees of freedom have been eliminated in this form. The most relevant quantities of the solution, such as its energy, can be expressed through certain holomorphic integrals on the curve. This allows for a classification of finite gap solutions analogous to the general solution of strings in flat space. The role of fermions in the context of the algebraic curve is clarified. Finally, we derive a set of integral equations which reformulates the algebraic curve as a Riemann-Hilbert problem. They agree with the planar, one-loop supersymmetric gauge theory proving the complete agreement of spectra in this approximation.

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References

  1. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: Gauge theory correlators from non-critical string theory. Phys. Lett. B428, 105 (1998)

    Google Scholar 

  3. Witten, E.: Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998)

    MATH  MathSciNet  Google Scholar 

  4. Metsaev, R.R., Tseytlin, A.A.: Type IIB superstring action in AdS 5× S 5 background. Nucl. Phys. B533, 109 (1998)

  5. Blau, M., Figueroa-O'Farrill, J., Hull, C., Papadopoulos, G.: A new maximally supersymmetric background of IIB superstring theory. JHEP 0201, 047 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  6. Blau, M., Figueroa-O'Farrill, J., Hull, C., Papadopoulos, G.: Penrose limits and maximal supersymmetry Class. Quant. Grav. 19, L87 (2002)

  7. Berenstein, D., Maldacena, J.M., Nastase, H.: Strings in flat space and pp waves from Super Yang Mills JHEP 0204, 013 (2002)

  8. Metsaev, R.R.: Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background. Nucl. Phys. B625, 70 (2002)

    Google Scholar 

  9. Metsaev, R.R., Tseytlin, A.A.: Exactly solvable model of superstring in plane wave Ramond- Ramond background. Phys. Rev. D65, 126004 (2002)

    Google Scholar 

  10. Callan, Jr., C.G., Lee, H.K., McLoughlin, T., Schwarz, J.H., Swanson, I., Wu, X.: Quantizing string theory in AdS 5× S 5: Beyond the pp-wave. Nucl. Phys. B673, 3 (2003)

  11. Parnachev, A., Ryzhov, A.V.: Strings in the near plane wave background and AdS/CFT. JHEP 0210, 066 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. Callan, Jr., C.G., McLoughlin, T., Swanson, I.: Holography beyond the Penrose limit. Nucl. Phys. B694, 115 (2004)

  13. Callan, Jr. C.G., McLoughlin, T., Swanson, I.: Higher impurity AdS/CFT correspondence in the near-BMN limit. Nucl. Phys. B700, 271 (2004)

    Google Scholar 

  14. McLoughlin, T., Swanson, I.: N-impurity superstring spectra near the pp-wave limit. Nucl. Phys. B702, 86 (2004)

  15. Gubser, S.S., Klebanov, I.R., Polyakov, A.M.: A semi-classical limit of the gauge/string correspondence. Nucl. Phys. B636, 99 (2002)

    Google Scholar 

  16. Frolov, S., Tseytlin, A.A.: Semiclassical quantization of rotating superstring in AdS 5 × S 5. JHEP 0206, 007 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  17. Russo, J.G.: Anomalous dimensions in gauge theories from rotating strings in AdS 5 × S 5. JHEP 0206, 038 (2002)

    Article  ADS  Google Scholar 

  18. Minahan, J.A.: Circular semiclassical string solutions on AdS 5× S 5. Nucl. Phys. B648, 203 (2003)

  19. Tseytlin, A.A.: Semiclassical quantization of superstrings: AdS 5× S 5 and beyond. Int. J. Mod. Phys. A18, 981 (2003)

  20. Frolov, S., Tseytlin, A.A.: Multi-spin string solutions in AdS 5× S 5. Nucl. Phys. B668, 77 (2003)

  21. Beisert, N., Minahan, J.A., Staudacher, M., Zarembo, K.: Stringing Spins and Spinning Strings. JHEP 0309, 010 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  22. Beisert, N., Frolov, S., Staudacher, M., Tseytlin, A.A.: Precision Spectroscopy of AdS/CFT. JHEP 0310, 037 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  23. Serban, D., Staudacher, M.: Planar gauge theory and the Inozemtsev long range spin chain. JHEP 0406, 001 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  24. Tseytlin, A.A.: Spinning strings and AdS/CFT duality. In: M. Shifman, A. Vainshtein J. Wheater (eds.) From Fields to Stings: Circumnavigating Theoretical Physics, Ian Kogan Memorial Volume, Singapore: World Scientific, 2005

  25. Tseytlin, A.A.: Semiclassical strings in AdS 5× S 5 and scalar operators in SYM theory. Comptes Rendus Physique 5, 1049 (2004)

    MathSciNet  Google Scholar 

  26. Beisert, N.: The Dilatation Operator of Super Yang-Mills Theory and Integrability. Phys. Rept. 405, 1 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  27. Beisert, N.: Higher-loop integrability in gauge theory. Comptes Rendus Physique 5, 1039 (2004)

    MathSciNet  Google Scholar 

  28. Zarembo, K.: Semiclassical Bethe ansatz and AdS/CFT. Comptes Rendus Physique 5, 1081 (2004)

    MathSciNet  Google Scholar 

  29. Kazakov, V.A., Marshakov, A., Minahan, J.A., Zarembo, K.: Classical/quantum integrability in AdS/CFT. JHEP 0405, 024 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  30. Arutyunov, G., Frolov, S., Russo, J., Tseytlin, A.A.: Spinning strings in AdS 5× S 5 and integrable systems. Nucl. Phys. B671, 3 (2003)

  31. Arutyunov, G., Russo, J., Tseytlin, A.A.: Spinning strings in AdS 5× S 5: New integrable system relations. Phys. Rev. D69, 086009 (2004)

  32. Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models in field theory integrable by the inverse problem technique. Sov. Phys. JETP 47, 1017 (1978), in Russian

    Google Scholar 

  33. Pohlmeyer, K.: Integrable Hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 46, 207 (1976)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  34. Lüscher, M., Pohlmeyer, K.: Scattering of massless lumps and nonlocal charges in the two-dimensional classical nonlinear sigma model. Nucl. Phys. B137, 46 (1978)

    Google Scholar 

  35. Brezin, E., Itzykson, C., Zinn-Justin, J., Zuber, J.B.: Remarks about the existence of nonlocal charges in two-dimensional models. Phys. Lett. B82, 442 (1979)

  36. Eichenherr, H., Forger, M.: Higher local conservation laws for nonlinear sigma models on symmetric spaces. Commun. Math. Phys. 82, 227 (1981)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  37. Minahan, J.A., Zarembo, K.: The Bethe-ansatz for super Yang-Mills. JHEP 0303, 013 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  38. Beisert, N., Staudacher, M.: The SYM Integrable Super Spin Chain. Nucl. Phys. B670, 439 (2003)

    Google Scholar 

  39. Beisert, N., Kristjansen, C., Staudacher, M.: The dilatation operator of conformal super Yang-Mills theory. Nucl. Phys. B664, 131 (2003)

    Google Scholar 

  40. Beisert, N.: The su(2/3) dynamic spin chain. Nucl. Phys. B682, 487 (2004)

  41. Staudacher, M.: The factorized S-matrix of CFT/AdS. JHEP 0505, 054 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  42. Beisert, N., Dippel, V., Staudacher, M.: A Novel Long Range Spin Chain and Planar Super Yang-Mills. JHEP 0407, 075 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  43. Sutherland, B.: Low-Lying Eigenstates of the One-Dimensional Heisenberg Ferromagnet for any Magnetization and Momentum. Phys. Rev. Lett. 74, 816 (1995)

    Article  ADS  Google Scholar 

  44. Kruczenski, M.: Spin chains and string theory. Phys. Rev. Lett. 93, 161602 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  45. Kruczenski, M., Ryzhov, A.V., Tseytlin, A.A.: Large spin limit of AdS 5× S 5 string theory and low energy expansion of ferromagnetic spin chains. Nucl. Phys. B692, 3 (2004)

  46. Kazakov, V.A., Zarembo, K.: Classical/quantum integrability in non-compact sector of AdS/CFT. JHEP 0410, 060 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  47. Beisert, N., Kazakov, V.A., Sakai, K.: Algebraic curve for the SO(6) sector of AdS/CFT. http://arXiv.org/list/hep-th/0410253, 2004, to appear in Commun. Math. Phys.

  48. Schäfer-Nameki, S.: The algebraic curve of 1-loop planar SYM. Nucl. Phys. B714, 3 (2005)

    Google Scholar 

  49. Arutyunov, G., Frolov, S.: Integrable Hamiltonian for classical strings on AdS 5× S 5. JHEP 0502, 059 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  50. Mikhailov, A.: Supersymmetric null-surfaces. JHEP 0409, 068 (2004)

    Article  ADS  Google Scholar 

  51. Hernández, R., López, E.: Spin chain sigma models with fermions. JHEP 0411, 079 (2004)

    Article  ADS  Google Scholar 

  52. Bena, I., Polchinski, J., Roiban, R.: Hidden symmetries of the AdS 5× S 5 superstring. Phys. Rev. D69, 046002 (2004)

  53. Beisert, N.: The Complete One-Loop Dilatation Operator of Super Yang-Mills Theory. Nucl. Phys. B676, 3 (2004)

    Google Scholar 

  54. Beisert, N., Kazakov, V.A., Sakai, K., Zarembo, K.: Complete Spectrum of Long Operators in SYM at One Loop. JHEP 0507, 030 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  55. Hatsuda, M., Yoshida, K.: Classical integrability and super Yangian of superstring on AdS 5× S 5. Int. J. Mod. Phys. A19, 4715 (2004)

  56. Das, A., Maharana, J., Melikyan, A., Sato, M.: The algebra of transition matrices for the AdS 5× S 5 superstring. JHEP 0412, 055 (2004)

    Article  ADS  Google Scholar 

  57. Kallosh, R., Rahmfeld, J., Rajaraman, A.: Near horizon superspace. JHEP 9809, 002 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  58. Berkovits, N., Bershadsky, M., Hauer, T., Zhukov, S., Zwiebach, B.: Superstring theory on AdS 2× S 2 as a coset supermanifold. Nucl. Phys. B567, 61 (2000)

  59. Roiban, R., Siegel, W.: Superstrings on AdS 5× S 5 supertwistor space. JHEP 0011, 024 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  60. Berkovits, N.: BRST cohomology and nonlocal conserved charges. JHEP 0502, 060 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  61. Berkovits, N.: Quantum consistency of the superstring in AdS 5× S 5 background. JHEP 0503, 041 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  62. Mikhailov, A.: Notes on fast moving strings. hep-th/0409040.

  63. Mikhailov, A.: Plane wave limit of local conserved charges. JHEP 0510, 093 (2005)

    Article  ADS  Google Scholar 

  64. Berkovits, N., Vafa, C., Witten, E.: Conformal field theory of AdS background with Ramond-Ramond flux. JHEP 9903, 018 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  65. Dolan, L., Witten, E.: Vertex operators for AdS 3 background with Ramond-Ramond flux. JHEP 9911, 003 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  66. Metsaev, R.R., Tseytlin, A.A.: Superparticle and superstring in AdS 3× S 3 Ramond-Ramond background in light-cone gauge. J. Math. Phys. 42, 2987 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  67. Arutyunov, G., Staudacher, M.: Matching Higher Conserved Charges for Strings and Spins. JHEP 0403, 004 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  68. Arutyunov, G., Staudacher, M.: Two-loop commuting charges and the string/gauge duality. In: ``Lie Theory and its Applications in Physics V'', Proceedings of the Fifth International Workshop, Varna, Bulgaria, 16-22 June 2003, H.-D. Doebner, V.K. Dobrev, (ed.) Singapore: World Scientific, 2004

  69. Arutyunov, G., Frolov, S., Staudacher, M.: Bethe ansatz for quantum strings. JHEP 0410, 016 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  70. Beisert, N.: Spin chain for quantum strings. Fortsch. Phys. 53, 852 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  71. Swanson, I.: Quantum string integrability and AdS/CFT. Nucl. Phys. B709, 443 (2005)

    Google Scholar 

  72. Beisert, N., Tseytlin, A.A., Zarembo, K.: Matching quantum strings to quantum spins: one-loop vs. finite-size corrections. Nucl. Phys. B715, 190 (2005)

  73. 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)

    Article  ADS  Google Scholar 

  74. Polyakov, A.M., Wiegmann, P.B.: Theory of nonabelian Goldstone bosons in two dimensions. Phys. Lett. B131, 121 (1983)

    Google Scholar 

  75. Polyakov, A.M., Wiegmann, P.B.: Goldstone fields in two-dimensions with multivalued actions. Phys. Lett. B141, 223 (1984)

  76. Faddeev, L.D., Reshetikhin, N.Y.: Integrability of the principal chiral field model in (1+1)-dimension. Ann. Phys. 167, 227 (1986)

    Article  MathSciNet  Google Scholar 

  77. Ogievetsky, E., Wiegmann, P., Reshetikhin, N.: The principal chiral field in two-dimensions on classical Lie algebras: The Bethe ansatz solution and factorized theory of scattering. Nucl. Phys. B280, 45 (1987)

  78. Novikov, S., Manakov, S.V., Pitaevsky, L.P., Zakharov, V.E.: ``Theory of Solitons. The Inverse Scattering Method''. Contemporary Soviet Mathematics. (1984) New York: Consultants Bureau, 276p

  79. Its, A.R., Matveev, V.B.: Schrödinger operators with finite-gap spectrum and N-soliton solutions of the Korteweg-de Vries equation. Theor. Math. Phys. 23, 343 (1975)

    Article  MathSciNet  Google Scholar 

  80. Dubrovin, B.A., Novikov, M.V.B., Novikov, S.P.: Non-linear equations of Korteweg-de Vries type, finite zone linear operators, and Abelian varieties. Russ. Math. Surv 31, 59 (1976)

    Article  MATH  Google Scholar 

  81. Krichever, I.M.: Elliptic solutions of KP equations and integrable systems of particles. Funk. Anal. App. 14, 282 (1980)

    Article  Google Scholar 

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Authors and Affiliations

  1. Joseph Henry Laboratories, Princeton University, Princeton, NJ, 08544, USA

    N. Beisert

  2. Laboratoire de Physique Théorique de l'Ecole Normale Supérieure et l'Université Paris-VI, 75231, Paris, France

    V.A. Kazakov & K. Sakai

  3. Department of Theoretical Physics, Uppsala University, 751 08, Uppsala, Sweden

    K. Zarembo

  4. ITEP, Moscow, Russia

    K. Zarembo

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  1. N. Beisert
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  2. V.A. Kazakov
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  3. K. Sakai
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Correspondence to N. Beisert.

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Communicated by N.A. Nekrasov

Membre de l'Institut Universitaire de France

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Beisert, N., Kazakov, V., Sakai, K. et al. The Algebraic Curve of Classical Superstrings on AdS 5×S 5 . Commun. Math. Phys. 263, 659–710 (2006). https://doi.org/10.1007/s00220-006-1529-4

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