Euclidean E-models

Abstract

We develop the theory of the first-order dynamical systems, called the Euclidean $\mathcal E$-models, which naturally give rise in their second-order formulation to non-unitary nonlinear $σ$-models with real Euclidean actions. We establish the Euclidean version of Poisson--Lie T-duality, formulate sufficient conditions for Lax integrability, and describe the one-loop renormalization flow of the Euclidean $\mathcal E$-operator. For perfect Drinfeld doubles, we introduce the $\mathcal E$-Wick rotation, which canonically associates a Euclidean $\mathcal E$-model with every Lorentzian one. In the families studied here, this construction induces natural analytic continuations relating the Lorentzian and Euclidean Lax representations and renormalization-group flows, while preserving Poisson--Lie duality. As the principal example, we construct the Euclidean bi-Yang--Baxter model on the Lu--Weinstein double and determine explicitly its action, Lax representation, Poisson--Lie dual and one-loop renormalization-group flow. We identify a one-parameter family of one-loop RG fixed points for which the target-space geometry develops singularities on the maximal torus of $K$, whereas the dual model with target $K^{\mathbb C}/K$ has an everywhere regular background. This regular dual model turns out to be a one-parameter deformation of the non-unitary hyperbolic Wess--Zumino--Witten model.