Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 005, 15 pages      arXiv:2508.04158      https://doi.org/10.3842/SIGMA.2026.005

On Integrable Structure of Null String in (Anti-)de Sitter Space

Dmytro V. Uvarov
NSC Kharkiv Institute of Physics and Technology, 1 Academichna Str., Kharkiv, Ukraine

Received August 07, 2025, in final form January 06, 2026; Published online January 16, 2026

Abstract
Presently integrability turned out to be the key property in the study of duality between superconformal gauge theories and strings in anti-de Sitter superspaces. Complexity of the study of integrable structure in string theory is caused by complicated dependence of background fields of the Type II supergravity multiplets, with which strings interact, on the superspace coordinates. This explains an interest to study of limiting cases, in which superstring equations simplify. In the present work, we considered the limiting case of zero tension corresponding to null string. The representation in the form of the Lax equation of null-string equations in (anti-)de Sitter space realized as a coset manifold is obtained. Proposed is twistor interpretation of the Lagrangian of (null) string in anti-de Sitter space expressed in terms of group variables.

Key words: tensionless string; (anti-)de Sitter space; classical integrability; Lax pair; twistor.

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References

  1. Adamo T., Skinner D., Williams J., Twistor methods for ${\rm AdS}_5$, J. High Energy Phys. 2016 (2016), no. 8, 167, 24 pages, arXiv:1607.03763.
  2. Aharony O., Bergman O., Jafferis D.L., Maldacena J., ${\mathcal N}=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, J. High Energy Phys. 2008 (2008), no. 10, 091, 38 pages, arXiv:0806.1218.
  3. Arutyunov G., Frolov S., Superstrings on ${\rm AdS}_4\times\mathbb{CP}^3$ as a coset sigma-model, J. High Energy Phys. 2008 (2008), no. 9, 129, 27 pages, arXiv:0806.4940.
  4. Arvanitakis A.S., Barns-Graham A.E., Townsend P.K., Twistor variables for anti-de Sitter (super)particles, Phys. Rev. Lett. 2017 (118), 141601, 5 pages, arXiv:1608.04380.
  5. Arvanitakis A.S., Barns-Graham A.E., Townsend P.K., Twistor description of spinning particles in AdS, J. High Energy Phys. 2018 (2018), no. 1, 059, 47 pages, arXiv:1710.09557.
  6. Bandos I., Twistor/ambitwistor strings and null-superstrings in spacetime of $D=4$, $10$ and $11$ dimensions, J. High Energy Phys. 2014 (2014), no. 9, 086, 26 pages, arXiv:1404.1299.
  7. Bandos I., Zheltukhin A., Twistors, harmonics, and zero-super-$p$-branes, JETP Lett. 51 (1990), 618-621.
  8. Bandos I.A., Zheltukhin A.A., Null super $p$-brane. Hamiltonian dynamics and quantization, Phys. Lett. B 261 (1991), 245-250.
  9. Bandos I.A., Zheltukhin A.A., Spinor Cartan moving $n$-hedron, Lorentz-harmonic formulations of superstrings, and $\kappa$ symmetry, JETP Lett. 54 (1991), 421-424.
  10. Bandos I.A., Zheltukhin A.A., Lorentz harmonics and new formulations of the superstrings in $D=10$ and supermembranes in $D=11$, Phys. Atom. Nucl. 56 (1993), 113-121.
  11. Bandos I.A., Zheltukhin A.A., Null super $p$-branes quantum theory in $4$-dimensional space-time, Fortschr. Phys. 41 (1993), 619-676.
  12. Bandos I.A., Zheltukhin A.A., Twistor-like approach in the Green-Schwarz ${D=10}$ superstring theory, Phys. Part. Nucl. 25 (1994), 453-477.
  13. Barbashov B.M., Nesterenko V.V., Relativistic string model in a space-time of a constant curvature, Comm. Math. Phys. 78 (1980), 499-506.
  14. Bars I., Twistor superstring in two-time physics, Phys. Rev. D 70 (2004), 104022, 16 pages, arXiv:hep-th/0407239.
  15. Bars I., Twistors and 2T-physics, in Fundamental Interactions and Twistor-Like Methods, AIP Conf. Proc., Vol. 767, American Institute of Physics, Melville, NY, 2005, 3-27, arXiv:hep-th/0502065.
  16. Bena I., Polchinski J., Roiban R., Hidden symmetries of the ${\rm AdS}_ 5\times S^5$ superstring, Phys. Rev. D 69 (2004), 046002, 7 pages, arXiv:hep-th/0305116.
  17. Berkovits N., Infinite tension limit of the pure spinor superstring, J. High Energy Phys. 2014 (2014), no. 3, 017, 7 pages, arXiv:1311.4156.
  18. Blair C.D.A., Lahnsteiner J., Obers N.A.J., Yan Z., Unification of decoupling limits in string and M theory, Phys. Rev. Lett. 132 (2024), 161603, 7 pages, arXiv:2311.10564.
  19. Bombardelli D., Cagnazzo A., Frassek R., Levkovich-Maslyuk F., Loebbert F., Negro S., Szécsényi I.M., Sfondrini A., van Tongeren S.J., Torrielli A., An integrability primer for the gauge-gravity correspondence: an introduction, J. Phys. A 49 (2016), 320301, 10 pages, arXiv:1606.02945.
  20. Bonelli G., On the covariant quantization of tensionless bosonic strings in AdS spacetime, J. High Energy Phys. 2003 (2003), no. 11, 028, 15 pages, arXiv:hep-th/0309222.
  21. Brink L., di Vecchia P., Howe P., A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. B 65 (1976), 471-474.
  22. Burrington B.A., Gao P., Minimal surfaces in AdS space and integrable systems, J. High Energy Phys. 2010 (2010), no. 4, 060, 33 pages, arXiv:0911.4551.
  23. Cagnazzo A., Sorokin D., Wulff L., More on integrable structures of superstrings in ${\rm AdS}_4\times \mathbb{CP}^3$ and ${\rm AdS}_2\times S^2\times T^6$ superbackgrounds, J. High Energy Phys. 2012 (2012), no. 1, 004, 29 pages, arXiv:1111.4197.
  24. Casali E., Tourkine P., On the null origin of the ambitwistor string, J. High Energy Phys. 2016 (2016), no. 11, 036, 22 pages, arXiv:1606.05636.
  25. Cederwall M., Geometric construction of AdS twistors, Phys. Lett. B 483 (2000), 257-263, arXiv:hep-th/0002216.
  26. Cederwall M., AdS twistors for higher spin theory, in Fundamental Interactions and Twistor-Like Methods, AIP Conf. Proc., Vol. 767, American Institute of Physics, Melville, NY, 2005, 96-105, arXiv:hep-th/0412222.
  27. Claus P., Rahmfeld J., Zunger Y., A simple particle action from a twistor parametrization of ${\rm AdS}_5$, Phys. Lett. B 466 (1999), 181-189, arXiv:hep-th/9906118.
  28. Dąbrowski M.P., Larsen A.L., Null strings in Schwarzschild spacetime, Phys. Rev. D 55 (1997), 6409-6414, arXiv:hep-th/9610243.
  29. Dąbrowski M.P., Próchnicka I., Null string evolution in black hole and cosmological spacetimes, Phys. Rev. D 66 (2002), 043508, 10 pages, arXiv:hep-th/0201180.
  30. de Vega H.J., Nicolaidis A., Strings in strong gravitational fields, Phys. Lett. B 295 (1992), 214-218.
  31. de Vega H.J., Sanchez N., Exact integrability of strings in $D$-dimensional de Sitter spacetime, Phys. Rev. D 47 (1993), 3394-3404.
  32. Demulder S., Driezen S., Knighton B., Oling G., Retore A.L., Seibold F.K., Sfondrini A., Yan Z., Exact approaches on the string worldsheet, J. Phys. A 57 (2024), 423001, 275 pages, arXiv:2312.12930.
  33. Deser S., Zumino B., A complete action for the spinning string, Phys. Lett. B 65 (1976), 369-373.
  34. Duff M.J., Howe P.S., Inami T., Stelle K.S., Superstrings in $D=10$ from supermembranes in $D=11$, Phys. Lett. B 191 (1987), 70-74.
  35. Eichenherr H., Forger M., On the dual symmetry of the nonlinear sigma models, Nuclear Phys. B 155 (1979), 381-393.
  36. Fursaev D.V., Physical effects of massless cosmic strings, Phys. Rev. D 96 (2017), 104005, 14 pages, arXiv:1707.02438.
  37. Gamboa J., Ramírez C., Ruiz-Altaba M., Quantum null (super)strings, Phys. Lett. B 225 (1989), 335-339.
  38. Gamboa J., Ramírez C., Ruiz-Altaba M., Null spinning strings, Nuclear Phys. B 338 (1990), 143-187.
  39. Gomis J., Sorokin D., Wulff L., The complete ${\rm AdS}_4\times {\mathbb C}{\rm P}^3$ superspace for the type IIA superstring and D-branes, J. High Energy Phys. 2009 (2009), no. 3, 015, 41 pages, arXiv:0811.1566.
  40. Gubser S.S., Klebanov I.R., Polyakov A.M., Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998), 105-114, arXiv:hep-th/9802109.
  41. Hoare B., Tseytlin A.A., Pohlmeyer reduction for superstrings in AdS space, J. Phys. A 46 (2013), 015401, 52 pages, arXiv:1209.2892.
  42. Ilyenko K., Twistor variational principle for null strings, Nuclear Phys. B Proc. Suppl. 102 (2001), 83-86, arXiv:hep-th/0104117.
  43. Ilyenko K., Twistor representation of null two-surfaces, J. Math. Phys. 43 (2002), 4770-4789, arXiv:hep-th/0109036.
  44. Joung E., Oh T., Manifestly covariant worldline actions from coadjoint orbits. II. Twistorial descriptions, Phys. Rev. D 111 (2025), 025004, 24 pages, arXiv:2408.14783.
  45. Kallosh R., Rahmfeld J., Rajaraman A., Near horizon superspace, J. High Energy Phys. 1998 (1998), no. 9, 002, 9 pages, arXiv:hep-th/9805217.
  46. Kar S., Schild's null strings in flat and curved backgrounds, Phys. Rev. D 53 (1996), 6842-6846, arXiv:hep-th/9511103.
  47. Karlhede A., Lindström U., The classical bosonic string in the zero tension limit, Classical Quantum Gravity 3 (1986), L73-L75.
  48. Kato M., Ogawa K., Covariant quantization of string based on BRS invariance, Nucl. Phys. B 212 (1983), 443-460.
  49. Katsinis D., Novel aspects of integrability for nonlinear sigma models in symmetric spaces, Phys. Rev. D 105 (2022), 126008, 21 pages, arXiv:2201.10554.
  50. Klose T., Review of AdS/CFT integrability, Chapter IV.3: ${\mathcal N}=6$ Chern-Simons and strings on ${\rm AdS}_4\times {\rm CP}^3$, Lett. Math. Phys. 99 (2012), 401-423, arXiv:1012.3999.
  51. Koning N.E., Kuzenko S.M., Embedding formalism for AdS superspaces in five dimensions, J. High Energy Phys. 2025 (2025), no. 6, 016, 37 pages, arXiv:2406.10875.
  52. Koning N.E., Kuzenko S.M., Raptakis E.S.N., Embedding formalism for $\mathcal{N} $-extended AdS superspace in four dimensions, J. High Energy Phys. 2023 (2023), no. 11, 063, 42 pages, arXiv:2308.04135.
  53. Kuzenko S.M., Tartaglino-Mazzucchelli G., Supertwistor realisations of $AdS$ superspaces, Eur. Phys. J. C 82 (2022), 146, 21 pages, arXiv:2108.03907.
  54. Lindstrom U., Sundborg B., Theodoridis G., The zero tension limit of the superstring, Phys. Lett. B 253 (1991), 319-323.
  55. Lüscher M., Pohlmeyer K., Scattering of massless lumps and non-local charges in the two-dimensional classical non-linear $\sigma $-model, Nuclear Phys. B 137 (1978), 46-54.
  56. Maldacena J., The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252, arXiv:hep-th/9711200.
  57. Mason L., Skinner D., Ambitwistor strings and the scattering equations, J. High Energy Phys. 2014 (2014), no. 7, 048, 34 pages, arXiv:1311.2564.
  58. Metsaev R.R., Tseytlin A.A., Type IIB superstring action in ${\rm AdS}_5\times S^5$ background, Nuclear Phys. B 533 (1998), 109-126, arXiv:hep-th/9805028.
  59. Minahan J.A., Zarembo K., The Bethe-ansatz for ${\mathcal N}=4$ super Yang-Mills, J. High Energy Phys. 2003 (2003), no. 3. 013, 29 pages, arXiv:hep-th/0212208.
  60. Penrose R., Twistor algebra, J. Math. Phys. 8 (1967), 345-366.
  61. Penrose R., MacCallum M.A.H., Twistor theory: an approach to the quantisation of fields and space-time, Phys. Rep. 6 (1973), 241-315.
  62. Penrose R., Rindler W., Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 1986.
  63. Pohlmeyer K., Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221.
  64. Polyakov A.M., Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), 207-210.
  65. Porfyriadis P.I., Papadopoulos D., Null strings in Kerr spacetime, Phys. Lett. B 417 (1998), 27-32, arXiv:hep-th/9707183.
  66. Roshchupkin S.N., Zheltukhin A.A., Friedmann universes and exact solutions in string cosmology, Classical Quantum Gravity 12 (1995), 2519-2524, arXiv:hep-th/9507006.
  67. Roshchupkin S.N., Zheltukhin A.A., Variational principle and a perturbative solution of non-linear string equations in curved space, Nuclear Phys. B 543 (1999), 365-386, arXiv:hep-th/9806054.
  68. Schild A., Classical null strings, Phys. Rev. D 16 (1977), 1722-1726.
  69. Sorokin D., Wulff L., Evidence for the classical integrability of the complete ${\rm AdS}_4\times \mathbb{CP}^3$ superstring, J. High Energy Phys. 2010 (2010), no. 11, 143, 26 pages, arXiv:1009.3498.
  70. Stefański Jr. B., Green-Schwarz action for type IIA strings on ${\rm AdS}_4\times{\mathbb C}{\rm P}^3$, Nuclear Phys. B 808 (2009), 80-87, arXiv:0806.4948.
  71. Tseytlin A.A., Review of AdS/CFT integrability, Chapter II.1: Classical ${\rm AdS}_5\times S^5$ string solutions, Lett. Math. Phys. 99 (2012), 103-125, arXiv:1012.3986.
  72. Uvarov D.V., Kaluza-Klein gauge and minimal integrable extension of ${\rm OSp}(4|6)/({\rm SO}(1,3)\times {\rm U}(3))$ sigma-model, Internat. J. Modern Phys. A 27 (2012), 1250118, 26 pages, arXiv:1203.3041.
  73. Uvarov D.V., Lagrangian mechanics of massless superparticle on ${\rm AdS}_4\times\mathbb{CP}^3$ superbackground, Nuclear Phys. B 867 (2013), 354-369, arXiv:1205.5388.
  74. Uvarov D.V., On integrability of massless ${\rm AdS}_4×\mathbb{CP}^3$ superparticle equations, Modern Phys. Lett. A 29 (2014), 1350183, 11 pages, arXiv:1212.3157.
  75. Uvarov D.V., Supertwistor formulation for massless superparticle in ${\rm AdS}_5\times S^5$ superspace, Nuclear Phys. B 936 (2018), 690-713, arXiv:1807.08318.
  76. Uvarov D.V., Multitwistor mechanics of massless superparticle on ${\rm AdS}_5\times S^5$ superbackground, Nuclear Phys. B 950 (2020), 114830, 26 pages, arXiv:1907.13613.
  77. Volkov D.V., Zheltukhin A.A., String description in the space and superspace, Ukrain. Phys. J. 30 (1985), 809-813.
  78. Witten E., Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998), 253-291, arXiv:hep-th/9802150.
  79. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of fieldtheory which are integrable by means of the inverse scattering problem method, Sov. Phys. JETP 47 (1978), 1017-1027.
  80. Zarembo K., Integrability in sigma-models, in Integrability: From Statistical Systems to Gauge Theory, Oxford University Press, Oxford, 2019, 205-247, arXiv:1712.07725.
  81. Zheltukhin A.A., On relation between a relativistic string and two-dimensional field models, Sov. J. Nuclear Phys. 33 (1981), 927-930.
  82. Zheltukhin A.A., Classical relativistic string as an exactly solvable sector of the ${\rm SO}(1,1)\times {\rm SO}(2)$ gauge model, Phys. Lett. B 116 (1982), 147-150.
  83. Zheltukhin A.A., Gauge descriptions and nonlinear string equations in $d$-dimensional space-time, Theoret. and Math. Phys. 56 (1983), 785-795.
  84. Zheltukhin A.A., Hamiltonian structure of the antisymmetric action of a string, JETP Lett. 46 (1987), 262-265.
  85. Zheltukhin A.A., Hamiltonian formulation for an antisymmetric representation of the action of strings, Theoret. and Math. Phys. 77 (1988), 1264-1273.
  86. Zheltukhin A.A., A Hamiltonian of null strings. An invariant action of null (super)membranes, Sov. J. Nuclear Phys. 48 (1988), 375-379.
  87. Zheltukhin A.A., Tension as a perturbative parameter in nonlinear string equations in curved spacetime, Classical Quantum Gravity 13 (1996), 2357-2360, arXiv:hep-th/9606013.
  88. Zheltukhin A.A., Roshchupkin S.N., On an approximate description of string and $p$-brane dynamics in curved space-time, Teoret. and Math. Phys. 111 (1997), 714-722.

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