scholarly journals Hitchin and Calabi–Yau Integrable Systems via Variations of Hodge Structures

2020 ◽  
Vol 71 (4) ◽  
pp. 1345-1375
Author(s):  
Florian Beck
Keyword(s):  
Integrable System ◽  
Lie Group ◽  
Integrable Systems ◽  
Hodge Structures ◽  
Langlands Duality ◽  
Hitchin Systems

Abstract Since its discovery by Hitchin in 1987, G-Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G-Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.

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

2017 ◽  
Vol 72 (4) ◽  
pp. 331-337 ◽  
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.

scholarly journals Folding of Hitchin Systems and Crepant Resolutions

2021 ◽  
Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Trace Back ◽  
Dynkin Diagrams ◽  
Or Groups ◽  
Hitchin Systems ◽  
Singular Fibers ◽  
Crepant Resolutions ◽  
Intermediate Jacobian

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.

scholarly journals A new hierarchy of integrable system of1+2dimensions: from Newton's law to generalized Hamiltonian system. Part II

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
Xuncheng Huang ◽  
Guizhang Tu
Keyword(s):  
Integrable System ◽  
Integrable Systems ◽  
Hamiltonian Equation ◽  
Hamiltonian Form ◽  
Fundamental Interest ◽  
Newton's Law ◽  
Newton’S Law ◽  
System Part ◽  
Generalized Hamiltonian System ◽  
Physical Laws

The Hamiltonian equation provides us an alternate description of the basic physical laws of motion, which is used to be described by Newton's law. The research on Hamiltonian integrable systems is one of the most important topics in the theory of solitons. This article proposes a new hierarchy of integrable systems of1+2dimensions with its Hamiltonian form by following the residue approach of Fokas and Tu. The new hierarchy of integrable system is of fundamental interest in studying the Hamiltonian systems.

scholarly journals Geometric Langlands duality and the equations of Nahm and Bogomolny

2010 ◽  
Vol 140 (4) ◽  
pp. 857-895 ◽  
Author(s):  
Edward Witten
Keyword(s):  
Gauge Theory ◽  
Lie Group ◽  
Moduli Space ◽  
Dual Group ◽  
Langlands Duality ◽  
Geometric Langlands

Geometric Langlands duality relates a representation of a simple Lie group Gv to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of Gv makes an unexpected appearance. This can be explained using gauge theory, as this paper will show, with the help of the equations of Nahm and Bogomolny.

scholarly journals Quantum footprints of Liouville integrable systems

2020 ◽  
pp. 2060014
Author(s):  
San Vũ Ngọc
Keyword(s):  
Integrable System ◽  
Integrable Systems ◽  
Degree Of Freedom ◽  
Quantum Integrable System ◽  
Geometric Objects ◽  
Liouville Integrable

We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labelings of asymptotic lattices.

scholarly journals LEVEL CORRELATIONS IN INTEGRABLE SYSTEMS

2002 ◽  
Vol 16 (30) ◽  
pp. 4649-4654 ◽  
Author(s):  
R. A. SEROTA ◽  
J. M. A. S. P. WICKRAMASINGHE
Keyword(s):  
Correlation Function ◽  
Integrable System ◽  
Excellent Agreement ◽  
Integrable Systems ◽  
Semiclassical Theory ◽  
Simple Analytical Expression ◽  
Analytical Expression ◽  
Global Rigidity ◽  
Number Conservation ◽  
Level Number

We derive a simple analytical expression for the level correlation function of an integrable system. It accounts for both the lack of correlations at smaller energy scales and for global rigidity (level number conservation) at larger scales. We apply our results to a rectangle with incommensurate sides and show that they are in excellent agreement with the limiting cases established in the semiclassical theory of level rigidity.

scholarly journals On collective complete integrability according to the method of Thimm

1983 ◽  
Vol 3 (2) ◽  
pp. 219-230 ◽  
Author(s):  
Victor Guillemin ◽  
Shlomo Sternberg
Keyword(s):  
Integrable System ◽  
Symplectic Manifold ◽  
Lie Group ◽  
Geodesic Flow ◽  
Necessary Conditions ◽  
Complete Integrability ◽  
Moment Map ◽  
The Real ◽  
Completely Integrable System ◽  
The Moment

AbstractLet G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

scholarly journals Langlands duality for Hitchin systems

2012 ◽  
Vol 189 (3) ◽  
pp. 653-735 ◽  
Author(s):  
R. Donagi ◽  
T. Pantev
Keyword(s):  
Langlands Duality ◽  
Hitchin Systems

Localization and N=2 supersymmetric field theory

2019 ◽  
pp. 501-540
Author(s):  
Vasily Pestun
Keyword(s):  
Field Theory ◽  
Integrable System ◽  
Gauge Theories ◽  
Supersymmetric Gauge Theories ◽  
Kahler Geometry ◽  
Hitchin Systems ◽  
Supersymmetric Gauge

The lectures give an introduction to supersymmetric gauge theories from a mathematical perspective. Basic notions about Kähler and special Kähler geometry, and electric–magnetic duality are introduced. Supersymmetry and N = 1 and N = 2 supersymmetric gauge theories are defined and described in detail. The last section deals with the Seiberg–Witten integrable system and Hitchin systems

Introduction

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Lie Group ◽  
Hodge Structure ◽  
Algebraic Groups ◽  
Complex Multiplication ◽  
Symmetry Groups ◽  
Hodge Structures ◽  
Fundamental Symmetry ◽  
Variation Of Hodge Structure ◽  
Geometric Origin

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of level one and those constructed from this case. In this book, Mumford-Tate groups M will be reductive algebraic groups over ℚ such that the derived or adjoint subgroup of the associated real Lie group M ℝ contains a compact maximal torus. In order to keep the statements of the results as simple as possible, the book emphasizes the case when M ℝ itself is semi-simple. The discussion covers period domains and Mumford-Tate domains, the Mumford-Tate group of a variation of Hodge structure, Hodge representations and Hodge domains, Hodge structures with complex multiplication, arithmetic aspects of Mumford-Tate domains, classification of Mumford-Tate subdomains, and arithmetic of period maps of geometric origin.

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