/ 95 07 02 0 v 1 4 J ul 1 99 5 A Brief Introduction to Poisson σ - Models

The close interplay between geometry, topology and algebra turned out to be a most crucial point in the analysis of low dimensional field theories. This is particularly true for the large class of topological and almost topological two-dimensional field theories which can be treated comprehensively in the framework of Poisson-σ models. Examples are provided by pure YangMills and gravity theories and, to some extent, by the G/G gauged WZW-model. It turned out [1, 2, 3] that the common mathematical structure behind these theories is the one of Poisson manifolds. In the present contribution we review how this structure leads to the formulation of the Poisson-σ models. We will demonstrate that the general features of Poisson manifolds provide powerful tools for the analysis of these models. And we will apply the results to the examples mentioned above. For pedagogical reasons we will, in the presentation of the examples, restrict ourselves to SU(2) as gauge groups of YM and G/G and, in the case of gravity theories, to R2-gravity (R being the Ricci scalar). This has the advantage that the considered target spaces will turn out to be threedimensional then, opening possiblities for a simple visualization and explicitness of formulas. The method is, however, by no means restricted to these cases; for YM and G/G, e.g., even arbitrary non-compact gauge groups are comprised by the treatment. Let us start by reviewing some features of Poisson manifolds [4], which finally will become the target space of our 2D field theories. (To avoid any confusion, let us note right away that the notion of a Poisson manifold is somewhat more general than the one of a symplectic manifold). Denote by N a manifold equipped with a Poisson bracket relation {., .} between functions on N . By definition the Poisson bracket is bilinear and antisymmetric. It is, moreover, subject to the Leibnitz rule {f, gh} = {f, g}h+ g{f, h} and the Jacoby identity {f, {g, h}}+ cycl. = 0. In local coordinates X i on N a Poisson bracket may be expressed in terms of a two-tensor P (X) := {X , Xj}. With the Leibnitz rule this relation determines the Poisson structure uniquely: {f, g} = P f,ig,j . (1)