Infinite Dimensional Kahler Manifolds

by Huckleberry, Alan T.; Wurzbacher, Tilman

• ISBN13: 9783764366025
• ISBN10: 3764366028
• Format: Paperback
• Copyright: 2001-09-01
• Publisher: Birkhauser

Summary

Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest. On the one hand this is a collection of closely related articles on infinite dimensional Ka'hler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas. The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold.

Table of Contents

Preface
Introduction to Group Actions in Symplectic and Complex Geometry
Finite-dimensional manifolds	p. 1
Element of Lie groups and their actions	p. 20
Manifolds with additional structure	p. 40
Symplectic manifolds with symmetry	p. 82
Kahlerian structures on coadjoint orbits of compact groups and associated representations	p. 95
Infinite-dimensional Groups and their Representations
Calculus in locally convex spaces	p. 132
Dual spaces of locally convex spaces	p. 144
Topologies on function spaces	p. 153
Representations of infinite-dimensional groups	p. 164
Generalized coherent state representations	p. 168
Borel-Weil Theory for Loop Groups
Compact groups	p. 180
Loop groups and their central extensions	p. 181
Root decompositions	p. 197
Representations of loop groups	p. 201
Representations of involutive semigroups	p. 209
Borel-Weil theory	p. 213
Consequences for general representations	p. 226
Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on Tensor-densities
Coadjoint representation of Virasoro group and Sturm-Liouville operators; Schwarzian derivative as a 1-cocycle	p. 233
Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivative	p. 238
Kirillov's method of Lie superalgebras	p. 241
Invariants of coadjoint representation of the Virasoro group	p. 244
Extension of the Lie algebra of first order linear differential operators on S[superscript 1] and matrix analogue of the Sturm-Liouville operator	p. 247
Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-product	p. 249
From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques
Renormalization techniques	p. 260
The first Chern form on a class of hermitian vector bundles	p. 266
The geometry of gauge orbits	p. 270
The geometry of determinant bundles	p. 275
An example: the action of diffeomorphisms on complex structures	p. 278
Fermionic Second Quantization and the Geometry of the Restricted Grassmannian
Fermionic second quantization	p. 290
Bogoliubov transformations and the Schwinger term	p. 298
The restricted Grassmannian of a polarized Hilbert space	p. 326
The non-equivariant moment map of the restricted Grassmannian	p. 336
The determinant line bundle on the restricted Grassmannian	p. 358
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