On multidimensional analogs of Melvin's solution for classical series of Lie algebras

A.A. Golubtsova1 and V.D. Ivashchuk2


A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is presented. The gravitational model contains n 2-forms and l n scalar fields, where n is the rank of G. The solution is governed by a set of n functions obeying n ordinary differential equations with certain boundary conditions. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). A program (in Maple) for calculating these polynomials for classical series of Lie algebras is suggested. The polynomials corresponding to the Lie algebra D4 are obtained. It is conjectured that the polynomials for An-,Bn- and Cn- series may be obtained from polynomials for Dn+1-series by using certain reduction formulas.


  1. M. A. Melvin, Pure magnetic and electric geons, Phys. Lett. 8, 65 (1964).
  2. V. D. Ivashchuk, Composite fluxbranes with general intersections, Class. Quant. Grav. 19 3033-3048 (2002); hep-th/0202022.
  3. D.V. Gal'tsov and O.A. Rytchkov, Generating branes via sigma models, Phys. Rev. D 58, 122001 (1998); hep-th/9801180.
  4. C.-M. Chen, D. V. Gal'tsov and S. A. Sharakin, Intersecting M-fluxbranes, Grav. Cosmol. 5, 45-48 (1999); hep-th/9908132.
  5. M. S. Costa and M. Gutperle, The Kaluza-Klein Melvin solution in M-theory, JHEP 0103, 027 (2001); hep-th/0012072.
  6. M. Gutperle and A. Strominger, Fluxbranes in string theory, JHEP 0106, 035 (2001); hep-th/0104136.
  7. C. M. Chen, D. V. Gal'tsov and P. M. Saffin, Supergravity fluxbranes in various dimensions, Phys. Rev. D 65, 084004 (2002); hep-th/0110164.
  8. I.S. Goncharenko, V. D. Ivashchuk and V.N. Melnikov, Fluxbrane and S-brane solutions with polynomials related to rank-2 Lie algebras, Grav. Cosmol. 13, 262-266 (2007); math-ph/0612079.
  9. V.D. Ivashchuk and S.-W. Kim, Solutions with intersecting p-branes related to Toda chains, J. Math. Phys. 41, 444-460 (2000); hep-th/9907019.
  10. V.D. Ivashchuk and V.N. Melnikov, Exact solutions in multidimensional gravity with antisymmetric forms, topical review, Class. Quantum Grav. 18 R87-R152 (2001); hep-th/0110274.
  11. J. Fuchs and C. Schweigert, Symmetries, Lie algebras and Representations. A graduate course for physicists (Cambridge University Press, Cambridge, 1997).
  12. V. D. Ivashchuk and V. N. Melnikov, P-brane black Holes for general intersections, Grav. Cosmol. 5, 313-318 (1999); gr-qc/0002085.
  13. B. Kostant, Adv. in Math. 34, 195 (1979).
  14. M. A. Olshanetsky and A. M. Perelomov, Invent. Math. 54, 261 (1979).
  15. A. A. Golubtsova and V. D. Ivashchuk, On calculation of fluxbrane polynomials corresponding to classical series of Lie algebras, arxiv: 0804.0757 [nlin.SI].
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