Skip to main content
SpringerLink
Log in

Lévy Laplacians, Holonomy Group and Instantons on 4-Manifolds

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

A connection between the Yang–Mills gauge fields on 4-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if the connection corresponding to the parallel transport satisfies the Yang–Mills anti-self-duality equations. An analogous connection between the Laplace equation for the Lévy Laplacian and the Yang–Mills equations was previously known.

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. Accardi, L., Gibilisco, P., Volovich, I.V.: The Lévy Laplacian and the Yang–Mills equations. Rend. Lincei, Sci. Fis. Nat. 4, 201–206 (1993). https://doi.org/10.1007/BF03001574

    Article  Google Scholar 

  2. Accardi, L., Gibilisco, P., Volovich, I.V.: Yang–Mills gauge fields as harmonic functions for the Lévy-Laplacians. Russ. J. Math. Phys. 2, 235–250 (1994)

    MATH  Google Scholar 

  3. Léandre, R., Volovich, I.V.: The stochastic Lévy Laplacian and Yang–Mills equation on manifolds. Infin. Dimens. Anal. Quantum Probab. Relat. Top 4, 161–172 (2001). https://doi.org/10.1142/S0219025701000449

    Article  MathSciNet  MATH  Google Scholar 

  4. Belavin, A.A., Polyakov, A.M., Schwartz, A.S., Tyupkin, Y.S.: Pseudo-particle solutions of the Yang–Mills equations. Phys. Lett. B 59, 85–87 (1975). https://doi.org/10.1016/0370-2693(75)90163-X

    Article  MathSciNet  Google Scholar 

  5. Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravitation, gauge theories and differential geometry. Phisics Rep. 66, 213–393 (1980). https://doi.org/10.1016/0370-1573(80)90130-1

    Article  MathSciNet  Google Scholar 

  6. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond. A 362, 425–461 (1978). https://doi.org/10.1098/rspa.1978.0143

    Article  MathSciNet  MATH  Google Scholar 

  7. Taubes, C.H.: Self-dual connections on non-self-dual manifolds. J. Diff. Geom. 17, 139–170 (1982). https://doi.org/10.4310/jdg/1214436701

    MathSciNet  MATH  Google Scholar 

  8. Atiyah, M.F., Drinfeld, V.G., Hitchin, N.J., Manin, Y.I.: Construction of instantons. Phys. Lett. A 362, 425–461 (1978). https://doi.org/10.1016/0375-9601(78)90141-X

    MathSciNet  MATH  Google Scholar 

  9. Bourguignon, J.-P., Lawson, H.B.: Stability and isolation phenomena for Yang–Mills fields. Comm. Math. Phys. 79, 189–230 (1981). https://doi.org/10.4310/jdg/1214436701

    Article  MathSciNet  MATH  Google Scholar 

  10. Accardi, L., Smolyanov, O.G.: Feynman formulas for evolution equations with Lévy Laplacians on infinite-dimensional manifolds. Dokl. Math. 73, 252–257 (2006). https://doi.org/10.1134/S106456240602027X

    Article  MATH  Google Scholar 

  11. Volkov, B.O.: Stochastic Lévy differential operators and Yang–Mills equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20, 1750008 (2017). https://doi.org/10.1142/S0219025717500084

    Article  MathSciNet  MATH  Google Scholar 

  12. Volkov, B.O.: Lévy differential operators and gauge invariant equations for Dirac and Higgs fields. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22, 1750008 (2019). https://doi.org/10.1142/S0219025719500012

    Article  Google Scholar 

  13. Lévy, P.: Problèmes Concrets D’analyse Fonctionnelle. Gautier-Villars, Paris (1951)

    MATH  Google Scholar 

  14. Accardi, L.: Yang–Mills equations and Lévy Laplacians. In: Dirichlet Forms and Stochastic Processes, Beijing, 1993, pp 593–660. de Gruyter, Berlin (1995)

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Volkov, B.O.: Lévy Laplacians and instantons on manifolds. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23, 2050008 (2020). https://doi.org/10.1142/S0219025720500083

    Article  MathSciNet  MATH  Google Scholar 

  17. Atiyah, M.F.: Instantons in two and four dimensions. Comm. Math. Phys. 93, 437–451 (1984). https://doi.org/10.1007/BF01212288

    Article  MathSciNet  MATH  Google Scholar 

  18. Donaldson, S.K.: Instantons and geometric invariant theory. Comm. Math. Phys. 93, 453–460 (1984). https://doi.org/10.1007/BF01212289

    Article  MathSciNet  MATH  Google Scholar 

  19. Sergeev, A.G.: Harmonic spheres conjecture. Theoret. Math. Phys. 164, 1140–1150 (2010). https://doi.org/10.1007/s11232-010-0092-5

    Article  MATH  Google Scholar 

  20. Polyakov, A.M.: String representations and hidden symmetries for gauge fields. Phys. Lett. B 82, 247–250 (1979). https://doi.org/10.1016/0370-2693(79)90747-0

    Article  Google Scholar 

  21. Polyakov, A.M.: Gauge fields as rings of glue. Nuclear Physics B. 164, 171–188 (1980). https://doi.org/10.1016/0370-2693(79)90747-0

    Article  MathSciNet  Google Scholar 

  22. Polyakov, A.M.: Gauge Fields and Strings. Harwood Academic Publishers, London (1987)

    MATH  Google Scholar 

  23. Aref’eva, I.Y., Volovich, I.V.: Higher order functional conservation laws in gauge theories. In: Proc. Int. Conf. Generalized Functions and Their Applications in Mathematical Physics, pp 43–49. Academy of Sciences of the USSR, Moscow (1981)

  24. Gross, L.: A Poincarè lemma for connection forms. J. Funct. Anal. 63, 1–46 (1985). https://doi.org/10.1016/0022-1236(85)90096-5

    Article  MathSciNet  MATH  Google Scholar 

  25. Driver, B.: Classifications of bundle connection pairs by parallel translation and lassos. J. Funct. Anal. 83, 1–46 (1989). https://doi.org/10.1016/0022-1236(89)90035-9

    Article  MathSciNet  MATH  Google Scholar 

  26. Bauer, R.O.: Characterizing Yang–Mills fields by stochastic parallel transport. J. Funct. Anal. 155, 536–549 (1998). https://doi.org/10.1006/jfan.1997.3238

    Article  MathSciNet  MATH  Google Scholar 

  27. Bauer, R.O.: Random holonomy for Yang–Mills fields: Long–time asymptotics. Potential Anal. 18, 43–57 (2003). https://doi.org/10.1023/A:1020529721290

    Article  MathSciNet  MATH  Google Scholar 

  28. Arnaudon, M., Bauer, R.O., Thalmaier, A.: A probabilistic approach to the Yang–Mills heat equation. J. Math. Pures Appl. 81, 43–57 (2003). https://doi.org/10.1016/S0021-7824(02)01254-0

    MathSciNet  MATH  Google Scholar 

  29. Arnaudon, M., Thalmaier, A.: Yang–Mills fields and random holonomy along Brownian bridges. Ann. Probab. 31, 769–790 (2003). https://doi.org/10.1214/aop/1048516535

    Article  MathSciNet  MATH  Google Scholar 

  30. Volkov, B.O.: Lévy Laplacians in Hida Calculus and Malliavin Calculus. Proc. Steklov Inst. Math. 301, 11–24 (2018). https://doi.org/10.1134/S00815438180400283

    Article  MATH  Google Scholar 

  31. Accardi, L., Hasegawa, A., Ji, U.C., Saitô, K.: White noise delta functions and infinite dimensional Laplacians. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 23, 2050028 (2020). https://doi.org/10.1142/S0219025720500289

    Article  MathSciNet  MATH  Google Scholar 

  32. Freed, D., Uhlenbeck, K.: Instantons and Four-Manifolds. Springer, New York (1984)

    Book  MATH  Google Scholar 

  33. Klingenberg, W.: Riemannian Geometry. de Gruyter Studies in Mathematics, vol. 1. Berlin (1982)

  34. Klingenberg, W.: Lectures on Closed Geodesics. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

  35. Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  36. Volkov, B.O.: Lévy Laplacian on manifold and Yang–Mills heat flow. Lobachevskii J. Math. 40, 1615–1626 (2019). https://doi.org/10.1134/S1995080219100305

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to express his deep gratitude to L. Accardi, O. G. Smolyanov and I. V. Volovich for helpful discussions.

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology (National Research University), Institutskiy per. 9, Dolgoprudny, 141701, Moscow Region, Russia

    Boris O. Volkov

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

    Boris O. Volkov

  3. National University of Science and Technology “MISiS”, Leninsky Prosp. 4, Moscow, 119991, Russia

    Boris O. Volkov

Authors
  1. Boris O. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris O. Volkov.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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, Holonomy Group and Instantons on 4-Manifolds. Potential Anal 59, 1381–1397 (2023). https://doi.org/10.1007/s11118-022-10013-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-022-10013-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation