Advertisement

Chern—Simons Action and Disclinations

  • M. O. Katanaev
Article
  • 14 Downloads

Abstract

We review the main properties of the Chern—Simons and Hilbert—Einstein actions on a three-dimensional manifold with Riemannian metric and torsion. We show a connection between these actions that is based on the gauge model for the inhomogeneous rotation group. The exact solution of the Euler—Lagrange equations is found for the Chern—Simons action with the linear source. This solution is proved to describe one straight linear disclination in the geometric theory of defects.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe,” Phys. Rev. D 93 (6), 061501(R) (2016).MathSciNetCrossRefGoogle Scholar
  2. 2.
    B. A. Bilby, R. Bullough, and E. Smith, “Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry,” Proc. R. Soc. London A 231, 263–273 (1955).MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. Bouchard, B. Florea, and M. Mari˜no, “Counting higher genus curves with crosscaps in Calabi–Yau orientifolds,” J. High Energy Phys. 2004 (12), 035 (2004).MathSciNetCrossRefGoogle Scholar
  4. 4.
    S.-s. Chern and J. Simons, “Characteristic forms and geometric invariants,” Ann. Math., Ser. 2, 99 (1), 48–69 (1974).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A. K. Gushchin, “Solvability of the Dirichlet problem for an inhomogeneous second-order elliptic equation,” Mat. Sb. 206 (10), 71–102 (2015) [Sb. Math. 206, 1410–1439 (2015)].CrossRefGoogle Scholar
  6. 6.
    A. K. Gushchin, “Lp-estimates for the nontangential maximal function of the solution to a second-order elliptic equation,” Mat. Sb. 207 (10), 28–55 (2016) [Sb. Math. 207, 1384–1409 (2016)].CrossRefGoogle Scholar
  7. 7.
    M. O. Katanaev, “Wedge dislocation in the geometric theory of defects,” Teor. Mat. Fiz. 135 (2), 338–352 (2003) [Theor. Math. Phys. 135, 733–744 (2003)].MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. O. Katanaev, “One-dimensional topologically nontrivial solutions in the Skyrme model,” Teor. Mat. Fiz. 138 (2), 193–208 (2004) [Theor. Math. Phys. 138, 163–176 (2004)].MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. O. Katanaev, “Geometric theory of defects,” Usp. Fiz. Nauk 175 (7), 705–733 (2005) [Phys. Usp. 48 (7), 675–701 (2005)].CrossRefGoogle Scholar
  10. 10.
    M. O. Katanaev, “On homogeneous and isotropic universe,” Mod. Phys. Lett. A 30 (34), 1550186 (2015); arXiv: 1511.00991 [gr-qc].MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. O. Katanaev, “Lorentz invariant vacuum solutions in general relativity,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 290, 149–153 (2015) [Proc. Steklov Inst. Math. 290, 138–142 (2015)].MathSciNetzbMATHGoogle Scholar
  12. 12.
    M. O. Katanaev, “Geometric methods in mathematical physics,” arXiv: 1311.0733v3 [math-ph].Google Scholar
  13. 13.
    M. O. Katanaev, “Killing vector fields and a homogeneous isotropic universe,” Usp. Fiz. Nauk 186 (7), 763–775 (2016) [Phys. Usp. 59 (7), 689–700 (2016)]; arXiv: 1610.05628 [gr-qc].CrossRefGoogle Scholar
  14. 14.
    M. O. Katanaev, “Chern–Simons term in the geometric theory of defects,” Phys. Rev. D 96 (8), 084054 (2017); arXiv: 1705.07888 [math-ph].CrossRefGoogle Scholar
  15. 15.
    M. O. Katanaev and I. V. Volovich, “Theory of defects in solids and three-dimensional gravity,” Ann. Phys. 216 (1), 1–28 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    M. O. Katanaev and I. V. Volovich, “Scattering on dislocations and cosmic strings in the geometric theory of defects,” Ann. Phys. 271 (2), 203–232 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    H. Kleinert, Multivalued Fields in Condenced Matter, Electromagnetism, and Gravitation (World Scientific, Singapore, 2008).CrossRefzbMATHGoogle Scholar
  18. 18.
    K. Kondo, “On the geometrical and physical foundations of the theory of yielding,” in Proc. 2nd Japan Natl. Congr. for Applied Mechanics, 1952 (Japan Natl. Committee Theor. Appl. Mech. (Sci. Counc. Japan), Tokyo, 1953), pp. 41–47.Google Scholar
  19. 19.
    E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen (Springer, Berlin, 1958).CrossRefzbMATHGoogle Scholar
  20. 20.
    J. F. Nye, “Some geometrical relations in dislocated crystals,” Acta Metall. 1, 153–162 (1953).CrossRefGoogle Scholar
  21. 21.
    V. P. Pavlov and V. M. Sergeev, “Fluid dynamics and thermodynamics as a unified field theory,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 294, 237–247 (2016) [Proc. Steklov Inst. Math. 294, 222–232 (2016)].MathSciNetzbMATHGoogle Scholar
  22. 22.
    M. Schlichenmaier and O. K. Sheinman, “Wess–Zumino–Witten–Novikov theory, Knizhnik–Zamolodchikov equations, and Krichever–Novikov algebras,” Usp. Mat. Nauk 54 (1), 213–250 (1999) [Russ. Math. Surv. 54, 213–249 (1999)].MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1967; M. Dekker, New York, 1971).zbMATHGoogle Scholar
  24. 24.
    E. Witten, “2+1 dimensional gravity as an exactly soluble system,” Nucl. Phys. B 311 (1), 46–78 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    V. V. Zharinov, “Conservation laws, differential identities, and constraints of partial differential equations,” Teor. Mat. Fiz. 185 (2), 227–251 (2015) [Theor. Math. Phys. 185, 1557–1581 (2015)].MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    V. V. Zharinov, “Bäcklund transformations,” Teor. Mat. Fiz. 189 (3), 323–334 (2016) [Theor. Math. Phys. 189, 1681–1692 (2016)].CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  2. 2.N.I. Lobachevsky Institute of Mathematics and MechanicsKazan Federal UniversityKazanRussia

Personalised recommendations