On the notion of gauge symmetries of generic Lagrangian field theory

TL;DR

This paper develops a general framework for understanding gauge symmetries in Lagrangian field theories, including their algebraic structure and conditions for closure, with applications to various fundamental theories.

Contribution

It introduces a comprehensive approach to gauge symmetries and their algebraic properties in generic Lagrangian theories using the inverse second Noether theorem and Koszul-Tate complex.

Findings

Defines reducible gauge symmetries via inverse Noether theorem

Establishes conditions for algebraic closure of gauge symmetries

Provides characteristic examples including Yang-Mills and Chern-Simons theories

Abstract

General Lagrangian theory of even and odd fields on an arbitrary smooth manifold is considered. Its non-trivial reducible gauge symmetries and their algebra are defined in this very general setting by means of the inverse second Noether theorem. In contrast with gauge symmetries, non-trivial Noether and higher-stage Noether identities of Lagrangian theory can be intrinsically defined by constructing the exact Koszul-Tate complex. The inverse second Noether theorem that we prove associates to this complex the cochain sequence with the ascent operator whose components define non-trivial gauge and higher-stage gauge symmetries. These gauge symmetries are said to be algebraically closed if the ascent operator can be extended to a nilpotent operator. The necessary conditions for this extension are stated. The characteristic examples of Yang-Mills supergauge theory, topological Chern-Simons theory, gauge gravitation theory and topological BF theory are presented.