Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 034, 12 pages      arXiv:1010.0516

Natural and Projectively Invariant Quantizations on Supermanifolds

Thomas Leuther and Fabian Radoux
Institute of Mathematics, Grande Traverse 12, B-4000 Liège, Belgium

Received October 05, 2010, in final form March 23, 2011; Published online March 31, 2011

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the pgl(n+1|m)-equivariant quantization on Rn|m constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two.

Key words: supergeometry; differential operators; projective invariance; quantization maps.

pdf (368 Kb)   tex (16 Kb)


  1. Bordemann M., Sur l'existence d'une prescription d'ordre naturelle projectivement invariante, math.DG/0208171.
  2. Bouarroudj S., Projectively equivariant quantization map, Lett. Math. Phys. 51 (2000), 265-274, math.DG/0003054.
  3. Deligne P., Etingof P., Freed D.S., Jeffrey L.C., Kazhdan D., Morgan J.W., Morrison D.R., Witten E. (Editors), Quantum fields and strings: a course for mathematicians, Vols. 1, 2, American Mathematical Society, Providence, RI, 1999.
  4. Gargoubi Kh., Ovsienko V., Modules of differential operators on the line, Funct. Anal. Appl. 35 (2001), no. 1, 13-18.
  5. Dirac P.A.M., The fundamental equations of quantum mechanics, Proc. Roy. Soc. London 109 (1925), 642-653.
  6. George J., Projective connections and Schwarzian derivatives for supermanifolds, and Batalin-Vilkovisky operators, arXiv:0909.5419.
  7. Kirillov A.A., Geometric quantization, Current Problems in Mathematics. Fundamental Directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, 141-178.
  8. Kolár I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  9. Kosmann-Schwarzbach Y., Monterde J., Divergence operators and odd Poisson brackets, Ann. Inst. Fourier (Grenoble) 52 (2002), 419-456, math.QA/0002209.
  10. Lecomte P.B.A., Ovsienko V.Yu., Projectively equivariant symbol calculus, Lett. Math. Phys. 49 (1999), 173-196, math.DG/9809061.
  11. Lecomte P.B.A., Ovsienko V.Yu., Cohomology of the vector fields lie algebra and modules of differential operators on a smooth manifold, Compositio Math. 124 (2000), 95-110, math.DG/9905058.
  12. Lecomte P.B.A., Towards projectively equivariant quantization, Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132.
  13. Leites D.A., Introduction to the theory of supermanifolds, Russ. Math. Surveys 35 (1980), no. 1, 1-64.
  14. Manin Yu.I., Gauge field theory and complex geometry, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Vol. 289, Springer-Verlag, Berlin, 1997.
  15. Mathonet P., Radoux F., Projectively equivariant quantizations over the superspace Rp|q, arXiv:1003.3320.
  16. Woodhouse N.M.J., Geometric quantization, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992.

Previous article   Next article   Contents of Volume 7 (2011)