Boundary \(G/G\) theory and topological Poisson-Lie sigma model
type: Item
instance of: scholarly article
title: Boundary \(G/G\) theory and topological Poisson-Lie sigma model
zbMATH Open document ID: 1005.53059
DOI: 10.1023/A:1014477117077
author: Fernando Falceto
author: Krzysztof Gawȩdzki
published in: Letters in Mathematical Physics
publication date: 2002-06-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/0108206
review text: 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.
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 81T45
zbMATH DE Number: 1753730
zbMATH Keywords: topological field theories
zbMATH Keywords: Poisson-Lie groups
zbMATH Keywords: Wess-Zumino-Witten model
zbMATH Keywords: topological Poisson-Lie sigma models
zbMATH Keywords: \(r\)-matrices
reviewed by: Angela Gammella-Mathieu
MaRDI profile type: Publication
arXiv ID: hep-th/0108206
OpenAlex ID: W1600048071
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description: scientific article