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Integrable $\mathcal{E}$-Models, $4\mathrm{d}$ Chern–Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects
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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. 
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     author = {Sylvain Lacroix and Beno{\^\i}t Vicedo},
     title = {Integrable $\mathcal{E}${-Models,} $4\mathrm{d}$ {Chern{\textendash}Simons} {Theory} and {Affine} {Gaudin} {Models.} {I.~Lagrangian} {Aspects}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a57/}
}
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Sylvain Lacroix; Benoît Vicedo. Integrable $\mathcal{E}$-Models, $4\mathrm{d}$ Chern–Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a57/

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