Seminar talk, 18 May 2016

Speaker: Hovhannes Khudaverdian

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

Abstract:
Second order operator $\mathrm{\Delta }$ on half-densities can be uniquely defined by its principal symbol up to a 'potential'. If $\mathrm{\Delta }$ is an odd operator such that order of operator ${\mathrm{\Delta }}^2$ 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 $\mathrm{\Delta }$ operator. The modular class of an odd Poisson supermanifold is an obstacle to existence of operator $\mathrm{\Delta }$ defining the Poisson structure such that ${\mathrm{\Delta }}^2=0$. In the case of non-degenerate odd Poisson structure the modular class vanishes, and we come to canonical odd Laplacian, the operator $\mathrm{\Delta }$ such that its principal symbol defines symplectic structure, potential vanishes in Darboux coordinates and ${\mathrm{\Delta }}^2=0$.

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.

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