The Srní lectures on non-integrable geometries with torsion

By I. Agricola. 2006

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This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connec- tions with torsion, and to discuss recent aspects of mathematical physics—in particular superstring theory—where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable 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 moti- vating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenbock formulas for Dirac operators associ- ated 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.

Subfields: Applied Mathematics, Astronomy and Astrophysics

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