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

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 Qmanifolds 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.

Homotopy momentum sections on multisymplectic manifolds

“…Refer to [28] for a generalization of the twisted R-Poisson structure to a general Lie algebroid and a topological sigma model. ¶ An n-multivector field is denoted by J in this paper though it is denoted by R in [15].…”

Topological Dirac Sigma Models and the Classical Master Equation

“…It is worth mentioning that this is certainly not an isolated example. A semi-infinite class of further examples are topological field theories with twisted R-Poisson structure in any dimension ≥ 2 [24], whose BV action is also possible to identify at least in three dimensions [31], although in a technically demanding way. This includes for example 4-form-twisted (pre-)Courant sigma models.…”

“…Such a situation typically arises in presence of Wess-Zumino terms, which present obstructions to QP-ness, as e.g. in the H-twisted Poisson sigma model [13] and higher dimensional generalizations thereof [23][24][25]. The second and more radical reason is that the Q manifold at hand might not even admit a natural symplectic structure, for example when it is not a (graded) cotangent bundle.…”

Higher dimensional Lie algebroid sigma model with WZ term

“…Recently, Chatzistavrakidis has proposed a higher generalization of the twisted Poisson structure and the twisted Poisson sigma model by considering a higher dimensional topological sigma model. [33] It is a topological sigma model with WZ term on a (n + 1)-dimensional worldvolume. The twisted R-Poisson structure is defined by the following condition, [π, π] S = 0,…”

“…where E d is the Lie algebroid differential, J is an E-(n + 1)-form, ρ is the so called anchor map of a Lie algebroid and H is a closed (n + 2)-form. We analyze mathematical structures † In this paper, we denote a multivector field by J though it is denoted by R in the paper [33]. R is used for the curvature.…”