A Triplectic Bi-Darboux Theorem and Para-Hypercomplex Geometry

Details A Triplectic Bi-Darboux Theorem and Para-Hypercomplex Geometry A Triplectic Bi-Darboux Theorem and Para-Hypercomplex Geometry

2011-10-27

arxiv.org/abs/1110.6165v2

Physics

Mathematical Physics

30 pages, LaTeX. v2: Changed title; Added references

Scientific paper

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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