Symplectic leaves and the symplectic foliation

2021 ◽  
pp. 63-84
Keyword(s):  
Symplectic Leaves

scholarly journals (Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Marieke van Beest ◽  
Antoine Bourget ◽  
Julius Eckhard ◽  
Sakura Schäfer-Nameki
Keyword(s):  
Rain Forest ◽  
Tropical Rain Forest ◽  
Gauge Theories ◽  
Edge Coloring ◽  
Field Theories ◽  
Minkowski Sum ◽  
Coulomb Branch ◽  
Superconformal Field Theories ◽  
Minkowski Sums ◽  
Symplectic Leaves

Abstract We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

scholarly journals DONAGI–MARKMAN CUBIC FOR THE GENERALIZED HITCHIN SYSTEM

2014 ◽  
Vol 25 (02) ◽  
pp. 1450016 ◽  
Author(s):  
UGO BRUZZO ◽  
PETER DALAKOV
Keyword(s):  
Hamiltonian System ◽  
Cotangent Bundle ◽  
Integrable Hamiltonian System ◽  
Maximal Rank ◽  
Symmetric Power ◽  
Period Map ◽  
The Third ◽  
Hitchin System ◽  
Completely Integrable Hamiltonian System ◽  
Symplectic Leaves

Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi–Pantev formula holds on maximal rank symplectic leaves of the G-generalized Hitchin system.

scholarly journals TRI–HAMILTONIAN VECTOR FIELDS, SPECTRAL CURVES AND SEPARATION COORDINATES

2002 ◽  
Vol 14 (10) ◽  
pp. 1115-1163 ◽  
Author(s):  
L. DEGIOVANNI ◽  
G. MAGNANO
Keyword(s):  
Vector Fields ◽  
Geometric Approach ◽  
Casimir Function ◽  
Bivariate Polynomials ◽  
Spectral Curves ◽  
Hamiltonian Vector Fields ◽  
Lax Equations ◽  
Symplectic Leaves ◽  
The Common ◽  
Inverse Procedure

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate polynomials. These polynomials, which generalise those considered by E. Sklyanin in his algebro-geometric approach, are obtained from the knowledge of: (i) a common Casimir function for the two Poisson pencils (P1-λP0) and (P2-μP0); (ii) a suitable set of vector fields, preserving P0 but transversal to its symplectic leaves. The framework is applied to Lax equations with spectral parameter, for which not only it establishes a theoretical link between the separation techniques of Sklyanin and of Magri, but also provides a more efficient "inverse" procedure to obtain separation variables, not involving the extraction of roots.

scholarly journals Geometric Models for Lie–Hamilton Systems on ℝ2

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1053
Author(s):  
Julia Lange ◽  
Javier de Lucas
Keyword(s):  
Lie Algebras ◽  
Poisson Structure ◽  
Quotient Space ◽  
Geometric Description ◽  
Natural Manner ◽  
Geometric Models ◽  
Hamilton System ◽  
Higher Dimensional ◽  
Symplectic Leaves ◽  
Lie Systems

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

scholarly journals Stability of symplectic leaves

2010 ◽  
Vol 180 (3) ◽  
pp. 481-533 ◽  
Author(s):  
Marius Crainic ◽  
Rui Loja Fernandes
Keyword(s):  
Symplectic Leaves

scholarly journals PATH INTEGRAL QUANTIZATION OF THE SYMPLECTIC LEAVES OF THE SU(2)* POISSON–LIE GROUP

1999 ◽  
Vol 14 (06) ◽  
pp. 919-936 ◽  
Author(s):  
BOGDAN MORARIU
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Lie Group ◽  
Unitary Representations ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Path Integral Quantization ◽  
Symplectic Leaves

The Feynman path integral is used to quantize the symplectic leaves of the Poisson–Lie group SU(2)*. In this way we obtain the unitary representations of [Formula: see text]. This is achieved by finding explicit Darboux coordinates and then using a phase space path integral. I discuss the *-structure of SU(2)* and give a detailed description of its leaves using various parametrizations. I also compare the results with the path integral quantization of spin.

scholarly journals Global Geometry of Bayesian Statistics

Entropy ◽  
2020 ◽  
Vol 22 (2) ◽  
pp. 240
Author(s):  
Atsuhide Mori
Keyword(s):  
Bayesian Statistics ◽  
Cholesky Decomposition ◽  
Information Geometry ◽  
Covariance Matrices ◽  
Multivariate Normal ◽  
Hamiltonian Flow ◽  
Normal Distributions ◽  
Global Geometry ◽  
Hilbert Modular ◽  
Symplectic Leaves

In the previous work of the author, a non-trivial symmetry of the relative entropy in the information geometry of normal distributions was discovered. The same symmetry also appears in the symplectic/contact geometry of Hilbert modular cusps. Further, it was observed that a contact Hamiltonian flow presents a certain Bayesian inference on normal distributions. In this paper, we describe Bayesian statistics and the information geometry in the language of current geometry in order to spread our interest in statistics through general geometers and topologists. Then, we foliate the space of multivariate normal distributions by symplectic leaves to generalize the above result of the author. This foliation arises from the Cholesky decomposition of the covariance matrices.

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