String topology for stacks

Details String topology for stacks String topology for stacks

2007-12-22

arxiv.org/abs/0712.3857v2

Mathematics

Algebraic Topology

extended version, 152 pages

Scientific paper

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. In particular, we give a good notion of a free loop stack, and of a mapping stack $\map(Y,\XX)$, where $Y$ is a compact space and $\XX$ a topological stack, which is functorial both in $\XX$ and $Y$ and behaves well enough with respect to pushouts. We also 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 a BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. 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 loop product of almost complex is isomorphic to the orbifold intersection pairing twisted by a canonical class. Finally we gave some examples including the case of the classifying stacks $[*/G]$ of a compact Lie group.

Affiliated with

Also associated with

No associations

Rating

String topology for stacks does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with String topology for stacks, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and String topology for stacks will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-425955