Quantization of Hitchin’s Integrable System and the Geometric Langlands Conjecture

EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION

Acta Mathematica Scientia ◽

10.1016/s0252-9602(18)30080-8 ◽

1993 ◽

Vol 13 (4) ◽

pp. 449-456

Author(s):

Zhongding Li

Keyword(s):

Integrable System ◽

Mathematical Physics ◽

Higher Order ◽

Representation Of Solutions ◽

Classical Integrable System ◽

Classical Integrable

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On the Heisenberg Supermagnet Model in (2+1)-Dimensions

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0397 ◽

2017 ◽

Vol 72 (4) ◽

pp. 331-337 ◽

Cited By ~ 7

Author(s):

Zhao-Wen Yan

Keyword(s):

Integrable System ◽

Integrable Systems ◽

Bäcklund Transformations ◽

Quadratic Constraints ◽

Backlund Transformations

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

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

Letters in Mathematical Physics ◽

10.1007/s11005-020-01343-4 ◽

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.

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Limit Cycle Bifurcations of a Planar Near-Integrable System with Two Small Parameters

Acta Mathematica Scientia ◽

10.1007/s10473-021-0402-z ◽

2021 ◽

Vol 41 (4) ◽

pp. 1034-1056

Author(s):

Feng Liang ◽

Maoan Han ◽

Chaoyuan Jiang

Keyword(s):

Limit Cycle ◽

Integrable System ◽

Small Parameters

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Solution of tetrahedron equation and cluster algebras

Journal of High Energy Physics ◽

10.1007/jhep05(2021)103 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

P. Gavrylenko ◽

M. Semenyakin ◽

Y. Zenkevich

Keyword(s):

Integrable System ◽

Cluster Algebra ◽

Newton Polygon ◽

Cluster Algebras ◽

Weyl Groups ◽

Newton Polygons ◽

Tetrahedron Equation ◽

Affine Weyl Groups ◽

New Formalism ◽

Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

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SCALING INVARIANCE IN A TIME-DEPENDENT ELLIPTICAL BILLIARD

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412502070 ◽

2012 ◽

Vol 22 (09) ◽

pp. 1250207 ◽

Cited By ~ 5

Author(s):

DIEGO F. M. OLIVEIRA ◽

MARKO ROBNIK

Keyword(s):

Integrable System ◽

Average Velocity ◽

Moving Boundary ◽

Time Dependent ◽

Scaling Invariance ◽

Energy Growth ◽

Dynamical Properties ◽

Scaling Arguments

We study some dynamical properties of a classical time-dependent elliptical billiard. We consider periodically moving boundary and collisions between the particle and the boundary are assumed to be elastic. Our results confirm that although the static elliptical billiard is an integrable system, after introducing time-dependent perturbation on the boundary the unlimited energy growth is observed. The behavior of the average velocity is described using scaling arguments.

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