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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2021 Volume 17, 058, 45 pp. (Mi sigma1741)

This article is cited in 18 papers

Integrable $\mathcal{E}$-Models, $4\mathrm{d}$ Chern–Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

Sylvain Lacroixab, Benoît Vicedoc

a Zentrum für Mathematische Physik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
b II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
c Department of Mathematics, University of York, York YO10 5DD, UK

Abstract: We construct the actions of a very broad family of $2\mathrm{d}$ integrable $\sigma$-models. Our starting point is a universal $2\mathrm{d}$ action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on $4\mathrm{d}$ Chern–Simons theory. This $2\mathrm{d}$ action depends on a pair of $2\mathrm{d}$ fields $h$ and $\mathcal{L}$, with $\mathcal{L}$ depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for $\mathcal{L}$ in terms of $h$ this produces a $2\mathrm{d}$ integrable field theory for the $2\mathrm{d}$ field $h$ whose Lax connection is given by $\mathcal{L}(h)$. We construct a general class of solutions to this constraint and show that the resulting $2\mathrm{d}$ integrable field theories can all naturally be described as $\mathcal{E}$-models.

Keywords: $4\mathrm{d}$ Chern–Simons theory, $\mathcal E$-models, affine Gaudin models, integrable $\sigma$-models.

MSC: 17B80, 37K05, 37K10

Received: December 7, 2020; in final form May 31, 2021; Published online June 10, 2021

Language: English

DOI: 10.3842/SIGMA.2021.058



Bibliographic databases:
ArXiv: 2011.13809


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