Manchester Geometry Seminar 2015/2016

Second order operator $\Delta$ on half-densities is uniquely defined by its principal symbol $E$ up to a `potential' $U$. If $\Delta$ is an odd operator such that order of operator $\Delta^2$ is less than $3$, then the principal symbol $E$ of this operator defines an odd Poisson bracket. We recall the concept of the modular class of a Poisson structure for usual manifolds and give a definition of the modular class of an odd Poisson supermanifold in terms of $\Delta$ operator. In the case of a non-degenerate Poisson structure (odd symplectic case), the modular class vanishes, and we come to canonical odd Laplacian on half-densities, the main ingredient of the Batalin-Vilkovisky formalism. Then we consider examples of odd Poisson supermanifolds with non-trivial modular classes related with the Nijenhuis bracket. The talk is based on the joint paper with M. Peddie: arXiv:1509.05686 [math-ph].