Generalizing Geometry - Algebroids and Sigma Models

TL;DR

This paper reviews the mathematical structures of Lie algebroids and Dirac structures and explores their applications in topological and physical sigma models, including higher gauge theories and characteristic classes.

Contribution

It provides a comprehensive overview of the interplay between advanced geometric structures and sigma models, including detailed proofs and new insights into higher gauge theories.

Findings

Generalization of Poisson sigma models to higher dimensions

Introduction of Lie algebroid Yang Mills theories

Discussion of characteristic classes for Dirac structures

Abstract

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.