String topology for stacks

We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. We construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks : it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. Further we prove an excess formula in this context. We introduce oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack and the homology of hidden loops (sometimes called ghost loops) are a Frobenius algebra which are related by a natural morphism of Frobenius algebras. We also prove that the homology of free loop stack has a natural structure of BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. We also use our constructions to study an analogue of the loop product for stacks of maps of (n-dimensional) spheres to oriented stacks and compatible power maps in their homology. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the hidden product of almost complex orbifolds is isomorphic to the orbifold intersection pairing twisted by a canonical . Finally we gave some examples including the case of the classifying stacks $[*/G]$ of a compact Lie group