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.

Variational Sequences, Representation Sequences and Applications in Physics

“…Since X is parallelizable, we can easily find a vertical vector field Φ such that Φ F = π * (α) and [[Φ Σ]] dR = 2π * 1,0 (π * (α) ∧ F). For non compact manifolds this is, of course, no longer true; thus, for non compact 3-manifolds the question, whether the obstruction is sharp, depends on the existence of a suitable compactification, see also [88].…”

Variations by generalized symmetries of local Noether strong currents equivalent to global canonical Noether currents

“…where i = j r Ξ V p d V λi + ξ λ i is the usual canonical Noether current; the Noether-Bessel-Hagen current (λ i , Ξ) − β i is a local object and it is conserved along the solutions of Euler-Lagrange equations (local conservation law) [9], [17], [18], [30].…”

A cohomological obstruction in higher dimensional Chern--Simons gauge theories

“…While the conservation laws so derived remain global, the conserved quantities may not. There appears a topological obstruction, a cohomology class in the complex, which may not vanish, even if a global Lagrangian exists; see [14,15,22,23].…”