## Preface

Manifolds with special geometric structures, such as special Kähler, hyper-Kähler and quaternionic-Kähler manifolds play a prominent role in string theory. These structures occur as natural structures on certain moduli spaces or on target spaces of low energy effective limits of string theories. For example it is well known that supersymmetry requires various target spaces (for sigma models, scalar fields in supergravity theories, etc.) to have certain special geometric properties. In many cases these requirements have interpretation as restrictions on the holonomy group of the Riemannian target space metric. However, in some cases, they cannot be expressed in terms of the Riemannian holonomy group alone and give rise to new geometries previously unknown to mathematicians. Special Kähler manifolds, for instance, which arose in the pioneering papers of de Wit and Van Proeyen, play a crucial role as admissible target spaces for scalar and vector couplings in both rigid and local N=2 supersymmetric theories with vector multiplets. Other developments involving special Kähler geometry are Seiberg-Witten duality and the AdS/CFT correspondence. The latter provided further applications of special geometries, including the appearance of five-dimensional variants, very special manifolds, of relevance for brane-world scenarios.The above topics, from both mathematical as well as physical perspectives, were the focus of this workshop held in Bonn between 8th and 11th September, 2001, under the auspices of the DFG (German Science Foundation) priority programme in String Theory . Other topics included spinor geometry, geometries with torsion and Yang-Mills theory. The idea of the workshop was to promote dialogue across the interdisciplinary boundary by bringing together physicists and mathematicians working on areas of mutual interest. Eighteen talks were held (see the programme ). Nine of these appear in these proceedings.

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