First Class Constrained Systems and Twisting of Courant Algebroids by a Closed 4-form

TL;DR

This paper explores how three-dimensional sigma models with Wess-Zumino terms naturally lead to twisting Courant algebroid structures by closed 4-forms, connecting physics and advanced geometry.

Contribution

It demonstrates the emergence of twisted Courant algebroids from sigma models, providing a pedagogical bridge between physical models and mathematical structures.

Findings

Twisting of Courant algebroids by closed 4-forms H.

Connection between sigma models and Courant algebroid structures.

Relevance to string theory and generalized complex structures.

Abstract

We show that in analogy to the introduction of Poisson structures twisted by a closed 3-form by Park and Klimcik-Strobl, the study of three dimensional sigma models with Wess-Zumino term leads in a likewise way to twisting of Courant algebroid structures by closed 4-forms H. The presentation is kept pedagogical and accessible to physicists as well as to mathematicians, explaining in detail in particular the interplay of field transformations in a sigma model with the type of geometrical structures induced on a target. In fact, as we also show, even if one does not know the mathematical concept of a Courant algebroid, the study of a rather general class of 3-dimensional sigma models leads one to that notion by itself. Courant algebroids became of relevance for mathematical physics lately from several perspectives - like for example by means of using generalized complex structures in String Theory. One may expect that their twisting by the curvature H of some 3-form Ramond-Ramond gauge field will become of relevance as well.