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Séminaire de Physique Théorique
# The modular class of an odd Poisson supermanifold and second order operators on half-densities

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Amhithéâtre Léon Motchane (IHES)
### Amhithéâtre Léon Motchane

#### IHES

IHES
Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

` Second order operator $\Delta$ on half-densities can be 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 principal symbol $E$ of this operator defines an odd Poisson bracket. We define the modular class of an odd Poisson supermanifold in terms of $\Delta$ operator defining the odd Poisson structure. In the case of non-degenerate odd Poisson structure (odd symplectic case) the modular class vanishes, and we come to canonical odd Laplacian on half-densities, the main ingridient of 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: arXive: 1509.05686

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