Euclidean E-models

We study a class of $\mathcal{E}$-models, referred to as Euclidean $\mathcal{E}$-models, in which the operator $\mathcal{E}$ acting on the Drinfeld double squares to minus the identity rather than to the identity. This modification leads to significant structural differences from the standard $\mathcal{E}$-model framework. Most notably, the associated $\sigma$-models naturally possess Euclidean world-sheets and real Euclidean actions. Although for some Drinfeld doubles every Lorentzian $\mathcal{E}$-model admits a natural Euclidean counterpart, the duality, integrability, and renormalization properties of Euclidean $\mathcal{E}$-models are not determined by the Lorentzian theory and must be studied separately. We develop the basic formalism, provide the Euclidean version of Poisson--Lie T-duality, formulate the Euclidean analogue of the integrability criterion, and describe the Euclidean one-loop renormalization flow. The general constructions are illustrated by the example of the Euclidean bi-Yang--Baxter deformation.