A Triplectic Bi-Darboux Theorem and Para-Hypercomplex Geometry

TL;DR

This paper establishes necessary and sufficient conditions for bi-Darboux coordinates on triplectic manifolds, linking them to para-hypercomplex geometry and flat Obata connections, with implications for field-antifield formulations.

Contribution

It provides a comprehensive characterization of bi-Darboux theorems on triplectic manifolds and reveals a correspondence with para-hypercomplex structures, including conditions for their existence.

Findings

Characterization of bi-Darboux coordinates on triplectic manifolds.

Correspondence between triplectic manifolds and para-hypercomplex manifolds.

Existence of bi-Darboux coordinates linked to flat Obata connections.

Abstract

We provide necessary and sufficient conditions for a bi-Darboux Theorem on triplectic manifolds. Here triplectic manifolds are manifolds equipped with two compatible, jointly non-degenerate Poisson brackets with mutually involutive Casimirs, and with ranks equal to 2/3 of the manifold dimension. By definition bi-Darboux coordinates are common Darboux coordinates for two Poisson brackets. We discuss both the Grassmann-even and the Grassmann-odd Poisson bracket case. Odd triplectic manifolds are, e.g., relevant for Sp(2)-symmetric field-antifield formulation. We demonstrate a one-to-one correspondence between triplectic manifolds and para-hypercomplex manifolds. Existence of bi-Darboux coordinates on the triplectic side of the correspondence translates into a flat Obata connection on the para-hypercomplex side.