scholarly journals A Darboux theorem for derived schemes with shifted symplectic structure

2018 ◽  
Vol 32 (2) ◽  
pp. 399-443 ◽  
Author(s):  
Christopher Brav ◽  
Vittoria Bussi ◽  
Dominic Joyce
Keyword(s):  
Symplectic Structure ◽  
Darboux Theorem

scholarly journals ELECTROMAGNETIC EXCITATIONS OF HALL SYSTEMS ON FOUR-DIMENSIONAL SPACE

2011 ◽  
Vol 08 (07) ◽  
pp. 1465-1486
Author(s):  
MOHAMMED DAOUD ◽  
AHMED JELLAL ◽  
ABDELLAH OUELD GUEJDI
Keyword(s):  
Symplectic Structure ◽  
Electromagnetic Interaction ◽  
Dimensional Space ◽  
Quantum Hall ◽  
Point Of View ◽  
Edge Field ◽  
Quantum Hall Systems ◽  
Dimensional Phase Space ◽  
Darboux Theorem ◽  
Refined Version

The noncommutativity of a four-dimensional phase space is introduced from a purely symplectic point of view. We show that there is always a coordinate map to locally eliminate the gauge fluctuations inducing the deformation of the symplectic structure. This uses the Moser's lemma; a refined version of the celebrated Darboux theorem. We discuss the relation between the coordinates change arising from Moser's lemma and the Seiberg–Witten map. As illustration, we consider the quantum Hall systems on CP2. We derive the action describing the electromagnetic interaction of Hall droplets. In particular, we show that the velocities of the edge field, along the droplet boundary, are noncommutativity parameters-dependents.

scholarly journals Quantization of emergent gravity

2015 ◽  
Vol 30 (04n05) ◽  
pp. 1550016 ◽  
Author(s):  
Hyun Seok Yang
Keyword(s):  
Gauge Theory ◽  
Quantum Gravity ◽  
Equivalence Principle ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Space Time ◽  
Quantization Scheme ◽  
Emergent Gravity ◽  
Darboux Theorem ◽  
Matter Fields

Emergent gravity is based on a novel form of the equivalence principle known as the Darboux theorem or the Moser lemma in symplectic geometry stating that the electromagnetic force can always be eliminated by a local coordinate transformation as far as space–time admits a symplectic structure, in other words, a microscopic space–time becomes noncommutative (NC). If gravity emerges from U(1) gauge theory on NC space–time, this picture of emergent gravity suggests a completely new quantization scheme where quantum gravity is defined by quantizing space–time itself, leading to a dynamical NC space–time. Therefore the quantization of emergent gravity is radically different from the conventional approach trying to quantize a phase space of metric fields. This approach for quantum gravity allows a background-independent formulation where space–time and matter fields are equally emergent from a universal vacuum of quantum gravity.

scholarly journals Boundary electromagnetic duality from homological edge modes

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Philippe Mathieu ◽  
Nicholas Teh
Keyword(s):  
Gauge Theory ◽  
Symplectic Structure ◽  
Central Charge ◽  
Wilson Line ◽  
Boundary Term ◽  
Global Symmetry ◽  
Homotopy Pullback ◽  
Line Singularities ◽  
Topological Boundary

Abstract Recent years have seen a renewed interest in using ‘edge modes’ to extend the pre-symplectic structure of gauge theory on manifolds with boundaries. Here we further the investigation undertaken in [1] by using the formalism of homotopy pullback and Deligne- Beilinson cohomology to describe an electromagnetic (EM) duality on the boundary of M = B3 × ℝ. Upon breaking a generalized global symmetry, the duality is implemented by a BF-like topological boundary term. We then introduce Wilson line singularities on ∂M and show that these induce the existence of dual edge modes, which we identify as connections over a (−1)-gerbe. We derive the pre-symplectic structure that yields the central charge in [1] and show that the central charge is related to a non-trivial class of the (−1)-gerbe.

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 Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

Darboux chart on projective limit of weak symplectic Banach manifold

2015 ◽  
Vol 12 (07) ◽  
pp. 1550072 ◽  
Author(s):  
Pradip Mishra
Keyword(s):  
Banach Space ◽  
Symplectic Structure ◽  
Reflexive Banach Space ◽  
Projective Limit ◽  
Fréchet Space ◽  
Frechet Space ◽  
Banach Manifold ◽  
Boundedness Condition ◽  
Projective System ◽  
Banach Manifolds

Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

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.

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