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Trudy Mat. Inst. Steklova, 2015, Volume 290, Pages 226–238 (Mi tm3631)  

This article is cited in 9 scientific papers (total in 9 papers)

Lévy Laplacians and instantons

B. O. Volkov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: We describe dual and antidual solutions of the Yang–Mills equations by means of Lévy Laplacians. To this end, we introduce a class of Lévy Laplacians parameterized by the choice of a curve in the group $\mathrm {SO}(4)$. Two approaches are used to define such Laplacians: (i) the Lévy Laplacian can be defined as an integral functional defined by a curve in $\mathrm {SO}(4)$ 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 $\mathbb R^4$ 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.

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work is supported by the Russian Science Foundation under grant 14-50-00005.


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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 290:1, 210–222

Bibliographic databases:

UDC: 517.98
Received: March 15, 2015

Citation: B. O. Volkov, “Lévy Laplacians and instantons”, Modern problems of mathematics, mechanics, and mathematical physics, Collected papers, Trudy Mat. Inst. Steklova, 290, MAIK Nauka/Interperiodica, Moscow, 2015, 226–238; Proc. Steklov Inst. Math., 290:1 (2015), 210–222

Citation in format AMSBIB
\by B.~O.~Volkov
\paper L\'evy Laplacians and instantons
\inbook Modern problems of mathematics, mechanics, and mathematical physics
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2015
\vol 290
\pages 226--238
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\jour Proc. Steklov Inst. Math.
\yr 2015
\vol 290
\issue 1
\pages 210--222

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    This publication is cited in the following articles:
    1. B. O. Volkov, “Stochastic Levy differential operators and Yang–Mills equations”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 20:2 (2017), 1750008  crossref  mathscinet  zmath  isi  scopus
    2. B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Proc. Steklov Inst. Math., 301 (2018), 11–24  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. N. A. Gusev, “On the definitions of boundary values of generalized solutions to an elliptic-type equation”, Proc. Steklov Inst. Math., 301 (2018), 39–43  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. A. S. Trushechkin, “Finding stationary solutions of the Lindblad equation by analyzing the entropy production functional”, Proc. Steklov Inst. Math., 301 (2018), 262–271  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. B. O. Volkov, “Lévy Laplacians and annihilation process”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 160, no. 2, Izd-vo Kazanskogo un-ta, Kazan, 2018, 399–409  mathnet  mathscinet
    6. B. O. Volkov, “Primenenie differentsialnykh operatorov Levi v teorii kalibrovochnykh polei”, Kvantovaya veroyatnost, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 151, VINITI RAN, M., 2018, 21–36  mathnet  mathscinet
    7. B. O. Volkov, “Levy differential operators and gauge invariant equations for Dirac and Higgs fields”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22:1 (2019), 1950001  crossref  isi
    8. B. O. Volkov, “Levy Laplacian on Manifold and Yang-Mills Heat Flow”, Lobachevskii J. Math., 40:10, SI (2019), 1619–1630  mathnet  crossref  mathscinet  isi
    9. B. O. Volkov, “Levy Laplacians and instantons on manifolds”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 23:2 (2020), 2050008  crossref  mathscinet  isi
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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