2006/05/23 by G. Giachetta, Giachetta, G., L. Mangiarotti +3
Mathematics · #58A20 #58C50 #58J70 #70S05 #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.math/0605618
openalex publication_date 2006/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A generic degenerate Lagrangian system of even and odd variables on an arbitrary smooth manifold is examined in terms of the Grassmann-graded variational bicomplex. Its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. However, non-trivial higher-stage Noether identities are ill defined, unless a certain homology condition holds. We show that, under this condition, there exists the exact Koszul-Tate chain complex whose boundary operator produces all non-trivial Noether and higher-stage Noether identities of an original Lagrangian system. Noether's inverse second theorem that we prove associates to this complex a cochain sequence whose ascent operator provides all gauge and higher-stage gauge supersymmetries of an original Lagrangian.