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

TL;DR

This paper explores the conditions for gauging two-dimensional sigma models with nonclosed 3-forms, revealing structures like contact Courant algebroids and linking Jacobi structures to gauge symmetries, extending geometric frameworks in theoretical physics.

Contribution

It introduces the concept of gauged sigma models with nonclosed 3-forms, connecting them to contact Courant algebroids and Jacobi structures, and extends the geometric understanding of gauge symmetries.

Findings

Target space of gauged models forms contact Courant algebroids.

Gauge invariance constrains models to Dirac structures.

Constructs sigma models with Jacobi structure-controlled gauge symmetry.

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.