On the Zakharov–Mikhailov action: 4d Chern–Simons origin and covariant Poisson algebra of the Lax connection

Abstract

We derive the 2d Zakharov–Mikhailov action from 4d Chern–Simons theory. This 2d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the 2d level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “H=TrL2”.

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Item Type:	Article
Authors/Creators:	Caudrelier, V https://orcid.org/0000-0003-0129-6758 Stoppato, M https://orcid.org/0000-0002-2722-4931 Vicedo, B
Copyright, Publisher and Additional Information:	© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Integrable field theories; 4d Chern-Simons theory; Covariant Hamiltonian formalism; Zakharov–Mikhailov action; Covariant classical r-matrix
Dates:	Published: 22 June 2021 Published (online): 22 June 2021 Accepted: 5 June 2021
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User:	Symplectic Publications
Date Deposited:	13 Jul 2021 10:34
Last Modified:	13 Jul 2021 10:34
Status:	Published
Publisher:	Springer
Identification Number:	10.1007/s11005-021-01425-x
Related URLs:	Author
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:176133

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On the Zakharov–Mikhailov action: 4d Chern–Simons origin and covariant Poisson algebra of the Lax connection

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