Fractional Loop Group and Twisted K-Theory

Twisted higher index theory on good orbifolds and fractional quantum numbers

The twisted Connes-Moscovici higher index theorem is generalized to the case of good orbifolds. The higher index is shown to be a rational number, and in fact non-integer in specific examples of 2-orbifolds. This results in a non-commutative geometry model that predicts the occurrence of fractional quantum numbers in the Hall effect on the hyperbolic plane.

Exotic twisted equivariant cohomology of loop spaces, twisted Bismut-Chern character and T-duality

We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant co…

Twisted higher index theory on good orbifolds, II: fractional quantum numbers

This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher cyclic traces on $K$-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes-Kubo Hall conductance in the discrete model of the q…

Fractional structures on bundle gerbe modules and fractional classifying spaces

We study the homotopy aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan's rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a notion, which we call fractional U-structure serving as a universal framework to study the twisted Chern classes of torsion bundle gerbe modules from the perspective of classifying spaces. Based on this, we introduce and study higher fractional structures on torsion bundle gerbe modules parallel to the higher structures on ordinary vector bundles.

Applying geometric K-cycles to fractional indices

A geometric model for twisted $K$-homology is introduced. It is modeled after the Mathai-Melrose-Singer fractional analytic index theorem in the same way as the Baum-Douglas model of $K$-homology was modeled after the Atiyah-Singer index theorem. A natural transformation from twisted geometric $K$-homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted $K$-homology in this model is an isomorphism for torsion twists on a finite CW-complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforemention…

面积保持微分同胚环的中心扩张及其模糊球极限

Central extensions for loop groups of area-preserving diffeomorphisms and their fuzzy sphere limits

研究面积保持微分同胚环的中心扩张，并揭示其在大k极限下的模糊球极限性质。