Jacobi algebroids and Jacobi 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.

35 pages

References in corpus (23)

• A path integral approach to the Kontsevich quantization formula
• Relative formality theorem and quantisation of coisotropic submanifolds
• Higher vector bundles and multi-graded symplectic manifolds
• Lie algebroids and Poisson-Nijenhuis structures
• Tangent Lifts of Poisson and Related Structures
• Algebroids - general differential calculi on vector bundles
• Poisson sigma models and symplectic groupoids
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Jacobi structures revisited
• Graded contact manifolds and contact Courant algebroids
• Variational calculus with constraints on general algebroids
• Nonholonomic Constraints: a New Viewpoint
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Quasi-derivations and QD-algebroids
• Brackets
• Reductions: precontact versus presymplectic
• Jacobi sigma models
• On the relation of Lie algebroids to constrained systems and their BV/BFV formulation
• VB-structures and generalizations
• Topological Dirac Sigma Models and the Classical Master Equation
• Contactifications: a Lagrangian description of compact Hamiltonian systems
• Star products and branes in Poisson-Sigma models
• Contact geometric mechanics: the Tulczyjew triples

Cited by in corpus (1)

• A so(2,2) extension of JT gravity via the Virasoro-Kac-Moody semidirect product