Title: Jacobi algebroids and Jacobi sigma models Show Chinese title

Title: 雅可比代数胚和雅可比σ模型

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. Show Chinese abstract

Abstract: 对于已知的Jacobi括号函数的Jacobi sigma模型的动作泛函的定义，被推广到\emph{雅可比丛}，即（可能非平凡的）线丛截面的李括号，特别是接触流形的情况。提出了不同的方法，但它们都具有一个共同特征：存在一个\emph{同质结构}，作为李群$\mathbb{R}^{\times}=\mathrm{GL}(1;\mathbb{R})$的主作用。因此，运动方程的解是某些\emph{雅可比代数胚}的同态，即额外配备有兼容李代数结构的主$\mathbb{R}^{\times}$-丛。尽管我们提出了不同的方法，不同模型的解空间之间存在一一对应关系。该定义可以立即扩展到\emph{几乎泊松}和\emph{几乎雅可比括号}，即不满足雅可比恒等式的括号。我们的sigma模型是几何的且完全协变的。