Gravitation and Cosmology

, Volume 23, Issue 4, pp 337–342 | Cite as

On generalized Melvin’s solutions for Lie algebras of rank 2

  • S. V. Bolokhov
  • V. D. Ivashchuk


We consider a class of solutions in multidimensional gravity which generalize Melvin’s well-known cylindrically symmetric solution, originally describing the gravitational field of a magnetic flux tube. The solutions considered contain the metric, two Abelian 2-forms and two scalar fields, and are governed by two moduli functions H1(z) and H2(z) (z = ρ2, ρ is a radial coordinate) which have a polynomial structure and obey two differential (Toda-like) master equations with certain boundary conditions. These equations are governed by a certain matrix A which is a Cartan matrix for some Lie algebra. The models for rank-2 Lie algebras A2, C2 and G2 are considered. We study a number of physical and geometric properties of these models. In particular, duality identities are proved, which reveal a certain behavior of the solutions under the transformation ρ → 1/ρ; asymptotic relations for the solutions at large distances are obtained; 2-form flux integrals over 2-dimensional regions and the corresponding Wilson loop factors are calculated, and their convergence is demonstrated. These properties make the solutions potentially applicable in the context of some dual holographic models. The duality identities can also be understood in terms of the Z2 symmetry on vertices of the Dynkin diagram for the corresponding Lie algebra.


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  1. 1.
    M. A. Melvin, “Pure magnetic and electric geons,” Phys. Lett. 8, 65 (1964).ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    V. D. Ivashchuk, Class. Quantum Grav. 19, 3033–3048 (2002); hep-th/0202022.ADSCrossRefGoogle Scholar
  3. 3.
    K. A. Bronnikov, “Static, cylindrically symmetric Einstein-Maxwell fields.” In: Problems in Gravitation Theory and Particle Theory (PGTPT), 10th issue (Atomizdat, Moscow, 1979, in Russian), p.37.Google Scholar
  4. 4.
    K. A. Bronnikov and G. N. Shikin, “On interacting fields in general relativity,” Izv. Vuzov (Fizika) 9, 25–30 (1977); Russ. Phys. J. 20, 1138–1143 (1977).Google Scholar
  5. 5.
    G. W. Gibbons and D. L. Wiltshire, Nucl. Phys. B 287, 717–742 (1987); hep-th/0109093.ADSCrossRefGoogle Scholar
  6. 6.
    G. Gibbons and K. Maeda, Nucl. Phys. B 298, 741–775 (1988).ADSCrossRefGoogle Scholar
  7. 7.
    H. F. Dowker, J. P. Gauntlett, D. A. Kastor, and J. Traschen, Phys. Rev. D 49, 2909–2917 (1994); hep-th/9309075.ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    F. Dowker, J. P. Gauntlett, G. W. Gibbons, and G. T. Horowitz, Phys. Rev. D 53, 7115 (1996); hepth/9512154.ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    D. V. Gal’tsov and O. A. Rytchkov, Phys. Rev. D 58, 122001 (1998); hep-th/9801180.ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    C.-M. Chen, D. V. Gal’tsov, and S. A. Sharakin, Grav. Cosmol. 5, 45 (1999); hep-th/9908132.ADSGoogle Scholar
  11. 11.
    M. S. Costa and M. Gutperle, JHEP 0103, 027 (2001); hep-th/0012072.ADSCrossRefGoogle Scholar
  12. 12.
    P. M. Saffin, Phys. Rev. D 64, 024014 (2001); grqc/0104014.ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    M. Gutperle and A. Strominger, JHEP 0106, 035 (2001); hep-th/0104136.ADSCrossRefGoogle Scholar
  14. 14.
    M. S. Costa, C. A. Herdeiro, and L. Cornalba, Nucl. Phys. B 619, 155 (2001); hep-th/0105023.ADSCrossRefGoogle Scholar
  15. 15.
    R. Emparan, Nucl. Phys. B 610, 169 (2001); hepth/0105062.ADSCrossRefGoogle Scholar
  16. 16.
    J. M. Figueroa-O’Farrill and G. Papadopoulos, JHEP 0106, 036 (2001); hep-th/0105308.CrossRefGoogle Scholar
  17. 17.
    J. G. Russo and A. A. Tseytlin, JHEP 11, 065 (2001); hep-th/0110107.ADSCrossRefGoogle Scholar
  18. 18.
    C. M. Chen, D. V. Gal’tsov, and P. M. Saffin, Phys. Rev. D 65, 084004 (2002); hep-th/0110164.ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    V. D. Ivashchuk and V. N. Melnikov, “Multidimensional gravitational models: Fluxbrane and S-brane solutions with polynomials.” AIP Conference Proceedings 910, 411–422 (2007).ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    I. S. Goncharenko, V. D. Ivashchuk, and V. N. Melnikov, Grav. Cosmol. 13, 262 (2007); mathph/0612079.ADSGoogle Scholar
  21. 21.
    A. A. Golubtsova and V. D. Ivashchuk, Phys. of Part. and Nuclei 43, 720 (2012).ADSCrossRefGoogle Scholar
  22. 22.
    V. D. Ivashchuk and V. N. Melnikov, Grav. Cosmol. 20, 182 (2014).ADSCrossRefGoogle Scholar
  23. 23.
    A. A. Golubtsova and V. D. Ivashchuk, Grav. Cosmol. 15, 144 (2009); arXiv: 1009.3667.ADSCrossRefGoogle Scholar
  24. 24.
    J. Fuchs and C. Schweigert, Symmetries, Lie Algebras and Representations. A Graduate Course for Physicists (Cambridge University Press, Cambridge, 1997).MATHGoogle Scholar
  25. 25.
    B. Kostant, Adv. inMath. 34, 195 (1979).MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. A. Olshanetsky and A. M. Perelomov, Invent. Math., 54, 261 (1979).ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    V. D. Ivashchuk, J. Geom. and Phys. 86, 101 (2014).ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    A. A. Golubtsova and V. D. Ivashchuk, “On calculation of fluxbrane polynomials corresponding to classical series of Lie algebras,” arXiv: 0804.0757.Google Scholar
  29. 29.
    S. V. Bolokhov and V. D. Ivashchuk, “On generalized Melvin solution for the Lie algebra E 6,” arXiv: 1706.06621.Google Scholar
  30. 30.
    S. V. Bolokhov and V. D. Ivashchuk, in preparation.Google Scholar
  31. 31.
    M. E. Abishev, K. A. Boshkayev, and V. D. Ivashchuk, Eur.Phys. J. C 77, 180 (2017).ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Institute of Gravitation and CosmologyPeoples’ Friendship University of Russia (RUDN University)MoscowRussia
  2. 2.Center for Gravitation and Fundamental MetrologyVNIIMSMoscowRussia

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