Publication details

A triplectic bi-Darboux theorem and para-hypercomplex geometry

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Year of publication 2012
Type Article in Periodical
Magazine / Source Journal of Mathematical Physics
MU Faculty or unit

Faculty of Science

Field Theoretical physics
Keywords Poisson Bracket; Antibracket; Sp(2)-Symmetric Quantization; Darboux Theorem; Poincare Lemma.
Description 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.
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