SIGMA 6 (2010), 056, 12 pages      arXiv:1003.5618      https://doi.org/10.3842/SIGMA.2010.056
Contribution to the Special Issue “Noncommutative Spaces and Fields”

A Note on Dirac Operators on the Quantum Punctured Disk

Slawomir Klimek and Matt McBride
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA

Received March 30, 2010, in final form July 07, 2010; Published online July 16, 2010

Abstract
We study quantum analogs of the Dirac type operator −2z⁻∂/∂z⁻ on the punctured disk, subject to the Atiyah-Patodi-Singer boundary conditions. We construct a parametrix of the quantum operator and show that it is bounded outside of the zero mode.

Key words: operator theory; functional analysis; non-commutative geometry.

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References

1. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.
2. Booß-Bavnbek B., Wojciechowski K.P., Elliptic boundary problems for Dirac operators, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993.
3. Carey A.L., Klimek S., Wojciechowski K.P., Dirac operators on noncommutative manifolds with boundary, arXiv:0901.0123.
4. Carey A.L., Phillips J., Rennie A., A noncommutative Atiyah-Patodi-Singer index theorem in KK-theory, arXiv:0711.3028.
5. Connes A., Non-commutative differential geometry, Academic Press, 1994.
6. Conway J., Subnormal operators, Research Notes in Mathematics, Vol. 51, Pitman, Boston, Mass. - London, 1981.
7. Halmos P.R., Sunder V.S., Bounded integral operators on L² spaces, Results in Mathematics and Related Areas, Vol. 96, Springer-Verlag, Berlin - New York, 1978.
8. Klimek S., Lesniewski A., Quantum Riemann surfaces. III. The exceptional cases, Lett. Math. Phys. 32 (1994), 45-61.
9. Klimek S., McBride M., D-bar operators on quantum domains, arXiv:1001.2216.