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Kähler geometry of loop spaces. (English) Zbl 1191.58001

MSJ Memoirs 23. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-60-0/pbk). xv, 212 p. (2010).
This impressive monograph covers three classes of infinite-dimensional Kähler manifolds:
(1)
loop spaces of compact Lie groups,
(2)
Teichmüller spaces of complex structures on loop spaces,
(3)
Grassmannians of Hilbert spaces.
Any of these manifolds can be considered as an universal object in a certain category, containing all its finite-dimensional counterparts.
The content is divided into 5 parts: Preliminary concepts; Loop spaces of compact Lie groups; Spaces of complex structures; Quantization of finite-dimensional Kähler manifolds; Quantization of loop spaces. There is a total of 15 chapters, each ending with very useful bibliographic comments. The bibliography covers 82 titles.
In conclusion, the volume contains very interesting material on several topics of contemporary geometry and it can be recommended as a valuable introduction to the themes selected for this text.

MSC:

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
58D30 Applications of manifolds of mappings to the sciences
53Z05 Applications of differential geometry to physics
55P35 Loop spaces
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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