Integrable degenerate $\mathcal{E}$-models from 4d Chern-Simons theory

Abstract

We present a general construction of integrable degenerate $\mathcal{E}$-models on a 2d manifold $\Sigma$ using the formalism of Costello and Yamazaki based on 4d Chern-Simons theory on $\Sigma \times \mathbb{C}{P}^1$. We begin with a physically motivated review of the mathematical results of [arXiv:2008.01829] where a unifying 2d action was obtained from 4d Chern-Simons theory which depends on a pair of 2d fields $h$ and $\mathcal{L}$ on $\Sigma$ subject to a constraint and with $\mathcal{L}$ depending rationally on the complex coordinate on $\mathbb{C}{P}^1$. When the meromorphic 1-form $\omega$ entering the action of 4d Chern-Simons theory is required to have a double pole at infinity, the constraint between $h$ and $\mathcal{L}$ was solved in [arXiv:2011.13809] to obtain integrable non-degenerate $\mathcal{E}$-models. We extend the latter approach to the most general setting of an arbitrary 1-form $\omega$ and obtain integrable degenerate $\mathcal{E}$-models. To illustrate the procedure we reproduce two well known examples of integrable degenerate $\mathcal{E}$-models: the pseudo dual of the principal chiral model and the bi-Yang-Baxter $\sigma$-model.

arXiv comment39 pages. Minor updates, matches published version

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