Integrable Degenerate \(\varvec{\mathcal {E}}\)-Models from 4d Chern–Simons Theory

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 Benini et al. (Commun Math Phys 389(3):1417–1443, 2022. https://doi.org/10.1007/s00220-021-04304-7) 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 Lacroix and Vicedo (SIGMA 17:058, 2021. https://doi.org/10.3842/SIGMA.2021.058) 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.