scholarly journals Poly-symplectic geometry and the AKSZ formalism

2021 ◽  
pp. 2150030
Author(s):  
Ivan Contreras ◽  
Nicolás Martínez Alba
Keyword(s):  
Phase Space ◽  
General Theory ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Sigma Model ◽  
Type Theorem ◽  
Poisson Structures ◽  
Action Functional ◽  
Symplectic Structures ◽  
Poisson Sigma Model

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

scholarly journals Poisson Sigma Model and Semiclassical Quantization of Integrable Systems

2018 ◽  
Vol 30 (06) ◽  
pp. 1840004 ◽  
Author(s):  
Alberto S. Cattaneo ◽  
Pavel Mnev ◽  
Nicolai Reshetikhin
Keyword(s):  
Phase Space ◽  
Power Series ◽  
Path Integral ◽  
Integrable Systems ◽  
Sigma Model ◽  
Star Product ◽  
Feynman Diagrams ◽  
Target Space ◽  
Poisson Sigma Model ◽  
Formal Power

In this paper, we outline the construction of semiclassical eigenfunctions of integrable models in terms of the semiclassical path integral for the Poisson sigma model with the target space being the phase space of the integrable system. The semiclassical path integral is defined as a formal power series with coefficients being Feynman diagrams. We also argue that in a similar way one can obtain irreducible semiclassical representations of Kontsevich’s star product. Dedicated to the memory of L. D. Faddeev

scholarly journals Canonical multi-symplectic structure on the total exterior algebra bundle

2006 ◽  
Vol 462 (2069) ◽  
pp. 1531-1551 ◽  
Author(s):  
Thomas J Bridges
Keyword(s):  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Hamiltonian Dynamics ◽  
Partial Differential Operator ◽  
Exterior Algebra ◽  
Legendre Transform ◽  
Natural Setting ◽  
Elliptic Pde ◽  
Symplectic Structures ◽  
Partial Differential

The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n -dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M . On the fibre, which has dimension 2 n , the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n -symplectic structures define a partial differential operator, J ∂ , which turns out to be a Dirac operator with multi-symplectic structure. The operator J ∂ generalizes the product operator J (d/d t ) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J ∂ is elliptic, and the generalization of Hamiltonian systems, J ∂ Z =∇ S ( Z ), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S ( Z ) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k -forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.

Covariant statistical mechanics of non-Hamiltonian systems

2018 ◽  
Vol 15 (02) ◽  
pp. 1850017
Author(s):  
Vahid Hosseinzadeh ◽  
Kourosh Nozari
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Hamiltonian Systems ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Differential Forms ◽  
Volume Element ◽  
Covariant Formulation ◽  
Locally Conformal

In this paper, using the elegant language of differential forms, we provide a covariant formulation of the equilibrium statistical mechanics of non-Hamiltonian systems. The key idea of the paper is to focus on the structure of phase space and its kinematical and dynamical roles. While in the case of Hamiltonian systems, the structure of the phase space is a symplectic structure (a nondegenerate closed two-form), we consider an almost symplectic structure for the more general case of non-Hamiltonian systems. An almost symplectic structure is a nondegenerate but not necessarily closed two-form structure. Consequently, the dynamics becomes non-Hamiltonian and based on the fact that the structure is nondegenerate, we can also define a volume element. With a well-defined volume in hand, we derive the Liouville equation and find an invariant statistical state. Recasting non-Hamiltonian systems in terms of the almost symplectic geometry has at least two advantages: the formalism is covariant and therefore does not depend on coordinates and there is no confusion in the determination of the natural volume element of the system. For clarification, we investigate the application of the formalism in two examples in which the underlying geometry of the phase space is locally conformal symplectic geometry.

scholarly journals GENERALIZED COMPLEX GEOMETRY AND THE POISSON SIGMA MODEL

2005 ◽  
Vol 20 (13) ◽  
pp. 985-995 ◽  
Author(s):  
L. BERGAMIN
Keyword(s):  
Sigma Model ◽  
Complex Geometry ◽  
Complex Structure ◽  
Complex Structures ◽  
Almost Complex Structure ◽  
Poisson Structures ◽  
Generalized Complex Structures ◽  
Almost Complex ◽  
Generalized Complex ◽  
Poisson Sigma Model

The supersymmetric Poisson Sigma model is studied as a possible worldsheet realization of generalized complex geometry. Generalized complex structures alone do not guarantee non-manifest N = (2, 1) or N = (2, 2) supersymmetry, but a certain relation among the different Poisson structures is needed. Moreover, important relations of an additional almost complex structure are found, which have no immediate interpretation in terms of generalized complex structures.

scholarly journals An invitation to singular symplectic geometry

2019 ◽  
Vol 16 (supp01) ◽  
pp. 1940008 ◽  
Author(s):  
Roisin Braddell ◽  
Amadeu Delshams ◽  
Eva Miranda ◽  
Cédric Oms ◽  
Arnau Planas
Keyword(s):  
Celestial Mechanics ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Symplectic Manifolds ◽  
Symplectic Structures ◽  
Contact Manifolds ◽  
Symplectic Forms

In this paper, we analyze in detail a collection of motivating examples to consider [Formula: see text]-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every [Formula: see text]-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to [Formula: see text]-symplectic manifolds: [Formula: see text]-contact manifolds.

scholarly journals BASIC SYMPLECTIC GEOMETRY FOR p-BRANES WITH THICKNESS IN A CURVED BACKGROUND

2004 ◽  
Vol 19 (27) ◽  
pp. 2069-2081 ◽  
Author(s):  
ALBERTO ESCALANTE
Keyword(s):  
Phase Space ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Covariant Description ◽  
Relevant Role

We show that the Witten covariant phase space for p-branes with thickness in an arbitrary background is endowed of a symplectic potential, which although is not important to the dynamics of the system, plays a relevant role on the phase space, allowing us to generate a symplectic structure for the theory and therefore give a covariant description of canonical formalism for quantization.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

scholarly journals COVARIANT DESCRIPTION OF PARAMETRIZED NONRELATIVISTIC HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (15) ◽  
pp. 2473-2493 ◽  
Author(s):  
MAURICIO MONDRAGÓN ◽  
MERCED MONTESINOS
Keyword(s):  
Phase Space ◽  
Gauge Transformation ◽  
Hamiltonian Systems ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Algebra Of Observables ◽  
Covariant Description ◽  
Constraint Surface

The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed.

scholarly journals Gibbs' States for Moser-Calogero Potentials

1997 ◽  
Vol 11 (01n02) ◽  
pp. 203-211 ◽  
Author(s):  
K. L. Vaninsky
Keyword(s):  
Phase Space ◽  
Equation Of State ◽  
Gibbs States ◽  
Symplectic Structures ◽  
Classical Particles

We present two independent approaches for computing the thermodynamics for classical particles interacting via the Moser-Calogero potential Combining the results we conjecture the form of equation of state or, what is equivalent, the asymptotics of the Jacobian between volume elements corresponding to two symplectic structures on the phase space.

scholarly journals Quantum gravity, dynamical phase-space and string theory

2014 ◽  
Vol 23 (12) ◽  
pp. 1442006 ◽  
Author(s):  
Laurent Freidel ◽  
Robert G. Leigh ◽  
Djordje Minic
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
String Theory ◽  
Quantum Gravity ◽  
Momentum Space ◽  
Symplectic Geometry ◽  
Complex Geometry ◽  
Hamiltonian Dynamics ◽  
Natural Extension ◽  
Dynamical Phase

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

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