Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 058, 45 pages      arXiv:2011.13809      https://doi.org/10.3842/SIGMA.2021.058

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

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

Received December 07, 2020, in final form May 31, 2021; Published online June 10, 2021

Abstract
We construct the actions of a very broad family of 2d integrable $\sigma$-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d 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 2d integrable field theory for the 2d 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 2d integrable field theories can all naturally be described as $\mathcal{E}$-models.

Key words: 4d Chern-Simons theory; $\mathcal E$-models; affine Gaudin models; integrable $\sigma$-models.

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