String topology for stacks

Abstract

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 $[\ast /G]$ of a compact Lie group.

extended version, 152 pages

References in corpus (8)

• Foundations of Topological Stacks I
• A polarized view of string topology
• Higher string topology operations
• String Topology: Background and Present State
• String topology of Poincare duality groups
• String Bracket and Flat Connections
• A Homotopy Theoretic Proof of the BV Identity in Loop Homology
• The Floer homotopy type of the cotangent bundle

Cited by in corpus (16)

• G-gerbes, principal 2-group bundles and characteristic classes
• Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps
• On string topology of classifying spaces
• Instantons and framed sheaves on Kähler Deligne-Mumford stacks
• Group actions on stacks and applications to equivariant string topology for stacks
• Differentiable stratified groupoids and a de Rham theorem for inertia spaces
• The homotopy type of a topological stack
• Lie Algebroids over Differentiable Stacks
• Constructible functions and Lagrangian cycles on orbifolds
• Higher operations in string topology of classifying spaces
• String topology of finite groups of Lie type
• Topological T-duality for Stacks using a Gysin Sequence
• Orbifold Euler characteristics of non-orbifold groupoids
• Augmented Homotopical Algebraic Geometry
• Mapping stacks of topological stacks
• String topology on Gorenstein spaces