Abstract
We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions ≥ 2 whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree (1, p − 1, p) we determine a generic Wess-Zumino topological field theory in p + 1 dimensions with background data consisting of a Poisson 2-vector, a (p + 1)-vector R and a (p + 2)-form H satisfying a specific geometrical condition that defines a H-twisted R-Poisson structure of order p + 1. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted R-Poisson. This extends the twisted R-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle T*[p]T*[1]M is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.
Highlights
Such topological field theories can be extended by Wess-Zumino terms [17]
We would like to answer the question: given a Poisson or twisted Poisson manifold as a target space, which topological field theories in spacetime dimension ≥ 2 exist such that their gauge symmetry is compatible with the structure on the manifold? In other words we will be looking for topological field theories in any spacetime dimension with gauge symmetries such that their classical action functional is gauge invariant provided that the target space is equipped with some structure that contains a Poisson or twisted Poisson 2-vector
We presented a large class of topological field theories with Wess-Zumino term, induced by a twisted R-Poisson structure and extensions thereof
Summary
We begin our analysis with the H-twisted Poisson Sigma Model (HPSM), which is a topological field theory with Wess-Zumino term in two dimensions and a target space equipped. With the above definition of the field strength F i of Xi, which may be written in a basis-independent way as F = dX + Π(·, A), it is obvious that SHPSM may be rewritten as This rewriting of the HPSM action in terms of the lowest degree field strength looks like a triviality, a similar, non-trivial rewriting will turn out to be instrumental in establishing target space covariance for topological field theories having an underlying Poisson structure in general dimensions. We note in advance that the simplicity of passing to the target space covariant form of the action in the present case is lost in the more general topological field theories to be discussed below and one should reside in the alternative expressions that correspond to (2.7) in those cases, as explained in detail in the ensuing.
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