Geometry of the group Diff(S1) of diffeomorphisms of the circle. Vector fields with divergence zero on the group of diffeomorphisms of the two dimensional torus

2009 ◽  
Vol 57 (5-7) ◽  
pp. 466-471
Author(s):  
H. Airault
Keyword(s):  
Vector Fields ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Group Diff ◽  
Group Of Diffeomorphisms

DARBOUXIAN INTEGRABILITY OF POLYNOMIAL VECTOR FIELDS WITH SPECIAL EMPHASIS ON THE TWO-DIMENSIONAL SURFACES

2002 ◽  
Vol 12 (12) ◽  
pp. 2821-2833 ◽  
Author(s):  
JAUME LLIBRE ◽  
GERARDO RODRÍGUEZ
Keyword(s):  
Vector Fields ◽  
Two Dimensional ◽  
Polynomial Vector ◽  
Dimensional Torus ◽  
Real Polynomial ◽  
Polynomial Vector Fields

First, we present a survey on the Darbouxian theory of integrability for real polynomial vector fields in the plane. Then we extend this theory to real polynomial vector fields on two-dimensional surfaces, more specifically on the quadratics and on the two-dimensional torus.

scholarly journals Circular quiver gauge theories, isomonodromic deformations and $$W_N$$ fermions on the torus

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Giulio Bonelli ◽  
Fabrizio Del Monte ◽  
Pavlo Gavrylenko ◽  
Alessandro Tanzini
Keyword(s):  
Integrable System ◽  
Gauge Theories ◽  
The Self ◽  
Free Fermion ◽  
Two Dimensional ◽  
Isomonodromic Deformation ◽  
Specific Time ◽  
Dimensional Torus ◽  
Isomonodromic Deformations ◽  
Deformation Problem

AbstractWe study the relation between class $$\mathcal {S}$$ S theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual $$\Omega $$ Ω -background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding $$\tau $$ τ -function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to $$W_N$$ W N free fermion correlators on the torus.

scholarly journals Stable Arcs Connecting Polar Cascades on a Torus

2021 ◽  
Vol 17 (1) ◽  
pp. 23-37
Author(s):  
O. V. Pochinka ◽  
◽  
E. V. Nozdrinova ◽  
Keyword(s):  
Two Dimensional ◽  
Dimensional Torus ◽  
Positive Orientation ◽  
Stable Isotopic ◽  
Orientation Type

In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

A Characterization of Harmonic $$L^r$$-Vector Fields in Two-Dimensional Exterior Domains

2019 ◽  
Vol 30 (4) ◽  
pp. 3742-3759 ◽  
Author(s):  
Matthias Hieber ◽  
Hideo Kozono ◽  
Anton Seyfert ◽  
Senjo Shimizu ◽  
Taku Yanagisawa
Keyword(s):  
Vector Fields ◽  
Two Dimensional ◽  
Exterior Domains
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