# Manchester Geometry Seminar 2015/2016

**Thursday 12 May 2016 . ** *The Frank Adams Room 1 (Room 1.212), the Alan Turing Building. 4.15pm*
##
Odd Laplacians on Half-Densities and Modular Class of an Odd Poisson Supermanifold

Hovhannes Khudaverdian (University of Manchester)

`khudian@manchester.ac.uk`

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].