A triplectic bi-Darboux theorem and para-hypercomplex geometry

作者：Batalin Igor A^*; Bering Klaus

来源：Journal of Mathematical Physics, 2012, 53(12): 123507.

DOI：10.1063/1.4759501

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.

• 出版日期2012-12