Skip to main content
SpringerLink
Log in

Lévy Laplacians and instantons

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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).

    Article  Google Scholar 

  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).

    MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  4. L. Accardi and O. G. Smolyanov, “Lévy–Laplace operators in functional rigged Hilbert spaces,” Mat. Zametki 72 (1), 145–150 (2002)

    Article  Google Scholar 

  5. L. Accardi and O. G. Smolyanov, Math. Notes 72, 129–134 (2002)].

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  Google Scholar 

  7. L. Accardi and O. G. Smolyanov, Dokl. Math. 73 (2), 252–257 (2006)].

    Article  MATH  Google Scholar 

  8. 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. L. Gross, “A PoincarÈ lemma for connection forms,” J. Funct. Anal. 63, 1–46 (1985).

    Article  MathSciNet  Google Scholar 

  10. 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).

    Article  MathSciNet  Google Scholar 

  11. P. Lévy, ProblÈmes concrets d’analyse fonctionnelle (Gautier-Villars, Paris, 1951).

    Google Scholar 

  12. A. G. Sergeev, Harmonic Maps (Steklov Math. Inst., Moscow, 2008), Lekts. Kursy Nauchno-Obrazov. Tsentra 10.

    MATH  Google Scholar 

  13. B. O. Volkov, “Lévy-Laplacian and the gauge fields,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 15 (4), 1250027 (2012).

    Article  MathSciNet  Google Scholar 

  14. B. O. Volkov, “Quantum probability and Lévy Laplacians,” Russ. J. Math. Phys. 20 (2), 254–256 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  15. B. O. Volkov, “Hierarchy of Lévy-Laplacians and quantum stochastic processes,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (4), 1350027 (2013).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    B. O. Volkov

Authors
  1. B. O. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. O. Volkov.

Additional information

Original Russian Text © B.O. Volkov, 2015, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2015, Vol. 290, pp. 226–238.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Volkov, B.O. Lévy Laplacians and instantons. Proc. Steklov Inst. Math. 290, 210–222 (2015). https://doi.org/10.1134/S008154381506019X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S008154381506019X

Keywords

Search

Navigation