$\eta$ and $\lambda$ deformations as $\mathcal{E}$-models

Abstract

We show that the so called $\lambda$ deformed $\sigma$-model as well as the $\eta$ deformed one belong to a class of the $\mathcal{E}$-models introduced in the context of the Poisson-Lie-T-duality. The $\lambda$ and $\eta$ theories differ solely by the choice of the Drinfeld double; for the $\lambda$ model the double is the direct product $G\times G$ while for the $\eta$ model it is the complexified group $G^{\mathbb{C}}$. As a consequence of this picture, we prove for any $G$ that the target space geometries of the $\lambda$-model and of the Poisson-Lie T-dual of the $\eta$-model are related by a simple analytic continuation.

20 pages, final version accepted for publication in Nuclear Physics B

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