Jacobi algebroids and Jacobi sigma models

TL;DR

The paper extends Poisson sigma-models to Jacobi structures by working with Jacobi bundles, i.e. line-bundle brackets, and emphasizes a canonical homogeneous ${\mathbb{R}^\times}$-action that leads to Jacobi algebroid morphisms as the natural solutions. It develops three equivalent formulations—constrained, homogeneous Poisson, and Jacobi algebroid approaches—and a reduced data model, all yielding identical solution spaces and revealing a covariant, geometric framework for Jacobi brackets on sections of line bundles. A key insight is the one-to-one correspondence between Jacobi brackets and homogeneous Poisson tensors on the corresponding ${\mathbb{R}^\times}$-bundle $L^{\boxtimes}$, which induces Jacobi algebroid structures on $T^*L^{\boxtimes}$ and a reduction to the first jet bundle ${\mathsf J}^1L$. The results unify several prior proposals, demonstrate the role of symplectic/contact groupoids in the solution space, and permit extensions to almost Poisson/almost Jacobi brackets, with potential applications to gauge symmetries and nonholonomic systems. Overall, the work provides a covariant, geometric realization of Jacobi structures in two-dimensional sigma-models and a robust framework for future dual and higher-structure generalizations.

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.

Theorems & Definitions (38)

• Definition 3.1
• Theorem 3.2
• Remark 3.3
• Definition 3.4
• Definition 4.1
• Remark 4.2
• Theorem 4.3
• Definition 4.4
• Proposition 4.5
• Example 4.6