On the Canonical Form of a Pair of Compatible Antibrackets

TL;DR

This paper proves a triplectic analogue of the Darboux theorem, showing that compatible antibrackets can be brought to a weakly-canonical form under certain axioms, and classifies related vector fields.

Contribution

It establishes a weakly-canonical form for compatible antibrackets in triplectic quantization and classifies compatible odd vector fields, extending the understanding of triplectic structures.

Findings

Antibrackets can be brought to a weakly-canonical form under triplectic axioms.

Classification of triplectic odd vector fields compatible with weakly-canonical antibrackets.

Conditions under which the obstruction to canonical form vanishes and implications for Sp(2)-covariance.

Abstract

In the triplectic quantization of general gauge theories, we prove a `triplectic' analogue of the Darboux theorem: we show that the doublet of compatible antibrackets can be brought to a weakly-canonical form provided the general triplectic axioms of [BMS] are imposed together with some additional requirements that can be formulated in terms of marked functions of the antibrackets. The weakly-canonical antibrackets involve an obstruction to bringing them to the canonical form. We also classify the `triplectic' odd vectors fields compatible with the weakly-canonical antibrackets and construct the Poisson bracket associated with the antibrackets and the odd vector fields. We formulate the Sp(2)-covariance requirement for the antibrackets and the vector fields; whenever the obstruction to the canonical form of the antibrackets vanishes, the Sp(2)-covariance condition implies the canonical form of the triplectic vector fields.