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

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.

DOI: https://doi.org/10.1134/S037196851503019X

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

Bibliographic databases:

UDC: 517.98

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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3631
• https://doi.org/10.1134/S037196851503019X
• http://mi.mathnet.ru/eng/tm/v290/p226

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. B. O. Volkov, “Lévy Laplacians in Hida calculus and Malliavin calculus”, Proc. Steklov Inst. Math., 301 (2018), 11–24
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
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
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
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
7. Volkov B.O., “Levy Differential Operators and Gauge Invariant Equations For Dirac and Higgs Fields”, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22:1 (2019), 1950001
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