A comparison of definitions for the Schouten bracket on jet spaces

The Schouten bracket (or antibracket) plays a central role in the Poisson formalism and the Batalin-Vilkovisky quantization of gauge systems. There are several (in)equivalent ways to realize this concept on jet spaces. In this paper, we compare the definitions, examining in what ways they agree or disagree and how they relate to the case of usual manifolds.

💡 Research Summary

The paper provides a systematic comparison of the various ways the Schouten bracket (also known as the antibracket) can be defined on jet spaces, a setting that extends the usual finite‑dimensional manifold framework to infinite‑dimensional bundles of fields and their derivatives. After recalling the classical Schouten–Nijenhuis bracket on smooth manifolds, the authors identify three principal constructions that appear in the literature for jet spaces: an algebraic definition based on graded multivector fields and differential forms, a variational‑complex definition that uses the total differential (d_{\text{tot}}) on the variational bicomplex, and a geometric definition that exploits an underlying Poisson (or symplectic) structure and Hamiltonian vector fields.

For each construction the paper examines the underlying graded Lie‑algebraic properties, the handling of degree shifts, and the compatibility with the total differential. The algebraic approach reproduces the graded‑Leibniz and graded‑antisymmetry identities of the classical bracket and is shown to reduce to the usual Schouten–Nijenhuis bracket when the jet variables are frozen. The variational‑complex approach, by contrast, emphasizes cohomological aspects: the bracket is defined on cohomology classes of the variational bicomplex, and the crucial identity (