Generalizing Geometry - Algebroids and Sigma Models

Details Generalizing Geometry - Algebroids and Sigma Models Generalizing Geometry - Algebroids and Sigma Models

2010-04-05

arxiv.org/abs/1004.0632v1

Physics

High Energy Physics

High Energy Physics - Theory

52 pages, Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry, ed. V. Cortes

Scientific paper

In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but also complex structures. Proofs are given in detail. The second part deals with sigma models. Topological ones, in particular the AKSZ and the Dirac sigma models, as generalizations of the Poisson sigma models to higher dimensions and to Dirac structures, respectively, but also physical ones, that reduce to standard Yang Mills theories for the "flat" choice of a Lie algebra: Lie algebroid Yang Mills theories and possible action functionals for nonabelian gerbes and general higher gauge theories. Characteristic classes associated to Dirac structures and to higher principal bundles are also mentioned.

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