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Author: Gleb Arutyunov Publisher: Springer ISBN: 303024198X Category : Science Languages : en Pages : 420
Book Description
Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.
Author: Gleb Arutyunov Publisher: Springer ISBN: 303024198X Category : Science Languages : en Pages : 420
Book Description
Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.
Author: Anton Dzhamay Publisher: American Mathematical Soc. ISBN: 0821887475 Category : Mathematics Languages : en Pages : 363
Book Description
This volume contains the proceedings of the AMS Special Session on Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, held from January 6-7, 2012, in Boston, MA. The very wide range of topics represented in this volume illustrates
Author: Sergey Novikov Publisher: American Mathematical Soc. ISBN: 1470455919 Category : Education Languages : en Pages : 516
Book Description
This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.
Author: Martin A. Guest Publisher: OUP Oxford ISBN: 0191606960 Category : Mathematics Languages : en Pages : 336
Book Description
Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry. Certain kinds of differential equations (or D-modules) provide the key links between quantum cohomology and traditional mathematics; these links are the main focus of the book, and quantum cohomology and other integrable PDEs such as the KdV equation and the harmonic map equation are discussed within this unified framework. Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.
Author: Andrei Marshakov Publisher: World Scientific ISBN: 9789810236366 Category : Science Languages : en Pages : 268
Book Description
In the past few decades many attempts have been made to search for a consistent formulation of quantum field theory beyond perturbation theory. One of the most interesting examples is the Seiberg-Witten ansatz for the N=2 SUSY supersymmetric Yang-Mills gauge theories in four dimensions. The aim of this book is to present in a clear form the main ideas of the relation between the exact solutions to the supersymmetric (SUSY) Yang-Mills theories and integrable systems. This relation is a beautiful example of reformulation of close-to-realistic physical theory in terms widely known in mathematical physics ? systems of integrable nonlinear differential equations and their algebro-geometric solutions.First, the book reviews what is known about the physical problem: the construction of low-energy effective actions for the N=2 Yang-Mills theories from the traditional viewpoint of quantum field theory. Then the necessary background information from the theory of integrable systems is presented. In particular the author considers the definition of the algebro-geometric solutions to integrable systems in terms of complex curves or Riemann surfaces and the generating meromorphic 1-form. These definitions are illustrated in detail on the basic example of the periodic Toda chain.Several ?toy-model? examples of string theory solutions where the structures of integrable systems appear are briefly discussed. Then the author proceeds to the Seiberg-Witten solutions and show that they are indeed defined by the same data as finite-gap solutions to integrable systems. The complete formulation requires the introduction of certain deformations of the finite-gap solutions described in terms of quasiclassical or Whitham hierarchies. The explicit differential equations and direct computations of the prepotential of the effective theory are presented and compared when possible with the well-known computations from supersymmetric quantum gauge theories.Finally, the book discusses the properties of the exact solutions to SUSY Yang-Mills theories and their relation to integrable systems in the general context of the modern approach to nonperturbative string or M-theory.
Author: G. Giachetta Publisher: World Scientific ISBN: 9814313726 Category : Science Languages : en Pages : 405
Book Description
The geometric formulation of autonomous Hamiltonian mechanics in the terms of symplectic and Poisson manifolds is generally accepted. This book provides the geometric formulation of non-autonomous mechanics in a general setting of time-dependent coordinate and reference frame transformations.
Author: Ron Donagi Publisher: Cambridge University Press ISBN: 1108805337 Category : Mathematics Languages : en Pages : 537
Book Description
Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. The articles in this second volume discuss areas related to algebraic geometry, emphasizing the connections of this central subject to integrable systems, arithmetic geometry, Riemann surfaces, coding theory and lattice theory.
Author: Fabio Franchini Publisher: Springer ISBN: 3319484877 Category : Science Languages : en Pages : 186
Book Description
This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.
Author: Gleb Arutyunov
Author: Gleb Arutyunov
Author: Anton Dzhamay
Author: Sergey Novikov
Author: Martin A. Guest
Author: G. F. Helminck
Author: Andrei Marshakov
Author: G. Giachetta
Author: Ron Donagi
Author: Fabio Franchini
Author: G. F. Helminck