Sergeev, Armen 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. Reviewer: Mircea Crâşmăreanu (Iaşi) Cited in 10 Documents 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 Keywords:infinite-dimensional Kähler manifold; loop space; quantization PDFBibTeX XMLCite \textit{A. Sergeev}, Kähler geometry of loop spaces. Tokyo: Mathematical Society of Japan (2010; Zbl 1191.58001) Full Text: DOI Link