Holonomy for Gerbes over Orbifolds

In this paper we compute explicit formulas for the holonomy map for a gerbe with connection over an orbifold. We show that the holonomy descends to a transgression map in Deligne cohomology. We prove that this recovers both the inner local systems in Ruan's theory of twisted orbifold cohomology and the local system of Freed-Hopkins-Teleman in their work in twisted K-theory. In the case in which the orbifold is simply a manifold we recover previous results of Gawedzki and Brylinski.Comment: 36 page

Stacky Lie groups

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added; coherence relations for groups in 2-categories given (section 4

An algebraic approach to manifold-valued generalized functions

We discuss the nature of structure-preserving maps of varies function algebras. In particular, we identify isomorphisms between special Colombeau algebras on manifolds with invertible manifold-valued generalized functions in the case of smooth parametrization. As a consequence, and to underline the consistency and validity of this approach, we see that this generalized version on algebra isomorphisms in turn implies the classical result on algebras of smooth functions.Comment: 7 page

Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid. For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of Connes-Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai-Melrose-Singer. We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map $f:W\longrightarrow M/F$ and a twisting $\sigma$ on the holonomy groupoid $M/F$, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes-Skandalis when the twisting is trivial and of Carey-Wang for manifolds.Comment: 42 page

Isomorphisms of algebras of Colombeau generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the special algebra of Colombeau generalized functions on X, resp. Y is given by composition with a unique Colombeau generalized function from Y to X. We also identify the multiplicative linear functionals from the special algebra of Colombeau generalized functions on X to the ring of Colombeau generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page