Topological obstructions in Lagrangian field theories, with an application to 3D Chern-Simons gauge theory

TL;DR

This paper explores the topological obstructions in Lagrangian field theories, particularly in 3D Chern-Simons gauge theory, linking cohomology classes to the existence of global solutions and extremals.

Contribution

It establishes a cohomological framework connecting topological obstructions to the existence of global solutions in Lagrangian field theories, with specific application to Chern-Simons theory.

Findings

Cohomological obstructions correspond to characteristic classes.

Sharp obstruction in 3D Chern-Simons matches classical flatness conditions.

Parallelism between geometric and variational obstructions is proposed.

Abstract

We relate the existence of Noether global conserved currents associated with locally variational field equations to existence of global solutions for a local variational problem generating global equations. Both can be characterized as the vanishing of certain cohomology classes. In the case of a 3-dimensional Chern-Simons gauge theory, the variationally featured cohomological obstruction to the existence of global solutions is sharp and equivalent to the usual obstruction in terms of the Chern characteristic class for the flatness of a principal connection. We suggest a parallelism between the geometric interpretation of characteristic classes as obstruction to the existence of flat principal connections and the interpretation of certain de Rham cohomology classes to be the obstruction to the existence of global extremals for a local variational principle.