Noether’s theorems in a general setting. Reducible graded Lagrangians
Noether’s first and second theorems are formulated in a general setting of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of higher-stage Noether identities described by a Koszul–Tate chain complex. Noether’s second theorems associate to this complex a cochain sequence whose ascent operator defines higher-stage gauge symmetries of the Grassmann-graded Lagrangian system. This operator is extended to a nilpotent BRST operator that provides a BRST extension of the original Lagrangian theory. Noether’s first theorem is formulated as a straightforward corollary of the global variational formula. It associates to any gauge symmetry a conserved current which is proved to be a total differential on-shell.