Topological Field Theories induced by twisted R-Poisson structure in any dimension

We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions $\ge 2$ 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 $T^{\ast}[p]T^{\ast}[1]M$ is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.