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

The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare 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 K0-theory to even twisted cyclic homology is introduced which gives rise to a Chern-Connes pairing between equivariant K0-theory and twisted cyclic cohomology. The Chern-Connes pairing between the equivariant K0-group of the standard Podles 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 Podles sphere.

💡 Research Summary

The paper makes two major contributions to the non‑commutative geometry of the standard Podleś sphere, a quantum deformation of the 2‑sphere equipped with a Uq(su(2)) symmetry. In the first part the authors revisit the known spin‑geometric data (spectral triple, Dirac operator, real structure) and address two fundamental topological properties: Poincaré duality and orientability. Classical Hochschild homology of the Podleś algebra suffers from the well‑known “dimension‑drop” phenomenon, i.e. the top‑degree Hochschild group collapses to zero, which prevents a straightforward definition of an orientation cycle. To overcome this, the authors introduce a twisted Hochschild homology HC·σ(A) where the twist is given by the modular automorphism σ (or its inverse). They construct an explicit 2‑cycle χσ in HC2σ(A) that survives the twist and prove that it coincides with the volume 2‑form ω of the distinguished covariant differential calculus on the sphere. This twisted cycle therefore provides a genuine orientation in the non‑commutative setting without any loss of dimension.

The second part of the work is devoted to index theory. The authors define a “twisted” Chern character  chσ : K0Uq(su(2))(A) → HC2σ(A) from the equivariant K0‑group of the Podleś algebra to even twisted cyclic homology. The equivariant K0‑group is generated by the quantum line bundles Ln (n∈ℤ), which are the natural analogues of classical monopole bundles. Using the Haar state h on A, they turn the twisted Hochschild cycle χσ into a twisted cyclic 2‑cocycle  τσ(a) = h(a χσ), which plays the role of integration over the quantum volume form. The pairing  ⟨chσ(