Global and Local Noncommutative Geometry

Ohio State University, The, Columbus OH

Investigators

Abstract

A fundamental challenge of modern physics is to reconcile the treatment of space-time as a continuum at the macroscopic level with the discrete, quantum features observed at the subatomic scale. The search for a mathematical apparatus capable of handling these dual aspects in a unified manner motivated the foundation of noncommutative geometry. In this young field of mathematics, the classical concept of a space formed of points labeled by numerical coordinates is replaced by a qualitatively more general paradigm, in which the algebra of coordinates consists of bounded operators (observables) that do not necessarily commute under the natural multiplication. The geometry of a noncommutative space is encoded in an unbounded operator, whose inverse plays the role of the line element in classical geometry, which interacts with the observables in a bounded fashion. With innate spatial intuition rendered inoperative by the forfeit of commutativity, even the most basic geometric notions, such as locality, symmetry, and curvature, have to undergo a drastic conceptual metamorphosis. Helpful hints come from physics, where the first two appear in quantum field theory as the high-energy limit and gauge symmetry, respectively, while the last is the very manifestation of gravity in Einstein's general relativity. This project aims to deepen the conceptual understanding of the above triad of notions in the noncommutative framework, as well as to develop effective mathematical tools for their quantitative measurement. Besides relevance for physics, the envisaged developments hold a proven potential of having applications in geometry and topology. The first part of the project, building on recent work of the principal investigator and collaborators, will apply the enhanced pseudodifferential calculus for C*-dynamical systems to a host of problems involving curvature calculations for concrete noncommutative spaces, such as the noncommutative tori of arbitrary dimension. The second subproject will refine the higher index pairing between elliptic operators and Alexander-Spanier cocycles on a closed manifold and extend it to a pairing defined over the algebra of bounded pseudo-differential operators. In particular, this will yield an upgraded version of the Helton-Howe integral formula for the trace of the totally antisymmetric commutator of an n-tuple of Toeplitz operators, promoting it to a full-fledged index theorem. It will also extend Perrot's formula for the periodic cyclic cohomology class of the Radul cocycle, as well as the vanishing of the periodic cyclic cohomology class of the Wodzicki residue. The third project concerns a new Hopf algebra K(n) that presents the advantage of acting directly on the noncommutative space of leaves of any foliation rather than on its frame bundle. Its Hopf cyclic cohomology captures all transverse Chern classes, but misses the secondary classes. The goal is to construct an enhanced, topological version of the Hopf algebra K(n), whose Hopf cyclic cohomology will serve as a universal receptacle for all the geometric characteristic classes of foliations. In addition, this part of the project seeks to express these classes in terms of explicit Hopf cyclic cocycles and employ the concrete representations to derive geometric and topological consequences.