Integrable Degenerate E-Models from 4d Chern–Simons Theory

Abstract We present a general construction of integrable degenerate E E -models on a 2d manifold Σ using the formalism of Costello and Yamazaki based on 4d Chern–Simons theory on {C}P¹ 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 {L} L on Σ subject to a constraint and with {L} L depending rationally on the complex coordinate on {C}P¹ C P 1. When the meromorphic 1-form ω entering the action of 4d Chern–Simons theory is required to have a double pole at infinity, the constraint between h and {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 E E -models. We extend the latter approach to the most general setting of an arbitrary 1-form ω and obtain integrable degenerate E E -models. To illustrate the procedure, we reproduce two well-known examples of integrable degenerate E E -models: the pseudo-dual of the principal chiral model and the bi-Yang-Baxter σ -model.