Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology

TL;DR

This paper develops a comprehensive mathematical framework for analyzing Lagrangian contact supersymmetries that depend on derivatives, using cohomology of the variational bicomplex on graded manifolds, and derives conservation laws.

Contribution

It provides a general treatment of contact supersymmetries depending on derivatives and computes their cohomology, extending the understanding of variational principles in graded geometry.

Findings

Computed the cohomology of the variational bicomplex on graded manifolds.

Derived the first variational formula for Lagrangian systems with contact supersymmetries.

Established conservation laws for systems with graded contact supersymmetries.

Abstract

Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained.