Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids

TL;DR

This paper connects 2D Poisson Sigma Models on Poisson groupoids with 3D Courant Sigma Models via Lie-theoretic reductions, revealing new insights into the structure of Poisson groupoids and their symplectic realizations.

Contribution

It demonstrates that the 2D Poisson Sigma Model on a Poisson groupoid can be derived as an effective theory from the 3D Courant Sigma Model, linking field theory with Lie groupoid structures.

Findings

Establishes a Lie-theoretic reduction linking 2D and 3D sigma models.

Provides examples including symplectic groupoids and relates to Crainic-Marcut's symplectic realization.

Shows how gauge fixing in the 3D theory yields known structures in Poisson geometry.

Abstract

We show that the 2d Poisson Sigma Model on a Poisson groupoid arises as an effective theory of the 3d Courant Sigma Model associated to the double of the underlying Lie bialgebroid. This field-theoretic result follows from a Lie-theoretic one involving a coisotropic reduction of the odd cotangent bundle by a generalized space of algebroid paths. We also provide several examples, including the case of symplectic groupoids in which we relate the symplectic realization construction of Crainic-Marcut to a particular gauge fixing of the 3d theory.