Kähler geometry of loop spaces (Q974097)

This impressive monograph covers three classes of infinite-dimensional Kähler manifolds: {\parindent=7mm \begin{itemize}\item[(1)]loop spaces of compact Lie groups, \item[(2)]Teichmüller spaces of complex structures on loop spaces, \item[(3)]Grassmannians of Hilbert spaces. \end{itemize}} 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.