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Generalizing geometry – algebroids and sigma models. (English) Zbl 1200.53084
Cortés, Vicente (ed.), Handbook of pseudo-Riemannian geometry and supersymmetry. Papers based on the 77th meeting “Encounter between mathematicians and theoretical physicists”, Strasbourg, France, 2005. Zürich: European Mathematical Society (ISBN 978-3-03719-079-1/hbk). IRMA Lectures in Mathematics and Theoretical Physics 16, 209-262 (2010).
This paper is essentially devoted to the links between sigma models and geometrical structures related to algebroids. Let us recall that sigma models are action functionals with a target space equipped with a geometry. There are many examples of sigma models (the Poisson sigma model, the Riemannian sigma model, twisted Poisson sigma model). In this work, the authors consider such examples and show the interplay with the underlying geometry. For instance, they discuss the supersymmetric sigma models and bihermitian geometry. Giving a general definition of algebroid, the authors deal with generalized geometry and the corresponding sigma models. Finally, sigma models in arbitrary space-time dimension are studied.
For the entire collection see [Zbl 1190.53001].

53D55 Deformation quantization, star products
53D17 Poisson manifolds; Poisson groupoids and algebroids
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)