Kähler Geometry of Loop Spaces

In this book we study three important classes of infinite-dimensional Kähler manifolds — loop spaces of compact Lie groups, Teichmüller spaces of complex structures on loop spaces, and Grassmannians of Hilbert spaces. Each of these manifolds has a rich Kähler geometry, considered in the first part of the book, and may be considered as a universal object in a category, containing all its finite-dimensional counterparts.

On the other hand, these manifolds are closely related to string theory. This motivates our interest in their geometric quantization presented in the second part of the book together with a brief survey of geometric quantization of finite-dimensional Kähler manifolds.

The book is provided with an introductory chapter containing basic notions on infinite-dimensional Frechet manifolds and Frechet Lie groups. It can also serve as an accessible introduction to Kähler geometry of infinite-dimensional complex manifolds with special attention to the aforementioned three particular classes.

It may be interesting for mathematicians working with infinite-dimensional complex manifolds and physicists dealing with string theory.

Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets

• Preliminary Concepts:
  • Frechet Manifolds
  • Frechet Lie Groups
  • Flag Manifolds and Representations
  • Central Extensions and Cohomologies
  • Grassmannians of a Hilbert Space
  • Quasiconformal Maps
• Loop Spaces of Compact Lie Groups:
  • Loop Space
  • Central Extensions
  • Grassmann Realizations
• Spaces of Complex Structures:
  • Virasoro Group
  • Universal Techmüller Space
• Quantization of Finite-Dimensional Kähler Manifolds:
  • Dirac Quantization
  • Kostant–Souriau Prequantization
  • Blattner–Kostant–Sternberg Quantization
• Quantization of Loop Spaces:
  • Quantization of Ωℝ^d
  • Quantization of Ω_TG