Skip to main content
SpringerLink
Log in

Minimal surfaces in AdS space and integrable systems

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We consider the Pohlmeyer reduction for spacelike minimal area worldsheets in AdS5. The Lax pair for the reduced theory is found, and written entirely in terms of the A3 = D3 root system, generalizing the B2 affine Toda system which appears for the AdS4 string. For the B2 affine Toda system, we show that the area of the worlsheet is obtainable from the moduli space Kähler potential of a related Hitchin system. We also explore the Saveliev-Leznov construction for solutions of the B2 affine Toda system, and recover the rotationally symmetric solution associated to Painleve transcendent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].

    MATH  MathSciNet  ADS  Google Scholar 

  2. S. Kachru and E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. I.R. Klebanov and E. Witten, Superconformal field theory on threebranes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  4. S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semi-classical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and Anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  8. L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. L.F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  13. K. Pohlmeyer, Integrable hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  14. B.M. Barbashov and V.V. Nesterenko, Relativistic string model in a space-time of a constant curvature, Commun. Math. Phys. 78 (1981) 499 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. B.M. Barbashov, V.V. Nesterenko and A.M. Chervyakov, General solutions of nonlinear equations in the geometric theory of the relativistic string, Commun. Math. Phys. 84 (1982) 471 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. B.M. Barbashov, V.V. Nesterenko and A.M. Chervyakov, The solitons in some geometrical field theories, Theor. Math. Phys. 40 (1979) 572 [Teor. Mat. Fiz. 40 (1979) 15] [SPIRES].

    Article  MathSciNet  Google Scholar 

  17. B.M. Barbashov, V.V. Nesterenko and A.M. Chervyakov, Reduction in the relativistic string model for the d-dimensional space-time, Theor. Math. Phys. 59 (1984) 458 [Teor. Mat. Fiz. 59 (1984) 209] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  18. H.J. De Vega and N.G. Sanchez, Exact integrability of strings in D-dimensional de Sitter space-time, Phys. Rev. D 47 (1993) 3394 [SPIRES].

    ADS  Google Scholar 

  19. M. Grigoriev and A.A. Tseytlin, Pohlmeyer reduction of AdS 5 × S 5 superstring σ-model, Nucl. Phys. B 800 (2008) 450 [arXiv:0711.0155] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. R. Roiban and A.A. Tseytlin, UV finiteness of Pohlmeyer-reduced form of the AdS 5 × S 5 superstring theory, JHEP 04 (2009) 078 [arXiv:0902.2489] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. B. Hoare, Y. Iwashita and A.A. Tseytlin, Pohlmeyer-reduced form of string theory in AdS 5 × S 5 : semiclassical expansion, J. Phys. A 42 (2009) 375204 [arXiv:0906.3800] [SPIRES].

    MathSciNet  Google Scholar 

  22. J.L. Miramontes, Pohlmeyer reduction revisited, JHEP 10 (2008) 087 [arXiv:0808.3365] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. T.J. Hollowood and J.L. Miramontes, Magnons, their solitonic avatars and the pohlmeyer reduction, JHEP 04 (2009) 060 [arXiv:0902.2405] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  24. I. Bakas, Q.-H. Park and H.-J. Shin, Lagrangian formulation of symmetric space sine-Gordon models, Phys. Lett. B 372 (1996) 45 [hep-th/9512030] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  25. I. Bakas and K. Sfetsos, Universal aspects of string propagation on curved backgrounds, Phys. Rev. D 54 (1996) 3995 [hep-th/9604195] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  26. I. Bakas, Conservation laws and geometry of perturbed coset models, Int. J. Mod. Phys. A 9 (1994) 3443 [hep-th/9310122] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  27. I. Bakas, W(∞) symmetry of Nambu-Goto string in four-dimensions, Phys. Lett. B 319 (1993) 457 [hep-th/9310121] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  28. A. Jevicki and K. Jin, Series solution and minimal surfaces in AdS, JHEP 03 (2010) 028 [arXiv:0911.1107] [SPIRES].

    Article  Google Scholar 

  29. H. Dorn, G. Jorjadze and S. Wuttke, On spacelike and timelike minimal surfaces in AdS n , JHEP 05 (2009) 048 [arXiv:0903.0977] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. H. Dorn, Some comments on spacelike minimal surfaces with null polygonal boundaries in AdS m , JHEP 02 (2010) 013 [arXiv:0910.0934] [SPIRES].

    Article  Google Scholar 

  31. S. Ryang, Asymptotic AdS string solutions for null polygonal Wilson loops in R 1,2, arXiv:0910.4796 [SPIRES].

  32. L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, arXiv:0911.4708 [SPIRES].

  33. L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [SPIRES].

    Article  ADS  Google Scholar 

  34. K. Sakai and Y. Satoh, A note on string solutions in AdS 3, JHEP 10 (2009) 001 [arXiv:0907.5259] [SPIRES].

    Article  ADS  Google Scholar 

  35. H. Ooguri and C. Vafa, Summing up D-instantons, Phys. Rev. Lett. 77 (1996) 3296 [hep-th/9608079] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. N. Seiberg and S.H. Shenker, Hypermultiplet moduli space and string compactification to three dimensions, Phys. Lett. B 388 (1996) 521 [hep-th/9608086] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  37. D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].

  38. O. Babelon and L. Bonora, Conformal affine sl(2) Toda field theory, Phys. Lett. B 244 (1990) 220 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  39. H. Aratyn, L.A. Ferreira, J.F. Gomes and A.H. Zimerman, Kac-Moody construction of Toda type field theories, Phys. Lett. B 254 (1991) 372 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  40. A.N. Leznov and M.V. Savelev, Representation theory and integration of nonlinear spherically symmetric equations to gauge theories, Commun. Math. Phys. 74 (1980) 111 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  41. A. Bilal, V.V. Fock and I.I. Kogan, On the origin of W algebras, Nucl. Phys. B 359 (1991) 635 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  42. J.-L. Gervais and Y. Matsuo, Classical A(n) W geometry, Commun. Math. Phys. 152 (1993) 317 [hep-th/9201026] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. E. Aldrovandi and G. Falqui, Geometry of Higgs and Toda fields on Riemann surfaces, J. Geom. Phys. 17 (1995) 25 [hep-th/9312093] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  44. G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES].

  45. M.A.C. Kneipp, Hitchin’s equations and integrability of BPS Z(N) strings in Yang-Mills theories, JHEP 11 (2008) 049 [arXiv:0801.0720] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  46. A.V. Mikhailov, M.A. Olshanetsky and A.M. Perelomov, Two-dimensional generalized Toda lattice, Commun. Math. Phys. 79 (1981) 473 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  47. P. Mansfield, Solution of Toda systems, Nucl. Phys. B 208 (1982) 277 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  48. D.I. Olive and N. Turok, Local conserved densities and zero curvature conditions for Toda lattice field theories, Nucl. Phys. B 257 (1985) 277 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  49. D.I. Olive and N. Turok, The Toda lattice field theory hierarchies and zero curvature conditions in Kac-Moody algebras, Nucl. Phys. B 265 (1986) 469 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  50. D.I. Olive, N. Turok and J.W.R. Underwood, Affine Toda solitons and vertex operators, Nucl. Phys. B 409 (1993) 509 [hep-th/9305160] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  51. L. Bonora, C.P. Constantinidis, L.A. Ferreira and E.E. Leite, Construction of exact Riemannian instanton solutions, J. Phys. A 36 (2003) 7193 [hep-th/0208175] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  52. D.I. Olive, N. Turok and J.W.R. Underwood, Solitons and the energy momentum tensor for affine Toda theory, Nucl. Phys. B 401 (1993) 663 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  53. T.J. Hollowood, Solitons in affine Toda field theories, Nucl. Phys. B 384 (1992) 523 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. I.B. Frenkel and V.G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. G. Segal, Unitarity representations of some infinite dimensional groups, Commun. Math. Phys. 80 (1981) 301 [SPIRES].

    Article  MATH  ADS  Google Scholar 

  56. A. Fring, H.C. Liao and D.I. Olive, The mass spectrum and coupling in affine Toda theories, Phys. Lett. B 266 (1991) 82 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  57. P. Goddard, W. Nahm, D.I. Olive and A. Schwimmer, Vertex operators for nonsimply laced algebras, Commun. Math. Phys. 107 (1986) 179 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. M.A.C. Kneipp and D.I. Olive, Crossing and anti-solitons in affine Toda theories, Nucl. Phys. B 408 (1993) 565 [hep-th/9305154] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  59. A. Fring, P.R. Johnson, M.A.C. Kneipp and D.I. Olive, Vertex operators and soliton time delays in affine Toda field theory, Nucl. Phys. B 430 (1994) 597 [hep-th/9405034] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  60. M.A.C. Kneipp, Vertex operators, semiclassical limit for soliton S-matrices and the number of bound states in affine Toda field theories, Nucl. Phys. B 577 (2000) 390 [hep-th/9909128] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  61. A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models, Annals Phys. 120 (1979) 253 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  62. C.-N. Yang and C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969) 1115 [SPIRES].

    Article  MATH  ADS  Google Scholar 

  63. A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  64. P. Fendley and K.A. Intriligator, Scattering and thermodynamics in integrable N = 2 theories, Nucl. Phys. B 380 (1992) 265 [hep-th/9202011] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  65. P. Fendley and H. Saleur, N = 2 supersymmetry, Painleve III and exact scaling functions in 2 − D polymers, Nucl. Phys. B 388 (1992) 609 [hep-th/9204094] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  66. S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  67. A.B. Zamolodchikov, Painleve III and 2 − D polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  68. B.M. McCoy, C.A. Tracy and T.T. Wu, Painleve functions of the third kind, J. Math. Phys. 18 (1977) 1058 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  69. V. Belinski and E. Verdaguer, Gravitational solitons, Cambridge University Press, Cambridge U.K. (2001), pag. 258 [SPIRES].

    Book  MATH  Google Scholar 

  70. H. Elvang and P. Figueras, Black Saturn, JHEP 05 (2007) 050 [hep-th/0701035] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  71. V.A. Fateev and S.L. Lukyanov, The models of two-dimensional conformal quantum field theory with Z(n) symmetry, Int. J. Mod. Phys. A 3 (1988) 507 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  72. T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989) 373 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  73. S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  74. G.W. Delius, M.T. Grisaru and D. Zanon, Exact S matrices for nonsimply laced affine Toda theories, Nucl. Phys. B 382 (1992) 365 [hep-th/9201067] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  75. V.E. Zakharov and A.V. Mikhailov, Relativistically invariant two-dimensional models in field theory integrable by the inverse problem technique (In russian), Sov. Phys. JETP 47 (1978) 1017 [Zh. Eksp. Teor. Fiz. 74 (1978) 1953] [SPIRES].

    ADS  Google Scholar 

  76. M. Spradlin and A. Volovich, Dressing the giant magnon, JHEP 10 (2006) 012 [hep-th/0607009] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  77. C. Kalousios, M. Spradlin and A. Volovich, Dressing the giant magnon. II, JHEP 03 (2007) 020 [hep-th/0611033] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  78. A. Jevicki, C. Kalousios, M. Spradlin and A. Volovich, Dressing the giant gluon, JHEP 12 (2007) 047 [arXiv:0708.0818] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, University of Toronto, 60 St. George st, Toronto, ON, M5S 1A7, Canada

    Benjamin A. Burrington & Peng Gao

  2. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5, Canada

    Peng Gao

Authors
  1. Benjamin A. Burrington
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peng Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Benjamin A. Burrington.

Additional information

ArXiv ePrint: 0911.4551

Rights and permissions

Reprints and permissions

About this article

Cite this article

Burrington, B.A., Gao, P. Minimal surfaces in AdS space and integrable systems. J. High Energ. Phys. 2010, 60 (2010). https://doi.org/10.1007/JHEP04(2010)060

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP04(2010)060

Keywords

Search

Navigation