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Supersymmetry and Fredholm modules over quantized spaces

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Abstract

The purpose of this paper is to apply the framework of non-commutative differential geometry to quantum deformations of a class of Kähler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute an explicit formula for the Chern character on generators of the Toeplitz ℂ*-algebra.

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References

  1. Berezin, F.A.: General concept of quantization. Commun. Math. Phys.40, 153–174 (1975)

    Article  Google Scholar 

  2. Berezin, F.A.: Introduction to Superanalysis. Dordrecht: D. Reidel Publ. Co., (1987)

    Google Scholar 

  3. Born, M., Jordan, P.: Zur Quantenmechanik, Z. Physik34, 858–888 (1925)

    Article  Google Scholar 

  4. Born, M., Heisenberg, W., Jordan, P.: Zur Quantenmechanik II. Z. Physik35, 557–615 (1926)

    Article  Google Scholar 

  5. Borthwick, D., Klimek, S., Lesniewski, A., Rinaldi, M.: Super Toeplitz operators and non-perturbative deformation quantization of supermanifolds. Commun. Math. Phys.153, 49–76 (1993)

    Article  Google Scholar 

  6. Borthwick, D., Klimek, S., Lesniewski, A., Rinaldi, M.: Cartan superdomains, Super Toeplitz operators and deformation quantization. To appear

  7. Borthwick, D., Lesniewski, A., Upmeier, H.: Non-perturbative deformation quantization of Cartan domains. J. Funct. Anal.113, 153–176 (1993)

    Article  Google Scholar 

  8. Coburn, L.A.: Deformation estimates for the Berezin-Toeplitz quantization. Commun. Math. Phys.149, 415–424 (1992)

    Google Scholar 

  9. Connes, A.: Non-commutative differential geometry. Publ. Math. IHES62, 94–144 (1986)

    Google Scholar 

  10. Connes, A.: Entire cyclic cohomology of Banach algebras and characters of ϕ-summable Fredholm modules.K-Theory1, 519–548 (1988)

    Google Scholar 

  11. Connes, A., Moscovici, H.: Transgression and the Chern character in non-commutativeK-homology. Commun. Math. Phys.155, 103–122, 1993

    Google Scholar 

  12. Dirac, P.A.M.: The fundamental equations of quantum mechanics. Proc. Roy. Soc.A109, 642–653 (1926)

    Google Scholar 

  13. Ernst, K., Feng, P., Jaffe, A., Lesniewski, A.: Quantum K-theory, II. Homotopy invariance of the Chern character. J. Funct. Anal.90, 355–368 (1990)

    Article  Google Scholar 

  14. Getzler, E., Szenes, A.: On the Chern character of a theta-summable Fredholm module. J. Funct. Anal.84, 343–357 (1989)

    Article  Google Scholar 

  15. Heisenberg, W.: Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Physik33, 879–893 (1925)

    Article  Google Scholar 

  16. Jaffe, A., Lesniewski, A., Osterwalder, K.: Quantum K-theory, I. The Chern character. Commun. Math. Phys.118, 1–14 (1988)

    Article  Google Scholar 

  17. Klimek, S., Lesniewski, A.: Quantum Riemann surfaces. I. The unit disc. Commun. Math. Phys.146, 103–122 (1992)

    Google Scholar 

  18. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointress. New York, San Francisco, London: Academic Press, (1975)

    Google Scholar 

  19. Rieffel, M.A.: Deformation quantization of Heisenberg manifolds. Commun. Math. Phys.122, 531–562 (1989)

    Article  Google Scholar 

  20. Rieffel, M.A.: Non-commutative tori—a case study of non-commutative differentiable manifolds. Cont. Math.105, 191–211 (1990)

    Google Scholar 

  21. Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys.B202, 253–316 (1982)

    Article  Google Scholar 

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Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, MA, USA

    David Borthwick, Andrzej Lesniewski & Maurizio Rinaldi

  2. Department of Mathematics, IUPUI, 46205, Indianapolis, IN, USA

    Slawomir Klimek

Authors
  1. David Borthwick
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  2. Slawomir Klimek
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  3. Andrzej Lesniewski
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  4. Maurizio Rinaldi
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Additional information

Communicated by A. Jaffe

Supported in part by the National Science Foundation under grant DMS-9206936

Supported in part by the Department of Energy under grant DE-FG02-88ER25065

Supported in part by the Consiglio Nazionale delle Ricerche (CNR)

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Borthwick, D., Klimek, S., Lesniewski, A. et al. Supersymmetry and Fredholm modules over quantized spaces. Commun.Math. Phys. 166, 397–415 (1994). https://doi.org/10.1007/BF02112322

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  • DOI: https://doi.org/10.1007/BF02112322

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