Seminar talk, 18 May 2016

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Speaker: Hovhannes Khudaverdian

Title: The modular class of an odd Poisson (super)manifold and odd Laplacian

Second order operator on half-densities can be uniquely defined by its principal symbol up to a 'potential'. If is an odd operator such that order of operator is less than 3 then principal symbol of this operator, an odd contravariant rank 2 tensor field defines an odd Poisson bracket.

We define the modular class of an odd Poisson supermanifold in terms of operator. The modular class of an odd Poisson supermanifold is an obstacle to existence of operator defining the Poisson structure such that . In the case of non-degenerate odd Poisson structure the modular class vanishes, and we come to canonical odd Laplacian, the operator such that its principal symbol defines symplectic structure, potential vanishes in Darboux coordinates and .

This operator is the main ingredient of Batalin-Vilkovisky formalism.

Then we consider an example of an odd Poisson manifolds with non-vanishing modular classes related with the Nijenhuis bracket of form-valued vector fields.

Khudaverdian H.M., Peddie M. Odd Laplacians: geometrical meaning of potential, and modular class, arXiv:1509.05686