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

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.

arXiv comment18 pages, presentation improved, reference list updated

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Cited by in corpus (8)

• The Noether Bessel Hagen Symmetry Approach for Dynamical Systems
• Variational Sequences, Representation Sequences and Applications in Physics
• Geometric integration by parts and Lepage equivalents
• Symmetry transformations of extremals and higher conserved quantities: invariant Yang--Mills connections
• Higgs fields induced by Yang--Mills type Lagrangians on gauge-natural prolongations of principal bundles
• The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang-Mills theory on a Minkowskian background
• A cohomological obstruction in higher dimensional Chern--Simons gauge theories
• Geometrical setting for the correspondence between gravity and Chern-Simons field theories