scholarly journals Contact Isotropic Realisations of Jacobi Manifolds via Spencer Operators

2017 ◽  
Author(s):  
Maria Amelia Salazar ◽  
◽  
Daniele Sepe ◽  
Keyword(s):  
Jacobi Manifolds

scholarly journals Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Athanasios Chatzistavrakidis ◽  
Grgur Šimunić
Keyword(s):  
Sigma Models ◽  
Sigma Model ◽  
Target Space ◽  
Two Dimensional ◽  
Courant Algebroids ◽  
Dirac Structures ◽  
Courant Algebroid ◽  
Abelian Case ◽  
Jacobi Manifolds ◽  
Jacobi Structure

Abstract We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

scholarly journals Holomorphic Jacobi manifolds and holomorphic contact groupoids

2019 ◽  
Vol 294 (3-4) ◽  
pp. 1181-1225 ◽  
Author(s):  
Luca Vitagliano ◽  
Aïssa Wade
Keyword(s):  
Jacobi Manifolds ◽  
Holomorphic Contact

scholarly journals On the geometric quantization of Jacobi manifolds

1997 ◽  
Vol 38 (12) ◽  
pp. 6185-6213 ◽  
Author(s):  
Manuel de León ◽  
Juan C. Marrero ◽  
Edith Padrón
Keyword(s):  
Geometric Quantization ◽  
Jacobi Manifolds

scholarly journals Jacobi geometry and Hamiltonian mechanics: The unit-free approach

2020 ◽  
Vol 17 (12) ◽  
pp. 2030005
Author(s):  
Carlos Zapata-Carratalá
Keyword(s):  
Vector Bundles ◽  
Mathematical Framework ◽  
Poisson Geometry ◽  
Hamiltonian Mechanics ◽  
Systematic Treatment ◽  
Straightforward Manner ◽  
Jacobi Manifolds ◽  
Categorical Structure ◽  
Physical Quantities

We present a systematic treatment of line bundle geometry and Jacobi manifolds with an application to geometric mechanics that has not been noted in the literature. We precisely identify categories that generalize the ordinary categories of smooth manifolds and vector bundles to account for a lack of choice of a preferred unit, which in standard differential geometry is always given by the global constant function [Formula: see text]. This is what we call the “unit-free” approach. After giving a characterization of local Lie brackets via their symbol maps, we apply our novel categorical language to review Jacobi manifolds and related notions such as Lichnerowicz brackets and Jacobi algebroids. The main advantage of our approach is that Jacobi geometry is recovered as the direct unit-free generalization of Poisson geometry, with all the familiar notions translating in a straightforward manner. We then apply this formalism to the question of whether there is a unit-free generalization of Hamiltonian mechanics. We identify the basic categorical structure of ordinary Hamiltonian mechanics to argue that it is indeed possible to find a unit-free analogue. This paper serves as a prelude to the investigation of dimensioned structures, an attempt at a general mathematical framework for the formal treatment of physical quantities and dimensional analysis.

Jacobi structures with background

2020 ◽  
Vol 17 (04) ◽  
pp. 2050063
Author(s):  
Eugen-Mihaita Cioroianu ◽  
Cornelia Vizman
Keyword(s):  
Poisson Structure ◽  
Poisson Geometry ◽  
Dual Pairs ◽  
Theor Phys ◽  
Jacobi Manifolds ◽  
Jacobi Structure

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen. 39(33) (2006) 10449–10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145–154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a [Formula: see text]-form and a [Formula: see text]-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background.

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