Showing papers on "Integrable system published in 2004"

Classical/quantum integrability in AdS/CFT

Abstract: We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in = 4 super Yang-Mills and the energy of their dual semiclassical string states in AdS5 × S5. The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

The Dilatation Operator of N=4 Super Yang-Mills Theory and Integrability

Abstract: In this work we review recent progress in four-dimensional conformal quantum field theories and scaling dimensions of local operators. Here we consider the example of maximally supersymmetric gauge theory and present techniques to derive, investigate and apply the dilatation operator which measures the scaling dimensions. We construct the dilatation operator by purely algebraic means: Relying on the symmetry algebra and structural properties of Feynman diagrams we are able to bypass involved, higher-loop field theory computations. In this way we obtain the complete one-loop dilatation operator and the planar, three-loop deformation in an interesting subsector. These results allow us to address the issue of integrability within a planar four-dimensional gauge theory: We prove that the complete dilatation generator is integrable at one loop and present the corresponding Bethe ansatz. We furthermore argue that integrability extends to three loops and beyond. Assuming that it holds indeed, we finally construct a novel spin chain model at five loops and propose a Bethe ansatz which might be valid at arbitrary loop order!

Spinning strings in AdS5 × S5 and integrable systems

Abstract: We show that solitonic solutions of the classical string action on the AdS5 × S 5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system We derive equations which determine the energy of these solitons as a function of spins In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N = 4 super YM theory We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J1, J2

Integrable Hamiltonian Systems: Geometry, Topology, Classification

Abstract: BASIC NOTIONS Linear Symplectic Geometry Symplectic and Poisson Manifolds The Darboux Theorem Liouville Integrable Hamiltonian Systems. The Liouville Theorem Non-Resonant and Resonant Systems Rotation Number The Momentum Mapping of an Integrable System and Its Bifurcation Diagram Non-Degenerate Critical Points of the Momentum Mapping Main Types of Equivalence of Dynamical Systems THE TOPOLOGY OF FOLIATIONS ON TWO-DIMENSIONAL SURFACES Generated by Morse Functions Simple Morse Functions Reeb Graph of a Morse Function Notion of an Atom Simple Atoms Simple Molecules Complicated Atoms Classification of Atoms Symmetry Groups of Oriented Atoms and the Universal Covering Tree Notion of a Molecule Approximation of Complicated Molecules by Simple Ones Classification of Morse-Smale Flows on Two-Dimensional Surfaces by Means of Atoms and Molecules ROUGH LIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM Classification of Non-degenerate Critical Submanifolds on Isoenergy 3-Surfaces The Topological Structure of a Neighborhood of a Singular Leaf Topologically Stable Hamiltonian Systems Example of a Topologically Unstable Integrable System 2-Atoms and 3-Atoms Classification of 3-Atoms 3-Atoms as Bifurcations of Liouville Tori The Molecule of an Integrable System Complexity of Integrable Systems LIOUVILLE EQUIVALENCE OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM Admissible Coordinate Systems on the Boundary of a 3-Atom Gluing Matrices and Superfluous Frames Invariants (Numerical Marks) r, e, and n The Marked Molecule is a Complete Invariant of Liouville Equivalence The Influence of the Orientation Realization Theorem Simple Examples of Molecules Hamiltonian Systems with Critical Klein Bottles Topological Obstructions to Integrability of Hamiltonian Systems with Two Degrees of Freedom ORBITAL CLASSIFICATION OF INTEGRABLE SYSTEMS WITH TWO DEGREES OF FREEDOM Rotation Function and Rotation Vector Reduction of the Three-Dimensional Orbital Classification to the Two-Dimensional Classification up to Conjugacy General Concept of Constructing Orbital Invariants of Integrable Hamiltonian Systems CLASSIFICATION OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACES UP TO TOPOLOGICAL CONJUGACY Invariants of a Hamiltonian System on a 2-Atom Classification of Hamiltonian Flows with One Degree of Freedom up to Topological Conjugacy Classification of Hamiltonian Flows on 2-Atoms with Involution up to Topological Conjugacy The Pasting-Cutting Operation Description of the Sets of Admissible delta-Invariants and Z-Invariants SMOOTH CONJUGACY OF HAMILTONIAN FLOWS ON TWO-DIMENSIONAL SURFACES Constructing Smooth Invariants on 2-Atoms Theorem of Classification of Hamiltonian Flows on Atoms up to Smooth Conjugacy ORBITAL CLASSIFICATION OF INTEGRABLE HAMILTONIAN SYSTEMS WITH TWO DEGREES OF FREEDOM. THE SECOND STEP Superfluous t-Frame of a Molecule (Topological Case). The Main Lemma on t-Frames The Group of Transformations of Transversal Sections. Pasting-Cutting Operation The Action of GP on the Set of Superfluous t-Frames Three General Principles for Constructing Invariants Admissible Superfluous t-Frames and a Realization Theorem Construction of Orbital Invariants in the Topological Case. A t-Molecule Theorem on the Topological Orbital Classification of Integrable Systems with Two Degrees of Freedom A Particular Case: Simple Integrable Systems Smooth Orbital Classification LIOUVILLE CLASSIFICATION OF INTEGRABLE SYSTEMS WITH NEIGHBORHOODS OF SINGULAR POINTS l-Type of a Four-Dimensional Singularity The Loop Molecule of a Four-Dimensional Singularity Center-Center Case Center-Saddle Case Saddle-Saddle Case Almost Direct Product Representation of a Four-Dimensional Singularity Proof of the Classification Theorems Focus-Focus Case Almost Direct Product Representation for Multidimensional Non-degenerate Singularities of Liouville Foliations METHODS OF CALCULATION OF TOPOLOGICAL INVARIANTS OF INTEGRABLE HAMILTONIAN SYSTEMS General Scheme for Topological Analysis of the Liouville Foliation Methods for Computing Marks The Loop Molecule Method List of Typical Loop Molecules The Structure of the Liouville Foliation for Typical Degenerate Singularities Typical Loop Molecules Corresponding to Degenerate One-Dimensional Orbits Computation of r- and e-Marks by Means of Rotation Functions Computation of the n-Mark by Means of Rotation Functions Relationship Between the Marks of the Molecule and the Topology of Q3 INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES 409 Statement of the Problem Topological Obstructions to Integrability of Geodesic Flows on Two-Dimensional Surfaces Two Examples of Integrable Geodesic Flows Riemannian Metrics Whose Geodesic Flows are Integrable by Means of Linear or Quadratic Integrals. Local Theory Linearly and Quadratically Integrable Geodesic Flows on Closed Surfaces LIOUVILLE CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES The Torus The Klein Bottle The Sphere The Projective Plane ORBITAL CLASSIFICATION OF INTEGRABLE GEODESIC FLOWS ON TWO-DIMENSIONAL SURFACES Case of the Torus Case of the Sphere Examples of Integrable Geodesic Flows on the Sphere Non-triviality of Orbital Equivalence Classes and Metrics with Closed Geodesics THE TOPOLOGY OF LIOUVILLE FOLIATIONS IN CLASSICAL INTEGRABLE CASES IN RIGID BODY DYNAMICS Integrable Cases in Rigid Body Dynamics Topological Type of Isoenergy 3-Surfaces Liouville Classification of Systems in the Euler Case Liouville Classification of Systems in the Lagrange Case Liouville Classification of Systems in the Kovalevskaya Case Liouville Classification of Systems in the Goryachev-Chaplygin-Sretenskii Case Liouville Classification of Systems in the Zhukovskii Case Rough Liouville Classification of Systems in the Clebsch Case Rough Liouville Classification of Systems in the Steklov Case Rough Liouville Classification of Integrable Four-Dimensional Rigid Body Systems The Complete List of Molecules Appearing in Integrable Cases of Rigid Body Dynamics MAUPERTUIS PRINCIPLE AND GEODESIC EQUIVALENCE General Maupertuis Principle Maupertuis Principle in Rigid Body Dynamics Classical Cases of Integrability in Rigid Body Dynamics and Related Integrable Geodesic Flows on the Sphere Conjecture on Geodesic Flows with Integrals of High Degree Dini Theorem and the Geodesic Equivalence of Riemannian Metrics Generalized Dini-Maupertuis Principle Orbital Equivalence of the Neumann Problem and the Jacobi Problem Explicit Forms of Some Remarkable Hamiltonians and Their Integrals in Separating Variables EULER CASE IN RIGID BODY DYNAMICS AND JACOBI PROBLEM ABOUT GEODESICS ON THE ELLIPSOID. ORBITAL ISOMORPHISM Introduction Jacobi Problem and Euler Case Liouville Foliations Rotation Functions The Main Theorem Smooth Invariants Topological Non-Conjugacy of the Jacobi Problem and the Euler Case REFERENCES SUBJECT INDEX

On the Integrability of (2+1)-Dimensional Quasilinear Systems

Abstract: A (2+1)-dimensional quasilinear system is said to be ‘integrable’ if it can be decoupled in infinitely many ways into a pair of compatible n-component one-dimensional systems in Riemann invariants. Exact solutions described by these reductions, known as nonlinear interactions of planar simple waves, can be viewed as natural dispersionless analogues of n-gap solutions. It is demonstrated that the requirement of the existence of ‘sufficiently many’ n-component reductions provides the effective classification criterion. As an example of this approach we classify integrable (2+1)-dimensional systems of conservation laws possessing a convex quadratic entropy.

Matching Higher Conserved Charges for Strings and Spins

Abstract: We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N = 4 gauge theory and energies of spinning strings on AdS5 × S 5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.

Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms

Abstract: We consider the integrable open XX quantum spin chain with non-diagonal boundary terms. We derive an exact inversion identity, by which we obtain the eigenvalues of the transfer matrix and the Bethe ansatz equations. For generic values of the boundary parameters, the Bethe ansatz solution is formulated in terms of the Jacobian elliptic functions.

Global solutions to a new integrable equation with peakons

Abstract: We prove the existence and uniqueness of global strong solutions and global weak solutions for a new integrable equation with peakons.

Three-Point Functions in N=4 SYM Theory at One-Loop

Abstract: We analyze the one-loop correction to the three-point function coefficient of scalar primary operators in = 4 SYM theory. By applying constraints from the superconformal symmetry, we demonstrate that the type of Feynman diagrams that contribute depends on the choice of renormalization scheme. In the planar limit, explicit expressions for the correction are interpreted in terms of the hamiltonians of the associated integrable closed and open spin chains. This suggests that at least at one-loop, the planar conformal field theory is integrable with the anomalous dimensions and OPE coefficients both obtainable from integrable spin chain calculations. We also connect the planar results with similar structures found in closed string field theory.

Classically integrable field theories with defects

Abstract: Some ideas and remarks are presented concerning a possible Lagrangian approach to the study of internal boundary conditions relating integrable fields at the junction of two domains. The main example given in the article concerns single real scalar fields in each domain and it is found that these may be free, of Liouville type, or of sinh-Gordon type.

Evolution of slow variables in a priori unstable Hamiltonian systems

Abstract: We study diffusion phenomena in a priori unstable (initially hyperbolic) Hamiltonian systems These systems are perturbations of integrable ones, which have a family of hyperbolic tori We prove that in the case of two and a half degrees of freedom the action variable generically drifts (ie changes on a trajectory by a quantity of order one) Moreover, there exists a trajectory such that the velocity of this drift is e/loge, where e is the parameter of the perturbation

Hydrodynamic reductions and solutions of a universal hierarchy

Abstract: We investigate the diagonal hydrodynamic reductions of a hierarchy of integrable hydrodynamic chains and analyze their compatibility with previously introduced reductions of differential type.

Yang-Mills Correlation Functions from Integrable Spin Chains

Abstract: The relation between the dilatation operator of = 4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX1/2 chain.

Equivariant normal form for nondegenerate singular orbits of integrable hamiltonian systems

Abstract: We consider an integrable Hamiltonian system with n degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a G-equivariant way, to the linearized foliation in a neighborhood of a compact singular nondegenerate orbit. We also show that the nondegeneracy condition is not equivalent to the nonresonance condition for smooth systems.

Positive and Negative Hierarchies of Integrable Lattice Models Associated with a Hamiltonian Pair

Abstract: A difference Hamiltonian operator involving two arbitrary constants is presented, and it is used to construct a pair of nondegenerate Hamiltonian operators. The resulting Hamiltonian pair yields two difference hereditary operators, and the associated positive and negative hierarchies of nonlinear integrable lattice models are derived through the bi-Hamiltonian formulation. Moreover, the two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. The use of zero curvature equation leads us to conclude that all resulting integrable lattice models are local and that the integrable lattice models in the positive hierarchy are of polynomial type and the integrable lattice models in the negative hierarchy are of rational type.

A variational approach to the stability of periodic peakons

Abstract: The peakons are peaked traveling wave solutions of an integrable shallow water equation. We present a variational proof of their stability.

New integrable structures in large-N QCD

Abstract: We study the anomalous dimensions of single trace operators composed of field strengths $F_{\mu u}$ in large-N QCD. The matrix of anomalous dimensions is the Hamiltonian of a compact spin chain with two spin one representations at each vertex corresponding to the selfdual and anti-selfdual components of $F_{\mu u}$. Due to the special form of the interaction it is possible to study separately renormalization of purely selfdual components. In this sector the Hamiltonian is integrable and can be exactly solved by Bethe ansatz. Its continuum limit is described by the level two SU(2) WZW model.

Theoretical derivation of 1/f noise in quantum chaos.

Abstract: It was recently conjectured that 1/ƒ noise is a fundamental characteristic of spectral fluctuations in chaotic quantum systems. This conjecture is based on the power spectrum behavior of the excitation energy fluctuations, which is different for chaotic and integrable systems. Using random matrix theory, we derive theoretical expressions that explain without free parameters the universal behavior of the excitation energy fluctuations power spectrum. The theory gives excellent agreement with numerical calculations and reproduces to a good approximation the 1/ƒ (1/ƒ^(2)) power law characteristic of chaotic (integrable) systems. Moreover, the theoretical results are valid for semiclassical systems as well.

Superintegrability in Classical and Quantum Systems

Abstract: Superintegrable deformations of the Smorodinsky-Winternitz Hamiltonian by A. Ballesteros, F. J. Herranz, F. Musso, and O. Ragnisco Isochronous motions galore: Nonlinearly coupled oscillators with lots of isochronous solutions by F. Calogero and J.-P. Francoise Nambu dynamics, deformation quantization, and superintegrability by T. L. Curtright and C. K. Zachos Maximally superintegrable systems of Winternitz type by C. Gonera Cubic integrals of motion and quantum superintegrability by S. Gravel Superintegrability, Lax matrices and separation of variables by J. Harnad and O. Yemolayeva Maximally superintegrable Smorodinsky-Winternitz systems on the $N$-dimensional sphere and hyperbolic spaces by F. J. Herranz, A. Ballesteros, M. Santander, and T. Sanz-Gil Invariant Wirtinger projective connection and Tau-functions on spaces of branched coverings by A. Kokotov and D. Korotkin Dyon-oscillator duality. Hidden symmetry of the Yang-Coulomb monopole by L. G. Mardoyan Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions by P. Desrosiers, L. Lapointe, and P. Mathieu Complete sets of invariants for classical systems by W. Miller, Jr. Higher-order symmetry operators for Schrodinger equation by A. G. Nikitin Symmetries and Lagrangian time-discretizations of Euler equations by A. V. Penskoi Two exactly-solvable problems in one-dimensional quantum mechanics on circle by L. G. Mardoyan, G. S. Pogosyan, and A. N. Sissakian Higher-order superintegrability of a rational oscillator with inversely quadratic nonlinearities: Euclidean and non-Euclidean cases by M. F. Ranada and M. Santander A survey of quasi-exactly solvable systems and spin Calogero-Sutherland models by F. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M. A. Rodriguez, and R. Zhdanov On the classification of third-order integrals of motion in two-dimensional quantum mechanics by M. Sheftel Towards a classification of cubic integrals of motion by R. G. McLenaghan, R. G. Smirnov, and D. The Integrable systems whose spectral curves are the graph of a function by K. Takasaki On superintegrable systems in $E_2$: Algebraic properties and symmetry preserving discretization by P. Tempesta Perturbations of integrable systems and Dyson-Mehta integrals by A. V. Turbiner Separability and the Birkhoff-Gustavson normalization of the perturbed harmonic oscillators with homogeneous polynomial potentials by Y. Uwano Integrability and superintegrability without separability by J. Berube and P. Winternitz Applications of CRACK in the classification of integrable systems by T. Wolf The prolate spheroidal phenomenon as a consequence of bispectrality by G. A. Grunbaum and M. Yakimov On a trigonometric analogue of Atiyah-Hitchin bracket by O. Yermolayeva Separation of variables in time-dependent Schrodinger equations by A. Zhalij and R. Zhdanov New types of solvability in PT symmetric quantum theory by M. Znojil.

Quantization of the Gaudin System

Abstract: In this article we exploit the known commutative family in Y(gl(n)) - the Bethe subalgebra - and its special limit to construct quantization of the Gaudin integrable system. We give explicit expressions for quantum hamiltonians QI_k(u), k=1,..., n. At small order k=1,...,3 they coincide with the quasiclassic ones, even in the case k=4 we obtain quantum correction.

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

Abstract: A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important examples include the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics and its six-dimensional generalization arising in the context of sdiff(Σ2) self-dual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a four-dimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full (2+1)-dimensional dis...

Classical/quantum integrability in AdS/CFT

Abstract: We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

Yang-Mills Correlation Functions from Integrable Spin Chains

Abstract: The relation between the dilatation operator of N=4 Yang-Mills theory and integrable spin chains makes it possible to compute the one-loop anomalous dimensions of all operators in the theory. In this paper we show how to apply the technology of integrable spin chains to the calculation of Yang-Mills correlation functions by expressing them in terms of matrix elements of spin operators on the corresponding spin chain. We illustrate this method with several examples in the SU(2) sector described by the XXX_1/2 chain.

Integrable Open Spin Chains in Defect Conformal Field Theory

Abstract: We demonstrate that the one-loop dilatation generator for the scalar sector of a certain perturbation of N=4 Super Yang-Mills with fundamentals is the Hamiltonian of an integrable spin chain with open boundary conditions. The theory is a supersymmetric defect conformal field theory (dCFT) with the fundamentals in hypermultiplets confined to a codimension one defect. We obtain a K-matrix satisfying a suitably generalized form of the boundary Yang-Baxter equation, study the Bethe ansatz equations and demonstrate how Dirichlet and Neumann boundary conditions arise in field theory, and match to existing results in the plane wave limit.