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arXiv · 2409.14568

Jacobi algebroids and Jacobi sigma models

Fabio Di Cosmo·Katarzyna Grabowska·Janusz Grabowski
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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.

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Fabio Di Cosmo, Katarzyna Grabowska, Janusz Grabowski. 2024-09-22. Jacobi algebroids and Jacobi sigma models. https://doi.org/10.1142/s0129055x25500047

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