Communications in Mathematical Physics

, Volume 302, Issue 3, pp 755–788 | Cite as

Sugawara-Type Constraints in Hyperbolic Coset Models

  • Thibault Damour
  • Axel Kleinschmidt
  • Hermann Nicolai


In the conjectured correspondence between supergravity and geodesic models on infinite-dimensional hyperbolic coset spaces, and E 10/K(E 10) in particular, the constraints play a central role. We present a Sugawara-type construction in terms of the E 10 Noether charges that extends these constraints infinitely into the hyperbolic algebra, in contrast to the truncated expressions obtained in Damour et al. (Class. Quant. Grav. 24:6097, 2007) that involved only finitely many generators. Our extended constraints are associated to an infinite set of roots which are all imaginary, and in fact fill the closed past light-cone of the Lorentzian root lattice. The construction makes crucial use of the E 10 Weyl group and of the fact that the E 10 model contains both D = 11 supergravity and D = 10 IIB supergravity. Our extended constraints appear to unite in a remarkable manner the different canonical constraints of these two theories. This construction may also shed new light on the issue of ‘open constraint algebras’ in traditional canonical approaches to gravity.


Weyl Group Dynkin Diagram Level Decomposition High Weight Vector Maximal Supergravity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    DeWitt B.S.: Quantum Theory of Gravity. 1. The Canonical Theory. Phys. Rev. 160, 1113 (1967)zbMATHCrossRefADSGoogle Scholar
  2. 2.
    Kiefer, C.: Quantum gravity. Int. Ser. Monogr. Phys. 124, Oxford: Oxford University Press, 2004Google Scholar
  3. 3.
    Damour T., Henneaux M., Nicolai H.: E 10 and a ‘small tension expansion’ of M Theory. Phys. Rev. Lett. 89, 221601 (2002)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Damour T., Kleinschmidt A., Nicolai H.: Constraints and the E 10 Coset Model. Class. Quant. Grav. 24, 6097 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Sugawara H.: A Field theory of currents. Phys. Rev. 170, 1659 (1968)CrossRefADSGoogle Scholar
  6. 6.
    Bardakçi K., Halpern M. B.: New dual quark models. Phys. Rev. D 3, 2493 (1971)CrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Goddard P., Olive D.I.: Kac-Moody And Virasoro Algebras In Relation To Quantum Physics. Int. J. Mod. Phys. A 1, 303 (1986)zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Kleinschmidt A., Koehn M., Nicolai H.: Supersymmetric quantum cosmological billiards. Phys. Rev. D 80, 061701 (2009)CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Forte L.A.: Arithmetical Chaos and Quantum Cosmology. Class. Quant. Grav. 26, 045001 (2009)CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Goddard P., Thorn C.B.: Compatibility of the Dual Pomeron with Unitarity and the Absence of Ghosts in the Dual Resonance Model. Phys. Lett. B 40, 235 (1972)CrossRefADSGoogle Scholar
  11. 11.
    Kleinschmidt A., Nicolai H.: E 10 and SO(9,9) invariant supergravity. JHEP 0407, 041 (2004)CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Damour, T., Nicolai, H.: Eleven dimensional supergravity and the E 10/K(E 10) sigma-model at low A 9 levels. In: Pogoyan, G.S., Vicent, L.E., Wolf, K.B. (eds.) Group Theoretical Methods in Physics. IOP conference series no. 185, pp. 93–111. IOP Publishing (2005)Google Scholar
  13. 13.
    Kleinschmidt A., Nicolai H.: IIB supergravity and E 10. Phys. Lett. B 606, 391 (2005)CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Henneaux M., Jamsin E., Kleinschmidt A., Persson D.: On the E 10/Massive Type IIA Supergravity Correspondence. Phys. Rev. D 79, 045008 (2009)CrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Nicolai H., Samtleben H.A.J.: On K(E 9). Q.J. Pure Appl. Math. 1, 180 (2005)zbMATHMathSciNetGoogle Scholar
  16. 16.
    West P.C.: E 11 and M theory. Class. Quant. Grav. 18, 4443 (2001)zbMATHCrossRefADSGoogle Scholar
  17. 17.
    West P.C.: E(11), SL(32) and central charges. Phys. Lett. B 575, 333 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    Riccioni F., West P.: Local E 11. JHEP 0904, 051 (2009)CrossRefMathSciNetADSGoogle Scholar
  19. 19.
    Schnakenburg I., West P.C.: Kac-Moody symmetries of IIB supergravity. Phys. Lett. B 517, 421 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    Schnakenburg I., West P.C.: Massive IIA supergravity as a non-linear realisation. Phys. Lett. B 540, 137 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Kleinschmidt A., Schnakenburg I., West P.C.: Very-extended Kac-Moody algebras and their interpretation at low levels. Class. Quant. Grav. 21, 2493 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    West P.C.: The IIA, IIB and eleven-dimensional theories and their common E(11) origin. Nucl. Phys. B 693, 76 (2004)zbMATHCrossRefADSGoogle Scholar
  23. 23.
    Morozov A.Y., Perelomov A.M., Roslyi A.A., Shifman M.A., Turbiner A.V.: Quasiexactly Solvable Quantal Problems: One-Dimensional Analog of Rational Conformal Field Theories. Int. J. Mod. Phys. A 5, 803 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Halpern M.B., Kiritsis E.: General Virasoro Construction on Affine G. Mod. Phys. Lett. A 4, 1373 (1989)CrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Kac V.G.: Infinite dimensional Lie algebras. Cambridge University Press, Cambridge (1995)Google Scholar
  26. 26.
    Damour T., de Buyl S., Henneaux M., Schomblond C.: Einstein billiards and overextensions of finite-dimensional simple Lie algebras. JHEP 0208, 030 (2002)CrossRefADSGoogle Scholar
  27. 27.
    Gebert R.W., Nicolai H.: An affine string vertex operator construction at arbitrary level. J. Math. Phys. 38, 4435 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Gaberdiel M.R., Olive D.I., West P.C.: A class of Lorentzian Kac-Moody algebras. Nucl. Phys. B 645, 403 (2002)zbMATHCrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Nicolai, H., Fischbacher, T.: Low level representations for E 10 and E 11. Cont. Math. 343, Providence, RI: Amer. Math. Soc., 2004, p. 191Google Scholar
  30. 30.
    Damour T., Henneaux M., Nicolai H.: Cosmological billiards. Class. Quant. Grav. 20, R145 (2003)zbMATHCrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Damour T., Henneaux M., Julia B., Nicolai H.: Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models. Phys. Lett. B 509, 323 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Kac V., Moody R.V., Wakimoto M.: On E 10. In: Bleuler, K., Werner, M. (eds) “Differential geometrical methods in theoretical physics”, pp. 109–128. Kluwer, Dordrecht (1988)Google Scholar
  33. 33.
    Kleinschmidt A., Nicolai H., Palmkvist J.: K(E 9) from K(E 10). JHEP 0706, 051 (2007)CrossRefMathSciNetADSGoogle Scholar
  34. 34.
    Damour T., Nicolai H.: Symmetries, singularities and the de-emergence of space. Int. J. Mod. Phys. D 17, 525 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  35. 35.
    Romans L.J.: Massive N=2a supergravity in ten-dimensions. Phys. Lett. B 169, 374 (1986)CrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Brown J., Ganor O.J., Helfgott C.: M-theory and E 10: Billiards, branes, and imaginary roots. JHEP 0408, 063 (2004)CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Thibault Damour
    • 1
  • Axel Kleinschmidt
    • 2
  • Hermann Nicolai
    • 3
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Physique Théorique et MathématiqueUniversité Libre de Bruxelles & International Solvay InstitutesBruxellesBelgium
  3. 3.Max-Planck-Insitut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany

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