Contact reduction and groupoid actions

Abstract:

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map $J:M \rightarrow \Gamma _0$ from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of $\Gamma _0$ on $M$, and we show that the quotients of fibers $J^{-1}(x)$ by suitable Lie subgroups $\Gamma _x$ 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. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.