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

Abstract

AbstractWe present a general construction of integrable degenerate $$\mathcal {E}$$ E -models on a 2d manifold $$\Sigma $$ Σ using the formalism of Costello and Yamazaki based on 4d Chern–Simons theory on $$\Sigma \times {\mathbb {C}}{P}^1$$ Σ × 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 $${\mathcal {L}}$$ L on $$\Sigma $$ Σ subject to a constraint and with $${\mathcal {L}}$$ L depending rationally on the complex coordinate on $${\mathbb {C}}{P}^1$$ C P 1 . When the meromorphic 1-form $$\omega $$ ω entering the action of 4d Chern–Simons theory is required to have a double pole at infinity, the constraint between h and $${\mathcal {L}}$$ L was solved in Lacroix and Vicedo (SIGMA 17:058, 2021. https://doi.org/10.3842/SIGMA.2021.058) to obtain integrable non-degenerate $$\mathcal {E}$$ E -models. We extend the latter approach to the most general setting of an arbitrary 1-form $$\omega $$ ω and obtain integrable degenerate $$\mathcal {E}$$ E -models. To illustrate the procedure, we reproduce two well-known examples of integrable degenerate $$\mathcal {E}$$ E -models: the pseudo-dual of the principal chiral model and the bi-Yang-Baxter $$\sigma $$ σ -model.