Euclidean ℰ-models

A bstract We develop the theory of the first-order dynamical systems, called the Euclidean 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 ℰ-operator. For perfect Drinfeld doubles, we introduce the ℰ-Wick rotation, which canonically associates a Euclidean 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 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.