Boundary \(G/G\) theory and topological Poisson-Lie sigma model
In this paper, a version of the gauged Wess-Zumino-Witten (WZW) model with a Poisson-Lie group \(G\) as target for surfaces with boundary is defined and studied. It is already known [\textit{A. Yu. Alekseev, P. Schaller} and \textit{Th. Strobl}, Phys. Rev. D 52, 7146-7160 (1995)], when the target space is the Poisson-Lie group \(G^{\ast}\) dual to a simple complex Lie group \(G\) equipped with an \(r\)-matrix Poisson structure, the Poisson sigma model in the bulk is equivalent on the classical level to the gauged WZW model, the so-called G/G coset theory. Here, after recalling some basic facts about Poisson Lie-groups, the authors define a boundary version of the WZW model by restricting appropriately the target. Then they present a covariant description of the relation between the restricted G/G coset theory and the topological Poisson-Lie sigma models with a discrete quotient of \(G^{\ast}\) as the target. The canonical structure of the resulting classical field theory on a cylinder and on a trip are also described. For the cylinder, one gets essentially the space of conjugacy classes of commuting pairs of elements in \(G\). For the trip, the phase space of the theory is shown to be essentially the Heisenberg double of \(G\), introduced by \textit{M. A. Semenov-Tian-Shansky} in [Publ. Res. Inst. Math. Sci. 21, 1237-1260 (1985; Zbl 0673.58019)] -- a generalization of \(T^{\ast}G\) for Poisson Lie-groups. This last fact gives an example of a general construction of a Poisson-Lie groupoid over a Poisson manifold, obtained by \textit{A. S. Cattaneo} and \textit{G. Felder} in [Prog. Math. 198, 61-93 (2001; Zbl 1038.53074) and in arXiv: math SG/0003023]. The authors conclude with a brief discussion about the quantization of the theory for the Poisson structures induced by the standard classical \(r\)-matrices.
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