Contact reduction and groupoid actions

Abstract

We introduce a new method to perform reduction of contact manifolds that extends Willett's (math.SG/0104080) and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J$ from a contact manifold $M$ to a Jacobi manifold $Γ_0$. This naturally generates the action of the contact groupoid of $Γ_0$ on $M$, and we show that the quotients of fibers of $J$ by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits.