Jacobi algebroids and Jacobi sigma models

TL;DR

This paper generalizes Jacobi sigma models to Jacobi bundles, introducing a homogeneity structure and Jacobi algebroids, and explores their solutions, extending to almost Poisson and Jacobi brackets in a geometric, covariant framework.

Contribution

It introduces a unified approach to Jacobi sigma models on Jacobi bundles using homogeneity structures and Jacobi algebroids, extending to non-Jacobi brackets.

Findings

One-to-one correspondence between different models' solutions.

Extension to almost Poisson and Jacobi brackets.

Geometric and fully covariant sigma models.

Abstract

The definition of an action functional for the Jacobi sigma models, known for Jacobi brackets of functions, is generalized to \emph{Jacobi bundles}, i.e., Lie brackets on sections of (possibly nontrivial) line bundles, with the particular case of contact manifolds. Different approaches are proposed, but all of them share a common feature: the presence of a \emph{homogeneity structure} appearing as a principal action of the Lie group $\mathbb{R}^{\times}=\mathrm{GL}(1;\mathbb{R})$. Consequently, solutions of the equations of motions are morphisms of certain \emph{Jacobi algebroids}, i.e., principal $\mathbb{R}^{\times}$-bundles equipped additionally with a compatible Lie algebroid structure. Despite the different approaches we propose, there is a one-to-one correspondence between the space of solutions of the different models. The definition can be immediately extended to \emph{almost Poisson} and \emph{almost Jacobi brackets}, i.e., to brackets that do not satisfy the Jacobi identity. Our sigma models are geometric and fully covariant.