Separation of Variables : New Trends
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In this article, it is shown that the standard construction of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of SoV, and moreover, for many particular models it has a direct quantum counterpart.Abstract:
The review is based on the author’s papers since 1985 in which a new approach to the separation of variables (SoV) has being developed. It is argued that SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard construction of the action-angle variables from the poles of the Baker-Akhiezer function can be interpreted as a variant of SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinread more
Citations
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Quantum Inverse Scattering Method and Correlation Functions
TL;DR: One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References as discussed by the authors
Proceedings ArticleDOI
Quantization of Integrable Systems and Four Dimensional Gauge Theories
TL;DR: In this paper, the authors studied four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N = 2 super-Poincare invariance and explained how this gauge theory provided the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four-dimensional N =2 theory, and showed the thermodynamic-Betheansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory.
Proceedings ArticleDOI
Quantization of Integrable Systems and Four Dimensional Gauge Theories
TL;DR: In this article, the authors studied four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N = 2 super-Poincare invariance and explained how this gauge theory provided the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional n=2 theory.
Journal ArticleDOI
Quantum Geometry of Refined Topological Strings
TL;DR: In this paper, the authors consider branes in refined topological strings and show that their wave-functions satisfy a Schrodinger equation depending on multiple times and prove this in the case where the topological string has a dual matrix model description.
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Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories
Nikita Nekrasov,Vasily Pestun +1 more
TL;DR: In this article, the Seiberg-Witten geometry of mass deformed N = 2 superconformal ADE quiver gauge theories in four dimensions is determined, and the integrable systems underlying, or rather, overlooking the special geometry of M are identified.
References
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Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Book
Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
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Quantum Inverse Scattering Method and Correlation Functions
TL;DR: In this article, a detailed explanation of Bethe Ansatz, Quantum Inverse Scattering Method and Algebraic Bether Ansatz as well as main models are Nonlinear Schrodinger equation (one dimensional Bose gas), Sine-Gordon and Thiring models.
Book
Tata Lectures on Theta I
TL;DR: In this paper, theta functions in one variable and motivation: motivation and theta function in several variables are compared. But the results are limited to one variable, and motivation is not considered.
Journal ArticleDOI
Gaudin model, Bethe ansatz and critical level
TL;DR: In this paper, a new method of diagonalization of the Gaudin model associated to an arbitrary simple Lie algebra, which is based on the Wakimoto modules over affine algebras at the critical level, is proposed.