XXVIII Workshop on Geometric Methods in Physics 28.06-04.07.2009

Alberto S. Cattaneo

Reduction via Graded Geometry

Various geometric (e.g. Poisson, Courant, generalized complex) structures maybe rephrased in terms of super symplectic manifolds endowed with functions satisfying certain equations. From the latter point of view the most general reduction is just that of super presymplectic submanifolds compatible with the given functions. By translating this back to the language of ordinary differential geometry, we recover all the known reduction procedures plus new ones. For example, in the Poisson world we get various generalizations of the Marsden-Ratiu reduction. This is based on joint work with Bursztyn, Mehta, and Zambon.