The Srni lectures on non-integrable geometries with torsion

TL;DR

This review introduces non-integrable geometric structures with torsion on Riemannian manifolds, highlighting their role in mathematical physics and superstring theory, through examples, intrinsic torsion, and Dirac operators.

Contribution

It provides a comprehensive overview of non-integrable geometries with torsion, emphasizing their applications in superstring theory and the use of characteristic connections and Dirac operators.

Findings

Connections with skew-symmetric torsion are key to understanding non-integrable geometries.

Several Weitzenb"ock formulas relate torsion connections to spinorial field equations.

The review links geometric structures to superstring theory through examples and Dirac operators.

Abstract

This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections 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 motivating the discussion of geometric stabilizers, with emphasis on spinors and differential forms. Several Weitzenb\"ock 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.