A Local Index Formula for the Quantum Sphere

Abstract

For the Dirac operator D on the standard quantum sphere we obtain an asymptotic expansion of the SU_q(2)-equivariant entire cyclic cocycle corresponding to εD when evaluated on the element k^2\in U_q(su_2). The constant term of this expansion is a twisted cyclic cocycle which up to a scalar coincides with the volume form and computes the quantum as well as the classical Fredholm indices.

17 pages; minor corrections, references added

References in corpus (6)

• Dirac Operator on the Standard Podles Quantum Sphere
• Dirac operator and a twisted cyclic cocycle on the standard Podles quantum sphere
• Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas
• Dirac Operator on the Quantum Sphere
• Dirac Operators on Quantum Flag Manifolds
• Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere

Cited by in corpus (28)

• The Dirac operator on SU_q(2)
• The Dirac operator on compact quantum groups
• Twisted cyclic theory, equivariant KK theory and KMS States
• Twisted cyclic homology of all Podles quantum spheres
• The Noncommutative Geometry of the Quantum Projective Plane
• Connes-Landi Deformation of Spectral Triples
• Gauged Laplacians on quantum Hopf bundles
• The Podles sphere as a spectral metric space
• The Podles spheres converge to the sphere
• Geometry of Quantum Spheres
• Asymptotic and exact expansions of heat traces
• Heat trace and spectral action on the standard Podles sphere
• Dimensional reduction over the quantum sphere and non-abelian q-vortices
• Equivariant Fredholm modules for the full quantum flag manifold of $S{U}_q(3)$
• On the noncommutative spin geometry of the standard Podles sphere and index computations
• Twisted Homology of Quantum SL(2) - Part II
• On modular semifinite index theory
• A residue formula for the fundamental Hochschild class of the Podles sphere
• The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere
• Quantum Dimension and Quantum Projective Spaces
• An analogue of Weyl's law for quantized irreducible generalized flag manifolds
• On the Dolbeault--Dirac Operator of Quantized Symmetric Spaces
• Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds
• Noncommutative Geometry and Quantum Group Symmetries
• Twisted Cyclic Cohomology and Modular Fredholm Modules
• Non-commutative integration, zeta functions and the Haar state for $S{U}_q(2)$
• Twisted sigma-model solitons on the quantum projective line
• On pseudodifferential operators on filtered and multifiltered manifolds