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Quantization of integrable systems and four dimensional gauge theories. (English) Zbl 1214.83049

Exner, Pavel (ed.), XVIth international congress on mathematical physics, Prague, Czech Republic, August 3–8, 2009. With DVD. Hackensack, NJ: World Scientific (ISBN 978-981-4304-62-7/hbk). 265-289 (2010).
Here is an interesting 25-page paper comprising and covering quite a few topics and items of right relevance to the understanding of quantization and gauge theories. There are 52 references, 7 sections, 60 equations and 5 remarks in it. Some of the well known researchers’ names like Yang and Witten in theoretical physics of the fundamental particles are directly connected to the main themes pursued.
The primary sections are: Introduction, Review of Bethe/gauge correspondence, Four-dimensional gauge theory, Integrable systems, Examples, Superpotential/Yang-Mills functions and Discussion. Detailed information concerning the various aspects of the work are carefully given in the subsections. The main claim of the authors is: the supersymmetric vacua of the four-dimensional \(N=2\) supersymmetric gauge theory are the eigenstates of the quantum integrable system obtained by the quantization of the classical algebraic integrable system underlying the geometry of the moduli space \(M_v\), of the undeformed \(N=2\) theory. The Planck constant, the parameter of the quantization, is identified with the deformation parameter \(\varepsilon\).
On the whole the research reported in the paper is of high order.
For the entire collection see [Zbl 1192.00040].

MSC:

83E50 Supergravity
81V22 Unified quantum theories
83C10 Equations of motion in general relativity and gravitational theory
83C47 Methods of quantum field theory in general relativity and gravitational theory
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
81T13 Yang-Mills and other gauge theories in quantum field theory
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