Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

Abstract

We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

Topological and Dynamical Aspects of Jacobi Sigma Models

“…This is achieved by the immersion i : M → M × R through the identification of M with M × {0}. It applies in particular to the Poisson sigma model and it is the approach followed in the first part of [1] (also see [13,14]). However, as we will see, it is not the choice we have made for the present paper.…”

“…In this section, we shall analyse the Jacobi sigma model, first introduced in [1] (also see [13]) as a generalisation of the Poisson sigma model. Although the defining action functional may be justified in terms of a Poissonization of the target Jacobi manifold and further reduction of the correspondent Poisson sigma model living on M × R, it has been shown in [1] that an independent formulation can be given.…”

“…The Poisson sigma model is described in terms of fields (X, η) which are formally associated with a bundle map from the tangent bundle of a source space Σ, a two-dimensional Symmetry 2021, 13, 1205 2 of 26 orientable manifold possibly with boundary, to the cotangent bundle of the target Poisson manifold M. In particular, X is the base map, describing the embedding of Σ into M, while η is the fibre map, an auxiliary field which is, in particular, a one-form on Σ with values in the pull-back of the cotangent bundle over M. In general, it is not possible to integrate out such an auxiliary field, unless the target space is a symplectic manifold. In this case, the Poisson bi-vector can be inverted and the equations of motion can be solved for η.…”

“…The model may be related to a Poisson sigma model with target space M × R within a "Poissonization" procedure. The latter approach has been pursued in [13] in relation with non-closed fluxes, and [14] with reference to gauge symmetry.…”

Jacobi sigma models

“…[ , ]S is the Schouten-Nijenhuis bracket 2. While being in the process of submitting the manuscript we have been aware of a new submission on the archives[18] where similar ideas are explored.…”