Gauged Laplacians on Quantum Hopf Bundles
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We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.
KeywordsLine Bundle Hopf Algebra Quantum Group Principal Bundle Differential Calculus
This work was partially supported by the ‘Italian project Cofin06 - Noncommutative geometry, quantum groups and applications’. The research of AZ started at SISSA (Trieste, Italy) and went on at the IAM at Bonn University (Germany), thanks to a fellowship by the Alexander von Humboldt Stiftung; he thanks the Mathematical Physics Sector of SISSA and his host in Germany, Prof. Sergio Albeverio, for their warm hospitality. We thank Francesco D’Andrea for reading the comptu-script.
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- 1.Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin-Heidelberg-New York (1991)Google Scholar
- 2.Brzezinski, T., Majid, S.: Quantum group gauge theory on quantum spaces. Commun. Math. Phys. 157 591–638 (1993) ; Erratum 167, 235 (1995)Google Scholar
- 21.Varshalovic D.A., Moskalev A.N., Khersonskii V.K.: Quantum theory of angular momentum. World Scientific, Singapore (1988)Google Scholar
- 22.Wagner, E.: On the noncommutative spin geometry of the standard Podles sphere and index computations. http://arxiv.org/abs/0707.3403v2 [math. QA]