396 research outputs found
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
-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 -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 we consider a lattice gauge model given by the -Hamiltonian system which consists of the cotangent bundle of a power of 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 . 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 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
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