Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the $K$-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes--Moscovici and its extension to orbifolds.Comment: 59 pages, this is a major revision, orbifold analytic higher index is introduce

Quantization of Whitney functions

We propose to study deformation quantizations of Whitney functions. To this end, we extend the notion of a deformation quantization to algebras of Whitney functions over a singular set, and show the existence of a deformation quantization of Whitney functions over a closed subset of a symplectic manifold. Under the assumption that the underlying symplectic manifold is analytic and the singular subset subanalytic, we determine that the Hochschild and cyclic homology of the deformed algebra of Whitney functions over the subanalytic subset coincide with the Whitney--de Rham cohomology. Finally, we note how an algebraic index theorem for Whitney functions can be derived.Comment: 10 page

The transverse index theorem for proper cocompact actions of Lie groupoids

Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples the meaning of the index pairing and our index formula.Comment: 29 page

The index of geometric operators on Lie groupoids

We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the "topological side" of the index theorem. This results in index formulae for Lie groupoid analogues of the familiar geometric operators on manifolds such as the signature and Dirac operator expressed in terms of the usual characteristic classes in Lie algebroid cohomology.Comment: 15 page

Orbifold cup products and ring structures on Hochschild cohomologies

In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an $S^1$-equivariant version of the Chen--Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology

Deformation quantization and homological reduction of a lattice gauge model

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on $G$. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $G = \operatorname{SU} (2)$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the - in general singular - symplectically reduced phase space of the corresponding lattice gauge model carries a star product

Geometry of orbit spaces of proper Lie groupoids

In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives the associated Lie algebroid the structure of a singular riemannian foliation. With this metric, the orbit space inherits a natural length space structure whose properties are studied. Moreover, we show that the orbit space of a proper Lie groupoid can be triangulated. Finally, we prove a de Rham theorem for the complex of basic differential forms on a proper Lie groupoid.Comment: 35 pages, minor changes, added reference and remark 3.1