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.

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 Poisson Principal Bundles

2021 ◽  
Author(s):  
Shahn Majid ◽  
◽  
Liam Williams ◽  
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Poisson Structure ◽  
Differential Calculus ◽  
Principal Bundle ◽  
Poisson Manifold ◽  
Spin Connection ◽  
Poisson Geometry ◽  
Hopf Fibration ◽  
Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

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.

scholarly journals On the Standard Poisson Structure and a Frobenius Splitting of the Basic Affine Space

2019 ◽  
Author(s):  
Jun Peng ◽  
Shizhuo Yu
Keyword(s):  
General Theory ◽  
Poisson Structure ◽  
Borel Subgroup ◽  
Closed Field ◽  
Poisson Geometry ◽  
Characteristic P ◽  
Frobenius Splitting ◽  
Algebraic Torus ◽  
Poisson Varieties ◽  
Simply Connected

Abstract The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

scholarly journals Odd Jacobi Manifolds and Loday-Poisson Brackets

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Andrew James Bruce
Keyword(s):  
Vector Field ◽  
Poisson Bracket ◽  
Vector Fields ◽  
Leibniz Rule ◽  
Poisson Brackets ◽  
Smooth Functions ◽  
Hamiltonian Vector Fields ◽  
Jacobi Manifolds ◽  
Symmetric Version ◽  
Jacobi Structure

We construct a nonskew symmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.

scholarly journals POISSON GEOMETRY OF PARABOLIC BUNDLES ON ELLIPTIC CURVES

2008 ◽  
Vol 19 (03) ◽  
pp. 339-367
Author(s):  
DAVID BALDUZZI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Poisson Structure ◽  
Moduli Space ◽  
Loop Group ◽  
Poisson Geometry ◽  
Double Loop ◽  
Sheaf Cohomology ◽  
Parabolic Bundles

The moduli space of G-bundles on an elliptic curve with additional flag structure admits a Poisson structure. The bivector can be defined using double loop group, loop group and sheaf cohomology constructions. We investigate the links between these methods and for the case SL2 perform explicit computations, describing the bracket and its leaves in detail.

scholarly journals Morita Equivalence of Formal Poisson Structures

2021 ◽  
Author(s):  
Henrique Bursztyn ◽  
Inocencio Ortiz ◽  
Stefan Waldmann
Keyword(s):  
Integrability Condition ◽  
Morita Equivalence ◽  
Poisson Manifolds ◽  
Poisson Geometry ◽  
Poisson Structures ◽  
General Study ◽  
Dual Pairs ◽  
Noncommutative Algebra ◽  
Morita Equivalent ◽  
Formal Power

Abstract We extend the notion of Morita equivalence of Poisson manifolds to the setting of formal Poisson structures, that is, formal power series of bivector fields $\pi =\pi _0 + \lambda \pi _1 +\cdots $ satisfying the Poisson integrability condition $[\pi ,\pi ]=0$. Our main result gives a complete description of Morita equivalent formal Poisson structures deforming the zero structure ($\pi _0=0$) in terms of $B$-field transformations, relying on a general study of formal deformations of Poisson morphisms and dual pairs. Combined with previous work on Morita equivalence of star products [ 5], our results link the notions of Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

Geometry and Physics: Volume II

2018 ◽  
Keyword(s):  
Moduli Spaces ◽  
Mirror Symmetry ◽  
Geometric Analysis ◽  
Mathematical Physics ◽  
Poisson Geometry ◽  
Special Holonomy ◽  
Low Dimensional Topology ◽  
Generalized Complex Structures ◽  
Wide Range ◽  
Low Dimensional

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and Savilian Professor of Geometry at Oxford. The proceedings contain twenty-nine articles, including three by Fields medallists (Donaldson, Mori and Yau). The articles cover a wide range of topics in geometry and mathematical physics, including the following: Riemannian geometry, geometric analysis, special holonomy, integrable systems, dynamical systems, generalized complex structures, symplectic and Poisson geometry, low-dimensional topology, algebraic geometry, moduli spaces, Higgs bundles, geometric Langlands programme, mirror symmetry and string theory. These volumes will be of interest to researchers and graduate students both in geometry and mathematical physics.

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