Odd Laplacians: geometrical meaning of potential and modular class
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A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.
KeywordsOperators on half-densities Odd Poisson manifold Modular class Odd symplectic manifold Canonical odd Laplacian Compensating field
Mathematics Subject Classification53D17 58A50 81R99
We are happy to acknowledge A. Voronov and B. Kruglikov for encouraging us to begin the work on this article. It is a pleasure to also thank K. Mackenzie and T. Voronov for very useful discussions. One of us (H.Kh.) learnt much about the peculiarities of Poisson geometry through discussions with A. Bolsinov. Many thanks to him.
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