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Poisson-Lie \(T\)-duality and loop groups of Drinfeld doubles. (English) Zbl 1037.81576
Summary: A duality-invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both \(\sigma\)-models in a pair related by Poisson-Lie \(T\)-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant \(\sigma\)-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shanskii symplectic form on the Drinfeld double and it directly gives a generating function of the canonical transformation relating the \(\sigma\)-models from the dual pair.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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