2016/08/31 by Marcella Palese, Ekkehart Winterroth
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Chern–Simons theory #Classical field theory #Field (mathematics) #Field theory (psychology) #Gauge (firearms) #Gauge symmetry #Gauge theory #Homotopy and Cohomology in Algebraic Topology #Introduction to gauge theory #Lagrangian #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Pure mathematics #Quantum mechanics #Theoretical physics #Topological quantum field theory #Yang–Mills theory #hep-th #math-ph #math.MP #msc:53C05 #msc:53Z05 #msc:55N30 #msc:58A20 #msc:58J28
paper · pdf · doi:10.1063/1.4975336
published as Journal of Mathematical Physics 58, 023502 (2017) · 18 pages, presentation improved, reference list updated
arxiv created 2016/11/11 · openalex publication_date 2017/02/01 · arxiv updated 2017/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We relate the existence of Noether global conserved currents associated with locally variational field equations to the 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.