Advertisement

Lévy Laplacians and instantons

  • B. O. VolkovEmail author
Article

Abstract

We describe dual and antidual solutions of the Yang–Mills equations by means of L´evy Laplacians. To this end, we introduce a class of L´evy Laplacians parameterized by the choice of a curve in the group SO(4). Two approaches are used to define such Laplacians: (i) the Lévy Laplacian can be defined as an integral functional defined by a curve in SO(4) and a special form of the second-order derivative, or (ii) the Lévy Laplacian can be defined as the Cesàro mean of second-order derivatives along vectors from the orthonormal basis constructed by such a curve. We prove that under some conditions imposed on the curve generating the Lévy Laplacian, a connection in the trivial vector bundle with base R4 is an instanton (or an anti-instanton) if and only if the parallel transport generated by the connection is harmonic for such a Lévy Laplacian.

Keywords

Orthonormal Basis STEKLOV Institute Bianchi Identity Parallel Transport Dense Basis 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Accardi, P. Gibilisco, and I. V. Volovich, “The Lévy Laplacian and the Yang–Mills equations,” Rend. Lincei, Sci. Fis. Nat. 4 (3), 201–206 (1993).CrossRefGoogle Scholar
  2. 2.
    L. Accardi, P. Gibilisco, and I. V. Volovich, “Yang–Mills gauge fields as harmonic functions for the Lévy Laplacian,” Russ. J. Math. Phys. 2 (2), 235–250 (1994).zbMATHGoogle Scholar
  3. 3.
    L. Accardi, U. C. Ji, and K. Saitô, “The exotic (higher order Lévy) Laplacians generate the Markov processes given by distribution derivatives of white noise,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (3), 1350020 (2013).MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Accardi and O. G. Smolyanov, “Lévy–Laplace operators in functional rigged Hilbert spaces,” Mat. Zametki 72 (1), 145–150 (2002)CrossRefGoogle Scholar
  5. 4.
    L. Accardi and O. G. Smolyanov, Math. Notes 72, 129–134 (2002)].MathSciNetCrossRefzbMATHGoogle Scholar
  6. 5.
    L. Accardi and O. G. Smolyanov, “Feynman formulas for evolution equations with Lévy Laplacians on infinitedimensional manifolds,” Dokl. Akad. Nauk 407 (5), 583–588 (2006)MathSciNetGoogle Scholar
  7. 5.
    L. Accardi and O. G. Smolyanov, Dokl. Math. 73 (2), 252–257 (2006)].CrossRefzbMATHGoogle Scholar
  8. 6.
    I. Ya. Aref’eva and I. V. Volovich, “Higher order functional conservation laws in gauge theories,” in Generalized Functions and Their Application in Mathematical Physics: Proc. Int. Conf., Moscow, 1980 (Vychisl. Tsentr, Akad. Nauk SSSR, Moscow, 1981), pp. 43–49.Google Scholar
  9. 7.
    L. Gross, “A PoincarÈ lemma for connection forms,” J. Funct. Anal. 63, 1–46 (1985).MathSciNetCrossRefGoogle Scholar
  10. 8.
    R. Léandre and I. V. Volovich, “The stochastic Lévy Laplacian and Yang–Mills equation on manifolds,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2), 161–172 (2001).MathSciNetCrossRefGoogle Scholar
  11. 9.
    P. Lévy, ProblÈmes concrets d’analyse fonctionnelle (Gautier-Villars, Paris, 1951).Google Scholar
  12. 10.
    A. G. Sergeev, Harmonic Maps (Steklov Math. Inst., Moscow, 2008), Lekts. Kursy Nauchno-Obrazov. Tsentra 10.zbMATHGoogle Scholar
  13. 11.
    B. O. Volkov, “Lévy-Laplacian and the gauge fields,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (4), 1250027 (2012).MathSciNetCrossRefGoogle Scholar
  14. 12.
    B. O. Volkov, “Quantum probability and Lévy Laplacians,” Russ. J. Math. Phys. 20 (2), 254–256 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 13.
    B. O. Volkov, “Hierarchy of Lévy-Laplacians and quantum stochastic processes,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (4), 1350027 (2013).MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia

Personalised recommendations