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

SIGMA 7 (2011), 106, 12 pages      arXiv:1111.5409      https://doi.org/10.3842/SIGMA.2011.106

Classical and Quantum Dynamics on Orbifolds

Yuri A. Kordyukov
Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky Str., Ufa 450008, Russia

Received September 04, 2011, in final form November 20, 2011; Published online November 23, 2011

Abstract
We present two versions of the Egorov theorem for orbifolds. The first one is a straightforward extension of the classical theorem for smooth manifolds. The second one considers an orbifold as a singular manifold, the orbit space of a Lie group action, and deals with the corresponding objects in noncommutative geometry.

Key words: microlocal analysis; noncommutative geometry; symplectic reduction; quantization; foliation; orbifold; Hamiltonian dynamics; elliptic operators.

pdf (346 kb)   tex (16 kb)

References

  1. Adem A., Leida J., Ruan Y., Orbifolds and stringy topology, Cambridge Tracts in Mathematics, Vol. 171, Cambridge University Press, Cambridge, 2007.
  2. Bucicovschi B., Seeley's theory of pseudodifferential operators on orbifolds, math.DG/9912228.
  3. Dryden E.B., Gordon C.S., Greenwald S.J., Webb D.L., Asymptotic expansion of the heat kernel for orbifolds, Michigan Math. J. 56 (2008), 205-238.
  4. Girbau J., Nicolau M., Pseudodifferential operators on V-manifolds and foliations. I, Collect. Math. 30 (1979), 247-265.
  5. Girbau J., Nicolau M., Pseudodifferential operators on V-manifolds and foliations. II, Collect. Math. 31 (1980), 63-95.
  6. Guillemin V., Uribe A., Wang Z., Geodesics on weighted projective spaces, Ann. Global Anal. Geom. 36 (2009), 205-220, arXiv:0805.1003.
  7. Jakobson D., Strohmaier A., High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows, Comm. Math. Phys. 270 (2007), 813-833, math.SP/0607616.
  8. Kawasaki T., The signature theorem for V-manifolds, Topology 17 (1978), 75-83.
  9. Kawasaki T., The index of elliptic operators over V-manifolds, Nagoya Math. J. 84 (1981), 135-157.
  10. Kordyukov Yu.A., Noncommutative spectral geometry of Riemannian foliations, Manuscripta Math. 94 (1997), 45-73.
  11. Kordyukov Yu.A., Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow, Math. Phys. Anal. Geom. 8 (2005), 97-119, math.DG/0407435.
  12. Kordyukov Yu.A., The Egorov theorem for transverse Dirac-type operators on foliated manifolds, J. Geom. Phys. 57 (2007), 2345-2364, arXiv:0708.1660.
  13. Kordyukov Yu.A., Noncommutative Hamiltonian dynamics on foliated manifolds, J. Fixed Point Theory Appl. 7 (2010), 385-399, arXiv:0912.1923.
  14. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  15. Moerdijk I., Mrcun J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge University Press, Cambridge, 2003.
  16. Sjamaar R., Lerman E., Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), 375-422.
  17. Stanhope E., Uribe A., The spectral function of a Riemannian orbifold, Ann. Global Anal. Geom. 40 (2011), 47-65.

Previous article   Next article   Contents of Volume 7 (2011)