On the noncommutative spin geometry of the standard Podleś sphere and index computations

Abstract

The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podleś sphere are extended by discussing Poincaré duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q -winding numbers of quantum line bundles. Second, a “twisted” Chern character from equivariant K 0 -theory to even twisted cyclic homology is introduced which gives rise to a Chern–Connes pairing between equivariant K 0 -theory and twisted cyclic cohomology. The Chern–Connes pairing between the equivariant K 0 -group of the standard Podleś sphere and the generators of twisted cyclic cohomology relative to the modular automorphism and its inverse are computed. This includes the pairings with the twisted cyclic 2-cocycle associated to the volume form, and the one corresponding to the “no-dimension drop” case. From explicit index computations, it follows that the pairings with these cocycles give the q -indices of the known equivariant 0-summable Dirac operator on the standard Podleś sphere.