A triplectic bi-Darboux theorem and para-hypercomplex geometry
| Authors | |
|---|---|
| Year of publication | 2012 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Mathematical Physics |
| MU Faculty or unit | |
| Citation | |
| Web | |
| Doi | http://dx.doi.org/10.1063/1.4759501 |
| 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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