Igor A. Batalin, Klaus Bering
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.
View original:
http://arxiv.org/abs/1110.6165
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