Infinite Dimensional Groups and Manifolds

This book contains seven papers connected with lectures at two meetings of mathematicians and theoretical physicists at IRMA, Strasbourg and CIRM, Marseille-Luminy in 2002. The main topic is the theory of infinite dimensional groups and manifolds and their use in mathematical physics. The contribution by H. Glöckner describes basic facts on the group Γ(K,G) of germs of analytic maps from a compact subset K in a metrizable topological vector space to a Banach Lie group G. The flow completions are discussed in the paper by B. A. Khesin and P. W. Michor. They consider the universal (possibly non-Hausdorff) completion of a smooth manifold equipped with a given vector field. They use it for a class of partial differential equations interpreted as vector fields on infinite dimensional manifolds. M. Mariño presents a longer review of a conjecture on a large N duality between links in 3-dimensional manifolds and strings in 6-dimensional Calabi-Yau manifolds, including various numerical tests of the conjecture. A discussion of anomalies in quantum field theory is contained in the contribution by J. Mickelsson. He describes a relation of anomalies to projective bundles, Dixmier-Douady classes and associated gerbs. Twisted K-theory classes are related to families of supercharges in the supersymmetric WZW model.
The paper by K.-H. Neeb contains a comprehensive discussion of central extensions of two types of current group. He considers the case of smooth maps with compact support from a non-compact manifold with values in a (possibly infinite dimensional) Lie group and the case of smooth maps on a compact manifold vanishing to all orders on a closed subset. He shows that a central extension of a given type exists for these current groups only if the manifold is a circle. S. Paycha and S. Rosenberg study in their paper analogues of the Chern-Weil classes in an infinite dimensional setting. They specify the structure group of infinite dimensional vector bundles and construct the classes using a chosen connection and various traces. The paper by S. G. Rajeev describes an interpretation of the large N limit in Yang-Mills theory as a version of a classical limit. These ideas are nicely used in a discussion of modular forms. The book contains a lot of interesting material useful for readers interested in recent intensive exchange of ideas between mathematics and theoretical physics.

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