1,132 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
On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket
In our recent paper we proved the polynomiality of a Poisson bracket for a
class of infinite-dimensional Hamiltonian systems of PDE's associated to
semi-simple Frobenius structures. In the conformal (homogeneous) case, these
systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the
first Poisson structure of their hierarchy. Our approach was based on a very
involved computation of a deformation formula for the bracket with respect to
the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the
structure of that deformation formula. In particular, we reprove it using a
deformation formula for weak quasi-Miura transformation that relates our
hierarchy of PDE's with its dispersionless limit.Comment: 21 page
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