Gauging the Wess-Zumino term of a sigma model with boundary

TL;DR

This paper studies the conditions under which the Wess-Zumino term of a sigma model with boundary can be gauged, revealing new obstructions and their geometric interpretations, especially in the Wess-Zumino-Witten model.

Contribution

It derives obstructions to gauging the Wess-Zumino term with boundary and interprets them using Courant algebroids, extending understanding of boundary conditions in sigma models.

Findings

Obstructions to gauging are characterized by equivariant relative de Rham complexes.

Boundary conditions in the Wess-Zumino-Witten model always allow gauging of certain subgroups.

Gauging is compatible with boundary conditions given by conjugacy classes and cosets.

Abstract

We investigate the gauging of the Wess-Zumino term of a sigma model with boundary. We derive a set of obstructions to gauging and we interpret them as the conditions for the Wess-Zumino term to extend to a closed form in a suitable equivariant relative de Rham complex. We illustrate this with the two-dimensional sigma model and we show that the new obstructions due to the boundary can be interpreted in terms of Courant algebroids. We specialise to the case of the Wess-Zumino-Witten model, where it is proved that there always exist suitable boundary conditions which allow gauging any subgroup which can be gauged in the absence of a boundary. We illustrate this with two natural classes of gaugings: (twisted) diagonal subgroups with boundary conditions given by (twisted) conjugacy classes, and chiral isotropic subgroups with boundary conditions given by cosets.