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Department of Mathematical Sciences

Seminar Archives

Pure Maths Colloquium: Laplacians on half-densities; odd laplacians and modular class of an odd Poisson manifold

Presented by Hovhannes Khudaverdian , Manchester

14 March 2016 16:00 in CM221

Let E(x) be a second order contravariant symmetric tensor field on a manifold M. We consider second order operators, Laplacians such that their principal symbol is defined by the tensor E, $\Delta=E(x)\partial^2+\dots$. If a volume form is given on manifold M, then one can define an action of a Laplacian on function f as a divergence (with respect to the volume form) of gradient of the function f (with respect to the tensor field E). If a Laplacian acts not on functions but on half-densities, then one can construct a family of operators with principal symbol E which differ only on a potential. We analyze this case and consider geometrical meaning of potential.

Then instead of manifold we consider supermanifold with even and odd coordinates, and an odd Laplacians on supermanifold, with an odd principal symbol E. In supercase second order contravarian symmetric tensor field which takes odd values defines odd Laplacian and on the other hand it may define an odd Poisson bracket. The Jacobi identity on this Poisson bracket can be formulated as a condition that for a corresponding odd Laplacian, its square is a Lie derivative, a first order operator. Moreover this first order operator defines a modular class of this odd Poisson structure. We consider example of supermanifold with non-degenerate odd Poisson (symplectic structure), with vanishing modular class. This is a base of Batalin-Vilkovisky formalism. Then we consider an example of an odd Poisson supermanifold with non-trivial modular class which is related with the Nijenhuis bracket.

The talk is based on the joint paper with M. Peddie: arXiv: 1509.05686