On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras

In this paper, we study conformal points among the class of $\mathcal{E}$-models. The latter are $\sigma$-models formulated in terms of a current Poisson algebra, whose Lie-theoretic definition allows for a purely algebraic description of their dynamics and their 1-loop RG-flow. We use these results to formulate a simple algebraic condition on the defining data of such a model which ensures its 1-loop conformal invariance and the decoupling of its observables into two chiral Poisson algebras, describing the classical left- and right-moving fields of the theory. In the case of so-called non-degenerate $\mathcal{E}$-models, these chiral sectors form two current algebras and the model takes the form of a WZW theory once realised as a $\sigma$-model. The case of degenerate $\mathcal{E}$-models, in which a subalgebra of the current algebra is gauged, is more involved: the conformal condition yields a wider class of theories, which includes gauged WZW models but also other examples, seemingly different, which however sometimes turn out to be related to gauged WZW models based on other Lie algebras. For this class, we build non-local chiral fields of parafermionic-type as well as higher-spin local ones, forming classical $\mathcal{W}$-algebras. In particular, we find an explicit and efficient algorithm to build these local chiral fields. These results (and their potential generalisations discussed at the end of the paper) open the way for the quantisation of a large class of conformal $\mathcal{E}$-models using the standard operator formalism of two-dimensional CFT.Comment: v3: References added, 68+10 page