The Srni lectures on nonintegrable geometries with torsion
Abstract
This review article intends to introduce the reader to nonintegrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physicsin particular superstring theorywhere these naturally appear. Connections with skewsymmetric torsion are exhibited as one of the main tools to understand nonintegrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a $G$structure as unifying principles. % The General Holonomy Principle bridges over to parallel objects, thus motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenböck formulas for Dirac operators associated with torsion connections enable us to discuss spinorial field equations, such as those governing the common sector of type II superstring theory. They also provide the link to Kostant's cubic Dirac operator.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606705
 Bibcode:
 2006math......6705A
 Keywords:

 Mathematics  Differential Geometry;
 53C25;
 53D15;
 81T30
 EPrint:
 61 pages