Andreas Swerdlow: Morphisms of Derived Poisson Manifolds and Thick Morphisms

Abstract

Derived Poisson manifolds are a homotopy version of usual Poisson manifolds, arising, for example, when generalising the Lie–Poisson construction for Lie algebras to L-infinity algebroids. Instead of a single bracket operation taking two functions, they have a sequence of n-ary brackets for each n > 0, which assemble into an L-infinity algebra structure on smooth functions. Morphisms between derived Poisson manifolds are given by formal nonlinear pullback maps on functions, which define L-infinity morphisms between the L-infinity algebra structures on functions and satisfy a certain Leibniz-type property. We will show that morphisms satisfying the Leibniz property are equivalent to certain formal canonical relations, called microformal or thick morphisms, between the cotangent bundles of the manifolds in question.