Integrable Degenerate varvec{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 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.