Quantum Structures Workshop

June 28, 2007



Leipzig University

Elmar Wagner (U Trieste)

Noncommutative spin geometry of the standard Podles sphere

Currently, there is an increasing interest in studying quantum spaces arising in Quantum Group Theory from the Noncommutative Differential Geometry point of view. The first attempts revealed some unexpected features. For instance, well-known covariant differential calculi on quantum SU(2) cannot be described by spectral triples and the Hochschild dimension of the Podles' quantum 2-spheres does not coincide with the classical dimension of the undeformed spaces. Therefore, it has been suggested recently to reformulate the axioms of Noncommutative Differential Geometry for quantum spaces. The talk shows that Alain Connes' Noncommutative Geometry can be applied successfully to the standard Podles 2-sphere. Using the description of quantum line bundles given by the Hopf fibration of quantum SU(2) over the standard Podles 2-sphere, the Dirac operator can be described exactly as in the classical case. An explicit description of an equivariant real even spectral triple on the standard Podles 2-sphere is given by applying the representation theory of quantum SU(2). This approach provides simple proofs of the axioms defining the noncommutative spin geometry. Taking commutators with the Dirac operator gives the distinguished 2-dimensional covariant differential calculus found by Podles. The spectral triple is 0-summable which reflects the dimension drop of the Hochschild dimension. Replacing Hochschild and cyclic (co)homology by a twisted version, the correct dimension can be recovered. For instance, there exists a twisted Hochschild 2-cycle representing the volume form of the distinguished covariant differential calculus, and a non-trivial twisted cyclic 2-cocycle associated with volume form. As a Hilbert space operator, the volume form is a multiple of a q-deformed grading operator and so it defines an orientation. The twisted cyclic 2-cocycle pairs with equivariant K0-classes computing the q-winding number of the quantum line bundles which coincides with the q-index of the Dirac operator. Moreover, Poincare duality holds for this example.

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Rainer Verch
ITP, Universitaet Leipzig
Vor dem Hospitaltore 1
Phone: +49 341 97 32423

Manfred Salmhofer
ITP, Universitaet Leipzig
Vor dem Hospitaltore 1
Phone: +49 341 97 32468