On the deformation quantization of symplectic orbispaces

In the first part of this article we provide a geometrically oriented approach to the theory of orbispaces which originally had been introduced by Chen. We explain the notion of a vector orbibundle and characterize the good sections of a reduced vector orbibundle as the smooth stratified sections. In the second part of the article we elaborate on the quantizability of a symplectic orbispace. By adapting Fedosov's method to the orbispace setting we show that every symplectic orbispace has a deformation quantization. As a byproduct we obtain that every symplectic orbifold possesses a star product

The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs

In this paper, we study the formal solution space of a nonlinear PDE in a fiber bundle. To this end, we start with foundational material and introduce the notion of a pfd structure to build up a new concept of profinite dimensional manifolds. We show that the infinite jet space of the fiber bundle is a profinite dimensional manifold in a natural way. The formal solution space of the nonlinear PDE then is a subspace of this jet space, and inherits from it the structure of a profinite dimensional manifold, if the PDE is formally integrable. We apply our concept to scalar PDEs and prove a new criterion for formal integrability of such PDEs. In particular, this result entails that the Euler-Lagrange equation of a relativistic scalar field with a polynomial self-interaction is formally integrable

Holomorphic deformation of Hopf algebras and applications to quantum groups

In this article we propose a new and so-called holomorphic deformation scheme for locally convex algebras and Hopf algebras. Essentially we regard converging power series expansion of a deformed product on a locally convex algebra, thus giving the means to actually insert complex values for the deformation parameter. Moreover we establish a topological duality theory for locally convex Hopf algebras. Examples coming from the theory of quantum groups are reconsidered within our holomorphic deformation scheme and topological duality theory. It is shown that all the standard quantum groups comprise holomorphic deformations. Furthermore we show that quantizing the function algebra of a (Poisson) Lie group and quantizing its universal enveloping algebra are topologically dual procedures indeed. Thus holomorphic deformation theory seems to be the appropriate language in which to describe quantum groups as deformed Lie groups or Lie algebras.Comment: 40 page

Relative pairing in cyclic cohomology and divisor flows

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to all the essential features of the divisor flows, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.Comment: 43 pages; revision 5.22; expanded by a factor of 1.5, in particular even case adde

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